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Deductive reasoning is the mental process of drawing deductive inferences. An inference is deductively valid if its conclusion follows logically from its premises, i.e. if it is impossible for the premises to be true and the conclusion to be false.

For example, the inference from the premises "all men are mortal" and " Socrates is a man" to the conclusion "Socrates is mortal" is deductively valid. An argument is if it is valid and all its premises are true. Some theorists define deduction in terms of the intentions of the author: they have to intend for the premises to offer deductive support to the conclusion. With the help of this modification, it is possible to distinguish valid from invalid deductive reasoning: it is invalid if the author's belief about the deductive support is false, but even invalid deductive reasoning is a form of deductive reasoning.

Psychology is interested in deductive reasoning as a psychological process, i.e. how people actually draw inferences. Logic, on the other hand, focuses on the deductive relation of logical consequence between the premises and the conclusion or how people should draw inferences. There are different ways of conceptualizing this relation. According to the semantic approach, an argument is deductively valid if and only if there is no possible interpretation of this argument where its premises are true and its conclusion is false. The syntactic approach, on the other hand, holds that an argument is deductively valid if and only if its conclusion can be deduced from its premises using a valid rule of inference. A rule of inference is a schema of drawing a conclusion from a set of premises based only on their logical form. There are various rules of inference, like the modus ponens and the modus tollens. Invalid deductive arguments, which do not follow a rule of inference, are called formal fallacies. Rules of inference are definitory rules and contrast to strategic rules, which specify what inferences one needs to draw in order to arrive at an intended conclusion. Deductive reasoning contrasts with non-deductive or ampliative reasoning. For ampliative arguments, like inductive or abductive arguments, the premises offer weaker support to their conclusion: they make it more likely but they do not guarantee its truth. They make up for this drawback by being able to provide genuinely new information not already found in the premises, unlike deductive arguments.

Cognitive psychology investigates the mental processes responsible for deductive reasoning. One of its topics concerns the factors determining whether people draw valid or invalid deductive inferences. One factor is the form of the argument: for example, people are more successful for arguments of the form modus ponens than for modus tollens. Another is the content of the arguments: people are more likely to believe that an argument is valid if the claim made in its conclusion is plausible. A general finding is that people tend to perform better for realistic and concrete cases than for abstract cases. Psychological theories of deductive reasoning aim to explain these findings by providing an account of the underlying psychological processes. Mental logic theories hold that deductive reasoning is a language-like process that happens through the manipulation of representations using rules of inference. Mental model theories, on the other hand, claim that deductive reasoning involves models of possible states of the world without the medium of language or rules of inference. According to dual-process theories of reasoning, there are two qualitatively different cognitive systems responsible for reasoning.

The problem of deductive reasoning is relevant to various fields and issues. Epistemology tries to understand how justification is transferred from the belief in the premises to the belief in the conclusion in the process of deductive reasoning. Probability logic studies how the probability of the premises of an inference affects the probability of its conclusion. The controversial thesis of deductivism denies that there are other correct forms of inference besides deduction. Natural deduction is a type of proof system based on simple and self-evident rules of inference. In philosophy, the geometrical method is a way of philosophizing that starts from a small set of self-evident axioms and tries to build a comprehensive logical system using deductive reasoning.

Deductive reasoning is the psychological process of drawing deductive inferences. An inference is a set of premises together with a conclusion. This psychological process starts from the premises and reasons to a conclusion based on and supported by these premises. If the reasoning was done correctly, it results in a valid deduction: the truth of the premises ensures the truth of the conclusion. For example, in the syllogistic argument "all frogs are reptiles; no cats are reptiles; therefore, no cats are frogs" the conclusion is true because its two premises are true. But even arguments with wrong premises can be deductively valid if they obey this principle, as in "all frogs are mammals; no cats are mammals; therefore, no cats are frogs". If the premises of a valid argument are true, then it is called a sound argument.

The relation between the premises and the conclusion of a deductive argument is usually referred to as " logical consequence". According to Alfred Tarski, logical consequence has 3 essential features: it is necessary, formal, and knowable a priori. It is necessary in the sense that the premises of valid deductive arguments necessitate the conclusion: it is impossible for the premises to be true and the conclusion to be false, independent of any other circumstances. Logical consequence is formal in the sense that it depends only on the form or the syntax of the premises and the conclusion. This means that the validity of a particular argument does not depend on the specific contents of this argument. If it is valid, then any argument with the same logical form is also valid, no matter how different it is on the level of its contents. Logical consequence is knowable a priori in the sense that no empirical knowledge of the world is necessary to determine whether a deduction is valid. So it is not necessary to engage in any form of empirical investigation. Some logicians define deduction in terms of possible worlds: A deductive inference is valid if and only if, there is no possible world in which its conclusion is false while its premises are true. This means that there are no counterexamples: the conclusion is true in all such cases, not just in most cases.

It has been argued against this and similar definitions that they fail to distinguish between valid and invalid deductive reasoning, i.e. they leave it open whether there are invalid deductive inferences and how to define them. Some authors define deductive reasoning in psychological terms in order to avoid this problem. According to Mark Vorobey, whether an argument is deductive depends on the psychological state of the person making the argument: "An argument is deductive if, and only if, the author of the argument believes that the truth of the premises necessitates (guarantees) the truth of the conclusion". A similar formulation holds that the speaker claims or intends that the premises offer deductive support for their conclusion. This is sometimes categorized as a speaker-determined definition of deduction since it depends also on the speaker whether the argument in question is deductive or not. For speakerless definitions, on the other hand, only the argument itself matters independent of the speaker. One advantage of this type of formulation is that it makes it possible to distinguish between good or valid and bad or invalid deductive arguments: the argument is good if the author's belief concerning the relation between the premises and the conclusion is true, otherwise it is bad. One consequence of this approach is that deductive arguments cannot be identified by the law of inference they use. For example, an argument of the form modus ponens may be non-deductive if the author's beliefs are sufficiently confused. That brings with it an important drawback of this definition: it is difficult to apply to concrete cases since the intentions of the author are usually not explicitly stated.

Deductive reasoning is studied in logic, psychology, and the cognitive sciences. Some theorists emphasize in their definition the difference between these fields. On this view, psychology studies deductive reasoning as an empirical mental process, i.e. what happens when humans engage in reasoning. But the descriptive question of how actual reasoning happens is different from the normative question of how it should happen or what constitutes correct deductive reasoning, which is studied by logic. This is sometimes expressed by stating that, strictly speaking, logic does not study deductive reasoning but the deductive relation between premises and a conclusion known as logical consequence. But this distinction is not always precisely observed in the academic literature. One important aspect of this difference is that logic is not interested in whether the conclusion of an argument is sensible. So from the premise "the printer has ink" one may draw the unhelpful conclusion "the printer has ink and the printer has ink and the printer has ink", which has little relevance from a psychological point of view. Instead, actual reasoners usually try to remove redundant or irrelevant information and make the relevant information more explicit. The psychological study of deductive reasoning is also concerned with how good people are at drawing deductive inferences and with the factors determining their performance. Deductive inferences are found both in natural language and in formal logical systems, such as propositional logic.

Deductive arguments differ from non-deductive arguments in that the truth of their premises ensures the truth of their conclusion. There are two important conceptions of what this exactly means. They are referred to as the syntactic and the semantic approach. According to the syntactic approach, whether an argument is deductively valid depends only on its form, syntax, or structure. Two arguments have the same form if they use the same logical vocabulary in the same arrangement, even if their contents differ. For example, the arguments "if it rains then the street will be wet; it rains; therefore, the street will be wet" and "if the meat is not cooled then it will spoil; the meat is not cooled; therefore, it will spoil" have the same logical form: they follow the modus ponens. Their form can be expressed more abstractly as "if A then B; A; therefore B" in order to make the common syntax explicit. There are various other valid logical forms or rules of inference, like modus tollens or the disjunction elimination. The syntactic approach then holds that an argument is deductively valid if and only if its conclusion can be deduced from its premises using a valid rule of inference. One difficulty for the syntactic approach is that it is usually necessary to express the argument in a formal language in order to assess whether it is valid. But since the problem of deduction is also relevant for natural languages, this often brings with it the difficulty of translating the natural language argument into a formal language, a process that comes with various problems of its own. Another difficulty is due to the fact that the syntactic approach depends on the distinction between formal and non-formal features. While there is a wide agreement concerning the paradigmatic cases, there are also various controversial cases where it is not clear how this distinction is to be drawn.

The semantic approach suggests an alternative definition of deductive validity. It is based on the idea that the sentences constituting the premises and conclusions have to be interpreted in order to determine whether the argument is valid. This means that one ascribes semantic values to the expressions used in the sentences, such as the reference to an object for singular terms or to a truth-value for atomic sentences. The semantic approach is also referred to as the model-theoretic approach since the branch of mathematics known as model theory is often used to interpret these sentences. Usually, many different interpretations are possible, such as whether a singular term refers to one object or to another. According to the semantic approach, an argument is deductively valid if and only if there is no possible interpretation where its premises are true and its conclusion is false. Some objections to the semantic approach are based on the claim that the semantics of a language cannot be expressed in the same language, i.e. that a richer metalanguage is necessary. This would imply that the semantic approach cannot provide a universal account of deduction for language as an all-encompassing medium.

Deductive reasoning usually happens by applying rules of inference. A rule of inference is a way or schema of drawing a conclusion from a set of premises. This happens usually based only on the logical form of the premises. A rule of inference is valid if, when applied to true premises, the conclusion cannot be false. A particular argument is valid if it follows a valid rule of inference. Deductive arguments that do not follow a valid rule of inference are called formal fallacies: the truth of their premises does not ensure the truth of their conclusion.

In some cases, whether a rule of inference is valid depends on the logical system one is using. The dominant logical system is classical logic and the rules of inference listed here are all valid in classical logic. But so-called deviant logics provide a different account of which inferences are valid. For example, the rule of inference known as double negation elimination, i.e. that if a proposition is not not true then it is also true, is accepted in classical logic but rejected in intuitionistic logic.

Modus ponens (also known as "affirming the antecedent" or "the law of detachment") is the primary deductive rule of inference. It applies to arguments that have as first premise a conditional statement ( ) and as second premise the antecedent ( ) of the conditional statement. It obtains the consequent ( ) of the conditional statement as its conclusion. The argument form is listed below:

1. (First premise is a conditional statement)

2. (Second premise is the antecedent)

3. (Conclusion deduced is the consequent)

In this form of deductive reasoning, the consequent ( ) obtains as the conclusion from the premises of a conditional statement ( ) and its antecedent ( ). However, the antecedent ( ) cannot be similarly obtained as the conclusion from the premises of the conditional statement ( ) and the consequent ( ). Such an argument commits the logical fallacy of affirming the consequent.

The following is an example of an argument using modus ponens:

1. If it is raining, then there are clouds in the sky.

2. It is raining.

3. Thus, there are clouds in the sky.

Modus tollens (also known as "the law of contrapositive") is a deductive rule of inference. It validates an argument that has as premises a conditional statement (formula) and the negation of the consequent ( ) and as conclusion the negation of the antecedent ( ). In contrast to modus ponens, reasoning with modus tollens goes in the opposite direction to that of the conditional. The general expression for modus tollens is the following:

1. . (First premise is a conditional statement)

2. . (Second premise is the negation of the consequent)

3. . (Conclusion deduced is the negation of the antecedent)

The following is an example of an argument using modus tollens:

1. If it is raining, then there are clouds in the sky.

2. There are no clouds in the sky.

3. Thus, it is not raining.

A hypothetical syllogism is an inference that takes two conditional statements and forms a conclusion by combining the hypothesis of one statement with the conclusion of another. Here is the general form:

3. Therefore, .

In there being a subformula in common between the two premises that does not occur in the consequence, this resembles syllogisms in term logic, although it differs in that this subformula is a proposition whereas in Aristotelian logic, this common element is a term and not a proposition.

The following is an example of an argument using a hypothetical syllogism:

1. If there had been a thunderstorm, it would have rained.

2. If it had rained, things would have gotten wet.

3. Thus, if there had been a thunderstorm, things would have gotten wet.

Various formal fallacies have been described. They are invalid forms of deductive reasoning. An additional aspect of them is that they appear to be valid on some occasions or on the first impression. They may thereby seduce people into accepting and committing them. One type of formal fallacy is affirming the consequent, as in "if John is a bachelor, then he is male; John is male; therefore, John is a bachelor". This is similar to the valid rule of inference named modus ponens, but the second premise and the conclusion are switched around, which is why it is invalid. A similar formal fallacy is denying the antecedent, as in "if Othello is a bachelor, then he is male; Othello is not a bachelor; therefore, Othello is not male". This is similar to the valid rule of inference called modus tollens, the difference being that the second premise and the conclusion are switched around. Other formal fallacies include affirming a disjunct, denying a conjunct, and the fallacy of the undistributed middle. All of them have in common that the truth of their premises does not ensure the truth of their conclusion. But it may still happen by coincidence that both the premises and the conclusion of formal fallacies are true.

Rules of inferences are definitory rules: they determine whether an argument is deductively valid or not. But reasoners are usually not just interested in making any kind of valid argument. Instead, they often have a specific point or conclusion that they wish to prove or refute. So given a set of premises, they are faced with the problem of choosing the relevant rules of inference for their deduction to arrive at their intended conclusion. This issue belongs to the field of strategic rules: the question of which inferences need to be drawn to support one's conclusion. The distinction between definitory and strategic rules is not exclusive to logic: it is also found in various games. In chess, for example, the definitory rules state that bishops may only move diagonally while the strategic rules recommend that one should control the center and protect one's king if one intends to win. In this sense, definitory rules determine whether one plays chess or something else whereas strategic rules determine whether one is a good or a bad chess player. The same applies to deductive reasoning: to be an effective reasoner involves mastering both definitory and strategic rules.

Deductive arguments are evaluated in terms of their and .

An argument is " valid" if it is impossible for its premises to be true while its conclusion is false. In other words, the conclusion must be true if the premises are true. An argument can be "valid" even if one or more of its premises are false.

An argument is " sound" if it is valid and the premises are true.

It is possible to have a deductive argument that is logically valid but is not sound. Fallacious arguments often take that form.

The following is an example of an argument that is "valid", but not "sound":

1. Everyone who eats carrots is a quarterback.

2. John eats carrots.

3. Therefore, John is a quarterback.

The example's first premise is false - there are people who eat carrots who are not quarterbacks - but the conclusion would necessarily be true, if the premises were true. In other words, it is impossible for the premises to be true and the conclusion false. Therefore, the argument is "valid", but not "sound". False generalizations - such as "Everyone who eats carrots is a quarterback" - are often used to make unsound arguments. The fact that there are some people who eat carrots but are not quarterbacks proves the flaw of the argument.

In this example, the first statement uses categorical reasoning, saying that all carrot-eaters are definitely quarterbacks. This theory of deductive reasoning - also known as term logic - was developed by Aristotle, but was superseded by propositional (sentential) logic and predicate logic.[citation needed ]

Deductive reasoning can be contrasted with inductive reasoning, in regards to validity and soundness. In cases of inductive reasoning, even though the premises are true and the argument is "valid", it is possible for the conclusion to be false (determined to be false with a counterexample or other means).

Deductive reasoning is usually contrasted with non-deductive or ampliative reasoning. The hallmark of valid deductive inferences is that it is impossible for their premises to be true and their conclusion to be false. In this way, the premises provide the strongest possible support to their conclusion. The premises of ampliative inferences also support their conclusion. But this support is weaker: they are not necessarily truth-preserving. So even for correct ampliative arguments, it is possible that their premises are true and their conclusion is false. Two important forms of ampliative reasoning are inductive and abductive reasoning. Sometimes the term "inductive reasoning" is used in a very wide sense to cover all forms of ampliative reasoning. However, in a more strict usage, inductive reasoning is just one form of ampliative reasoning. In the narrow sense, inductive inferences are forms of statistical generalization. They are usually based on many individual observations that all show a certain pattern. These observations are then used to form a conclusion either about a yet unobserved entity or about a general law. For abductive inferences, the premises support the conclusion because the conclusion is the best explanation of why the premises are true.

The support ampliative arguments provide for their conclusion comes in degrees: some ampliative arguments are stronger than others. This is often explained in terms of probability: the premises make it more likely that the conclusion is true. Strong ampliative arguments make their conclusion very likely, but not absolutely certain. An example of ampliative reasoning is the inference from the premise "every raven in a random sample of 3200 ravens is black" to the conclusion "all ravens are black": the extensive random sample makes the conclusion very likely, but it does not exclude that there are rare exceptions. In this sense, ampliative reasoning is defeasible: it may become necessary to retract an earlier conclusion upon receiving new related information. Ampliative reasoning is very common in everyday discourse and the sciences.

An important drawback of deductive reasoning is that it does not lead to genuinely new information. This means that the conclusion only repeats information already found in the premises. Ampliative reasoning, on the other hand, goes beyond the premises by arriving at genuinely new information. One difficulty for this characterization is that it makes deductive reasoning appear useless: if deduction is uninformative, it is not clear why people would engage in it and study it. It has been suggested that this problem can be solved by distinguishing between surface and depth information. On this view, deductive reasoning is uninformative on the depth level, in contrast to ampliative reasoning. But it may still be valuable on the surface level by presenting the information in the premises in a new and sometimes surprising way.

A popular misconception of the relation between deduction and induction identifies their difference on the level of particular and general claims. On this view, deductive inferences start from general premises and draw particular conclusions, while inductive inferences start from particular premises and draw general conclusions. This idea is often motivated by seeing deduction and induction as two inverse processes that complement each other: deduction is top-down while induction is bottom-up. But this is a misconception that does not reflect how valid deduction is defined in the field of logic: a deduction is valid if it is impossible for its premises to be true while its conclusion is false, independent of whether the premises or the conclusion are particular or general. Because of this, some deductive inferences have a general conclusion and some also have particular premises.

Cognitive psychology studies the psychological processes responsible for deductive reasoning. It is concerned, among other things, with how good people are at drawing valid deductive inferences. This includes the study of the factors affecting their performance, their tendency to commit fallacies, and the underlying biases involved. A notable finding in this field is that the type of deductive inference has a significant impact on whether the correct conclusion is drawn. In a meta-analysis of 65 studies, for example, 97% of the subjects evaluated modus ponens inferences correctly, while the success rate for modus tollens was only 72%. On the other hand, even some fallacies like affirming the consequent or denying the antecedent were regarded as valid arguments by the majority of the subjects. An important factor for these mistakes is whether the conclusion seems initially plausible: the more believable the conclusion is, the higher the chance that a subject will mistake a fallacy for a valid argument.

An important bias is the matching bias, which is often illustrated using the Wason selection task. In an often-cited experiment by Peter Wason, 4 cards are presented to the participant. In one case, the visible sides show the symbols D, K, 3, and 7 on the different cards. The participant is told that every card has a letter on one side and a number on the other side, and that "[e]very card which has a D on one side has a 3 on the other side". Their task is to identify which cards need to be turned around in order to confirm or refute this conditional claim. The correct answer, only given by about 10%, is the cards D and 7. Many select card 3 instead, even though the conditional claim does not involve any requirements on what symbols can be found on the opposite side of card 3. But this result can be drastically changed if different symbols are used: the visible sides show "drinking a beer", "drinking a coke", "16 years of age", and "22 years of age" and the participants are asked to evaluate the claim "[i]f a person is drinking beer, then the person must be over 19 years of age". In this case, 74% of the participants identified correctly that the cards "drinking a beer" and "16 years of age" have to be turned around. These findings suggest that the deductive reasoning ability is heavily influenced by the content of the involved claims and not just by the abstract logical form of the task: the more realistic and concrete the cases are, the better the subjects tend to perform.

Another bias is called the "negative conclusion bias", which happens when one of the premises has the form of a negative material conditional, as in "If the card does not have an A on the left, then it has a 3 on the right. The card does not have a 3 on the right. Therefore, the card has an A on the left". The increased tendency to misjudge the validity of this type of argument is not present for positive material conditionals, as in "If the card has an A on the left, then it has a 3 on the right. The card does not have a 3 on the right. Therefore, the card does not have an A on the left".

Various psychological theories of deductive reasoning have been proposed. These theories aim to explain how deductive reasoning works in relation to the underlying psychological processes responsible. They are often used to explain the empirical findings, such as why human reasoners are more susceptible to some types of fallacies than to others.

An important distinction is between mental logic theories, sometimes also referred to as rule theories, and mental model theories. Mental logic theories see deductive reasoning as a language-like process that happens through the manipulation of representations. This is done by applying syntactic rules of inference in a way very similar to how systems of natural deduction transform their premises to arrive at a conclusion. On this view, some deductions are simpler than others since they involve fewer inferential steps. This idea can be used, for example, to explain why humans have more difficulties with some deductions, like the modus tollens, than with others, like the modus ponens: because the more error-prone forms do not have a native rule of inference but need to be calculated by combining several inferential steps with other rules of inference. In such cases, the additional cognitive labor makes the inferences more open to error.

Mental model theories, on the other hand, hold that deductive reasoning involves models or mental representations of possible states of the world without the medium of language or rules of inference. In order to assess whether a deductive inference is valid, the reasoner mentally constructs models that are compatible with the premises of the inference. The conclusion is then tested by looking at these models and trying to find a counterexample in which the conclusion is false. The inference is valid if no such counterexample can be found. In order to reduce cognitive labor, only such models are represented in which the premises are true. Because of this, the evaluation of some forms of inference only requires the construction of very few models while for others, many different models are necessary. In the latter case, the additional cognitive labor required makes deductive reasoning more error-prone, thereby explaining the increased rate of error observed. This theory can also explain why some errors depend on the content rather than the form of the argument. For example, when the conclusion of an argument is very plausible, the subjects may lack the motivation to search for counterexamples among the constructed models.

Both mental logic theories and mental model theories assume that there is one general-purpose reasoning mechanism that applies to all forms of deductive reasoning. But there are also alternative accounts that posit various different special-purpose reasoning mechanisms for different contents and contexts. In this sense, it has been claimed that humans possess a special mechanism for permissions and obligations, specifically for detecting cheating in social exchanges. This can be used to explain why humans are often more successful in drawing valid inferences if the contents involve human behavior in relation to social norms. Another example is the so-called dual-process theory. This theory posits that there are two distinct cognitive systems responsible for reasoning. Their interrelation can be used to explain commonly observed biases in deductive reasoning. System 1 is the older system in terms of evolution. It is based on associative learning and happens fast and automatically without demanding many cognitive resources. System 2, on the other hand, is of more recent evolutionary origin. It is slow and cognitively demanding, but also more flexible and under deliberate control. The dual-process theory posits that system 1 is the default system guiding most of our everyday reasoning in a pragmatic way. But for particularly difficult problems on the logical level, system 2 is employed. System 2 is mostly responsible for deductive reasoning.

The ability of deductive reasoning is an important aspect of intelligence and many tests of intelligence include problems that call for deductive inferences. Because of this relation to intelligence, deduction is highly relevant to psychology and the cognitive sciences. But the subject of deductive reasoning is also pertinent to the computer sciences, for example, in the creation of artificial intelligence.

Deductive reasoning plays an important role in epistemology. Epistemology is concerned with the question of justification, i.e. to point out which beliefs are justified and why. Deductive inferences are able to transfer the justification of the premises onto the conclusion. So while logic is interested in the truth-preserving nature of deduction, epistemology is interested in the justification-preserving nature of deduction. There are different theories trying to explain why deductive reasoning is justification-preserving. According to reliabilism, this is the case because deductions are truth-preserving: they are reliable processes that ensure a true conclusion given the premises are true. Some theorists hold that the thinker has to have explicit awareness of the truth-preserving nature of the inference for the justification to be transferred from the premises to the conclusion. One consequence of such a view is that, for young children, this deductive transference does not take place since they lack this specific awareness.

Probability logic is interested in how the probability of the premises of an argument affects the probability of its conclusion. It differs from classical logic, which assumes that propositions are either true or false but does not take into consideration the probability or certainty that a proposition is true or false. The probability of the conclusion of a deductive argument cannot be calculated by figuring out the cumulative probability of the argument's premises. Dr. Timothy McGrew, a specialist in the applications of probability theory, and Dr. Ernest W. Adams, a Professor Emeritus at UC Berkeley, pointed out that the theorem on the accumulation of uncertainty designates only a lower limit on the probability of the conclusion. So the probability of the conjunction of the argument's premises sets only a minimum probability of the conclusion. The probability of the argument's conclusion cannot be any lower than the probability of the conjunction of the argument's premises. For example, if the probability of a deductive argument's four premises is ~0.43, then it is assured that the probability of the argument's conclusion is no less than ~0.43. It could be much higher, but it cannot drop under that lower limit.

There can be examples in which each single premise is more likely true than not and yet it would be unreasonable to accept the conjunction of the premises. Professor Henry Kyburg, who was known for his work in probability and logic, clarified that the issue here is one of closure - specifically, closure under conjunction. There are examples where it is reasonable to accept P and reasonable to accept Q without its being reasonable to accept the conjunction (P&Q). Lotteries serve as very intuitive examples of this, because in a basic non-discriminatory finite lottery with only a single winner to be drawn, it is sound to think that ticket 1 is a loser, sound to think that ticket 2 is a loser,...all the way up to the final number. However, clearly, it is irrational to accept the conjunction of these statements; the conjunction would deny the very terms of the lottery because (taken with the background knowledge) it would entail that there is no winner.

Dr. McGrew further adds that the sole method to ensure that a conclusion deductively drawn from a group of premises is more probable than not is to use premises the conjunction of which is more probable than not. This point is slightly tricky, because it can lead to a possible misunderstanding. What is being searched for is a general principle that specifies factors under which, for any logical consequence C of the group of premises, C is more probable than not. Particular consequences will differ in their probability. However, the goal is to state a condition under which this attribute is ensured, regardless of which consequence one draws, and fulfilment of that condition is required to complete the task.

This principle can be demonstrated in a moderately clear way. Suppose, for instance, the following group of premises:

{P, Q, R}

Suppose that the conjunction ((P & Q) & R) fails to be more probable than not. Then there is at least one logical consequence of the group that fails to be more probable than not - namely, that very conjunction. So it is an essential factor for the argument to "preserve plausibility" (Dr. McGrew coins this phrase to mean "guarantee, from information about the plausibility of the premises alone, that any conclusion drawn from those premises by deductive inference is itself more plausible than not") that the conjunction of the premises be more probable than not.

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Aristotle, a Greek philosopher, started documenting deductive reasoning in the 4th century BC. René Descartes, in his book Discourse on Method, refined the idea for the Scientific Revolution. Developing four rules to follow for proving an idea deductively, Descartes laid the foundation for the deductive portion of the scientific method. Descartes' background in geometry and mathematics influenced his ideas on the truth and reasoning, causing him to develop a system of general reasoning now used for most mathematical reasoning. Similar to postulates, Descartes believed that ideas could be self-evident and that reasoning alone must prove that observations are reliable. These ideas also lay the foundations for the ideas of rationalism.

Deductivism is a philosophical position that gives primacy to deductive reasoning or arguments over their non-deductive counterparts. It is often understood as the evaluative claim that only deductive inferences are good or correct inferences. This theory would have wide-reaching consequences for various fields since it implies that the rules of deduction are "the only acceptable standard of evidence". This way, the rationality or correctness of the different forms of inductive reasoning is denied. Some forms of deductivism express this in terms of degrees of reasonableness or probability. Inductive inferences are usually seen as providing a certain degree of support for their conclusion: they make it more likely that their conclusion is true. Deductivism states that such inferences are not rational: the premises either ensure their conclusion, as in deductive reasoning, or they do not provide any support at all.

One motivation for deductivism is the problem of induction introduced by David Hume. It consists in the challenge of explaining how or whether inductive inferences based on past experiences support conclusions about future events. For example, a chicken comes to expect, based on all its past experiences, that the person entering its coop is going to feed it, until one day the person "at last wrings its neck instead". According to Karl Popper's falsificationism, deductive reasoning alone is sufficient. This is due to its truth-preserving nature: a theory can be falsified if one of its deductive consequences is false. So while inductive reasoning does not offer positive evidence for a theory, the theory still remains a viable competitor until falsified by empirical observation. In this sense, deduction alone is sufficient for discriminating between competing hypotheses about what is the case. Hypothetico-deductivism is a closely related scientific method, according to which science progresses by formulating hypotheses and then aims to falsify them by trying to make observations that run counter to their deductive consequences.

The term """"""""""""""" natural deduction" refers to a class of proof systems based on self-evident rules of inference.[70][71] The first systems of natural deduction were developed by Gerhard Gentzen and Stanislaw Jaskowski in the 1930s. The core motivation was to give a simple presentation of deductive reasoning that closely mirrors how reasoning actually takes place.[72] In this sense, natural deduction stands in contrast to other less intuitive proof systems, such as Hilbert-style deductive systems, which employ axiom schemes to express logical truths.[70] Natural deduction, on the other hand, avoids axioms schemes by including many different rules of inference that can be used to formulate proofs. These rules of inference express how logical constants behave. They are often divided into introduction rules and elimination rules. Introduction rules specify under which conditions a logical constant may be introduced into a new sentence of the proof.[70][71] For example, the introduction rule for the logical constant " (and) is ". It expresses that, given the premises " and " individually, one may draw the conclusion " and thereby include it in one's proof. This way, the symbol " is introduced into the proof. The removal of this symbol is governed by other rules of inference, such as the elimination rule ", which states that one may deduce the sentence " from the premise ". Similar introduction and elimination rules are given for other logical constants, such as the propositional operator ", the propositional connectives " and ", and the quantifiers " and ".[70][71]

The focus on rules of inferences instead of axiom schemes is an important feature of natural deduction. But there is no general agreement on how natural deduction is to be defined. Some theorists hold that all proof systems with this feature are forms of natural deduction. This would include various forms of sequent calculi or tableau calculi. But other theorists use the term in a more narrow sense, for example, to refer to the proof systems developed by Gentzen and Jaskowski. Because of its simplicity, natural deduction is often used for teaching logic to students.

The geometrical method is a method of philosophy based on deductive reasoning. It starts from a small set of self-evident axioms and tries to build a comprehensive logical system based only on deductive inferences from these first axioms. It was initially formulated by Baruch Spinoza and came to prominence in various rationalist philosophical systems in the modern era. It gets its name from the forms of mathematical demonstration found in traditional geometry, which are usually based on axioms, definitions, and inferred theorems. An important motivation of the geometrical method is to repudiate philosophical skepticism by grounding one's philosophical system on absolutely certain axioms. Deductive reasoning is central to this endeavor because of its necessarily truth-preserving nature. This way, the certainty initially invested only in the axioms is transferred to all parts of the philosophical system.

One recurrent criticism of philosophical systems build using the geometrical method is that their initial axioms are not as self-evident or certain as their defenders proclaim. This problem lies beyond the deductive reasoning itself, which only ensures that the conclusion is true if the premises are true, but not that the premises themselves are true. For example, Spinoza's philosophical system has been criticized this way based on objections raised against the causal axiom, i.e. that "the knowledge of an effect depends on and involves knowledge of its cause". A different criticism targets not the premises but the reasoning itself, which may at times implicitly assume premises that are themselves not self-evident.

Vincent F. Hendricks, Thought 2 Talk: A Crash Course in Reflection and Expression, New York: Automatic Press / VIP, 2005, ISBN 87-991013-7-8

Philip Johnson-Laird, Ruth M. J. Byrne, Deduction, Psychology Press 1991, ISBN 978-0-86377-149-1

Zarefsky, David, Argumentation: The Study of Effective Reasoning Parts I and II, The Teaching Company 2002

Bullemore, Thomas. The Pragmatic Problem of Induction.

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Deductive reasoning is the mental process of drawing deductive inferences. An inference is deductively valid if its conclusion follows logically from its premises, i.e. if it is impossible for the premises to be true and the conclusion to be false.

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The concept of the aesthetic descends from the concept of taste. Why the concept of taste commanded so much philosophical attention during the 18th century is a complicated matter, but this much is clear: the eighteenth-century theory of taste emerged, in part, as a corrective to the rise of rationalism, particularly as applied to beauty, and to the rise of egoism, particularly as applied to virtue. Against rationalism about beauty, the eighteenth-century theory of taste held the judgment of beauty to be immediate; against egoism about virtue, it held the pleasure of beauty to be disinterested.

Rationalism about beauty is the view that judgments of beauty are judgments of reason, i.e., that we judge things to be beautiful by reasoning it out, where reasoning it out typically involves inferring from principles or applying concepts. At the beginning of the 18th century, rationalism about beauty had achieved dominance on the continent, and was being pushed to new extremes by "les géomètres," a group of literary theorists who aimed to bring to literary criticism the mathematical rigor that Descartes had brought to physics. As one such theorist put it:

The way to think about a literary problem is that pointed out by Descartes for problems of physical science. A critic who tries any other way is not worthy to be living in the present century. There is nothing better than mathematics as propaedeutic for literary criticism. (Terrasson 1715, Preface, 65; quoted in Wimsatt and Brooks 1957, 258)

It was against this, and against more moderate forms of rationalism about beauty, that mainly British philosophers working mainly within an empiricist framework began to develop theories of taste. The fundamental idea behind any such theory-which we may call the immediacy thesis-is that judgments of beauty are not (or at least not canonically) mediated by inferences from principles or applications of concepts, but rather have all the immediacy of straightforwardly sensory judgments. It is the idea, in other words, that we do not reason to the conclusion that things are beautiful, but rather "taste" that they are. Here is an early expression of the thesis, from Jean-Baptiste Dubos's Critical Reflections on Poetry, Painting, and Music, which first appeared in 1719:

Do we ever reason, in order to know whether a ragoo be good or bad; and has it ever entered into any body's head, after having settled the geometrical principles of taste, and defined the qualities of each ingredient that enters into the composition of those messes, to examine into the proportion observed in their mixture, in order to decide whether it be good or bad? No, this is never practiced. We have a sense given us by nature to distinguish whether the cook acted according to the rules of his art. People taste the ragoo, and tho' unacquainted with those rules, they are able to tell whether it be good or no. The same may be said in some respect of the productions of the mind, and of pictures made to please and move us. (Dubos 1748, vol. II, 238-239)

And here is a late expression, from Kant's 1790 Critique of the Power of Judgment:

If someone reads me his poem or takes me to a play that in the end fails to please my taste, then he can adduce Batteux or Lessing, or even older and more famous critics of taste, and adduce all the rules they established as proofs that his poem is beautiful... . I will stop my ears, listen to no reasons and arguments, and would rather believe that those rules of the critics are false ... than allow that my judgment should be determined by means of a priori grounds of proof, since it is supposed to be a judgment of taste and not of the understanding of reason. (Kant 1790, 165)

But the theory of taste would not have enjoyed its eighteenth-century run, nor would it continue now to exert its influence, had it been without resources to counter an obvious rationalist objection. There is a wide difference-so goes the objection-between judging the excellence of a ragout and judging the excellence of a poem or a play. More often than not, poems and plays are objects of great complication. But taking in all that complication requires a lot of cognitive work, including the application of concepts and the drawing of inferences. Judging the beauty of poems and plays, then, is evidently not immediate and so evidently not a matter of taste.

The chief way of meeting this objection was first to distinguish between the act of grasping the object preparatory to judging it and the act of judging the object once grasped, and then to allow the former, but not the latter, to be as concept- and inference-mediated as any rationalist might wish. Here is Hume, with characteristic clarity:

[I]n order to pave the way for [a judgment of taste], and give a proper discernment of its object, it is often necessary, we find, that much reasoning should precede, that nice distinctions be made, just conclusions drawn, distant comparisons formed, complicated relations examined, and general facts fixed and ascertained. Some species of beauty, especially the natural kinds, on their first appearance command our affection and approbation; and where they fail of this effect, it is impossible for any reasoning to redress their influence, or adapt them better to our taste and sentiment. But in many orders of beauty, particularly those of the fine arts, it is requisite to employ much reasoning, in order to feel the proper sentiment. (Hume, 1751, Section I)

Hume-like Shaftesbury and Hutcheson before him, and Reid after him (Cooper 1711, 17, 231; Hutcheson 1725, 16-24; Reid 1785, 760-761)-regarded the faculty of taste as a kind of "internal sense." Unlike the five "external" or "direct" senses, an "internal" (or "reflex" or "secondary") sense is one that depends for its objects on the antecedent operation of some other mental faculty or faculties. Reid characterizes it as follows:

Beauty or deformity in an object, results from its nature or structure. To perceive the beauty therefore, we must perceive the nature or structure from which it results. In this the internal sense differs from the external. Our external senses may discover qualities which do not depend upon any antecedent perception... . But it is impossible to perceive the beauty of an object, without perceiving the object, or at least conceiving it. (Reid 1785, 760-761)

Because of the highly complex natures or structures of many beautiful objects, there will have to be a role for reason in their perception. But perceiving the nature or structure of an object is one thing. Perceiving its beauty is another.

Egoism about virtue is the view that to judge an action or trait virtuous is to take pleasure in it because you believe it to serve some interest of yours. Its central instance is the Hobbesian view-still very much on early eighteenth-century minds-that to judge an action or trait virtuous is to take pleasure in it because you believe it to promote your safety. Against Hobbesian egoism a number of British moralists-preeminently Shaftesbury, Hutcheson, and Hume-argued that, while a judgment of virtue is a matter of taking pleasure in response to an action or trait, the pleasure is disinterested, by which they meant that it is not self-interested (Cooper 1711, 220-223; Hutcheson 1725, 9, 25-26; Hume 1751, 218-232, 295-302). One argument went roughly as follows. That we judge virtue by means of an immediate sensation of pleasure means that judgments of virtue are judgments of taste, no less than judgments of beauty. But pleasure in the beautiful is not self-interested: we judge objects to be beautiful whether or not we believe them to serve our interests. But if pleasure in the beautiful is disinterested, there is no reason to think that pleasure in the virtuous cannot also be (Hutcheson 1725, 9-10).

The eighteenth-century view that judgments of virtue are judgments of taste highlights a difference between the eighteenth-century concept of taste and our concept of the aesthetic, since for us the concepts aesthetic and moral tend oppose one another such that a judgment's falling under one typically precludes its falling under the other. Kant is chiefly responsible for introducing this difference. He brought the moral and the aesthetic into opposition by re-interpreting what we might call the disinterest thesis -the thesis that pleasure in the beautiful is disinterested (though see Cooper 1711, 222 and Home 2005, 36-38 for anticipations of Kant's re-interpretation).

According to Kant, to say that a pleasure is interested is not to say that it is self-interested in the Hobbesian sense, but rather that it stands in a certain relation to the faculty of desire. The pleasure involved in judging an action to be morally good is interested because such a judgment issues in a desire to bring the action into existence, i.e., to perform it. To judge an action to be morally good is to become aware that one has a duty to perform the action, and to become so aware is to gain a desire to perform it. By contrast, the pleasure involved in judging an object to be beautiful is disinterested because such a judgment issues in no desire to do anything in particular. If we can be said to have a duty with regard to beautiful things, it appears to be exhausted in our judging them aesthetically to be beautiful. That is what Kant means when he says that the judgment of taste is not practical but rather "merely contemplative" (Kant 1790, 95).

By thus re-orienting the notion of disinterest, Kant brought the concept of taste into opposition with the concept of morality, and so into line, more or less, with the present concept of the aesthetic. But if the Kantian concept of taste is continuous, more or less, with the present-day concept of the aesthetic, why the terminological discontinuity? Why have we come to prefer the term 'aesthetic' to the term 'taste'? The not very interesting answer appears to be that we have preferred an adjective to a noun. The term 'aesthetic' derives from the Greek term for sensory perception, and so preserves the implication of immediacy carried by the term 'taste.' Kant employed both terms, though not equivalently: according to his usage, 'aesthetic' is broader, picking out a class of judgments that includes both the normative judgment of taste and the non-normative, though equally immediate, judgment of the agreeable. Though Kant was not the first modern to use 'aesthetic' (Baumgarten had used it as early as 1735), the term became widespread only, though quickly, after his employment of it in the third Critique. Yet the employment that became widespread was not exactly Kant's, but a narrower one according to which 'aesthetic' simply functions as an adjective corresponding to the noun "taste." So for example we find Coleridge, in 1821, expressing the wish that he "could find a more familiar word than aesthetic for works of taste and criticism," before going on to argue:

As our language ... contains no other useable adjective, to express coincidence of form, feeling, and intellect, that something, which, confirming the inner and the outward senses, becomes a new sense in itself ... there is reason to hope, that the termaesthetic, will be brought into common use. (Coleridge 1821, 254)

The availability of an adjective corresponding to "taste" has allowed for the retiring of a series of awkward expressions: the expressions "judgment of taste," "emotion of taste" and "quality of taste" have given way to the arguably less offensive 'aesthetic judgment,' 'aesthetic emotion,' and 'aesthetic quality.' However, as the noun 'taste' phased out, we became saddled with other perhaps equally awkward expressions, including the one that names this entry.

Much of the history of more recent thinking about the concept of the aesthetic can be seen as the history of the development of the immediacy and disinterest theses.

Artistic formalism is the view that the artistically relevant properties of an artwork-the properties in virtue of which it is an artwork and in virtue of which it is a good or bad one-are formal merely, where formal properties are typically regarded as properties graspable by sight or by hearing merely. Artistic formalism has been taken to follow from both the immediacy and the disinterest theses (Binkley 1970, 266-267; Carroll 2001, 20-40). If you take the immediacy thesis to imply the artistic irrelevance of all properties whose grasping requires the use of reason, and you include representational properties in that class, then you are apt to think that the immediacy thesis implies artistic formalism. If you take the disinterest thesis to imply the artistic irrelevance of all properties capable of practical import, and you include representational properties in that class, then you are apt to think that the disinterest thesis implies artistic formalism.

This is not to suggest that the popularity enjoyed by artistic formalism during the late 19th and early 20th centuries owed mainly to its inference from the immediacy or disinterest theses. The most influential advocates of formalism during this period were professional critics, and their formalism derived, at least in part, from the artistic developments with which they were concerned. As a critic Eduard Hanslick advocated for the pure music of Mozart, Beethoven, Schumann, and later Brahms, and against the dramatically impure music of Wagner; as a theorist he urged that music has no content but "tonally moving forms" (Hanslick 1986, 29). As a critic Clive Bell was an early champion of the post-Impressionists, especially Cezanne; as a theorist he maintained that the formal properties of painting-"relations and combinations of lines and colours"-alone have artistic relevance (Bell 1958, 17-18). As a critic Clement Greenberg was abstract expressionism's ablest defender; as a theorist he held painting's "proper area of competence" to be exhausted by flatness, pigment, and shape (Greenberg 1986, 86-87).

Not every influential defender of formalism has also been a professional critic. Monroe Beardsley, who arguably gave formalism its most sophisticated articulation, was not (Beardsley 1958). Nor is Nick Zangwill, who recently has mounted a spirited and resourceful defense of a moderate version of formalism (Zangwill 2001). But formalism has always been sufficiently motivated by art-critical data that once Arthur Danto made the case that the data no longer supported it, and perhaps never really had, formalism's heyday came to an end. Inspired in particular by Warhol's Brillo Boxes, which are (more or less) perceptually indistinguishable from the brand-printed cartons in which boxes of Brillo were delivered to supermarkets, Danto observed that for most any artwork it is possible to imagine both (a) another object that is perceptually indiscernible from it but which is not an artwork, and (b) another artwork that is perceptually indiscernible from it but which differs in artistic value. From these observations he concluded that form alone neither makes an artwork nor gives it whatever value it has (Danto 1981, 94-95; Danto 1986, 30-31; Danto 1997, 91).

But Danto has taken the possibility of such perceptual indiscernibles to show the limitations not merely of form but also of aesthetics, and he has done so on the grounds, apparently, that the formal and the aesthetic are co-extensive. Regarding a urinal Duchamp once exhibited and a perceptual indiscernible ordinary urinal, Danto maintains that

aesthetics could not explain why one was a work of fine art and the other not, since for all practical purposes they were aesthetically indiscernible: if one was beautiful, the other one had to be beautiful, since they looked just alike. (Danto 2003, 7)

But the inference from the limits of the artistically formal to the limits of the artistically aesthetic is presumably only as strong as the inferences from the immediacy and disinterest theses to artistic formalism, and these are not beyond question. The inference from the disinterest thesis appears to go through only if you employ a stronger notion of disinterest than the one Kant understands himself to be employing: Kant, it is worth recalling, regards poetry as the highest of the fine arts precisely because of its capacity to employ representational content in the expression of what he calls 'aesthetic ideas' (Kant 1790, 191-194; see Costello 2008 and 2013 for extended treatment of the capacity of Kantian aesthetics to accommodate conceptual art). The inference from the immediacy thesis appears to go through only if you employ a notion of immediacy stronger than the one Hume, for example, takes himself to be defending when he claims (in a passage quoted in section1.1) that "in many orders of beauty, particularly those of the fine arts, it is requisite to employ much reasoning, in order to feel the proper sentiment" (Hume 1751, 173). It may be that artistic formalism results if you push either of the tendencies embodied in the immediacy and disinterest theses to extremes. It may be that the history of aesthetics from the 18th century to the mid-Twentieth is largely the history of pushing those two tendencies to extremes. It does not follow that those tendencies must be so pushed.

Consider Warhol's Brillo Boxes. Danto is right to maintain that the eighteenth-century theorist of taste would not know how to regard it as an artwork. But this is because the eighteenth-century theorist of taste lives in the 18th century, and so would be unable to situate that work in its twentieth-century art-historical context, and not because the kind of theory he holds forbids him from situating a work in its art-historical context. When Hume, for instance, observes that artists address their works to particular, historically-situated audiences, and that a critic therefore "must place himself in the same situation as the audience" to whom a work is addressed (Hume 1757, 239), he is allowing that artworks are cultural products, and that the properties that works have as the cultural products they are are among the "ingredients of the composition" that a critic must grasp if she is to feel the proper sentiment. Nor does there seem to be anything in the celebrated conceptuality of Brillo Boxes, nor of any other conceptual work, that ought to give the eighteenth-century theorist pause. Francis Hutcheson asserts that mathematical and scientific theorems are objects of taste (Hutcheson 1725, 36-41). Alexander Gerard asserts that scientific discoveries and philosophical theories are objects of taste (Gerard 1757, 6). Neither argues for his assertion. Both regard it as commonplace that objects of intellect may be objects of taste as readily as objects of sight and hearing may be. Why should the present-day aesthetic theorist think otherwise? If an object is conceptual in nature, grasping its nature will require intellectual work. If grasping an object's conceptual nature requires situating it art-historically, then the intellectual work required to grasp its nature will include situating it art-historically. But-as Hume and Reid held (see section 1.1)-grasping the nature of an object preparatory to aesthetically judging it is one thing; aesthetically judging the object once grasped is another.

Though Danto has been the most influential and persistent critic of formalism, his criticisms are no more decisive than those advanced by Kendall Walton in his essay "Categories of Art." Walton's anti-formalist argument hinges on two main theses, one psychological and one philosophical. According to the psychological thesis, which aesthetic properties we perceive a work as having depends on which category we perceive the work as belonging to. Perceived as belonging to the category of painting, Picasso's Guernica will be perceived as "violent, dynamic, vital, disturbing" (Walton 1970, 347). But perceived as belonging to the category of "guernicas"-where guernicas are works with "surfaces with the colors and shapes of Picasso's Guernica, but the surfaces are molded to protrude from the wall like relief maps of different kinds of terrain"-Picasso's Guernica will be perceived not as violent and dynamic, but as "cold, stark, lifeless, or serene and restful, or perhaps bland, dull, boring" (Walton 1970, 347). That Picasso's Guernica can be perceived both as violent and dynamic and as not violent and not dynamic might be thought to imply that there is no fact of the matter whether it is violent and dynamic. But this implication holds only on the assumption that there is no fact of the matter which category Picasso's Guernica actually belongs to, and this assumption appears to be false given that Picasso intended that Guernica be a painting and did not intend that it be a Guernica, and that the category of paintings was well-established in the society in which Picasso painted it while the category of guernicas was not. Hence the philosophical thesis, according to which the aesthetic properties a work actually has are those it is perceived as having when perceived as belonging to the category (or categories) it actually belongs to. Since the properties of having been intended to be a painting and having been created in a society in which painting is well-established category are artistically relevant though not graspable merely by seeing (or hearing) the work, it seems that artistic formalism cannot be true. "I do not deny," Walton concludes, "that paintings and sonatas are to be judged solely on what can be seen or heard in them-when they are perceived correctly. But examining a work with the senses can by itself reveal neither how it is correct to perceive it, nor how to perceive it that way" (Walton 1970, 367).

But if we cannot judge which aesthetic properties paintings and sonatas have without consulting the intentions and the societies of the artists who created them, what of the aesthetic properties of natural items? With respect to them it may appear as if there is nothing to consult except the way they look and sound, so that an aesthetic formalism about nature must be true. Allen Carlson, a central figure in the burgeoning field of the aesthetics of nature, argues against this appearance. Carlson observes that Walton's psychological thesis readily transfers from works of art to natural items: that we perceive Shetland ponies as cute and charming and Clydesdales as lumbering surely owes to our perceiving them as belonging to the category of horses (Carlson 1981, 19). He also maintains that the philosophical thesis transfers: whales actually have the aesthetic properties we perceive them as having when we perceive them as mammals, and do not actually have any contrasting aesthetic properties we might perceive them to have when we perceive them as fish. If we ask what determines which category or categories natural items actually belong to, the answer, according to Carlson, is their natural histories as discovered by natural science (Carlson 1981, 21-22). Inasmuch as a natural item's natural history will tend not to be graspable by merely seeing or hearing it, formalism is no truer of natural items than it is of works of art.

The claim that Walton's psychological thesis transfers to natural items has been widely accepted (and was in fact anticipated, as Carlson acknowledges, by Ronald Hepburn (Hepburn 1966 and 1968)). The claim that Walton's philosophical thesis transfers to natural items has proven more controversial. Carlson is surely right that aesthetic judgments about natural items are prone to be mistaken insofar as they result from perceptions of those items as belonging to categories to which they do not belong, and, insofar as determining which categories natural items actually belong to requires scientific investigation, this point seems sufficient to undercut the plausibility of any very strong formalism about nature (see Carlson 1979 for independent objections against such formalism). Carlson, however, also wishes to establish that aesthetic judgments about natural items have whatever objectivity aesthetic judgments about works of art do, and it is controversial whether Walton's philosophical claim transfers sufficiently to support such a claim. One difficulty, raised by Malcolm Budd (Budd 2002 and 2003) and Robert Stecker (Stecker1997c), is that since there are many categories in which a given natural item may correctly be perceived, it is unclear which correct category is the one in which the item is perceived as having the aesthetic properties it actually has. Perceived as belonging to the category of Shetland ponies, a large Shetland pony may be perceived as lumbering; perceived as belonging to the category of horses, the same pony may be perceived as cute and charming but certainly not lumbering. If the Shetland pony were a work of art, we might appeal to the intentions (or society) of its creator to determine which correct category is the one that fixes its aesthetic character. But as natural items are not human creations they can give us no basis for deciding between equally correct but aesthetically contrasting categorizations. It follows, according to Budd, "the aesthetic appreciation of nature is endowed with a freedom denied to the appreciation of art" (Budd 2003, 34), though this is perhaps merely another way of saying that the aesthetic appreciation of art is endowed with an objectivity denied to the appreciation of nature.

The eighteenth-century debate between rationalists and theorists of taste (or sentimentalists) was primarily a debate over the immediacy thesis, i.e., over whether we judge objects to be beautiful by applying principles of beauty to them. It was not primarily a debate over the existence of principles of beauty, a matter over which theorists of taste might disagree. Kant denied that there are any such principles (Kant 1790, 101), but both Hutcheson and Hume affirmed their existence: they maintained that although judgments of beauty are judgments of taste and not of reason, taste nevertheless operates according to general principles, which might be discovered through empirical investigation (Hutcheson 1725, 28-35; Hume 1757, 231-233).

It is tempting to think of recent debate in aesthetics between particularists and generalists as a revival of the eighteenth-century debate between rationalists and theorists of taste. But the accuracy of this thought is difficult to gauge. One reason is that it is often unclear whether particularists and generalists take themselves merely to be debating the existence of aesthetic principles or to be debating their employment in aesthetic judgment. Another is that, to the degree particularists and generalists take themselves to be debating the employment of aesthetic principles in aesthetic judgment, it is hard to know what they can be meaning by 'aesthetic judgment.' If 'aesthetic' still carries its eighteenth-century implication of immediacy, then the question under debate is whether judgment that is immediate is immediate. If 'aesthetic' no longer carries that implication, then it is hard to know what question is under debate because it is hard to know what aesthetic judgment could be. It may be tempting to think that we can simply re-define 'aesthetic judgment' such that it refers to any judgment in which an aesthetic property is predicated of an object. But this requires being able to say what an aesthetic property is without reference to its being immediately graspable, something no one seems to have done. It may seem that we can simply re-define 'aesthetic judgment' such that it refers to any judgment in which any property of the class exemplified by beauty is predicated of an object. But which class is this? The classes exemplified by beauty are presumably endless, and the difficulty is to specify the relevant class without reference to the immediate graspability of its members, and that is what no one seems to have done.

However we are to sort out the particularist/generalist debate, important contributions to it include, on the side of particularism, Arnold Isenberg's "Critical Communication" (1949) Frank Sibley's "Aesthetic Concepts" (in Sibley 2001) and Mary Mothersill's Beauty Restored (1984) and, on the side of generalism, Monroe Beardsley's Aesthetics (1958) and "On the Generality of Critical Reasons" (1962), Sibley's "General Reasons and Criteria in Aesthetics" (in Sibley 2001), George Dickie's Evaluating Art (1987), Stephen Davies's "Replies to Arguments Suggesting that Critics' Strong Evaluations Could not be Soundly Deduced" (1995), and John Bender's "General but Defeasible Reasons in Aesthetic Evaluation: The Generalist/Particularist Dispute" (1995). Of these, the papers by Isenberg and Sibley have arguably enjoyed the greatest influence.

Isenberg concedes that we often appeal to descriptive features of works in support of our judgments of their value, and he allows that this may make it seem as if we must be appealing to principles in making those judgments. If in support of a favorable judgment of some painting a critic appeals to the wavelike contour formed by the figures clustered in its foreground, it may seem as if his judgment must involve tacit appeal to the principle that any painting having such a contour is so much the better. But Isenberg argues that this cannot be, since no one agrees to any such principle:

There is not in all the world's criticism a single purely descriptive statement concerning which one is prepared to say beforehand, 'If it is true, I shall like that work so much the better' (Isenberg 1949, 338).

But if in appealing to the descriptive features of a work we are not acknowledging tacit appeals to principles linking those features to aesthetic value, what are we doing? Isenberg believes we are offering "directions for perceiving" the work, i.e., by singling out certain its features, we are "narrow[ing] down the field of possible visual orientations" and thereby guiding others in "the discrimination of details, the organization of parts, the grouping of discrete objects into patterns" (Isenberg 1949, 336). In this way we get others to see what we have seen, rather than getting them to infer from principle what we have so inferred.

That Sibley advances a variety of particularism in one paper and a variety of generalism in another will give the appearance of inconsistency where there is none: Sibley is a particularist of one sort, and with respect to one distinction, and a generalist of another sort with respect to another distinction. Isenberg, as noted, is a particularist with respect to the distinction between descriptions and verdicts, i.e., he maintains that there are no principles by which we may infer from value-neutral descriptions of works to judgments of their overall value. Sibley's particularism and generalism, by contrast, both have to do with judgments falling in between descriptions and verdicts. With respect to a distinction between descriptions and a set of judgments intermediate between descriptions and verdicts, Sibley is straightforwardly particularist. With respect to a distinction between a set of judgments intermediate between descriptions and verdicts and verdicts, Sibley is a kind of generalist and describes himself as such.

Sibley's generalism, as set forth in "General Reasons and Criteria in Aesthetics," begins with the observation that the properties to which we appeal in justification of favorable verdicts are not all descriptive or value-neutral. We also appeal to properties that are inherently positive, such as grace, balance, dramatic intensity, or comicality. To say that a property is inherently positive is not to say that any work having it is so much the better, but rather that its tout court attribution implies value. So although a work may be made worse on account of its comical elements, the simple claim that a work is good because comical is intelligible in a way that the simple claims that a work is good because yellow, or because it lasts twelve minutes, or because it contains many puns, are not. But if the simple claim that a work is good because comical is thus intelligible, comicality is a general criterion for aesthetic value, and the principle that articulates that generality is true. But none of this casts any doubt on the immediacy thesis, as Sibley himself observes:

I have argued elsewhere that there are no sure-fire rules by which, referring to the neutral and non-aesthetic qualities of things, one can infer that something is balanced, tragic, comic, joyous, and so on. One has to look and see. Here, equally, at a different level, I am saying that there are no sure-fire mechanical rules or procedures for deciding which qualities are actual defects in the work; one has to judge for oneself. (Sibley 2001, 107-108)

The "elsewhere" referred to in the first sentence is Sibley's earlier paper, "Aesthetic Concepts," which argues that the application of concepts such as 'balanced,' 'tragic,' 'comic,' or 'joyous' is not a matter of determining whether the descriptive (i.e., non-aesthetic) conditions for their application are met, but is rather a matter of taste. Hence aesthetic judgments are immediate in something like the way that judgments of color, or of flavor, are:

We see that a book is red by looking, just as we tell that the tea is sweet by tasting it. So too, it might be said, we just see (or fail to see) that things are delicate, balanced, and the like. This kind of comparison between the exercise of taste and the use of the five senses is indeed familiar; our use of the word 'taste' itself shows that the comparison is age-old and very natural (Sibley 2001, 13-14).

But Sibley recognizes-as his eighteenth-century forebears did and his formalist contemporaries did not-that important differences remain between the exercise of taste and the use of the five senses. Central among these is that we offer reasons, or something like them, in support of our aesthetic judgments: by talking-in particular, by appealing to the descriptive properties on which the aesthetic properties depend-we justify aesthetic judgments by bringing others to see what we have seen (Sibley 2001, 14-19).

It is unclear to what degree Sibley, beyond seeking to establish that the application of aesthetic concepts is not condition-governed, seeks also to define the term 'aesthetic' in terms of their not being so. It is clearer, perhaps, that he does not succeed in defining the term this way, whatever his intentions. Aesthetic concepts are not alone in being non-condition-governed, as Sibley himself recognizes in comparing them with color concepts. But there is also no reason to think them alone in being non-condition-governed while also being reason-supportable, since moral concepts, to give one example, at least arguably also have both these features. Isolating the aesthetic requires something more than immediacy, as Kant saw. It requires something like the Kantian notion of disinterest, or at least something to play the role played by that notion in Kant's theory.

Given the degree to which Kant and Hume continue to influence thinking about aesthetic judgment (or critical judgment, more broadly), given the degree to which Sibley and Isenberg continue to abet that influence, it is not surprising that the immediacy thesis is now very widely received. The thesis, however, has come under attack, notably by Davies (1990) and Bender (1995). (See also Carroll (2009), who follows closely after Davies (1990), and Dorsch (2013) for further discussion.)

Isenberg, it will be recalled, maintains that if the critic is arguing for her verdict, her argumentation must go something as follows:

Since the critical principle expressed in premise 1 is open to counter-example, no matter what property we substitute for p, Isenberg concludes that we cannot plausibly interpret the critic as arguing for her verdict. Rather than defend the principle expressed in premise 1, Davies and Bender both posit alternative principles, consistent with the fact that no property is good-making in all artworks, which they ascribe to the critic. Davies proposes that we interpret the critic as arguing deductively from principles relativized to artistic type, that is, from principles holding that artworks of a specific types or categories-Italian Renaissance paintings, romantic symphonies, Hollywood Westerns, etc.-having p are better for having it (Davies 1990, 174). Bender proposes that we interpret the critic as arguing inductively from principles expressing mere tendencies that hold between certain properties and artworks-principles, in other words, holding that artworks having p tend to be better for having it (Bender 1995, 386).

Each proposal has its own weaknesses and strengths. A problem with Bender's approach is that critics do not seem to couch their verdicts in probabilistic terms. Were a critic to say that a work is likely to be good, or almost certainly good, or even that she has the highest confidence that it must be good, her language would suggest that she had not herself experienced the work, perhaps that she had judged the work on the basis of someone else's testimony, and that she was, therefore, no critic at all. We would therefore have good reason to prefer Davies's deductive approach if only we had good reason for thinking that relativizing critical principles to artistic type removed the original threat of counterexample. Though it is clear that such relativizing reduces the relative number of counterexamples, we need good reason for thinking that it reduces that number to zero, and Davies provides no such reason. Bender's inductive approach, by contrast, cannot be refuted by counterexample, but only by counter-tendency.

If the critic argues from the truth of a principle to the truth of a verdict-as Davies and Bender both contend-it must be possible for her to establish the truth of the principle before establishing the truth of the verdict. How might she do this? It seems unlikely that mere reflection on the nature of art, or on the natures of types of art, could yield up the relevant lists of good- and bad-making properties. At least the literature has yet to produce a promising account as to how this might be done. Observation therefore seems the most promising answer. To say that the critic establishes the truth of critical principles on the basis of observation, however, is to say that she establishes a correlation between certain artworks she has already established to be good and certain properties she has already established those works to have. But then any capacity to establish that works are good by inference from principles evidently depends on some capacity to establish that works are good without any such inference, and the question arises why the critic should prefer to do by inference what she can do perfectly well without. The answer cannot be that judging by inference from principle yields epistemically better results, since a principle based on observations can be no more epistemically sound than the observations on which it is based.

None of this shows that aesthetic or critical judgment could never be inferred from principles. It does however suggest that such judgment is first and foremost non-inferential, which is what the immediacy thesis holds.

The Kantian notion of disinterest has its most direct recent descendents in the aesthetic-attitude theories that flourished from the early to mid 20th century. Though Kant followed the British in applying the term 'disinterested' strictly to pleasures, its migration to attitudes is not difficult to explain. For Kant the pleasure involved in a judgment of taste is disinterested because such a judgment does not issue in a motive to do anything in particular. For this reason Kant refers to the judgment of taste as contemplative rather than practical (Kant 1790, 95). But if the judgment of taste is not practical, then the attitude we bear toward its object is presumably also not practical: when we judge an object aesthetically we are unconcerned with whether and how it may further our practical aims. Hence it is natural to speak of our attitude toward the object as disinterested.

To say, however, that the migration of disinterest from pleasures to attitudes is natural is not to say that it is inconsequential. Consider the difference between Kant's aesthetic theory, the last great theory of taste, and Schopenhauer's aesthetic theory, the first great aesthetic-attitude theory. Whereas for Kant disinterested pleasure is the means by which we discover things to bear aesthetic value, for Schopenhauer disinterested attention (or "will-less contemplation") is itself the locus of aesthetic value. According to Schopenhauer, we lead our ordinary, practical lives in a kind of bondage to our own desires (Schopenhauer 1819, 196). This bondage is a source not merely of pain but also of cognitive distortion in that it restricts our attention to those aspects of things relevant to the fulfilling or thwarting of our desires. Aesthetic contemplation, being will-less, is therefore both epistemically and hedonically valuable, allowing us a desire-free glimpse into the essences of things as well as a respite from desire-induced pain:

When, however, an external cause or inward disposition suddenly raises us out of the endless stream of willing, and snatches knowledge from the thralldom of the will, the attention is now no longer directed to the motives of willing, but comprehends things free from their relation to the will ... Then all at once the peace, always sought but always escaping us ... comes to us of its own accord, and all is well with us. (Schopenhauer 1819, 196)

The two most influential aesthetic-attitude theories of the 20th century are those of Edward Bullough and Jerome Stolnitz. According to Stolnitz's theory, which is the more straightforward of the two, bearing an aesthetic attitude toward an object is a matter of attending to it disinterestedly and sympathetically, where to attend to it disinterestedly is to attend to it with no purpose beyond that of attending to it, and to attend to it sympathetically is to "accept it on its own terms," allowing it, and not one's own preconceptions, to guide one's attention of it (Stolnitz 1960, 32-36). The result of such attention is a comparatively richer experience of the object, i.e., an experience taking in comparatively many of the object's features. Whereas a practical attitude limits and fragments the object of our experience, allowing us to "see only those of its features which are relevant to our purposes,.... By contrast, the aesthetic attitude 'isolates' the object and focuses upon it-the 'look' of the rocks, the sound of the ocean, the colors in the painting." (Stolnitz 1960, 33, 35).

Bullough, who prefers to speak of "psychical distance" rather than disinterest, characterizes aesthetic appreciation as something achieved

by putting the phenomenon, so to speak, out of gear with our actual practical self; by allowing it to stand outside the context of our personal needs and ends-in short, by looking at it 'objectively' ... by permitting only such reactions on our part as emphasise the 'objective features of the experience, and by interpreting even our 'subjective' affections not as modes of our being but rather as characteristics of the phenomenon. (Bullough 1995, 298-299; emphasis in original).

Bullough has been criticized for claiming that aesthetic appreciation requires dispassionate detachment:

Bullough's characterization of the aesthetic attitude is the easiest to attack. When we cry at a tragedy, jump in fear at a horror movie, or lose ourselves in the plot of a complex novel, we cannot be said to be detached, although we may be appreciating the aesthetic qualities of these works to the fullest... . And we can appreciate the aesthetic properties of the fog or storm while fearing the dangers they present. (Goldman 2005, 264)

But such a criticism seems to overlook a subtlety of Bullough's view. While Bullough does hold that aesthetic appreciation requires distance "between our own self and its affections" (Bullough 1995, 298), he does not take this to require that we not undergo affections but quite the opposite: only if we undergo affections have we affections from which to be distanced. So, for example, the properly distanced spectator of a well-constructed tragedy is not the "over-distanced" spectator who feels no pity or fear, nor the "under-distanced" spectator who feels pity and fear as she would to an actual, present catastrophe, but the spectator who interprets the pity and fear she feels "not as modes of [her] being but rather as characteristics of the phenomenon" (Bullough 1995, 299). The properly distanced spectator of a tragedy, we might say, understands her fear and pity to be part of what tragedy is about.

The notion of the aesthetic attitude has been attacked from all corners and has very few remaining sympathizers. George Dickie is widely regarded as having delivered the decisive blow in his essay "The Myth of the Aesthetic Attitude" (Dickie 1964) by arguing that all purported examples of interested or distanced attention are really just examples of inattention. So consider the case of the spectator at a performance of Othello who becomes increasingly suspicious of his own wife as the action proceeds, or the case of the impresario who sits gauging the size of the audience, or the case of the father who sits taking pride in his daughter's performance, or the case of the moralist who sits gauging the moral effects the play is apt to produce in its audience. These and all such cases will be regarded by the attitude theorist as cases of interested or distanced attention to the performance, when they are actually nothing but cases of inattention to the performance: the jealous husband is attending to his wife, the impresario to the till, the father to his daughter, the moralist to the effects of the play. But if none of them is attending to the performance, then none of them is attending to it disinterestedly or with distance (Dickie 1964, 57-59).

The attitude theorist, however, can plausibly resist Dickie's interpretation of such examples. Clearly the impresario is not attending to the performance, but there is no reason to regard the attitude theorist as committed to thinking otherwise. As for the others, it might be argued that they are all attending. The jealous husband must be attending to the performance, since it is the action of the play, as presented by the performance, that is making him suspicious. The proud father must be attending to the performance, since he is attending to his daughter's performance, which is an element of it. The moralist must be attending to the performance, since he otherwise would have no basis by which to gauge its moral effects on the audience. It may be that none of these spectators is giving the performance the attention it demands, but that is precisely the attitude theorist's point.

But perhaps another of Dickie's criticisms, one lesser known, ultimately poses a greater threat to the ambitions of the attitude theorist. Stolnitz, it will be recalled, distinguishes between disinterested and interested attention according to the purpose governing the attention: to attend disinterestedly is to attend with no purpose beyond that of attending; to attend interestedly is to attend with some purpose beyond that of attending. But Dickie objects that a difference in purpose does not imply a difference in attention:

Suppose Jones listens to a piece of music for the purpose of being able to analyze and describe it on an examination the next day and Smith listens to the same music with no such ulterior purpose. There is certainly a difference in the motives and intentions of the two men: Jones has an ulterior purpose and Smith does not, but this does not mean Jones's listening differs from Smith's ... . There is only one way tolisten to (to attend to) music, although there may be a variety of motives, intentions, and reasons for doing so and a variety of ways of being distracted from the music. (Dickie 1964, 58).

There is again much here that the attitude theorist can resist. The idea that listening is a species of attending can be resisted: the question at hand, strictly speaking, is not whether Jones and Smith listen to the music in the same way, but whether they attend in the same way to the music they are listening to. The contention that Jones and Smith are attending in the same way appears to be question-begging, as it evidently depends on a principle of individuation that the attitude theorist rejects: if Jones's attention is governed by some ulterior purpose and Smith's is not, and we individuate attention according to the purpose that governs it, their attention is not the same. Finally, even if we reject the attitude theorist's principle of individuation, the claim that there is but one way to attend to music is doubtful: one can seemingly attend to music in myriad ways-as historical document, as cultural artifact, as aural wallpaper, as sonic disturbance-depending on which of the music's features one attends to in listening to it. But Dickie is nevertheless onto something crucial to the degree he urges that a difference in purpose need not imply a relevant difference in attention. Disinterest plausibly figures in the definition of the aesthetic attitude only to the degree that it, and it alone, focuses attention on the features of the object that matter aesthetically. The possibility that there are interests that focus attention on just those same features implies that disinterest has no place in such a definition, which in turn implies that neither it nor the notion of the aesthetic attitude is likely to be of any use in fixing the meaning of the term 'aesthetic.' If to take the aesthetic attitude toward an object simply is to attend to its aesthetically relevant properties, whether the attention is interested or disinterested, then determining whether an attitude is aesthetic apparently requires first determining which properties are the aesthetically relevant ones. And this task seems always to result either in claims about the immediate graspability of aesthetic properties, which are arguably insufficient to the task, or in claims about the essentially formal nature of aesthetic properties, which are arguably groundless.

But that the notions of disinterest and psychical distance prove unhelpful in fixing the meaning of the term 'aesthetic' does not imply that they are mythic. At times we seem unable to get by without them. Consider the case of The Fall of Miletus -a tragedy written by the Greek dramatist Phrynicus and staged in Athens barely two years after the violent Persian capture of the Greek city of Miletus in 494 BC. Herodotus records that

[the Athenians] found many ways to express their sorrow at the fall of Miletus, and in particular, when Phrynicus composed and produced a play called The Fall of Miletus, the audience burst into tears and fined him a thousand drachmas for reminding them of a disaster that was so close to home; future productions of the play were also banned. (Herodotus,The Histories, 359)

How are we to explain the Athenian reaction to this play without recourse to something like interest or lack of distance? How, in particular, are we to explain the difference between the sorrow elicited by a successful tragedy and the sorrow elicited in this case? The distinction between attention and inattention is of no use here. The difference is not that the Athenians could not attend to The Fall whereas they could attend to other plays. The difference is that they could not attend to The Fall as they could attend to other plays, and this because of their too intimate connection to what attending to The Fall required their attending to.

Theories of aesthetic experience may be divided into two kinds according to the kind of feature appealed to in explanation of what makes experience aesthetic. Internalist theories appeal to features internal to experience, typically to phenomenological features, whereas externalist theories appeal to features external to the experience, typically to features of the object experienced. (The distinction between internalist and externalist theories of aesthetic experience is similar, though not identical, to the distinction between phenomenal and epistemic conceptions of aesthetic experience drawn by Gary Iseminger (Iseminger 2003, 100, and Iseminger 2004, 27, 36)). Though internalist theories-particularly John Dewey's (1934) and Monroe Beardsley's (1958)-predominated during the early and middle parts of the 20th century, externalist theories-including Beardsley's (1982) and George Dickie's (1988)-have been in the ascendant since. Beardsley's views on aesthetic experience make a strong claim on our attention, given that Beardsley might be said to have authored the culminating internalist theory as well as the founding externalist one. Dickie's criticisms of Beardsley's internalism make an equally strong claim, since they moved Beardsley-and with him most everyone else-from internalism toward externalism.

According to the version of internalism Beardsley advances in his Aesthetics (1958), all aesthetic experiences have in common three or four (depending on how you count) features, which "some writers have [discovered] through acute introspection, and which each of us can test in his own experience" (Beardsley 1958, 527). These are focus ("an aesthetic experience is one in which attention is firmly fixed upon [its object]"), intensity, and unity, where unity is a matter of coherence and of completeness (Beardsley 1958, 527). Coherence, in turn, is a matter of having elements that are properly connected one to another such that

[o]ne thing leads to another; continuity of development, without gaps or dead spaces, a sense of overall providential pattern of guidance, an orderly cumulation of energy toward a climax, are present to an unusual degree. (Beardsley 1958, 528)

Completeness, by contrast, is a matter having elements that "counterbalance" or "resolve" one another such that the whole stands apart from elements without it:

The impulses and expectations aroused by elements within the experience are felt to be counterbalanced or resolved by other elements within the experience, so that some degree of equilibrium or finality is achieved and enjoyed. The experience detaches itself, and even insulates itself, from the intrusion of alien elements. (Beardsley 1958, 528)

Dickie's most consequential criticism of Beardsley's theory is that Beardsley, in describing the phenomenology of aesthetic experience, has failed to distinguish between the features we experience aesthetic objects as having and the features aesthetic experiences themselves have. So while every feature mentioned in Beardsley's description of the coherence of aesthetic experience-continuity of development, the absence of gaps, the mounting of energy toward a climax-surely is a feature we experience aesthetic objects as having, there is no reason to think of aesthetic experience itself as having any such features:

Note that everything referred to [in Beardsley's description of coherence] is a perceptual characteristic ... and not an effect of perceptual characteristics. Thus, no ground is furnished for concluding that experience can be unified in the sense of being coherent. What is actually argued for is that aesthetic objects are coherent, a conclusion which must be granted, but not the one which is relevant. (Dickie 1965, 131)

Dickie raises a similar worry about Beardsley's description of the completeness of aesthetic experience:

One can speak of elements being counterbalanced in the painting and say that the painting is stable, balanced and so on, but what does it mean to say theexperience of the spectator of the painting is stable or balanced? ... Looking at a painting in some cases might aid some persons in coming to feel stable because it might distract them from whatever is unsettling them, but such cases are atypical of aesthetic appreciation and not relevant to aesthetic theory. Aren't characteristics attributable to the painting simply being mistakenly shifted to the spectator? (Dickie 1965, 132)

Though these objections turned out to be only the beginning of the debate between Dickie and Beardsley on the nature of aesthetic experience (See Beardsley 1969, Dickie 1974, Beardsley 1982, and Dickie 1987; see also Iseminger 2003 for a helpful overview of the Beardsley-Dickie debate), they nevertheless went a long way toward shaping that debate, which taken as whole might be seen as the working out of an answer to the question "What can a theory of aesthetic experience be that takes seriously the distinction between the experience of features and the features of experience?" The answer turned out to be an externalist theory of the sort that Beardsley advances in the 1982 essay "The Aesthetic Point of View" and that many others have advanced since: a theory according to which an aesthetic experience just is an experience having aesthetic content, i.e., an experience of an object as having the aesthetic features that it has.

The shift from internalism to externalism has not been without costs. One central ambition of internalism-that of fixing the meaning of 'aesthetic' by tying it to features peculiar to aesthetic experience-has had to be given up. But a second, equally central, ambition-that of accounting for aesthetic value by tying it to the value of aesthetic experience-has been retained. Indeed most everything written on aesthetic experience since the Beardsley-Dickie debate has been written in service of the view that an object has aesthetic value insofar as it affords valuable experience when correctly perceived. This view-which has come to be called empiricism about aesthetic value, given that it reduces aesthetic value to the value of aesthetic experience-has attracted many advocates over the last several years (Beardsley 1982, Budd 1985 and 1995, Goldman 1995 and 2006, Walton 1993, Levinson 1996 and 2006, Miller 1998, Railton 1998, and Iseminger 2004), while provoking relatively little criticism (Zangwill 1999, Sharpe 2000, D. Davies 2004, and Kieran 2005). Yet it can be wondered whether empiricism about aesthetic value is susceptible to a version of the criticism that has done internalism in.

For there is something odd about the position that combines externalism about aesthetic experience with empiricism about aesthetic value. Externalism locates the features that determine aesthetic character in the object, whereas empiricism locates the features that determine aesthetic value in the experience, when one might have thought that the features that determine aesthetic character just are the features that determine aesthetic value. If externalism and empiricism are both true, there is nothing to stop two objects that have different, even wholly disparate, aesthetic characters from nevertheless having the very same aesthetic value-unless, that is, the value-determining features of an experience are bound logically to the character-determining features of the object that affords it such that only an object with those features could afford an experience having that value. But in that case the value-determining features of the experience are evidently not simply the phenomenological features that might have seemed best suited to determine the value of the experience, but perhaps rather the representational or epistemic features of the experience that it has only in relation to its object. And this is what some empiricists have been urging of late:

Aesthetic experience ... aims first at understanding and appreciation, at taking in the aesthetic properties of the object. The object itself is valuable for providing experience that could only be an experience of that object... . Part of the value of aesthetic experience lies in experiencing the object in the right way, in a way true to its nonaesthetic properties, so that the aim of understanding and appreciation is fulfilled. (Goldman 2006, 339-341; see also Iseminger 2004, 36)

But there is an unaddressed difficulty with this line of thought. While the representational or epistemic features of an aesthetic experience might very plausibly contribute to its value, such features very implausibly contribute to the value of the object affording such an experience. If the value of the experience of a good poem consists, in part, in its being an experience in which the poem is properly understood or accurately represented, the value of a good poem cannot consist, even in part, in its capacity to afford an experience in which it is properly understood or accurately represented, because, all things being equal, a bad poem presumably has these capacities in equal measure. It is of course true that only a good poem rewards an understanding of it. But then a good poem's capacity for rewarding understanding is evidently to be explained by the poem's already being good; it is evidently in virtue of its already being good that a poem rewards us on condition that we understand it.

Other empiricists have taken a different tack. Instead of trying to isolate the general features of aesthetic experience in virtue of which it and its objects are valuable, they simply observe the impossibility, in any particular case, of saying much about the value of an aesthetic experience without also saying a lot about the aesthetic character of the object. So, for example, referring to the values of the experiences that works of art afford, Jerrold Levinson maintains that

if we examine more closely these goods ... we see that their most adequate description invariably reveals them to involve ineliminably the artworks that provide them... . The cognitive expansion afforded us by Bartok's Fourth String Quartet, similarly, is not so much a generalized effect of that sort as it is a specific state of stimulation undetachable from the particular turns and twists of Bartok's carefully crafted essay... . even the pleasure we take in the Allegro of Mozart's Symphony no. 29 is, as it were, the pleasure of discovering the individual nature and potential of its thematic material, and the precise way its aesthetic character emerges from its musical underpinnings... . there is a sense in which the pleasure of the Twenty-Ninth can be had only from that work. (Levinson 1996, 22-23; see also Budd 1985, 123-124)

There is no denying that when we attempt to describe, in any detail, the values of experiences afforded by particular works we quickly find ourselves describing the works themselves. The question is what to make of this fact. If one is antecedently committed to empiricism, it may seem a manifestation of the appropriately intimate connection between the aesthetic character of a work and the value of the experience that the work affords. But if one is not so committed, it may seem to manifest something else. If, when attempting to account for the aesthetic value of Bartok's Fourth String Quartet in terms of the value of the experience it affords, we find ourselves unable to say much about the value of that experience without saying something about the quartet's "particular turns and twists," this may be because the value resides in those twists and turns and not in the experience of them. To affirm such a possibility, of course, is not to deny that the value the quartet has in virtue of its particular twists in turns is a value that we experience it as having. It is rather to insist on sharply distinguishing between the value of experience and the experience of value, in something like the way Dickie insisted on sharply distinguishing between the unity of experience and the experience of unity. When the empiricist maintains that that value of Bartok's Fourth String Quartet, with its particular twists and turns, consists in the value of the experience that it affords, which experience is valuable, at least in part, because it is an experience of a quartet with those twists and turns, one may wonder whether a value originally belonging to the quartet has been transferred to the experience, before being reflected back, once again, onto the quartet.

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# data[].attributes.attachment_link.description
The concept of the aesthetic descends from the concept of taste. Why the concept of taste commanded so much philosophical attention during the 18th century is a complicated matter, but this much is clear: the eighteenth-century theory of taste emerged, in part, as a corrective to the rise of rationalism, particularly as applied to beauty, and to the rise of egoism, particularly as applied to virtue.

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Oxford Research Encyclopedia of Education

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The concept of imagination, with its potential to contribute to education, is attracting increasing interest as humanity faces major challenges such as migration and climate change.

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The Definition of Art (Stanford Encyclopedia of Philosophy)

# data[].attributes.attachment_link.description
Any definition of art has to square with the following
uncontroversial facts: (i) entities (artifacts or performances)
intentionally endowed by their makers with a significant degree of
aesthetic interest, often greatly surpassing that of most everyday
objects, first appeared hundreds of thousands o…

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1. Infinity in philosophy: some historical remarks

In Greek, 'to apeiron' means 'the infinite': 'a' denotes privation and 'peras' the notion of 'limit' or 'bound'. Etymologically, the English word 'infinite' comes from the Latin word 'infinitas': 'in' = 'not' and 'finis' = 'end', 'boundary', 'limit', 'termination', or 'determining factor'. In contemporary English, there is a range of uses of the word 'infinite':

1. In a loose or hyperbolic sense, 'infinite' means 'indefinitely or exceedingly great', 'exceeding measurement or calculation', 'immense', or 'vast'.

2. In a strict but non-mathematical sense that reflects its etymological history, 'infinite' means 'having no limit or end', 'boundless', 'unlimited', 'endless', 'immeasurably great in extent (or duration, or some other respect)'. This strict, non-mathematical sense is often applied to God and divine attributes, and to space, time and the universe.

3. There is also a strict, mathematical sense, according to which 'infinite' quantities or magnitudes are those that are measurable but that have no finite measure; and 'infinite' lines or surfaces or volumes are measurable lines or surfaces or volumes that have no finite measure.

Related to the distinction between meanings (2) and (3) is a distinction between metaphysical and mathematical meanings of infinity. This has been usefully employed in some of the most encompassing accounts of infinity, such as Moore (1990/2019; for another recent treatment that includes extensive discussion of the history of infinity see Zellini (2005)). Moore sees the metaphysical notion as bound up with the notions of 'totality', 'absoluteness' and 'perfection'. While our entry is focused on the strict mathematical sense of 'infinity', one cannot cleanly separate the various meanings in the historical development of the subject, especially in the first stages. In addition, treating infinity as a 'perfection' in theology from the outset does not mirror the complexity of the historical development; for instance, we find traces in the 13th century of thinkers who attributed finiteness to God or in any case denied God's infinity even when not explicitly stating the finiteness of God (see Coté 2002, 127-144).

The infinite has been of central concern to Western thought since the very first pre-Socratic fragment. It concerned the philosopher Anaximander (who flourished in the 6th century BCE), who identified the principle and origin of existing things as to apeiron. In Anaximander, the principle has both an ontological and an ethical significance. The Pythagoreans (6th century BCE) saw the infinite negatively and emphasized the lack of definiteness associated with it; they also gave it spatial connotations. Indeed, in the 5th century BCE the Pythagorean Archytas of Tarentum (see Huffman 2005, 540-550) gave the following argument for the spatial infinitude of the cosmos based on the contradiction that postulating a boundary to it would seem to entail. If the cosmos is bounded, then one could extend one's hand or a stick beyond its boundary to find either empty space or matter. And this would be part of the world, which thus cannot be bounded on pain of contradiction. So the world is unbounded. Archytas identified this with the world being infinite. Kant similarly identified the unbounded and the infinite in his cosmological antinomy. In Section 8 we will see that these notions should be distinguished, but a mathematically precise articulation of the distinction had to wait until the development of new conceptions of space in the 19th century.

The Eleatics (Parmenides and Melissus, 5th century BCE) held a monist conception of reality, the One, and Melissus declared it to be infinite. Such a monistic conception of reality sees change (or becoming) as appearance, and Zeno's famous paradoxes of infinity (see the entry on Zeno's paradoxes) emerge in this context. Suffice here to say that Zeno's paradoxes (the Achilles, the arrow, and others) involved the infinitely small and were aimed at buttressing Parmenides' monism. Working across the 5th and the 4th century BCE, Democritus defended an atomistic theory with an infinite void and infinitely many atoms. The infinite by this time had shown some of its major aspects, taken as substance by some and as plurality (of atoms, times, geometrical points, etc.) by others.

If the urgency of problems related to the infinite reached Greek consciousness with Zeno's paradoxes, the most influential discussion was due to Aristotle. In order to put Aristotle's discussion in perspective, we need to list a number of ways in which mathematical infinity had emerged not only in philosophy, as we have described, but also in mathematics. We have already seen with Archytas the notion of spatial infinitude of the cosmos. But in number theory, the natural numbers were considered infinite, at least in the sense that given any natural number a greater one could be found. In geometry, we find both the infinite by addition (any segment can be extended) and by division (any segment can be halved). Thus, mathematics presented processes of iteration without limit. The most sophisticated technique for dealing with iterated processes in the measurements of plane and solid figures was developed by Eudoxus (4th century BCE), and we discuss it in Section 2.1.

By the time Aristotle (4th century BCE) developed his discussion of the infinite, this concept had thus made its presence felt in philosophy, mathematics, and natural philosophy (including cosmology, astronomy, and physics). It would be hard to exaggerate the role played by Aristotle in the history of infinity. He articulated some essential conceptual distinctions that were to influence all subsequent discussions. He was a finitist in the sense that in his universe, everything is finite. The cosmos is finite, bodies are finite, geometrical segments are finite, each number is finite, etc. However, there are processes that can be iterated indefinitely, giving rise to what he called 'potential infinity'. He claimed in fact that "in a sense [the infinite] is and in a sense it is not." ( Phys. 3.6, 206a13-14).

Any arbitrary segment can be extended in length (subject to cosmological restrictions mentioned below) or halved without limit, but at each stage we remain within the finite. Time is also potentially infinite in both directions and can be divided without limit.

This conception stands in opposition to that of 'actual infinity', which would result if some infinite processes could be completed, carried out 'all at once', as it were. If actual infinity were real, then one could have infinitely long bodies, infinitely long or infinitely small segments, the totality of natural numbers, an infinite number, infinitely many instants of time, etc. Aristotle rejected the notion of the infinite as a primordial substance, as we have encountered in Anaximander, and most of his discussion of the infinite takes place within a physical context, namely one relating to spatio-temporal features of reality. As a consequence, Aristotle's discussion of the infinite fell squarely in what we have characterized as the 'mathematical' notion of infinity, where infinity applies first of all to magnitudes (continuous or discrete) and what is quantifiable (time, extension, numbers etc.). His Physics discusses the infinitely large, excluded because the world is finite; and the infinitely small, excluded because the division of matter can only be potentially infinite and thus finite at each stage, never reaching an infinitesimal quantity-one that is less than any finite quantity, while being something. The exclusion of the infinitely large also has as a consequence that Aristotle cannot allow a potential infinity by addition in an unqualified manner (for otherwise any finite extension could be added to itself sufficiently many times to become larger than the size of the world). Infinity by addition, then, is to be conceptualized as a sort of inverse operation to infinity by division which gives us the primary evidence for the existence of the potential infinite. This is the implicit force of the contrastive "but" in the following quote. Aristotle writes (our emphasis):

'To be', then, may mean 'to be potentially' or 'to beactually'; and the infinite is either inaddition or in division. It has been stated that magnitude is not in actual operation infinite; but it is infinite in division - it is not hard to refute indivisible lines - so that it remains for the infinite to be potentially. (Physics 3.6, 206a14-24)

The Aristotelian distinction between potential and actual infinity has had a major influence up to contemporary times. (For further discussion of Aristotle on infinity see Hintikka (1966), Lear (1980), Kouremenos (1995), Coope (2012), Nawar (2015), Cooper (2016), Ugaglia (2018), and Hussey's commentary to Aristotle (1983).)

Aristotle's conception had, in addition to issues related to the constitution of the physical continuum, important consequences in cosmology. While he considered the cosmos to be finite, he thought that the movement of the celestial spheres had no beginning and no end (thus time for him, as we have noted, is potentially infinite in both directions). The issue of the "eternity of the world" was to exercise some of the best theological and philosophical minds after Aristotle, especially in connection to theological issues. For instance, Johannes Philoponus (6th century CE; see Philoponus 2004) argued in favor of a beginning of the world by claiming that the contrary thesis would lead to a paradox of infinity (we discuss this in Section 2.4).

Philoponus presented another paradox of infinity concerning infinite time that we will discuss in the version formulated by al-Ghazālī (11th century CE)-see the

Supplement on al-Ghazālī's objection

Of even more pressing significance was the abandonment of Aristotle's view on the finiteness of the cosmos and the Renaissance move from the finite to the infinite universe described in the classic text by Koyré (1957; see also Jammer 1993). While Copernicus (1473-1543) put the sun at the center of the universe, he still worked with a finite model of the universe. Foreshadowed by Epicurus (341-270), Hasdai Crescas (1340-1412), and Nicolaus Cusanus (1401-1464), Giordano Bruno (1548-1600) defended the idea of infinitely many worlds, each of infinite size, existing simultaneously. Bruno is a good example of how mathematical and theological notions of infinity were used simultaneously in the history of the concept. For instance, in On the Infinite, the Universe, and Worlds (1584) he argued from God's infinite power to the infinitude of the universe.

By contrast, Kepler and Galileo did not think that the issue of whether the world was infinite in size could be settled either way. Kepler thought that the notion of an infinite universe was a metaphysical one and not founded on empirical evidence. Galileo claimed, in a famous letter to Francesco Ingoli written in 1624, that mankind would never be able to know whether the universe is finite or infinite. The progressive geometrization of space (see De Risi 2015) led to Newton's gravitational theory in which the universe is infinitely extended spatially and temporally. Physical space became identified with the space of Euclid's geometry and in this way physical space was geometrized.

Theological elements were still present when Newton identified space with the "sensorium Dei" ("God's sensorium"). For the next two centuries cosmology was developed according to Newtonian theory: an infinite Euclidean space, flat and absolute, which provides the receptacle for all physical objects whose relations are structured by universal gravitation.

With Riemann in the mid-19th Century, and then with relativistic cosmology, one went back to a finite universe, but cosmologists are now fully aware that the issue of the finitude of the world is very much an open question that depends crucially on the curvature and the topology of space (see Section 8.2).

Our discussion above indicates a few essential aspects of the concept of infinity that will be useful in the later discussion. There are obviously many areas of contact and/or intersection between the more mathematical notion of infinity and the qualitative notion of infinity. Qualitative notions of infinity cannot be easily characterized directly but in general they appeal to features that do not seem to have a clear quantitative aspect. For instance, God might be defined as infinite because it has none of the limitations of finite creatures; this property was accounted for in some Scholastic philosophy by claiming that God, unlike finite creatures, is that unique entity in which essence and existence coincide. Often coupled with this was the claim that God's infinity is incomprehensible, and this might be a good indicator that we cannot achieve a positive account of qualitative infinity. At the same time, claims concerning infinite divine power or goodness offer a possible connection to quantitative conceptions, and this explains why the boundary line between quantitative and qualitative conceptions is not so sharp.

Indeed, according to some authors the qualitative and mathematical conceptions are inextricably tied. Consider for instance Pascal's use of infinite distance both in projective geometry and in his Pensées where he muses on the infinite distance (and disproportion) between finite human beings and the infinite God (see Cortese 2015). The following passage is representative of the powerful and suggestive role that appeal to finiteness and infinity plays in Pascal's apologetics:

For in the end what is humanity [l'homme] in nature? A nothingness compared to the infinite, everything [un tout] compared to a nothingness, a mid-point between nothing and everything, infinitely far from understanding the extremes; the end of things and their beginning [principe] are insuperably hidden for him in an impenetrable secret. \(\langle\)What can he therefore imagine? He is\(\rangle\) equally incapable of seeing the nothingness from where he came, and the infinite in which he is covered [englouti]. [...] (Pascal 2008: 70; we have added the French original where the translation seems less than faithful).

Moreover, Pascal's pari (wager) is also intimately tied to the notion of infinity in the form of an infinite reward. (See Section 7.3 on Pascal's wager) These topics are of great importance for philosophy of religion, decision theory, and philosophical anthropology.

However, this entry does not concern those conceptions of infinity that are connected to infinite divine power, infinite modes, and in general about those conceptions of infinity that are not of a mathematical kind. We do not intend to downplay the importance of those aspects of the history of infinity to which giants such as Plotinus, Cusanus, Descartes, Pascal, Spinoza, Fichte, Hegel, and Kierkegaard contributed, among others. Leibniz and Kant also belong to that list, but we will say more about them later on. But our entry would lose focus if we were to try to pursue all these developments even at a superficial level, and the treatment of qualitative infinity is worthy of an article in its own right. Thus, we content ourselves with a list of bibliographical references through which the reader can reconstruct the contributions to the topic.

For overviews of the history of infinity which include both mathematical and metaphysical aspects, see Moore (1990/2019) and Zellini (2005). For further discussion of Aristotle's views on infinity see the entries on: Aristotle; Aristotle and mathematics; and Aristotle and metaphysics. For ancient and medieval conceptions of infinity see Sweeney (1972), Sweeney (1992), Kretzmann (1982), Coté (2002), Biard and Celeyrette (2005), Duhem (1987), Dewender (2002), Davenport (1999), Murdoch (1982), Uckelman (2015); for the early modern period see Nachtomy and Winegar (2018); for infinity in Kant and the idealist period see Kreis (2015); Monnoyeur (1992) spans all periods.

For more on infinity in philosophy of religion, see the following references.

1. on divine infinity: Koetsier and Bergmans (2005), Göcke and Tapp (2018), the papers in the final section of Heller and Woodin (2011), and various entries including God and other ultimates,ontological arguments,Nicolaus Cusanus,Robert Grosseteste,John Duns Scotus, andIbn Arabi;

2. on infinity in God's creation, apart from our subsequent discussion of whether space and time are infinite: the entries cosmology and theology,cosmological argument,fine-tuning,infinite regress arguments,principle of sufficient reason, andbeing and becoming in modern physics; and

3. on 'heavenly infinity', apart from our subsequent discussion of Pascal's Wager: the entries on Pascal's wager,the meaning of life, andreligion and morality.

It is worth noting that Cantor's development of set theory was influenced by theological considerations: see, for example, Dauben (1990) and Tapp (2005).

As we have said, we are mostly excluding the topic of infinity in science and the social sciences from our purview, although see the

Supplement on Infinite Idealizations.

To the extent that we discuss infinity in science (notably in Section 8), our focus is primarily on the mathematical machinery involved, which has a venerable history. This brings us to the topic of the next section.

2. Infinity in mathematics: a brief historical overview

In this section we will begin by showing how Greek mathematics studiously avoided the use of infinity in the presentation of its results by making use of the method of exhaustion (3.1). Then we will look at the widespread use of infinitary objects and procedures in 17th-century mathematics (theory of indivisibles and points at infinity in geometry (3.2), infinitesimals in the calculus (3.3)) and Galileo's problem of extending counting to infinite collections (3.4). By the early 18th-century mathematics had undergone its first "infinitistic revolution" (the second is associated with the name of Cantor, see section 3). Infinity had become a pressing foundational problem, and this will lead us to section 3.

2.1 The method of exhaustion

We have already mentioned that the potential infinite occurs in Greek mathematics from the outset, most obviously in the natural number series and in the geometrical operations of addition and division of segments and other geometrical magnitudes. The Greek mathematicians, starting with Eudoxus, developed a technique for measuring plane and solid figures that avoided recourse to the infinite even where an infinite "limit" process would seem to be forced by the situation. This technique, known today as the method of exhaustion (the expression was coined in the 17th century by Gregory of Saint Vincent), is found in Euclid's Elements, book XII, and then in some of the most spectacular results by Archimedes (3rd century BCE). The idea is to replace an infinite approximation by a double reductio ad absurdum. That means that one shows the equality in area or volume of two figures, say a circle \(C\) and an associated triangle \(T\), by noting that \(C \lt T, C \gt T\) or \(C = T\) and then showing that the assumptions \(C \lt T\) and \(C \gt T\) both lead to a contradiction. (Here '\(C\)' and '\(T\)' refer with systematic ambiguity to the figures and their areas/volumes.) Further discussion can be found in the

Supplement on Quadratures of the Circle by Exhaustion and by Indivisibles.

Greek mathematics generally avoids any recourse to the actual infinite, and scholars have spoken of a "horror of infinity" typical of Greek mathematics. This is in general correct with respect to the way mathematical results are presented in their final and public presentation. However, one should keep in mind that no such "horror of infinity" is to be found when one looks at the heuristic strategies pursued by Greek mathematicians. In the case of Archimedes, this was made evident by the fortunate rediscovery of his method (found in 1906; see Netz and Noel 2007) where we see him using infinitary and mechanical considerations as tools he exploited for the discovery of geometrical theorems (see Knorr 1982, 1986 and Jullien 2015). For instance, in his description of the method for finding the proportion between the area of a parabolic segment and that of a related triangle, Archimedes thinks of geometric figures (the parabolic segment and the related triangle, in this case) as composed of infinitely many one-dimensional segments and then exploits the law of the lever to gain the determination of the relation between the areas in question. In a portion of the text of the method that has only recently become available (a section of proposition XIV, see Netz and Noel 2007), Archimedes explicitly operates with infinite collections.

2.2 The theory of indivisibles and points at infinity

Early modern mathematicians were impressed by the Euclidean and Archimedean rigor, but there was widespread suspicion (confirmed in 1906) that Archimedes must have had a less rigorous heuristic method that he used to discover his surprising results.

Supplement on Quadratures of the Circle by Exhaustion and by Indivisibles

for an explanation of how to give the quadrature of the circle with the indivisibilist method, and how this courts infinite collections.

Cavalieri's applications of the theory of indivisibles were limited to finite figures and thus did not go beyond the geometrical boundaries typical of Greek mathematics. However, Torricelli broke new ground with the determination of the volume of an infinitely long ( infinite longum) solid (Torricelli 1644). Up to then, all the results concerning finite figures obtained through indivisibles could easily be proved by finitary Archimedean techniques and by avoiding any mention of infinity-just as in the case of the quadrature of the circle presented in the

Supplement on Quadratures of the Circle by Exhaustion and by Indivisibles.

However, infinity figured explicitly in Torricelli's result that an infinitely long solid (FEOBMDC in the diagram) had a finite volume (the volume of the cylinder ACIH in the diagram).

This was the first infinitary result in Western mathematics, for the infinite was not eliminable using some alternative finitary technique but rather showed up as a feature of the very object that had to be measured. Torricelli's infinitary result put enormous pressure on empiricist conceptions of infinity. The heuristic fruitfulness of the indivisibilist method was also accompanied by paradoxes that threatened its foundations. Among them was Tacquet's proof using indivisibles that all triangles have the same area. The indivisibilists were able to deal with such paradoxes in various ways, but the foundations of the system remained shaky (for a detailed discussion of the foundations of the theory of indivisibles and the mathematical and philosophical issues connected to Torricelli's result see Mancosu (1996) and Jullien (2015)).

Another area in which the infinite made its appearance in 17th century geometry is in the work of Desargues (see Sakarovitch and Dhombres 1994 and Desargues 1636). Whereas in Euclidean geometry parallel lines do not meet, Desargues entertained the idea of having parallel lines meet at a point at infinity. This was a very fruitful idea that led to the development of projective geometry.

2.3 The calculus

The most fruitful development in the use of infinity in 17th century mathematics was that of the calculus.

From a geometrical point of view, the calculus provides techniques for drawing tangents at an arbitrary point of a curve and for measuring the area under a portion of a curve. The differential calculus treats the first problem and the integral calculus the second. The fundamental theorem of the calculus states that these problems are inverses of each other. The calculus was developed independently by Newton and Leibniz, but its spread owed much to a significant number of mathematicians throughout Europe. The first textbook of the differential calculus was published in 1696 by the Marquis de l'Hôpital (1696; see Bradley et al. 2015 for a translation, which we follow below, with commentary). It is worthwhile to consider its axiomatized structure, for it will help us see immediately the infinitary foundations on which the new discipline presented itself to the international community. We first have two definitions:

Definition I. Those quantities are called variable which increase or decrease continually, as opposed to constant quantities that remain the same while others change.

Definition II. The infinitely small portion by which a variable quantity continually increases or decreases is called the Differential.

The two postulates are as follows.

Postulate I. We suppose that two quantities that differ by an infinitely small quantity may be used interchangeably, or (what amounts to the same thing) that a quantity which is increased or decreased by another quantity that is infinitely smaller than it is, may be considered as remaining the same.

Postulate II. We suppose that a curved line may be considered as an assemblage of infinitely many straight lines, each one being infinitely small, or (what amounts to the same thing) as a polygon with an infinite number of sides, each being infinitely small, which determine the curvature of the line by the angles formed amongst themselves.

We see in the above the explicit infinitary characterization of some of the basic entities appealed to in the new calculus. Both postulates require something that the Greeks had studiously avoided, namely the consideration of infinitely small quantities and the reduction of curves to infinilateral polygons. While l'Hôpital and a number of French mathematicians were enthusiastic about going "infinitary", Leibniz himself developed a fictionalist account of the appeal to infinitely small quantities (foreshadowed already in his early De Quadratura which did not see the light of day until 1993; see Leibniz (1993)). Also note the use of geometrical and kinematic (i.e. based on movement, as implied by the notions of continual increase or decrease) concepts. Much of the 19th century work on the calculus was devoted to removing geometrical and kinematic notions from the foundations of the discipline.

The literature in this area is enormous and we refer to Goldenbaum and Jesseph (2008) for a recent collection of essays on Leibnizian infinitesimals. The debates on the foundations of the calculus led to some lively contributions, such as Berkeley's The Analyst (1734) and more mathematical work. But even after infinitesimals were eliminated from the calculus through the combined work of Cauchy, Bolzano, Dedekind, and Weierstrass in the 19th century, they were widely employed in geometry. Moreover, contemporary alternative theories of analysis (non-standard analysis, infinitesimal analysis etc.) have led to rigorous theories that, taken with a grain of salt, can be seen as vindicating some of the 17th century intuitions. We will come back to these developments below.

2.4 Counting infinite collections

There is one final aspect of 17th century discussions of infinity that is relevant for later considerations: the problem of extending the concept of counting from the finite to the infinite. This problem is related to the issue of whether there is only one infinity or whether there might be different sizes of infinity. As we have mentioned, Philoponus argued that the eternity of the world led to a contradiction. In particular, he claimed, if the world has no beginning in the past, then the number of individuals up to Socrates would be infinite; but then by adding the number of individuals from Socrates to now, one would obtain an infinity larger than the previous one, and this, he concluded, is "one of the most impossible things" (see Sorabji 1983). It was typical of Greek thought to reject the idea that there can be different sizes of infinity.

The Islamic mathematician Ibn Qurrah (9th century CE) took a decidedly infinitistic attitude and argued, against the Aristotelian commentators, that there can be different sizes of infinity (see Rashed 2009). He claimed, for instance, that the odd and the even numbers have the same size, but that the multiples of three are 1/3 of the total number of natural numbers. Contrary to what has been claimed in the literature, his intuition was not that the even numbers and the odd numbers have the same size because there is a one-to-one correspondence between them. Rather we have a "frequentist" intuition: every even number is followed by an odd number; multiples of three appear every three numbers, etc. We find a similar position in Grosseteste's treatise De Luce (see Mancosu 2009, 2016 for an overview of the historical developments and further references).

Galileo Galilei epitomized the paradoxical situation we run into when trying to generalize counting from the finite to the infinite. In Two New Sciences (1638; Galilei 1974), he presented a paradox of infinity. On the one hand, there is an intuition that there are fewer square numbers than natural numbers, since the first collection is properly contained in the second (the former has some but not all of the latter's members). On the other hand, there is an intuition that there are the same number of squares and natural numbers, since there is a one-to-one correspondence-a bijection-between the natural numbers and their squares. Galileo's own conclusion, following Oresme and Albert of Saxony who had discussed similar issues in the 14th century, was to claim that one cannot apply the relations of equality, greater than, and smaller than to infinite collections. Much subsequent theorizing about infinity can be regarded as respecting one intuition at the expense of the other.

The intuition that if one set is a proper subset of another, the former is smaller than the latter, traces back to Euclid-call this the part-whole intuition. Bolzano (1851) was sympathetic to it, and he tried to develop a theory of infinite sets that preserved it. He was not successful, but he warned his readers not to conflate one property of an infinite set-that it can be put in one-to-one correspondence with a proper subset of itself-with a criterion of 'size' (what he called the "multiplicity" of a collection). Cantor (see Hallett 1986), by contrast, later used one-to-one correspondence as the defining characteristic of cardinal numbers: the numbers that answer 'how many?' questions, and that generalize counting from the finite to the infinite in his set theory. He thus sided with the intuition that if there is a bijection between two sets, they have the same size-call this the bijection intuition. The intuition is clearly correct for finite sets. For example, the set of fingers on a normal human hand can be paired up with the set of toes on a normal human foot, and vice versa: there is a bijection between these two sets. And of course, the two sets have the same size (five). A central question is whether the intuition is correct also for infinite sets. We will discuss Cantor's theory and, by contrast, some recent implementations of counting, known as theories of numerosities, that preserve the part-whole intuition also for infinite sets-see the

Supplement on Theories of Numerosities.

In conclusion, the "infinitistic revolution" in the 17th and the early 18th century left an important legacy for philosophy and mathematics. The theory of indivisibles introduced new magnitudes characterized infinitarily (the collection of all the indivisibles of a figure), and new infinitary geometrical objects extended the classical geometrical universe. Moreover, the debates on the calculus were focused on the nature of the infinitely small and the infinitely large. Finally, the issues emerging from Galileo's paradox were a prelude to the problem of extending counting from finite to infinite collections.

These problems were gradually addressed in the 19th and 20th centuries, and out of these discussions there emerged different mathematical notions of infinity. We will work our way in stages to these conceptions, via a discussion of some landmarks in the contemporary mathematics of infinity.

Among the numerous general treatments of the use of infinity in mathematics we recommend Lévy (1987), Zellini (2005), Moore (1990/2019), Vilenkin (1995), Barrow (2006). For more detailed accounts of the history of the calculus, see Kline (1990), Boyer (1959), Edwards (1979), and Grattan-Guinness (1980). The most recent scholarship on the theory of indivisibles is to be found in Jullien (2015). For recent collections on Leibnizian infinitesimals see Goldenbaum and Jesseph (2008) and Goethe, Beeley and Rabouin (2015). On concepts of mathematical infinity in the 19th century see König (1990).

3. Mathematics: number systems, Cantor's paradise, and beyond

To deal with some of the issues concerning the infinite raised in Section 2, mathematicians have developed various different structures that explicitly include infinities. These structures ascribe different properties to infinities that are appropriate for different applications. In some cases, there are multiple kinds of structure that can be developed for an application. Explicitly countenancing infinities has opened up an enormous range of choices and possibilities, which has been a wellspring of development in modern mathematics.

We now give a very quick tour of infinity in modern mathematics. Section 3.1 reminds the reader of several familiar number systems: the natural numbers, integers, rational numbers, and real numbers. Section 3.2 discusses the infinite operations of limits and sums that underlie calculus, and introduces the "extended real numbers" \(+\infty\) and \(-\infty\). The material in these two sections is covered in most textbooks on real analysis, and even many calculus textbooks, so some readers might already be familiar with it, while others might benefit from having such a textbook on hand to expand on some of the points.

Sections 3.3 and 3.4 are more mathematically advanced. Section 3.3 introduces Cantor's more mathematically sophisticated "cardinals and ordinals", which are probably the mathematical developments that have done the most to untangle many of the conceptual confusions around infinity. This material is covered in greater detail in most textbooks on set theory, and parts of it are discussed in many logic textbooks as well. It can be read largely independently of the other sections.

Section 3.4 discusses a more recent mathematical theory of infinitely large and infinitesimal numbers that provides an alternate setting for calculus. This theory of "non-standard analysis" has not become as central a part of the mathematics curriculum as real analysis and set theory. It may thus be unfamiliar to most readers, and it is harder to find accessible introductions elsewhere. Although non-standard analysis is not as central a part of the cultural understanding of infinity in mathematics as cardinals are, we include it both because it is a topic of growing interest in mathematics research, and because it can help make mathematically rigorous sense of both many intuitive thoughts about infinity and some of the early work on calculus in the 17th and 18th centuries.

Up to a point, various philosophical applications and puzzles involving infinity can be understood without much understanding of the mathematics of infinity. However, the mathematics helps us formulate and tackle them rigorously. Mathematics-shy readers could skip parts of our tour (particularly section 3.4) and still benefit from the later sections, but we encourage them to make the effort and read on. The mathematical understanding of infinity is a great achievement in its own right.

The natural numbers form the most elementary number system. (Some mathematicians count \(0\) as a natural number as well, but some others do not.) \(1\) is a natural number. For any natural number \(n\), \(n+1\)-the successor of \(n\)-is also a natural number. The natural numbers-\(1, 2, 3, \dots\)-are closed under addition: if \(n_1\) and \(n_2\) are natural numbers, then so is \(n_1+n_2\). And they are closed under multiplication: if \(n_1\) and \(n_2\) are natural numbers, then so is \(n_1 \cdot n_2\). We use the natural numbers for counting 'how many' of something there are, though they clearly fail when applied to infinite sets, for instance the set of squares of natural numbers or the set of natural numbers themselves. This is what 'infinities of counting' will extend, in section 3.3.

The integers consist of the natural numbers, their additive inverses (a number and its additive inverse sum to \(0\)), and \(0\):

\[\dots , -3, -2, -1, 0, 1, 2, 3, \dots.\]

They form the most elementary number system that is also closed under subtraction: If \(j_1\) and \(j_2\) are integers, then so is \(j_1-j_2\).

The rational numbers can be expressed in the form \(j_1/j_2\), where \(j_1\) and \(j_2\) are integers, \(j_2\neq 0\). They form the most elementary number system that includes the integers and that is closed under division, except by 0. If \(q_1\) and \(q_2\) are rational numbers, \(q_1/q_2\) is also a rational number if \(q_2\neq 0\).

The rational numbers are dense: for any two rational numbers \(q_1\) and \(q_2\) such that \(q_1 < q_2\), there is at least one rational number \(q_3\) such that \(q_1 < q_3 < q_2\)-for example, the arithmetic mean of \(q_1\) and \(q_2\), \((q_1+ q_2)/2\), lies between them. Indeed, for any two rational numbers \(q_1\) and \(q_2\) such that \(q_1\) is strictly less than \(q_2\), for any natural number \(n\), there are more than \(n\) distinct rational numbers that lie between \(q_1\) and \(q_2\). Where the integers spread infinitely 'outward' in both directions, the rationals also divide infinitely 'inward'.

However, the rational numbers still have "gaps". For instance, if we consider the equation, \(y=x^3-2\), we can verify that there are values of \(x\) where \(y\) is negative, and values of \(x\) where \(y\) is positive. However, there is no rational number \(x\) for which \(y\) is exactly equal to 0. To fill these gaps, we construct the "real numbers".

The real numbers can be constructed out of the rational numbers by defining each real number to be a Dedekind cut of the rationals. A Dedekind cut of the rationals is a pair of sets \(L\) and \(R\) such that:

1. every rational number belongs to exactly one of \(L\) and \(R\);

2. every member of \(L\) is less than each member of \(R\); and

3. \(L\) has no largest element.

We refer to \(L\) as the 'left set' of the cut and \(R\) as the 'right set'.

For any rational number \(q\), there is a Dedekind cut corresponding to it, where \(L\) consists of the numbers strictly less than \(q\), while \(R\) consists of \(q\) and all larger numbers. However, there are also other partitions, where \(R\) does not contain a smallest element. For instance, we can let \(L\) include all the rational numbers whose cube is less than 2, while \(R\) includes all the rational numbers whose cube is greater than 2. Since there is no rational number whose cube is exactly equal to 2, this pair of sets forms a partition, representing the real number we think of as the cube root of 2.

If \(x\) and \(y\) are two real numbers, represented by Dedekind cuts with left sets \(x_L\) and \(y_L\), and right sets \(x_R\) and \(y_R\), we can define operations of addition and multiplication of real numbers in terms of operations on the members of these sets. The left set of \(x+y\) is the set of all rational numbers that result from adding a member of \(x_L\) and a member of \(y_L\), while the right set is the set of all rational numbers that result from adding a member of \(x_R\) and a member of \(y_R\). (It takes a little work to check that every rational number is in fact in one of these two sets, but the other conditions for being a Dedekind cut are straightforward to check.) If \(x\) and \(y\) are both positive, then we can define the right set of \(x\cdot y\) as the set of all rational numbers that result from multiplying a member of \(x_R\) by a member of \(y_R\), with the left set defined as the set of all other rational numbers. (Some modifications of this definition are needed if either \(x\) or \(y\) is negative.) Subtraction and division can then be defined as the inverses of these operations, just as for the rationals.

The real numbers are closed under addition, subtraction, multiplication, and division by all real numbers except 0. They have the further feature that there are no "gaps": for any bounded set of real numbers, there is a least upper bound, and for any continuous function from real numbers to real numbers, if the function is negative at one point and positive at another, there must be some point at which it is exactly equal to 0. For further discussion, see the entry on Dedekind's contributions to the foundations of mathematics.

There are several uses for which we want a number system with no gaps, and thus we use the real numbers. If you try to measure the amount of water in a large vessel by counting out a specific number of small cups, there's no guarantee the number of cups will be an integer. If you try to measure a long distance by counting out the length of your foot, there's no guarantee the number of feet will be an integer. We might know that something is more than 4 of the units and less than 5. By moving to fractions of these units, we can get more precise-4 cups and 5 to 6 ounces, or 4 feet and 5 to 6 inches-but there's still no guarantee that a specific rational number will give the precise amount. But we can generate a sequence of approximations that get closer and closer by using smaller and smaller fractions of these units. So in measuring 'how much' of something there is, or giving coordinates to describe the location of a point in geometric space, we use the real numbers, to guarantee, as we show in the next section, that there is some precise value the sequence of approximations converges to.

3.2 Limits, infinite sums, and the extended real numbers; \(+\infty\) and \(-\infty\)

The mathematical property of 'lacking gaps' is referred to as 'completeness' - the formal statement is that an ordered set is complete if every bounded, increasing sequence of elements has a 'limit'. These limits are the first infinite operation that we define on numbers.

3.2.1 Limits of sequences

A sequence of numbers is an ordered list of numbers, which we may symbolize:

\[a_1, a_2, \dots\]

or

\[\langle a_n\rangle.\]

The members of a sequence are indexed by the natural numbers, \(n=1, 2, \dots\).

The formal definition of a limit says the sequence \(\langle a_n\rangle\) converges to \(l\), or has limit \(l\) if and only if the terms of the sequence eventually stay arbitrarily close to \(l\). Formally:

\[\lim_{n\to\infty} a_n=l\]

iff

for each real number \(\epsilon>0\), there exists a natural number \(N\), such that for every natural number \(n>N\), \(|a_n-l|<\epsilon\).

We will say a bit more about the symbol '\(\infty\)' that appears in this definition later, but for now it just indicates the behavior of the sequence beyond any particular finite point.

As we will demonstrate shortly, not every sequence has a limit, but we can define an important class of sequences that do. A sequence is increasing if each term in the sequence is at least as large as the previous term. A sequence is bounded if there is some real number that is larger than every term in the sequence. It turns out that every bounded, increasing sequence has a limit. The successive approximations to the measurement of some physical quantity with a finer and finer measuring unit will amount to a bounded, increasing sequence of numbers, and thus this definition of a limit allows us to give a numerical representation of any physical quantity.

To show that every bounded, increasing sequence has a limit, consider the Dedekind cuts defining the individual real numbers in the sequence. Let us define a new Dedekind cut by taking its left set to contain every rational in the left set of at least one of these terms of the sequence, and taking its right set to contain every rational that is in all of the right sets of these terms. Because the sequence is bounded, we know that the right set is non-empty, and the rest of the properties of a Dedekind cut are not hard to check.

It is not hard to check that the real number constructed in this way is the limit of the sequence. To see how this works in a specific case, we can consider the sequence \(1/2, 3/4, 7/8, \dots, 1-1/2^n, \dots\) This sequence is increasing, since each term is greater than the one that came before, and it is bounded, since 2 is a number that is strictly greater than every term in the sequence. The Dedekind cut constructed as above will correspond to the number 1. To see this, note that for any rational number q less than 1, we can let \(\epsilon=1-q\). Then there is some \(N\) such that \(1/2^N<\epsilon\). The \(N\)th term in the sequence will then be greater than \(q\), so \(q\) will be in its left set, and thus in the left set we constructed above. But 1 itself, and every rational number greater than it, are all in the right set constructed above. This reasoning also shows that the sequence converges to 1 according to the definition of a limit. For any \(\epsilon\), and an \(N\) such that \(1/2^N<\epsilon\), the \(N\)th term in the sequence is within \(\epsilon\) of 1, and all later terms of the sequence are greater than the \(N\)th term, but still less than 1, so they must also all be within \(\epsilon\) of 1.

This fact about bounded, increasing sequences also underlies the use of infinite decimal notation for real numbers. When we say that the number \(\pi=3.1415926\dots\), we just mean that \(\pi\) is the limit of the sequence \(3, 3.1, 3.14, 3.141, \dots\). One fact about this notation that many people find surprising is that the decimal notation \(0.99999\dots\) is the limit of the sequence \(.9, .99, .999, \dots\), and thus is precisely 1. Some people feel the intuition that \(0.9999\dots\) should somehow denote a number 'infinitely close' to 1, but not equal to it. We will be able to make sense of an idea like this in section 3.4, but it turns out that decimal notation is not the way to do it. For a useful demonstration of this point, see Vi Hart's video 9.999... reasons that .999... = 1.

Many sequences that are not increasing have limits as well. For instance, the sequence \(1, -1/2, 1/3, -1/4, \dots, (-1)^n/n, \dots\) can be seen to converge to the value 0, even though it is not increasing. However, if a sequence is not bounded, like the sequence \(1, 2, 3, \dots\), it does not have a limit-if it had a limit, there would have to be some values \(l\), \(\epsilon\), and \(N\), such that all terms in the sequence beyond the first \(N\) are within \(\epsilon\) of \(l\). But then any number that is larger than the first \(N\) terms of the sequence, and also larger than \(l+\epsilon\), would be a bound for the sequence. And some sequences that are not increasing also fail to have a limit-for instance, the sequence \(1, 0, 1, 0, 1, 0, \dots\) does not have a limit, because there is no value such that all terms of the sequence are eventually within \(1/3\) of that value.

3.2.2 Infinite sums and products

With the definition of the limit of a sequence, we can now also often define infinitary versions of the operations of addition and multiplication. For a finite number of terms, we define the 'partial sum' \(\sum_{i=1}^n a_i=a_1+\dots+a_n\) and 'partial product' \(\prod_{i=1}^{n} a_i=a_1\cdot\dots\cdot a_n\). For an infinite sequence of numbers \(a_n\), their infinite sum or product (when it is defined) is the limit of the partial sums or products:

\[\sum_{i=1}^\infty a_i=\lim_{n\to\infty}\sum_{i=1}^n a_i\]

and

\[\prod_{i=1}^\infty a_i=\lim_{n\to\infty}\prod_{i=1}^n a_i.\]

Thus, \(\sum_{i=1}^\infty\frac{1}{2^i}=\lim_{n\to\infty}\sum_{i=1}^n\frac{1}{2^i}=\lim_{n\to\infty}(1-\frac{1}{2^n})=1\). We can show that if an infinite sequence of terms has a sum that converges, then the terms themselves must converge to 0. That's because the terms \(\sum_{i=1}^n a_i\) must converge, so that for any \(\epsilon\), there must be \(N\), such that every one of the \(\sum_{i=1}^n a_i\) is within \(\epsilon\) of the limit whenever \(n>N\). Thus, every such \(a_n\) must have absolute value less than \(2\epsilon\), so that both successive partial sums can be within \(\epsilon\) of this limit. (A similar condition holds also for infinite products, but from now on we will focus only on infinite sums.)

However, just having the terms \(a_n\) converge to 0 is not sufficient for the sum to converge. A deep and important fact about infinite sums is that \(\sum_{i=1}^\infty\frac{1}{i}\) fails to converge, because the partial sums eventually exceed any finite bound. To see this, note that the first term is greater than \(1/2\), the next two terms are both greater than \(1/4\), the next four terms are all greater than \(1/8\), and in general there are \(2^{n-1}\) terms that are each greater than \(1/2^n\). So to get a partial sum greater than \(n\), it is sufficient to add the first \(2^{2n}\) terms.

But if the terms \(a_n\) converge to 0, and each one is smaller in absolute value and has the opposite sign as the previous term, then the infinite sum must converge. If the first term in the sequence is positive, this is because the even numbered partial sums form a bounded, increasing sequence, and the odd numbered partial sums form a bounded, decreasing sequence, and the difference between successive terms of these two sequences is a term of the original sequence and thus converges to 0. Thus, the sum \(\sum_{i=1}^\infty\frac{(-1)^{i+1}}{i}\) converges (in particular, to the natural logarithm of 2). But a somewhat surprising fact, known as the Riemann rearrangement theorem, states that if the positive terms of a sequence have no finite sum, and the negative terms of the sequence have no finite sum, but the individual terms themselves converge to 0, then for any real number \(x\), the terms of the sequence can be put into some order so that \(x\) is the sum of the sequence in that order! To prove this, just rearrange the terms by beginning with enough positive terms to bring a partial sum above \(x\), then enough negative terms to bring a partial sum below \(x\), then enough partial sums to bring a partial sum above \(x\) again, and so on. This process must be able to be carried out, because the partial sum of the positive terms eventually exceeds any finite bound, and similarly with the partial sum of the negative terms. Since the individual terms of the sequence are all eventually within \(\epsilon\) of 0, these partial sums must eventually never differ from \(x\) by more than \(\epsilon\), and so the sum of this arrangement of the terms must converge to \(x\).

Thus, infinite summation has some importantly different features from finite summation. For any finite set of real numbers, the sum of those numbers is well-defined, and doesn't depend on the order you add them. But with an infinite sequence of real numbers, there may be no number that is the sum of that sequence in that order. And even if there is, it may be possible to rearrange the terms of the sequence so that they sum to another value.

3.2.3 Limits of functions and the extended real numbers \(\pm\infty\)

Infinite sequences and sums aren't the only ways that limits appear in mathematics. Functions of a real value can also have limits. The limit of a function of a real value \(x\) as \(x\) goes to infinity is defined in a similar way to the limit of a sequence indexed by natural numbers. To say

\[\lim_{x\to+\infty} f(x)=l\]

is to say that for every \(\epsilon\), there is an \(N\), such that whenever \(x>N\), \(f(x)\) is within \(\epsilon\) of \(l\). For example, \(\lim_{x\to+\infty}e^{-x}=0\), because \(e^{-x}\) can be made as small as one likes by choosing large enough \(x\). (Note that the inputs to a function can be positive or negative, so we need to specify that \(x\) approaches positive infinity to distinguish this from the limit at the other end of the axis.)

But it is also often useful to be able to define the limit of a function at some particular finite real-valued input. For instance, we might be interested in the function \(f(x)=\frac{x^2-9}{x-3}\) as \(x\) approaches 3. (As we will see in section 3.4, this sort of calculation is particularly important in defining the concept of 'derivative' of a function, giving the slope of a continuous curve at a point.) This particular function is defined for all real numbers other than 3, and at any such input \(x\) it takes the value \(x+3\). We would like to be able to say that the limit of this function as \(x\) approaches 3 is 6. The way we make this precise is to say that

\[\lim_{x\to a} f(x)=l\]

iff

for each real number \(\epsilon>0\), there exists a \(\delta\), such that for every \(x\) with \(0<|x-a|<\delta\), \(|f(x)-l|<\epsilon.\)

That is, for any degree of approximation to the limit that we want, there is some degree of approximation for the input that suffices to guarantee that the function is that close. In the initial definition of a limit as \(x\) goes to \(+\infty\), we required \(x\) to 'approximate' \(+\infty\) by being sufficiently large, but now we require it to approximate \(a\) by having a difference from \(a\) that is sufficiently small, just as the values of the sequence or function approximate the limit \(l\).

We can do the reverse to put \(\infty\) on the right side of the limit as well. That is, we say

\[\lim_{x\to a} f(x)=+\infty\]

iff

for every \(M\), there exists a \(\delta\), such that for every \(x\) with \(0<|x-a|<\delta\), \(f(x)>M\),

and similarly

\[\lim_{x\to+\infty} f(x)=+\infty\]

iff

for every \(M\), there exists an \(N\), such that for every \(x>N\), \(f(x)>M\).

Since \(\infty\) (or more precisely '\(+\infty\)'-similar methods work for '\(-\infty\)') can appear in each place where a real number can appear in this limit notation, it is natural to see if we can extend the definition of real numbers so that it can be included.

And in fact, if we take the definition of a Dedekind cut, and relax the requirement that the left and right sets be non-empty, we get two new elements-the one with an empty right set is greater than every rational number, and called '\(+\infty\)', while the one with an empty left set is less than every rational number, and called '\(-\infty\)'. Within these 'extended real numbers', not only does every bounded increasing sequence have a limit, but every increasing sequence has a limit.

Just as the real numbers emerge naturally as the tools to measure finite quantities, as the limits of rational approximations, the extended real numbers emerge naturally as the tools to measure potentially infinite quantities that can be approximated by finite quantities. \(+\infty\) can be taken to be the area of an infinite region, the length of an infinite line, the limit of \(1/x^2\) as \(x\) goes to 0, and so on. Although we are used to thinking of lengths and areas as positive numbers, it is sometimes useful to consider them as negative as well, when we care about the direction that they are pointing, and in this sort of situation \(-\infty\) is useful as well. Just as the real number operations of addition, multiplication, subtraction, and division correspond to certain operations on the quantities they measure, these operations can often be extended to these extended real numbers, as long as we are careful about a few cases.

Adding or subtracting a finite area from an infinite area leaves it infinite. Adding a shape with infinite area to another shape of the same infinite area leaves the total area unchanged, and subtracting a negative infinite one from a positive one or vice versa is similar. But \((+\infty) + (-\infty)\) cannot be meaningfully evaluated; nor can \((+\infty) - (+\infty)\). If you take one infinitely large region, and remove an infinitely large region, you might be left with nothing, or a positive region, but you might still be left with an infinitely large region; or if the region you subtracted was larger than the original region, you might be left with a negative region-even an infinite negative region.

These restrictions also apply to the use of these extended real numbers as the limits of sequences or functions. Whenever two sequences or functions both have a finite limit, their sum or difference will have a limit that is the sum or difference of their limits. When one has a finite limit and the other is infinite, then their sum or difference will be determined by the one that is infinite. But when both are infinite, there are problems. We can see that \(1/x^2\), \(2+1/x^2\), and \(1/x^4\) are all functions that go to \(+\infty\) as \(x\) goes to 0. If we add or subtract any of these functions from a function with a finite limit, the resulting function still has limit of \(+\infty\). If we add them to each other in any combination, the result still has limit \(+\infty\). But if we consider their differences, we see that \(1/x^2 - 1/x^2\) has 0 as its limit, \(1/x^2 - 1/x^4\) has \(-\infty\) as its limit, and \(1/x^2 -(2+1/x^2)\) has \(-2\) as its limit. So "\(\infty - \infty\)" is said to be an "indeterminate form" that can't be evaluated.

Multiplying or dividing an infinite number by a finite positive number leaves it unchanged, and multiplying or dividing by a finite negative number reverses its sign. Similarly for multiplying the infinite numbers by themselves or each other. But an infinite number divided by an infinite number, or an infinite number multiplied by 0, are also indeterminate forms. As \(x\) approaches 0, the function \(\frac{1/x^2}{1/x^2}\) has limit 1, while the function \(\frac{1/x^2}{1/x^4}\) has limit \(+\infty\). If we take the function \(1/x^2\) whose limit is \(+\infty\) and multiply it by the function \(x\) whose limit is 0, we get the function \(1/x\), that has no limit as \(x\) approaches 0 (since it takes on both large positive and large negative values in any small interval around 0-this is why we have used \(1/x^2\) and \(1/x^4\) as the paradigms of functions with limit \(+\infty\), rather than \(1/x\) or \(1/x^3\)). For similar reasons, these extended real numbers don't provide a way to divide by 0. So although the extended real numbers have some nice properties, and can be used for measurement in various cases, arithmetic involving them is not as nice as the standard real numbers.

3.2.4 Related infinities

The Dedekind cut construction was done as a way to make sense of limits of the rational numbers. This first created the real numbers, which can be thought of as the limits of bounded infinite sequences of rational numbers. We then considered all limits that made sense, including towards the ends of the real line, yielding the extended reals, including the standard reals as well as \(+\infty\) and \(-\infty\).

Versions of this process can be carried out with other mathematical entities as well. Projective geometry adds additional points 'at infinity' to the Euclidean plane, one for each family of parallel lines, to help explain features of visual geometry, like the way that parallel railroad tracks appear to meet at the horizon, infinitely far away. Riemannian geometry extends the complex numbers by adding just a single number \(\infty\) that one can approach simultaneously 'in all directions' in the complex plane. These alternate geometries provide foundations for material discussed in section 2.2, and also section 8.2. Several of these are discussed in the entry on nineteenth century geometry, and others are discussed in topology textbooks under the topic of 'compactification'.

Because these infinities are inherently considered as limits of finite approximations, there is no way for one infinite element to lie "beyond" another-at most it can lie in a "different direction", the way that the points of convergence of different families of parallel lines do, or the way \(+\infty\) and \(-\infty\) do in the extended reals.

But as we will shortly see, there are other notions of infinity for which one infinity can lie "beyond" another. In section 3.3, we will discuss the ideas of infinity that we get from generalizing the use of the natural numbers for counting, rather than generalizing the use of the real numbers for measuring. And in section 3.4, we will discuss yet another mathematical theory of infinity, which arises from an alternate formulation of calculus, where the \(\epsilon\)'s and \(\delta\)'s are treated as actually being infinitely small, rather than just being arbitrary finite measures of smallness.

3.3 Infinities of counting

3.3.1 Preliminaries

As mentioned above, this section is more mathematically dense than the previous two. However, we need this level of mathematical rigor to develop Cantor's theory of ordinals and cardinals, which are widely regarded as the most significant mathematical advance in our understanding of the infinite.

An insomniac, counting imaginary sheep to try to get to sleep, will never run out of natural numbers to do the job: 1, 2, 3, .... There is no bound on the set of natural numbers. This is our first infinite set. It is perhaps a natural thought that there is just one infinity for counting infinite sets, which we might symbolise '\(\infty\)'. The thought may seem even more natural when we define an infinite set as one that has the same size (in a sense to be made precise shortly) as a proper subset of it. In fact, the thought could hardly be more mistaken: as we will soon see, according to mathematical orthodoxy-namely contemporary set theory and the attendant notion of cardinality of a set-there are infinitely many infinities. This prompts a series of questions: Is there a smallest one? Yes, as we will see. Is there a largest one? No, as we will see. What can be said about the spacing between the infinities and how far infinities extend? Well, we will see. We can also ask parallel questions about the infinitely small.

Recall Galileo's paradox of infinity, based on the collision of the part-whole intuition and the bijection intuition, and his conclusion that one cannot apply the relations of 'less than', 'equality', and 'greater than' to infinite collections. Modern mathematical orthodoxy, embodied in contemporary set theory, rejects Galileo's conclusion. That orthodoxy is grounded in the bijection intuition, following Cantor rather than Euclid and Bolzano. When there is a bijection between two sets, we say that they have the same cardinality. The notion of cardinality does not respect the part-whole intuition. For example, the squares of natural numbers are a proper subset of the natural numbers, but they have the same cardinality since they can be put in one-to-one correspondence.

Foundational programs such as neo-logicism also start from a notion of "equinumerosity" based on the Cantorian bijection intuition. It was not until the early 2000s that a group of mathematicians working on non-standard analysis (Benci, di Nasso, and Forti) developed a theory of "numerosities" that agrees on finite sets with Cantorian cardinality but that also upholds the part-whole intuition for infinite sets, and thus diverges from Cantorian cardinalities. (See Benci and Di Nasso 2003, 2019; Benci, Di Nasso and Forti 2006, 2007). One can consider numerosities as a refinement of Cantorian cardinalities. Every two sets that have the same numerosity have the same cardinality, but the converse does not hold. For instance, in this approach the set of squares has numerosity strictly less than the set of natural numbers. See the

Supplement on Theories of Numerosities.

3.3.2 Set theory: \(\omega\) and \(\aleph\)

The sort of infinities that are most familiar to philosophers are infinities used for counting. In the early 1820s, Bolzano arrived at the idea that an infinite set is one for which there is a bijection between the set and a proper subset of it. (Recall Galileo's paradox.) Dedekind (1884) gave this as a definition of being infinite. It is easy to prove that a Dedekind-infinite set must contain a set that is just as large as the natural numbers. See the

Supplement on Proofs of Theorems.

Dedekind's definition is only one among several alternative definitions of infinite set (and correspondingly of finite set) that have been proposed by him and after him. If one assumes the Axiom of Choice, these alternative definitions turn out to be equivalent. (This axiom says that for every set \(A\) of pairwise-disjoint non-empty sets, there exists a function that selects exactly one element from each set in \(A\).) But without the Axiom of Choice, the definitions can be shown not to be equivalent and the foundational situation is rather delicate but well understood (see Moore 1982).

The modern theory of infinities of counting derives from Cantor (1932). He observed that a natural way to set up a bijection between finite sets is to order the elements of each set, and pair the first element of one set with the first of the other, the second of one set with the second of the other, and so on. This sometimes works with infinite sets-for example, it gives the one-to-one correspondence between the natural numbers and the squares (considered under their standard ordering). When there is a one-to-one correspondence between two sets, such that every pair of elements of one set bears the same ordering as the corresponding pair of elements of the other set, the two ordered sets are said to have the same order type.

We can reorder the integers, alternating between positive and negative: 0, -1, 1, -2, 2, -3, 3, .... This ordering has the same order type as the natural numbers, and thus enables a one-to-one correspondence between the natural numbers and the integers. One of Cantor's most striking early observations is that the same is possible with the positive rational numbers. Every positive rational number can be written uniquely in lowest terms as some fraction \(p/q\), where \(p\) and \(q\) are positive integers with no common factor. We can then order these fractions by first comparing the sum \(p+q\) of numerator and denominator, and then if the sum is equal for two fractions, put the one with lower numerator \(p\) first. This ordering begins 1, 1/2, 2, 1/3, 3, 1/4, 2/3, 3/2, 4, 1/5, 5, 1/6, 2/5, 3/4, 4/3, 5/2, 6, .... (Note that fractions like 2/2, 2/4, 3/3, 4/2, are missing from this list, because they are not written in lowest terms.) Every positive rational number must appear in this list (because it can be written in lowest terms with some particular finite sum of numerator and denominator), and only has finitely many predecessors (because there are at most \(n+1\) fractions whose sum of numerator and denominator is \(n\)).

The ordering: 1, 1/2, 2, 1/3, 3, 1/4, 2/3, 3/2, .... Every rational number is eventually included.

However, Cantor also observed that different well-orderings of the same infinite sets will produce a different order type. For instance, we can define an ordering on the integers where every non-negative integer comes before every negative integer, with any two integers of the same sign being ordered by their absolute value. To approximately represent this ordering we could write it as \(0, 1, 2, 3, \dots, -1, -2, -3, \dots\). In this ordering, every non-empty subset still has a first element (the non-negative element of lowest absolute value if it contains any non-negative elements, and the negative element of lowest absolute value if it only contains negative elements). If we pair up the first element of this ordering with the first of the standard ordering on the natural numbers, and the second with the second, and the third with the third, and so on, then the negative integers would not be paired with any natural number. But we can put the natural numbers into the same order type by declaring the odd numbers to be before the even numbers, and ordering them by size within these two sets: \(1, 3, 5, \dots, 0, 2, 4, \dots\). A single infinite set can be given orderings of many different order types, and also different orderings of the same order type (for instance if we put the even numbers first and the odd numbers second).

Cantor noted that for any two well-ordered sets, the initial positions in one ordering (the first, the second, the third, etc.) correspond to the initial positions in the other, the way that they do for finite sets. In fact, he showed that all of the positions of one well-ordering must correspond to initial positions in the other. (If this weren't true, then the set of positions in one that don't correspond to positions in the other would be non-empty for each set, and the first elements of these sets would correspond, which would contradict the claim that these positions don't correspond.) Thus, there is a single list of all the possible positions in well-ordered sets, beginning with the first, second, third, and so on, and these positions are called the ordinal numbers. A well-ordered set can be said to have its own ordinal number, which is the first ordinal number that does not correspond to a position in that set.

Cantor used lowercase Greek letters to represent infinite ordinal numbers, with \(\omega\) representing the order type of the standard ordering of the natural numbers. Addition of ordinal numbers corresponds to the order type that results from taking an ordering of the first type, followed by an ordering of the second type. Thus, \(\omega+\omega\) represents the order type of the integers with the non-negative numbers first and the negative numbers afterwards, while \(\omega+1\) represents the order type of the natural numbers with just one element put at the end. Note that \(1+\omega\) is the order type of a single element, followed by a copy of the natural numbers, which is in fact the same as the order type of the natural numbers! Thus, \(1+\omega=\omega\), which is not equal to \(\omega+1\). So ordinal addition is not commutative.

Von Neumann (1923) defined a canonical representation of ordinal numbers, using the fact that the ordinals are themselves well-ordered. Each ordinal is represented by the set consisting of all smaller ordinals. Thus 0 is represented by the empty set \(\emptyset\), 1 by the set containing the empty set \(\{\emptyset\}\), 2 by the set containing both of those \(\{\emptyset,\{\emptyset\}\}\), and so on. \(\omega\) is then the set containing all of these finite ordinals \(\{\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\},\dots\}\), \(\omega+1\) is the set containing it as well \(\{\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\},\dots,\omega\}\), and so on.

Multiplication of ordinal numbers corresponds to replacing each element of the second ordering by an entire ordering of the first type. \(\omega\cdot\omega\) represents the order type of an \(\omega\)-sequence of \(\omega\)-sequences, which we can have by taking the positive rational numbers and sorting them first by denominator, and then by numerator: 1, 2, 3, ... 1/2, 3/2, 5/2, ..., 1/3, 2/3, 4/3, 5/3, ..., 1/4, 3/4, 5/4, .... Note that \(\omega\cdot 2\) is 2 copies of \(\omega\), while \(2\cdot\omega\) is \(\omega\) copies of 2. \(\omega\cdot 2\) is thus \(\omega+\omega\), while \(2\cdot\omega\) is \(2+2+2+\dots\), which is \(\omega\). (This is the difference between the orderings 1, 2, 3, ..., −1, −2, −3, ... and 1, −1, 2, −2, 3, −3, ....) So again, ordinal multiplication is not commutative.

But for any ordinal, one can generate another ordinal by putting one more element at the end. And for any increasing sequence of ordinals, there is a limit of that sequence. Cantor also defined a notion of exponentiation for ordinals, and this gives many different ordinal numbers: 0, 1, 2, 3, ... ,\(\omega,\) \(\omega+1,\) \(\omega+2,\) ..., \(\omega+\omega(=\omega\cdot 2),\) \(\omega\cdot 2+1,\) \(\omega\cdot 2+2, \dots,\) \(\omega\cdot 3,\) \(\omega\cdot 4,\) ..., \(\omega\cdot \omega(=\omega^2),\) ..., \(\omega^3,\) ..., \(\omega^4,\) ..., \(\omega^\omega,\) \(\omega^{\omega+1},\) \(\omega^{\omega+2},\) ..., \(\omega^{\omega\cdot\omega}=\omega^{\omega^2},\) ..., \(\omega^{\omega^3},\) ..., \(\omega^{\omega^\omega},\) \(\epsilon_0\) (defined as the limit ordinal for the sequence \(\omega,\) \(\omega^\omega,\) \(\omega^{\omega^\omega},\) ...). But these ordinal numbers all correspond to the same cardinal number, which is that of the natural numbers.

Supplement on Proofs of Theorems.

Thus, just as there is an infinite hierarchy of infinite ordinal numbers, which Cantor represented with lowercase Greek letters, there is also an infinite hierarchy of infinite cardinal numbers, which Cantor represented with Hebrew letters, and in particular aleph, "\(\aleph\)". The finite cardinals are 0, 1, 2, 3, .... The first infinite cardinal, that of the natural numbers (and all countable sets), is \(\aleph_0\). Cantor's "well-ordering principle", stating that every set can be put into some well-ordered form (and equivalent to the Axiom of Choice), implies that every cardinal number can be represented by an ordinal number, and the definition of ordinal numbers ensures that for any non-empty set of ordinal numbers there is always a first. So the cardinals must in fact be well-ordered. So Cantor used ordinal numbers to designate each cardinal's position in their ordering. \(\aleph_1\) is the first cardinal beyond \(\aleph_0\), \(\aleph_2\) is the next beyond that, and after \(\aleph_3\), \(\aleph_4\), and so on, we eventually reach \(\aleph_\omega\), \(\aleph_{\omega+1}\), \(\aleph_{\omega+2}\), and so on with one cardinal for every ordinal.

Because we have this one-to-one correspondence between the cardinals and the ordinals, one might be tempted to say that the set of cardinals and the set of ordinals have the same order type, and then ask what the ordinal of this order type (and its cardinality) is. However, if there were such an ordinal, there would be a paradox-it would have to contain, and thus be larger than, all ordinals, including itself! This is the Burali-Forti paradox (see entry paradoxes and contemporary logic).

Relatedly, since every set has a cardinality less than that of its power set, there can be no set that contains everything (since such a set would already include all its subsets, and thus be at least as large as its power set). The two results imply that there cannot be a set of all ordinals or a set of all sets. In a similar way one argues that there cannot be a set of all cardinals. As a consequence of the above results we can also answer some of the questions we raised at the beginning: there are infinitely many cardinal numbers and infinitely many ordinal numbers. However, there is neither a set of all cardinal numbers nor a set of all ordinal numbers. Thus, the infinity of the cardinals and of the ordinals cannot be measured by a cardinal or an ordinal, for otherwise a paradox would ensue. (See also entry on Russell's paradox.)

Although it took several decades from Cantor's work to find a system of axioms for set theory that avoids these paradoxes (see the entries on the early development of set theory and set theory), Cantor already saw, in this unreachability of the totality of all ordinals or cardinals, a notion of "absolute infinity". Although in his system, there are many infinite sets that are tractable and graspable at many different levels, starting with the natural numbers (which Aristotle had thought was merely potential rather than actual), he discovered an even greater Aristotelian potential infinity. This led to the distinction between a "set" as a totality that can be grasped in the relevant sense, and a "proper class", which is too large even for a system that encompasses each of Cantor's many infinities.

We can define addition for cardinal numbers as the cardinality of a set that is the union of two disjoint sets of those cardinalities. We can define multiplication for a pair of cardinal numbers as the cardinality of the set of all ordered pairs whose first elements are drawn from a set of the first cardinality and whose second elements are drawn from a set of the second cardinality. But it turns out that these operations are relatively trivial once we get beyond the finite cardinals-just as we saw that the sum or product of two countable ordinals was still countable, the sum or product of two infinite cardinals is equal to whichever of the two is larger! (At least this operation is commutative.) Thus, one can't get to \(\aleph_1\) from \(\aleph_0\) by means of addition or multiplication (as we saw when we considered order types achieved by addition and multiplication).

However, cardinal exponentiation is more powerful. It turns out to be natural to define \(2^\kappa\) as the cardinality of the power set of a set whose cardinality is \(\kappa\). (Greek letters \(\kappa\) and \(\lambda\) are traditionally used as variables to represent infinite cardinals, with letters \(\alpha\) and \(\beta\) used for ordinals.) It turns out that the cardinality of the set of real numbers (otherwise called the "continuum" for its role in representing continuous space) is the same as that of the power set of the natural numbers, \(2^{\aleph_0}\), and Cantor showed that \(2^{\aleph_0}>\aleph_0\). Cantor conjectured that \(2^{\aleph_0}\) was in fact equal to \(\aleph_1\), and this conjecture was called the "Continuum hypothesis". (See the entry on the continuum hypothesis for more on this conjecture and why it hasn't been settled.)

Just as \(\aleph_1\) is a cardinality that can't be reached by the operations of ordinal addition, multiplication, or exponentiation, even in the limit, one might conjecture that even before one reaches the Cantorian absolute infinity, there are further cardinalities that can still be grasped in some sense, but can't be reached even by the stronger operation of cardinal exponentiation, even in the limit. Such conjectures have turned out to be surprisingly fruitful for the study of set theory and mathematical logic generally. (See the entry on large cardinals and determinacy.)

Although we have defined addition and multiplication for both ordinals and cardinals, their features make it hard to make sense of subtraction or division. First, the non-commutativity of the ordinal operations, and the triviality of the cardinal operations, makes it hard to define meaningful inverses of these operations. (If it's possible to find an ordinal or cardinal that can be added or multiplied to a first one to get a second one, then it is usually possible to find infinitely many that can.) But more importantly, the conceptual ideas of counting (whether by order type or by bijection) don't really allow for negatives or fractions, the way that the conceptual ideas of measuring and geometry do (considered in section 3.2). Counting involves treating each element as discrete and unique, and there is no way for multiple elements to combine to yield nothing (as subtraction requires) or to yield a unit (as division does). However, measurement (e.g., of distance) involves a sort of structure on the thing being measured so that some measurements can be fractions of others, or can point in the opposite direction, which yields meaningful notions of division and subtraction.

This section has given just an introductory taste of the mathematics of ordinal and cardinal numbers-it has come to be called Cantor's Paradise. But we have already seen many characteristic features of these notions of infinity that differentiate them starkly from those discussed in the previous section. Just as the differences between the natural numbers and the real numbers demonstrate the differences between counting elements of sets, and measuring lengths, areas, and so on, the differences between Cantor's cardinals and the extended reals further demonstrate differences between counting and measuring.

For further discussion of the kind of material presented in this section, see the SEP entries on: set theory, the early development of set theory, and the axiom of choice. An excellent introduction to basic set theory is Enderton (1977); an informal but still rigorous introduction is Sheppard (2014); more advanced texts include Devlin (1993), Kunen (1983), and Jech (2006). For the higher reaches, to which we will come back in section 4, see Kanamori (2003).

Set theory provides a theory of cardinality that implements the idea that "sameness of size" upholds the bijection intuition. The recent theory of numerosities, developed by Benci, Di Nasso, and Forti, by contrast, upholds the part-whole intuition. See the

Supplement on Theories of Numerosities.

3.4 Infinitesimals and hyperreals

We have already discussed a notion of \(+\infty\) and of \(-\infty\) that is designed to provide values for real-valued functions to take as limits at special points. But the understanding of limits themselves was originally thought to require a notion of "infinitesimally small" distances. While these quantities were considered problematic for several centuries, in recent decades some mathematical entities with their properties have been rigorously studied.

3.4.1 Newton and Leibniz

An infinitesimal is a number smaller in absolute magnitude than any positive finite number, and yet not zero. Infinitesimals have had a chequered history. Early work in the calculus, as we have seen when presenting the structure of L'Hôpital's 1696 treatise, was mostly based on a geometrical or kinematic (i.e. based on motion) understanding of infinitesimals. Here we treat infinitesimals arithmetically, i.e. within the context of the real number system, while conveying the key features of how the early analysts made use of them. To figure out the slope of the function \(f(x)=x^2\) precisely at one point, one considered an "infinitesimally small number" \(\epsilon\) and considered the slope of the straight line through \(f(x)\) and \(f(x+\epsilon)\).

To find the slope of a function, one considers the slope of a straight line that intersects it at two "infinitesmally close" points.

This slope is equal to \(\frac{(x+\epsilon)^2-x^2}{\epsilon}\). In order for this fraction to make sense, \(\epsilon\) must be non-zero. However, we can calculate that this value is \(\frac{2x\epsilon+\epsilon^2}{\epsilon}\), or \(2x+\epsilon\). At this point, we no longer need \(\epsilon\) to be non-zero, so the slope can be said to be just \(2x\). This sort of slippage between non-zero and zero for these infinitesimals is what made Berkeley refer to them as "the ghosts of departed quantities". However, engineers, scientists and mathematicians who actually made use of the calculus rested content in the knowledge that the calculus delivered the goods.

In the 19th century, Cauchy, Bolzano, Weierstrass, Dedekind and Cantor sought to establish foundations for real analysis that gave no role to infinitesimals: in what became the canonical account of the calculus, Cantorian set theory and the \(\epsilon\)-\(\delta\) formalization of the notion of a limit described in section 3.2 allowed for a fully rigorous development of real analysis.

Instead of taking particular infinitesimal values, one quantifies over values of real-valued variables \(\epsilon\) and \(\delta\). For example, the claim that the slope of the squaring function at \(x\) is \(2x\) is interpreted as saying that for any desired degree \(\epsilon\) of approximation, there is some finite \(\delta\) such that for any \(x'\) within \(\delta\) of \(x\), the slope of the line from \((x,x^2)\) to \((x', x'^2)\) approximates \(2x\) to within \(\epsilon\). A single infinitesimal is replaced by a relation involving two nested quantifiers. The process of elimination of infinitesimal quantities from the calculus was a central part of a larger process known as the "arithmetization of analysis", which aimed at removing kinematical and geometrical notions from the calculus in favour of purely arithmetical notions. (These are broadly construed to include the arithmetic of real and complex numbers. For a recent account of the history of real and complex analysis in the 19th century that also pays attention to foundational issues see Gray (2015).)

This infinitesimal-free program accomplished its goals successfully-among its major accomplishments were the rigorous definition of a continuous function at a point and the definition of the Riemann integral. However, one should keep in mind that other areas of mathematics, such as geometry, continued exploiting infinitesimal considerations and studied extensively non-Archimedean number systems. Archimedes' axiom states:

given any two areas, or two distances, or any two quantities of the same sort, say \(A\) and \(B\), it is possible to add \(A\) to itself finitely many times so that the quantity obtained is greater than \(B\).

Non-Archimedean systems are ones in which this axiom does not hold. If the 17th century engagement with infinity, and Cantor's work in set theory, can be seen as revolutions, the study of non-Archimedean mathematics in the second half of the 19th century can be likened to an infinitary uprising.

Many of the quantities considered in 17th century calculus, such as Leibnizian infinitesimals, do not obey the Archimedean axiom. An infinitesimal can be added to itself any finite number of times but the outcome of that process will never be greater than any finite quantity, however small. A pervasive historiographical tradition has argued that with the elimination of infinitely small quantities from the calculus, non-Archimedean quantities were relegated to engineering practice for a long time. According to the standard account, it was only in the 1960s that infinitesimals came back when Abraham Robinson presented his theory of non-standard analysis, which has received a lot of attention from philosophers and mathematicians (see section 3.4.2). Robinson's theory gave a legitimate mathematical status to infinitely small and infinitely large quantities in the reconstruction of the infinitesimal calculus, now developed accordingly to rigorous model-theoretic techniques. For a long time, Robinson's work was hailed as the first successful effort to develop a system of non-Archimedean quantities. But Philip Ehrlich, in a series of fundamental papers (including 1994, 2012), has argued that this widespread perception needs serious questioning. Indeed, he has shown convincingly that interest in non-Archimedean mathematics emerged in the 1870s and continued to grow in the hands of mathematicians such as Veronese, du Bois-Reymond, Levi-Civita, Hahn, Stolz, Hardy, and others.

It would be out of place in this entry to attempt even a small survey of the developments mentioned above. We simply refer to the reader to Ehrlich (1994) and (2006). The interconnection with many important related issues such as Conway's surreal numbers (see section 3.4.3) and other alternative approaches to the construction of the real numbers, such as smooth infinitesimal analysis mentioned at the end of section 3.4.2, cannot be properly addressed here. See Salanskis and Sinaceur (1992), Ehrlich (1994), Berger, Osswald, and Schuster (2001), and Ehrlich (2012).

3.4.2 Non-standard analysis and infinitesimal analysis

As a result of the rigorous definitions in the calculus mentioned above, from the mid-19th century most mathematicians working in analysis abandoned infinitesimals. However, in the mid-20th century, Robinson (1966) showed that it is possible to give a rigorous definition of infinitesimals, and that infinitesimals can be used in a non-standard development of real analysis (D. Laugwitz also did similar work around the same time, but Robinson's system has been more widely discussed). While he developed his non-standard analysis using model theory, subsequent developments have also been grounded in algebra and topology. Robinson's approach supplies an extended number system-the hyperreal number system-that contains the standard real number system, plus further 'infinitesimal' numbers whose absolute values are greater than 0, but less than any positive standard real number. Robinson's construction of the hyperreals provides a set with the same cardinality as the standard reals. Simple modifications of the construction can create sets of hyperreals of larger cardinality.

Importantly, due to the logical techniques used in its construction, Robinson's system behaves exactly like the standard finite real numbers for any sentence expressible within the algebraic language of addition and multiplication. Thus, every number other than 0 has a multiplicative inverse, and if \(x > y\), then \(1/y > 1/x\). In particular, this means that if \(\epsilon\) is a positive infinitesimal number, then \(1/\epsilon\) is an infinitely large number! Unlike Cantorian infinities of counting from section 3.3 these infinitely large numbers are subject to subtraction and division as well as addition and multiplication, and unlike the infinities of the extended real line from section 3.2, they behave just as nicely as the finite numbers with respect to them. For instance, statements such as these hold of standard real numbers as well as of the new infinitely small and infinitely large numbers:

\(x+y=y+x\) (commutativity of addition);
\(x\cdot y=y\cdot x\) (commutativity of multiplication);
\(x(y+z)=xy+xz\) (distributivity of multiplication over addition).

In fact, Robinson's hyperreals satisfy a "transfer principle"-if statements are formulated entirely within a first-order language for the reals, then they are true of the standard reals if and only if they are true for the hyperreals. So any proof of such a theorem in one system can be transferred to the other. This sometimes greatly simplifies calculations and proofs of theorems.

Consider the limit of the quantity \(((x+h)^3-x^3)/h\) as \(h\) approaches 0. In the standard reals, to show that this is \(3x^2\), we need to show that for every \(\epsilon\) there is a \(\delta\) such that for any value of \(h\) less than \(\delta\), the corresponding value of this function is less than \(\epsilon\) away from \(3x^2\). In this case, it turns out that choosing \(\delta < \epsilon/4x\) works when \(x\) is sufficiently large, and choosing \(\delta < \epsilon\) works when \(x\) is sufficiently small, but figuring out these choices is difficult.

For the hyperreals though, it is sufficient to show that this value is infinitesimally close to \(3x^2\) whenever h is infinitesimal.

\[\begin{align} \frac{(x+h)^3-x^3}{h} &= \frac{x^3+3x^2h+3xh^2+h^3-x^3}{h} \\ &= 3x^2+3xh+h^2, \end{align}\]

and for any real \(x\), \(3xh+h^2\) is infinitesimal whenever \(h\) is infinitesimal.) For any particular real \(\epsilon\), this shows that there is some hyperreal \(\delta\) that works (namely, any infinitesimal), and by the transfer principle we can conclude that for this real \(\epsilon\) there is some real \(\delta\) that works, and we no longer need to worry about the details of how to find it. Thus, we can validate the reasoning of Newton and Leibniz that allows them to treat infinitesimals as non-zero for calculations until we get to the final result, and then treat them as zero at the end. They really work like the "ghosts of departed quantities" that Berkeley satirized! (The extent to which Robinson's system is a vindication of Leibniz and Newton was the subject of extended discussion in articles by Robinson and others. See Bos (1974) for a classic source of this debate.)

For results stated in a first-order logical language, the hyperreals and the standard reals satisfy the transfer principle. But for results about sets, they behave differently. Every bounded set of standard reals has a least upper bound. However, for instance, the set of infinitesimal hyperreals is bounded (every member is less than .00001, among other bounds), but there is no least upper bound (no infinitesimal is an upper bound for all of the others, and every finitely large upper bound can be decreased by some infinitesimal amount to give a smaller one). Edward Nelson (1977) pioneered an alternative approach-Internal Set Theory-on which the basic language of mathematics is enriched in order to allow us to distinguish between standard and non-standard real numbers, as well as "internal" and "external" sets. On Nelson's approach, infinitesimals are non-standard real numbers that are smaller in absolute value than any positive standard real numbers. "Internal sets" are those that can be defined in the basic language, and they behave just the same as standard sets of standard reals-for instance, bounded internal sets always have a least upper bound. But the set of all infinitesimals, just like the set of all standard real numbers, is an "external set" of the theory that can't be defined within the language, and thus doesn't necessarily have a least upper bound.

The approaches pioneered by Robinson and Nelson do not allow us to prove results about the standard real numbers that cannot be proved using standard real analysis. However, these approaches do provide simpler-and, in some sense, more intuitive-proofs of many theorems of standard real analysis. (On the pedagogical benefits of non-standard analysis, see, for example, Keisler (1976)). And there are cases of results in real analysis that were first proven using non-standard real analysis (see, for example, Bernstein and Robinson (1966).) Moreover, these approaches clearly show that we do not need to adopt the \(\epsilon\)-\(\delta\) formalization of the notion of a limit in order to have access to a fully rigorous development of real analysis.

The literature on non-standard analysis is very rich. See Dauben (1995) for a biography of Robinson with special emphasis on non-standard analysis. See also Goldblatt (1998) for a recent formal introduction and Cutland, di Nasso, and Ross (2006) for recent mathematical developments. The reader is referred to the extensive bibliographies contained in those volumes for further references.

An interesting alternative to nonstandard analysis, which allows for a development of substantive parts of mathematics, goes under the name of (smooth) infinitesimal analysis. This differs from both ordinary and nonstandard analysis by allowing nilpotent infinitesimals, namely 'linelets' \(dx\) such that \(dx\neq 0\) but \(dx\cdot dx = 0\). The consistency of such a theory is proved using toposes in category theory. The best exposition of the topic is Bell (1998b) (see also Bell 1988a, 2019); philosophical aspects of the theory are discussed in Hellman and Shapiro (2018). Arthur (2013) discusses infinitesimal analysis in connection to Leibniz and makes points similar to those made by Bos (1974) on Leibniz and non-standard analysis. Constructive interpretations of non-standard analysis are developed in Salanskis (1999), which includes discussions of Nelson's approach as well as the French school of non-standard analysis (Reeb, Harthong). For further discussion of infinitesimals, see Davis (1977), Thomason (1982), Bell (2005), and the entry on continuity and infinitesimals.

3.4.3 Surreal numbers

Dedekind showed how to fill the gaps between rational numbers; Cantor showed how to extend (ordinal and cardinal) numbers beyond the existing finite numbers. John Conway (1976) integrated both ideas. He developed a very different system that generalizes von Neumann's representation of Cantor's ordinals, as well as Dedekind's representation of the real numbers, to generate a much larger field that has become known as the "surreal numbers". It contains a copy of each ordinal and cardinal number, while defining operations that work just like addition, subtraction, multiplication, division, exponentiation, and the taking of roots, on the standard reals. In particular, even the infinite and infinitesimal surreal numbers are amenable to these operations. Thus, as well as familiar numbers, we now have numbers such as \(\sqrt\omega, \omega/2, -\omega, 1/\omega, -\omega^\omega,\) and so on. Indeed, as Ehrlich (2001, 2012) observes, the surreal numbers may plausibly be regarded as including "all numbers great and small"! The surreal numbers can apparently be applied in cases where there is no straightforward way to use hyperreals, as for example in the treatment of Pascal's Wager, discussed in section 7.3 -see Hájek (2003a).

Because the field of surreal numbers contains copies of all the ordinals, it is too big to form a set. But because the operations behave like the operations on the standard reals, these copies of the ordinals don't represent counting. See the

Supplement on the Construction of Surreal Numbers

for a summary of Conway's construction of them. Other constructions of this same structure have been carried out by Knuth (1974) and Gonshor (1986) in more introductory texts.

3.5 Wrapup

Let us take stock. In response to worries that infinities in mathematics are suspect (section 2), rigorous mathematical theories of infinity have been developed (this section). But one might worry that, even if we can talk with mathematical rigor about infinities, they do not correspond to or apply to anything in the real world (as we think finite quantities do, however they do). Infinities might just be castles in the sky. Furthermore, one might suspect that we can, by some further mathematical developments, remove any reference to infinities in any practically important mathematics. The following section places this dialectic in the context of general questions about mathematical ontology, canvassing some important historical attempts to expunge the infinite from mathematics. It then explains the very difficult-perhaps insurmountable-challenges that any such attempt faces.

4. Mathematical ontology

Various questions about infinity naturally arise in the course of theorizing about ontology. If mathematical objects exist, are there infinitely many of them? Do individual infinite objects like the ones mentioned above exist, in addition to the infinitely many individual finite numbers? This article will not directly discuss the question of whether and in what sense mathematical objects exist. Instead, we will focus on the question of whether the infinities discussed above exist in the same sense as the finite integers. For more on the general questions of mathematical existence, see the entries on: logic and ontology, philosophy of mathematics, platonism in the philosophy of mathematics, nominalism in the philosophy of mathematics, fictionalism in the philosophy of mathematics, naturalism in the philosophy of mathematics, and logicism and neologicism.

Most viewpoints in the philosophy of mathematics accept the existence of all of the finite and infinite objects mentioned so far in exactly the same way that they accept the existence of finite integers. (Platonists might accept that this is literal existence, while fictionalists accept this as some sort of fictional existence, and others might have a different idea of what this means.) Standard set theories can prove the existence of all these objects, and for most mathematicians and philosophers, this is all that is needed. Logicist and neologicist accounts of mathematics may obtain the existence of infinite sets or infinitely many numbers by explicit postulation (as in the case of the axiom of infinity in Whitehead and Russell's Principia Mathematica) or as the outcome of an implicit postulation (such as Hume's Principle in Scottish neologicism, see Hale and Wright 2001, Heck 1997, 2011). While the axiom of infinity is easily stated and understood, Hume's Principle has a peculiar form, for it postulates the existence of a function \(\#\) that sends concepts into objects while respecting an equivalence relation \(\approx\) among concepts. Formally it is stated as follows:

\[\tag{HP} \forall B\forall C\, \# B = \# C \text{ iff } B \approx C \]

where \(B \approx C\) is short-hand for one of the many equivalent formulas of pure second order logic expressing the equivalence relation "there is a one to one correlation between the objects falling under \(B\) and those falling under \(C\)". Informally, it can be read as saying that two concepts \(B\) and \(C\) have the same 'number' if and only if there is a one to one correspondence among the objects that fall under \(B\) and those that fall under \(C\). Principles like HP that define a function from an equivalence relation are called abstraction principles. By presupposing the existence of a function that sends concepts into objects, Hume's Principle exploits the possibility of defining infinitely many concepts that do not stand in the equivalence relation mentioned in its right-hand side to generate infinitely many natural numbers. There are other varieties of neologicism that do not postulate Hume's Principle or an axiom of infinity at the outset but yield infinitely many natural numbers by means of other logical principles (see for instance Linsky and Zalta 2006). In addition, all these varieties of neologicism generate at least an infinite cardinal numbers and what is of philosophical relevance here is the different resources they use to establish these results.

Incidentally, Frege's logicism and the neologicist program use one-to-one correspondence to state identity criteria for "concepts" even when infinitely many objects fall under the concepts. For alternative criteria for assigning numbers to infinite concepts see Mancosu (2015) and (2016).

In light of the paradoxes for early set theories (Russell's paradox, the Burali-Forti paradox, and others), some mathematicians and philosophers worried that standard set theory might be inconsistent as well. One alternative viewpoint on mathematics is intuitionism, which only accepts the existence of mathematical objects whose construction can be carried out in some sense by the human mind. Intuitionism requires a revision of logic, since this limitation invalidates the Law of Excluded Middle-there are cases in which we can prove that the non-existence of a certain type of object would lead to a contradiction, but don't have any method of constructing such an object, so that there might be a truth-value gap. Intuitionists often accept the Aristotelian limitation to "potential infinities", rather than "actual infinities", but there is also sophisticated intuitionistic reasoning about what types of infinite entities might exist. (For more, see the entries on intuitionistic logic and intuitionism in the philosophy of mathematics.)

Another viewpoint, associated with David Hilbert, is called finitism (see Hilbert 1926). Most finitists accept classical logic, but worry about the consistency of theories of infinite objects. Hilbert's worries about consistency were fueled by the paradoxes that the new infinitary set-theoretic mathematics was giving rise to (Cantor's inconsistent sets; Burali-Forti Paradox; Russell's paradox etc.) Hilbert was convinced that quantification over such infinite totalities was at the root of the troubles. Finite objects, like configurations of strokes corresponding to the natural numbers, and finite sentences of a formal language, are taken by the Hilbertian finitist to be unproblematic, since these objects can in some sense be grasped individually and thus in their (potential) totality. But infinite objects are taken to be problematic: this includes Cantor's higher ordinals and cardinals, and all the geometric, algebraic, and topological objects of which mathematicians were starting to develop detailed theories at the turn of the 20th century.

Hilbert's proposed project (sometimes taken to be the starting point for formalism in the philosophy of mathematics) was to replace talk of these infinitary entities themselves with talk of the finitely long sentences that we ordinarily interpret as being about the entities. His goal was to axiomatize the theories of these infinite objects, and then to prove, using finitary means of syntactic reasoning about the language, that these theories are consistent. While this idea doesn't deny the existence of the infinite objects, it suggests a methodological approach of only literally accepting the finitary objects, whether strokes standing for the integers or sentences. (See the entry on Hilbert's program.)

The detailed foundational work carried out in set theory and other foundational areas has in many ways dispelled the fear of impending doom that characterized the reaction to the paradoxes at the beginning of the twentieth century. As a consequence, most mathematicians today are perfectly happy to work with completed infinities. But there are still some finitists and intuitionists.

An intermediate position is that defended by classical "predicativists" such as Poincaré and Weyl. The theory, presented in a satisfactory logical way by Feferman and others, accepts the excluded middle on the natural numbers (and as such it is arguably committed to the existence of the set of natural numbers and in any case to accepting bivalence on the natural numbers) but does not accept the existence of the power set of the natural numbers. According to predicativism (see Feferman 2005), sets exist only if they are definable in some non-circular linguistic way. In accepting the excluded middle on the natural numbers and in making the existence of sets depend on our definitional abilities, this position is a sort of a compromise between a classical and a constructive viewpoint. In 1918, Hermann Weyl (Weyl 1918; see Kaufmann 1930 for a related program) presented the foundations of analysis within this framework and showed that a great part of classical analysis can be carried out within the framework by replacing talk of arbitrary sets of real numbers with arithmetical sequences of real numbers. Feferman 1988 gave a detailed formal presentation of the theory and proved that, on a certain reconstruction, the theory is a conservative extension of Peano Arithmetic. In addition, he also used the theory to state an important conjecture concerning how much mathematics is needed in physics. In Feferman 1984 and 1987, he proposed that all of mathematics used in physical theory can be recaptured in a predicative system of analysis. Using the metatheoretical result of conservativity mentioned above, he also exploited the argument to claim that Quine and Putnam's indispensability arguments (see the entry on indispensability arguments in philosophy of mathematics) at best commit us to what Peano Arithmetic commits us to.

In contrast to the possibility of eliminating infinity as just described stand a number of results that show that some finitary statements can only be proved through infinitary considerations. These results originally emerge with Gödel's incompleteness theorems (Gödel 1931) but have been recently refined by displaying statements of mathematical interest (whereas Gödel's statements are of metamathematical interest but have no obvious mathematical interest). In order to understand the conceptual distinctions required, let us grant -as most logicians do-that all finitistic modes of reasoning are included in first-order Peano Arithmetic (henceforth PA).

A consequence of Gödel's incompleteness theorems is that under the assumption that PA is consistent one can find a finitistic statement such that neither it nor its negation is provable from Peano Arithmetic. Gödel showed, through subtle coding of metamathematical notions in the language of arithmetic, how to express in the language of arithmetic a formula \(G\) that "says" of itself that it is not provable. One can also ascertain that the formula is true on the natural numbers. Since all the finitary reasoning is assumed to be included in PA, establishing the truth of the Gödel sentence and of the new incompleteness results requires appeal to some "infinitary" principles (when the truth of the Gödel sentence \(G\) is established through appeal to the statement expressing the consistency of PA, it is establishing the latter that requires some portion of infinitary reasoning, such as induction up to an infinite ordinal called \(\varepsilon_0)\).

The situation is the same for the statement Con(PA) expressing the consistency of PA. Gödel's second incompleteness theorem shows that neither it nor its negation can be proved from PA but an appeal to some infinitary reasoning shows it to hold in the natural numbers. While perfectly fine for the logician's need and central to the evaluation of Hilbert's program, Gödel's sentences appear concocted from the point of view of the practicing mathematician. Within Hilbert's program statements of PA expressible either without quantifiers or with a string of universal quantifiers followed by a non-quantificational formula count as finitistic statements. The statements \(G\) and Con(PA) mentioned above also belong to this class. Finitistic statements with obvious mathematical relevance include basic properties such as the commutativity of addition as well as the statement of Fermat's last theorem (whose proof has been established using higher mathematics but logicians are convinced that it can also be carried out in PA). Logicians have not been able to find finitistic statements with obvious mathematical significance that require a detour through the infinite, but they have been able to do the next best thing. They have found statements that have the form \((\forall x) (\exists y) A(x, y),\) which express a certain functional connection between numbers and have shown that such statements, although true, cannot be proved using only the resources of PA. Among the most famous such results are a modification of Ramsey's finite theorem provided by Paris and Harrington (1977), and the proof that a theorem by Goodstein (1944) cannot be proved in PA (Kirby and Paris 1982). There are stronger results that are independent of even stronger systems that are studied within the context of reverse mathematics (e.g., Kruskal's theorem is independent of predicative analysis-see Simpson 1985, 2002).

Such results show that even an arithmetical theory such as PA can express statements of mathematical significance (as opposed to statements concocted for logical purposes) that require some detour through the infinite to be proved, even though they can be stated purely arithmetically. In contrast to arithmetic, the mathematical incompleteness of set theory was shown by Gödel and Cohen for important statements such as the Axiom of Choice, the continuum hypothesis, etc. It is important to emphasize here that logicians working in set theory, recursion theory and proof theory probe the mysterious role of the infinite in proving results about the finite. It could be said that set theorists are mainly concerned with understanding how the demonstrable mathematical incompletability of Zermelo-Fraenkel (with Choice, i.e. ZFC) set theory, which is a consequence of results by Gödel and Cohen, can be addressed by finding new principles that will allow us to solve some of the most pressing questions concerned with the structure of the real numbers. In other words, since ZFC cannot be taken as a sufficient basis for the mathematics of infinity much of contemporary set theory is trying to solve the problem by finding new principles, which often take the form of assuming the existence of very large cardinals (see the entry on independence and large cardinals). The hope is that this work will lead to settling the Continuum Hypothesis and other major problems about the projective sets (on projective sets see entry on set theory).

Recursion theorists are also trying to understand the role that infinitary principles or compactness arguments play in our determination of results about the finite. And proof theorists would like to know when certain infinitistic theories can be justified through finitary means. Obviously, a more precise description of these developments goes well beyond the technical knowledge that we can presuppose here.

Most working mathematicians don't worry about the existence of infinitely large sets and other objects. There are some other ontological worries about particular infinite sets, related to the Axiom of Choice, and some of the larger cardinalities mentioned above in the section on Cantor. But bigger worries arise in the context of whether there can be physical infinities.

For collections of sources on the classical foundational positions (finitism, intuitionism, predicativism) see van Heijenoort (1967), Ewald (1996), and Mancosu (1998). On finitism and intuitionism see the entries Hilbert's program and intuitionism in mathematics. On predicativity see Feferman (2005). On Paris-Harrington see Katz and Reimann 2018; on Goodstein's theorem see the friendly presentation in Stillwell (2010). Stillwell (2010) also has a chapter on large cardinals; for recent directions see Woodin (2011) and Steel (2015). On the interplay between finite and infinite in recursion theory see Hirschfeldt (2015).

5. Paradise lost? Paradoxes and puzzles involving infinity

The latter part of this entry will explore selected applications of mathematical concepts of infinity in theories of probability, decision, and spacetime, and some associated paradoxes. Before we turn to those theories, we warm up with some paradoxes and puzzles that link mathematics, metaphysical possibility, and physical possibility. There are many different paradoxes and puzzles that we might have included in this section. We consider a small sample of paradoxes and puzzles that some-e.g. Pruss (2018a)-have thought might motivate a return to Aristotle's views on the impossibility of actual infinites.

In the

Supplement on al-Ghazālī's objection,

we discuss a puzzle due to al-Ghazālī that is of historical interest. For more see, for example, Rucker (1982), Moore (1990/2019), Oppy (2006), and Huemer (2016).

5.1 Hilbert's Hotel

Hilbert's Hotel has infinitely many rooms, labelled 1, 2, 3, ..., each of which is currently occupied by a guest. Despite the fact that the hotel is already full, a new guest who turns up at reception is readily accommodated: for each n, the guest in room \(n\) is moved to room \(n+1\), and the new guest is installed in room 1. Indeed, despite the fact that the hotel is already full, it can accommodate infinitely many new guests: for each \(n\), the guest in room \(n\) is moved to room \(2n\), and the new guests are installed in the odd-numbered rooms. Of course, if the infinitely many people in the odd-numbered rooms check out, there are infinitely many people left in Hilbert's Hotel; but if infinitely many people check out from all but the first three rooms, only three people remain.

Some philosophers have thought that Hilbert's Hotel supports an argument against the possibility of physically realized infinities:

1. If there could be physically realized infinities, then there could be a hotel with infinitely many rooms.

2. But if there could be a hotel with infinitely many rooms, then the events described in the preceding paragraph could occur.

3. But it is absurd to suppose that the events described in the preceding paragraph could occur.

So there cannot be physically realized infinities. (See, e.g., Craig (1979).)

This argument faces various challenges, depending on one's views about what physical possibility amounts to. The first premise may be challenged: perhaps some kinds of physical infinities can be realized even though other kinds of physical infinities cannot: for example, perhaps there can be infinitely many stars even though there cannot be a hotel with infinitely many rooms. The second premise may also be challenged: even if there could be a hotel with infinitely many rooms, perhaps the events described in the story could not occur-the story was told at a high level of abstraction, and the details may matter. And the third premise is also questionable: it is not clearly absurd to suppose that there could be an infinite hotel in which guests come and go in the manner described.

The first public discussion of Hilbert's Hotel is Gamow (1946), who cites earlier unpublished lectures by Hilbert and Courant. Gamow's discussion of the aleph numbers (see Section 3.3) erroneously assumes that the Continuum Hypothesis has been established. For further discussion of Hilbert's Hotel, see Huby (1971), Rucker (1982), Moore (1990/2019), Oppy (2006), Kragh (2014), Huemer (2016), and the entries on supertasks and cosmology and theology.

5.2 Thomson's Lamp

Suppose that we have a lamp and a means of turning the lamp off and on. Suppose that the lamp is initially off. In the first minute, we change the state of the lamp from off to on. In the next half minute, we change the state of the lamp from on to off. ... In the next \(1/2^n\) minute we change the state of the lamp to the other state ... . The question that we are invited to answer is: what is the state of the lamp at the end of the second minute?

The scenario is under-described. We can imagine that the means of turning the lamp off and on requires a spacetime location at which at least one physical quantity is infinite. If so, it is plausible to say that the case is impossible: there could not be such a lamp, and so there is no question to answer. Suppose, for example, that there is a switch that moves the same distance back and forth to turn the lamp on and off. Consider the speed at which the tip of the switch is travelling at the end of the second minute.

We can also imagine that the means of turning the lamp off and on does not involve any spacetime location at which at least one physical quantity is infinite; Grünbaum (1968) describes a scenario that fits this specification. In that case, the means of turning the lamp off and on converges to a specified state at the end of the two minutes, and there is an answer to the question that lies in the details of the specified state. But that answer is underdetermined by the brief description that we were initially given, as Benacerraf (1962) argues: we can have the lamp on at the end of the two minutes, or off at the end of the two minutes, depending upon the details of the implementation of Grünbaum's proposal. Huemer (2016: 198-201) points out that if we hold fixed enough physics, then, before the end of the two minutes, the activation of the mechanism will stop changing the state of the lamp. So, depending upon your views about the range of what is possible, you may regard as impossible even cases in which there is no spacetime location at which at least one physical quantity is infinite.

Thomson's lamp is an example of a supertask (Thomson coined this term): a process that involves infinitely many steps completed in a finite amount of time. The trick is that the steps are completed in shorter and shorter periods of time, corresponding to a convergent series. The lamp is one of many examples of supertasks that various authors have found paradoxical, while other authors have been less troubled by them. See the entry on supertasks.

For further discussion of Thomson's lamp, see: Thomson (1954, 1967), Benacerraf (1962), Chihara (1965), Grünbaum (1968, 1973), Craig (1979), Berresford (1981), Moore (1990/2019), Earman and Norton (1996), McLaughlin (1998), Oppy (2006), Huemer (2016), and Pruss (2018a).

5.3 Paralysis

Suppose that Achilles wants to run from \(A\) to \(B\) but there are infinitely many gods who-unbeknownst to one another and to Achilles-each have reason to stop him from getting to \(B\). God 1 resolves to instantaneously paralyse Achilles if and when he reaches halfway between \(A\) and \(B\). God 2 resolves to instantaneously paralyse Achilles if and when he reaches one quarter of the way from \(A\) to \(B\). ... God \(n\) resolves to instantaneously paralyse Achilles if and when he reaches \(1/2^n\) of the way from \(A\) to \(B\). ... Since all of the Gods are able to act on their resolve, Achilles is unable to move: for, if he moved, he would violate the intentions of infinitely many Gods. But, until he does move, none of the Gods act on their intentions. So what actually stops him from moving? Isn't it absurd to suppose that someone can be rendered immobile by a nested sequence of conditional intentions upon which no one acts?

Suppose instead that each of the Gods erects a force field, placed in a parallel manner to the previous case, that it is impossible for Achilles to cross. Then Achilles is completely immobilized. On the assumption that it is possible for an infinite number of Gods to collectively create such a force field in the manner described, there is a straightforward explanation for Achilles' inability to move. Of course, granted this assumption, there is no single God whose force field immobilizes Achilles; indeed, there is no finite collection of Gods whose collective force field does so; indeed, no force field touches him at all. Is this possible? Presumably one should come to the same verdict regarding the conditional intentions case as in this case.

Depending on your views about the range of possibility, there is much in this story that you may think is impossible. You may think that it is impossible for there to be Gods who can act as required; for example, depending on your conception of the Gods and their actions, you might think that the story requires instantaneous action at a distance. You may think that it is impossible for force fields to be positioned with unbounded degrees of accuracy. And so on. However, if there is nothing in the set-up that makes you baulk, and if further elaboration of the set-up does not introduce any singularities, then it seems that you should just accept the conclusions with equanimity: Achilles is rendered immobile by conditional intentions on which no one acts, or by a set of force fields none of which he is in direct contact with. Bizarrely counterfactual circumstances have bizarre consequences.

For further discussion of this case, see: Benardete (1964, which introduces it), Moore (1990/2019), Priest (1999), Hawthorne (2000), Perez-Laraudogoitia (2000, 2003), Yablo (2000), Oppy (2006), Koons (2014), and Huemer (2016), Caie (2018).

We have started to see that infinity seems to be both friend and foe-it features in powerful mathematics, but also in some vexing conundrums. We will see more of its Manichean nature in the following sections on probability, decision theory, and space and time. We will also see how sophisticated methods for reclaiming it have been developed.

6. Probability

Probability theory runs relatively smoothly in the finite realm, but puzzles emerge when infinities are afoot. There are multiple sources of infinitude, arising both in the mathematics and the interpretation of probability. We will firstly discuss more informally these sources, and then progress to more advanced issues.

6.1 Infinitude in the mathematics of probability: basics

Let us begin with the mathematics. Probability theory assumes that we have a set of "possibilities" or "outcomes", called a sample space, regarded as ways the world could be, or the possible outcomes of a random experiment. For many purposes, an infinite set is assumed. For example, we may toss a coin repeatedly and be interested in how many tosses it takes until we see the first heads. The number could be 1, or 2, or 3, or ... Here, the sample space is denumerable. Or we might consider picking a point at random from the [0, 1] interval of the real line-e.g., we might imagine throwing an idealized dart at a representation of that interval, and consider the point on which it lands. Here, the sample space is uncountable, because it is infinitely divisible and has limits of sequences, but bounded. Or we might consider sampling a quantity that is governed by a normal distribution, the bell-shaped distribution that is used to model various quantities in the real world. Here, infinitude enters twice over: the sample space is both uncountable and unbounded, being the entire real line.

Orthodox probability theory assigns real numbers between 0 and 1 (inclusive) to subsets of the sample space, and again we encounter infinitude: there are uncountably many possible probability values. We will soon see how we encounter it again in the way that these values are additive.

6.2 Infinitude in the interpretations of probability

Infinitary considerations also enter into certain interpretations of probability-attempts to explain what probabilities are and what probability statements mean. (See the entry on interpretations of probability for more details on what follows.) Hypothetical frequentism regards probabilities as limiting relative frequencies in hypothetical infinite sequences of trials. For example, we may toss a coin repeatedly, generating a sequence of outcomes-e.g.

heads, tails, heads, heads, tails, tails, tails, heads, ...

We can keep track after each trial of the relative frequency of heads so far: the ratio of the number of heads to the total number of tosses. In our example, the sequence of relative frequencies is

\[ \frac{1}{1}, \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{3}{5}, \frac{3}{6}, \frac{3}{7}, \frac{4}{8}, \ldots \]

We may then imagine this sequence extended indefinitely, and identify the probability of heads with the limit of this sequence. However, the very same results may be reordered, one way or another, to generate any limiting relative frequency in [0, 1] whatsoever, if there are infinitely many heads and infinitely many tails. Infinitude rears its ugly head here-for a finite sequence, reordering can make no difference to the relative frequencies of its outcomes.

6.3 Infinitude in the mathematics of probability: more advanced issues

To state some of the thornier puzzles generated by the mathematics of probability, we need a more formal presentation. Kolmogorov's (1933/1950) axiomatization begins with a finite set \(\Omega\) and an algebra \(F\) of subsets of \(\Omega\): a set closed under complementation and union. The members of \(\Omega\) are known as states while the members of \(F\) are known as events. A probability function is a function from \(F\) to the real numbers. It is non-negative, assigns 1 to \(\Omega\), and it is (finitely) additive-the probability that one of two mutually exclusive events occurs is the sum of their individual probabilities:

Finite additivity
If \(A\) and \(B\) are disjoint sets in \(F\), then \(P(A \cup B) = P(A) + P(B)\).

Countable additivity
If \(\{A_i\}\) is a countably infinite collection of (pairwise) disjoint sets, each in \(F\), then \[ P(\bigcup_{n=1}^{\infty} A_n) = \sum_{n=1}^{\infty} P(A_n) \]

Some have felt that restricting additivity to merely countable sums is arbitrary, and is merely an artifact of the summation technique introduced in section 3.2. An alternate technique for summing infinite sets of non-negative numbers takes advantage of the fact that a sum of nonnegative numbers as defined earlier doesn't depend on the order of the terms. We consider the set of all partial sums of arbitrary finite subsets of the set, and take the least upper bound of this set to represent the sum of the set as a whole. If this sum is some positive finite value \(k\), then we can see that at most \(nk\) of the terms in the set being summed can be greater than 1/\(n\). Since every positive real number is greater than 1/\(n\) for some \(n\), this means that the set of positive elements of the set is a countable union of finite sets, and thus must be countable. That is, if a set being summed in this way has uncountably many non-zero elements, the sum must be infinite.

So if we require full (unrestricted) additivity, rather than merely countable additivity, then we can see that at most countably many events have positive probability, and their probabilities sum to 1. A probability distribution with these features, where events of probability 0 have been removed, is known as a discrete distribution (such as the Poisson, geometric, or negative binomial distributions). For such a distribution, the probabilities of the individual states determine the probabilities of all events through the use of additivity.

However, many applications of probability require what are known as continuous distributions (such as the uniform/rectangular, normal, and beta distributions), and thus require a restriction to countable additivity. In a continuous distribution, there are uncountably many states, usually named by real numbers. Each individual state has probability 0, even though events containing uncountably many states often have non-zero probability. (This violates full additivity.) However, in the common continuous distributions, there is usually a way to define a probability density for each state, such that the probability of any event is the integral of the density over the states that make it up. In finite and discrete distributions, it is standard to treat events of probability 0 as if they do not happen, while in continuous distributions there is always some event of probability 0 that occurs.

For finite and discrete distributions, there is a straightforward definition of a concept of conditional probability. For any two events \(A\) and \(B\), the probability of \(A\) conditional on \(B\), notated as \(P(A\mid B)\), is defined as \(P(A \amp B)/P(B)\), if the probability of B is non-zero, and undefined otherwise. For any fixed \(B\), the function \(P(\_ \mid B)\) is another probability function on the same space. We can prove the Law of Total Probability. If \(B_1, B_2, B_3,\ldots\) form a partition (that is, every outcome is in exactly one of the \(B_i)\) then:

\[ P(A) = \sum_i P(B_i)P(A \mid B_i). \]

This tells us that the unconditional probability \(P(A)\) is a weighted average of the conditional probabilities \(P(A\mid B_i)\).

However, for distributions that are not discrete, so that the set of states is uncountable in an essential way, and events of probability 0 often occur, we can't use this ratio definition of conditional probability, since it would involve dividing by 0. However, Kolmogorov notes (1933/1950, Ch. 5) that for any suitably nice partition, it is still possible to come up with a definition of conditional probability conditional on events in this partition, satisfying a generalization of the Law of Total Probability, replacing the sum with an integral:

\[ P(A) = \int P(A \mid B)dP(B) \]

(The possibility of finding a conditional probability satisfying this integral formula is known as 'disintegrability', and it is equivalent to a principle known as "conglomerability". For philosophical arguments in favor of this, see Easwaran (2013b, 2019), Rescorla (2018).) For more on determining when conditional probabilities that satisfy this rule exist, see Hoffmann-Jørgensen (1971), and for more on how to use probability densities to calculate these conditional probabilities, see Chang and Pollard (1997).

However, there are some difficulties with this account of conditional probability. Kolmogorov notes that if the original probability space is the uniform distribution of points on a sphere, and if \(B\) ranges over the set of longitudes (great circles through the poles), then probability conditional on a line of longitude will not be uniform, but instead will be concentrated near the equator. (This fact is known as the "Borel paradox", because Emile Borel investigated it even before Kolmogorov's work.) Since every great circle on a sphere can be viewed as a line of longitude with an appropriate choice of pole, this makes the probability conditional on an event depend not only on which event was chosen, but also which family of alternatives it is contrasted with. (We can view each great circle as a longitudinal line through multiple different poles, each of which disagrees about where the equator is.)

Some have found this consequence troubling enough that they have endorsed an alternative account of conditional probability that gives up the Law of Total Probability, and instead insists that \(P(A|B)\) has a unique value regardless of which alternatives to \(B\) are under consideration. However, this also has some unpalatable consequences. Since \(P(A)\) is no longer the average of \(P(A|B)\) where \(B\) ranges over the elements of a partition, this means that there are some partitions such that every element of the partition is positively correlated with \(A\). Furthermore, the conditional probability functions generated in this way no longer satisfy countable additivity (Kadane, Schervish, and Seidenfeld 1996, Seidenfeld, Schervish, and Kadane 2001, 2013).

But some, starting with de Finetti (1937, 1972, 1974) have argued on other grounds that we should give up even countable additivity and only accept finite additivity, with a correspondingly broader class of probability distributions. One of de Finetti's chief arguments involves an infinite lottery with each natural number appearing on exactly one ticket. We would like to assign each ticket the same probability of being drawn. Under countable additivity, this is not possible. For if we assign probability 0 to each number's being picked, then the sum of all these probabilities is again 0; yet the union of all of these events has probability 1 (since it is guaranteed that some number will be picked), and \(1 \ne 0\). On the other hand, if we assign some (real-valued) probability \(\varepsilon \gt 0\) to each number being picked, then the sum of these probabilities diverges to \(\infty\), and \(1 \ne \infty\). If we drop countable additivity, however, then we may assign 0 to each event and 1 to their union without contradiction. In the

Supplement on God's Lottery,

we explore how an alternative approach to Kolmogorov's, a non-Archimedean probability theory (NAP), can account for de Finetti's lottery by assigning an infinitesimal probability to each ticket.

However, a probability function that satisfies finite additivity without satisfying countable additivity is mathematically much more complicated than one that satisfies countable additivity. To even prove the existence of such a function over the algebra of subsets of a countable set of states requires the Axiom of Choice. With countable additivity, it is possible to specify a discrete probability function by enumerating the probabilities of the countably many states, and it is possible to specify a continuous probability function by enumerating the probabilities of the countably many rational open sets. But if merely finite additivity is assumed, specifying a probability function even on a countable state space may require specifying the probabilities of uncountably many events, rather than calculating the probabilities of these events from the countably many probabilities of the states. Furthermore, with such probability functions, many standard convergence results like the Strong Law of Large Numbers fail.

For more on infinite probability spaces where only finitely additive probability holds, see Bartha (2004), Bingham (2010), de Finetti (1937/1989), Dubins (1975), Easwaran (2013b), Hill and Lane (1985), Howson (2008), Kadane, Schervish, and Seidenfeld (1986), Schervish, Seidenfeld, and Kadane (1984), Seidenfeld (2001), Seidenfeld, Schervish, and Kadane, (2014).

A lively debate concerns a further constraint on probabilities that may be regarded as desirable: anything that is possible should be assigned positive probability. This is known as regularity:

Regularity
If \(X\) is a non-empty subset of \(\Omega\), then \(P(X) \gt 0\).

Folk thinking about probability seems to be committed to regularity-"if it has probability zero, it can't happen!", as one might say.

For several further puzzles involving probability in infinite spaces, see Arntzenius, Elga, and Hawthorne (2004) and Bartha and Hitchcock (1999). For more on infinitesimal probabilities in philosophical applications, see Benci, Horsten and Wenmackers (2012, 2018), Easwaran (2014), Halpern (2010), Hofweber (2014a, 2014b), Howson (2018), Kremer (2014), Lauvers (2017), Pruss (2012, 2013, 2014, 2018a, 2018b), van Fraassen (1976), and Wenmackers and Horsten (2013).

Infinitesimal probabilities are also appealed to in game theory. For example, the concept of trembling hand perfect equilibrium assumes that each player in a game may make a mistake with positive but negligible probability, which may be regarded as infinitesimal-see Halpern and Moses (2017). We will see further use of infinitesimal probabilities in decision theory, to which we now turn.

7. Decision

\[ \text{EU}(A_i) = \sum_j p_j u_{ij} \]

(We ignore complications and variations that are irrelevant here-see the entries on normative theories of rational choice: expected utility and decision theory.)

In standard cases, it is assumed that

1. there are finitely many possible actions,

2. finitely many states of the world,

and that

the utilities are finite.

However, we may drop each of these assumptions, yielding three different sources of infinitude in a decision problem. Accordingly, we will present some well-known problems that arise when one or more of these assumptions are violated. We begin with a decision under certainty.

7.1 Infinitely many possible actions: Ever-better wine

Pollock (1983) offers the following puzzle. You have a bottle of Ever-better wine, which keeps improving as it ages: the later you open it, the better it will be. When should you open it? There's a good sense in which any time is too soon: opening it slightly later would be better. But the worst option is never to open it, and to avoid this it must be opened at some time. This decision problem has uncountably many possible actions, but we could easily make them denumerable by adding that the bottle can only be opened at discrete times-e.g. on the hour. You would gladly perform an action that has maximal utility, but here there is no such action! This problem displays an intriguing feature that Bartha, Barker and Hájek (2013) call discontinuity at infinity: "an infinite sequence of choices, each apparently sanctioned by plausible principles, converges ... to a 'limit choice' whose utility is distinct from, and typically much lower than, the limit approached by the utilities of the choices in the sequence" (630). Their paper discusses other decision problems with this feature. For more discussion of this kind of phenomenon, see Chow, Robbins, and Siegmund (1971) and Seidenfeld (1981).

7.2 Infinitely many states: the St. Petersburg paradox

A fair coin is tossed. If it comes up heads, you receive $2. If it comes up tails, the coin is tossed for a second time. If it then comes up heads, you receive $4. If it comes up tails, the coin is tossed for a third time. If it then comes up heads, you receive $8. If it comes up tails, the coin is tossed for a fourth time. And so on. We continue until the coin comes up heads. If this takes n tosses, then you win $\(2^n\).

How much should you be prepared to pay to play this game? You have a 1/2 chance of winning $2; and a 1/4 chance of winning $4; and a 1/8 chance of winning $8; ....; and a \(1/2^n\) chance of winning $\(2^n\); and ... . Hence, your expected payoff from playing the St. Petersburg game is infinite:

\[\begin{align} (\frac{1}{2}\times 2) &+ (\frac{1}{4} \times 4) + (\frac{1}{8} \times 8) + \cdots + (\frac{1}{2^n} \times 2^n) + \cdots \\ &= 1 + 1 + 1 + \cdots \end{align}\]

If we identify dollars won with utilities, the game has infinite expected utility.

Decision theory seems to say that you should be prepared to pay any finite amount to play this game. But most people think this is crazy; indeed, most would only pay a few dollars to play (Neugebauer 2010). And decision theory seems to say that you should be prepared to pay any finite amount for a ticket in any finite lottery whose payoff is a single play of this game. That may seem really crazy.

You might object that the utility of money decreases as you obtain more of it: if the rate of this decrease is sufficiently large, then the expected value of playing the game is finite. Daniel Bernoulli argued that utility goes by the logarithm of the amount of money, and indeed replacing the dollar amounts with their logarithms yields a finite expected utility. However, we can retell the story in terms of utilities themselves. And we can retell it with super-exponential escalation of the value of the payoffs: taking logarithms then gives us exactly the original expected utility. (See Menger 1967/1934.) In fact, as long as utilities are unbounded, we can fashion a version of the game that has infinite expected utility.

So you might object that the utilities are bounded. (See e.g., Hardin 1982.) However, unbounded quantities abound-length, volume, mass, curvature, temperature, and so on. Why is utility unlike them in this regard? Moreover, one might imagine a case in which utilities are intimately linked to another such quantity-e.g., the further away you get from some undesirable place, the better-and an unbounded function might link them. Moreover, as we have noted, probability theory is already shot through with infinitude; we need a principled reason why this kind of infinitude should be shunned. (See Nover and Hájek 2004 for further discussion.) And perhaps it is not crazy after all to value the St. Petersburg game infinitely. After all, it dominates each truncation of the game, which pays nothing if heads has not come up after \(n\) trials (for each \(n\)): the St. Petersburg game's outcome is equally good in finitely many states, and strictly better in infinitely many. Plausibly, then, it should be preferred to all these truncations of the game (Hájek and Nover 2006, 2008)-its desirability is greater than \(n\), for each \(n\).

For further discussion of the St. Petersburg game, see, for example: Samuelson (1977), Jeffrey (1983), Weirich (1984), Cowen and High (1988), Jordan (1994), Chalmers (2002), Peters (2011) and the entry on the St. Petersburg paradox.

Related but different problems arise in the Pasadena game, a St. Petersburg-like game in which the expected payoff is apparently undefined (rather than infinite). Then, it seems that decision theory goes silent regarding the value of the game. And yet various choices regarding the game seem to be rationally required-e.g. preferring the game plus a $1 to the game itself. For further discussion, see e.g. Nover and Hájek (2004), Hájek and Nover (2006, 2008), Hájek (2014), Easwaran (2008), Bartha (2016) and Colyvan and Hájek (2016).

7.3 Infinite utility: Pascal's Wager

In the St. Petersburg game, each possible payoff is finite; it is the way in which they are averaged by the expected utility formula that yields the infinitude. We now turn to a classic decision problem in which a possible payoff itself is infinite.

Pascal maintains that we cannot know whether God exists or not, but he argues that we can solve the decision problem of whether or not to 'wager for God'-roughly, to cultivate belief in God. There are two available courses of action: wager for God, or fail to wager for God. There are two relevant conceivable states of the world: God exists, or God does not exist. The probability that God exists is \(p\), whence the probability assigned that God does not exist is \(1 - p\). The utility of wagering for God, if God exists-salvation forever-is infinite. All of the other utilities-of an earthly life of some finite duration-are finite. We may formulate the resulting decision table as follows:

We may now do the expected utility calculations:

The expected utility of wagering for God is

\[ p\cdot \infty + (1 - p)\cdot f_1 = \infty. \]

The expected utility of failing to wager for God is

\[ p \cdot f_2 + (1 - p)\cdot f_3 = \text{ a finite value}. \]

In order to maximize expected utility, one ought to wager for God.

Among the many objections that have been levelled at Pascal's Wager, several focus on the role that '\(\infty\)' plays in the argument. Can utilities be infinite? There is a considerable literature that considers possible extensions of our decision rule, and possible modifications to the framework within which the decision problem is framed. However, to date, there is no widely accepted alternative formulation of Pascal's Wager that avoids all these difficulties that focus on the role that '\(\infty\)' plays in the argument. And once infinite utilities are countenanced, it seems that we should be open to infinitesimal probabilities also. But then there is the prospect that when an infinite utility and an infinitesimal probability are multiplied in the expected utility formula, the product may be a finite number. Will wagering for God still maximize expected utility? These issues and more are discussed in the entry on Pascal's wager.

For further discussions of the treatment of infinity in Pascal's Wager, see, for example: Duff (1986), Oppy (1991, 2018), Hájek (2003a, 2018), Bartha (2007, 2018), Bartha and Pasternack (2018), Monton (2011), and Wenmackers (2018).

7.4 Infinite utility streams

So far we have been considering decisions in which one's payoff comes in a single 'hit': a reward (or punishment) comes all at once. However, we can also consider cases in which one is to choose between different infinite utility streams -e.g., streams of finite daily utility that accumulate over an infinite future. There is an obvious candidate for evaluating the utility of a finite stream: add the utilities along the stream. But this method is not available when we have an infinite stream; we require additional principles to help us evaluate such streams, and it is not obvious what those principles should be.

Suppose that a day spent in Heaven has utility 1 and a day spent in Hell has utility \(-1\). Suppose further that, for all \(n\), \(n\) days in Heaven have utility \(n\), and \(n\) days in Hell have utility \(-n\). Suppose, finally, that, for all \(m\) and \(n\), any combination of \(m\) days in Heaven and \(n\) days in Hell has utility \(m - n\).

Here are some candidate principles for the comparison of alternative possible future utility streams:

1. One should prefer the possible future utility stream that has maximal total utility (if there is one).

2. If more than one possible future utility stream has divergent utility-i.e. is such that there is no finite value to which the total utility of the stream converges as the number of days increases-one should prefer the divergent utility stream whose partial sum is dominant (if there is one). This means that on some days, the sum of utility to that day is greater than for any other stream, and on no days, the sum of utility to that day is less than for some other stream.

3. If more than one possible future utility stream has divergent utility, one should be indifferent between divergent future utility streams that are permutations of one another.

Consider the choice between the following two infinite utility streams:

1. An infinite number of days in Heaven.

2. An infinite number of days in Heaven preceded by a finite number of days in Hell.

Principle 2 says correctly that we should prefer (a) to (b).

However, consider the choice between the following two options:

1. An infinite number of alternating days, first in Heaven, and then in Hell.

2. An infinite number of alternating days, first in Hell, and then in Heaven.

Principle 2 says, incorrectly presumably, that we should prefer (c) to (d).

Now consider the choice between the following two options:

1. An infinite number of alternating days, first in Heaven, and then in Hell.

2. An infinite number of alternating days, first one in Heaven and one in Hell, then two in Heaven and one in Hell, then three in Heaven and one in Hell, and so on.

While Principle 3 says, (perhaps) correctly that we should be indifferent between (c) and (d), it also says, (surely) incorrectly, that we should be indifferent between (e) and (f).

In the face of these difficulties, you might consider weakening the principles:

1. If more than one possible future utility stream has divergent utility, prefer the divergent possible future utility stream that is step-by-step dominant (if there is one).

2. If more than one possible future utility stream has divergent utility, maintain indifference between divergent possible future utility streams that are finite permutations of one another (i.e. that can be derived from one another by finitely many exchanges at neighbouring steps).

But this pair of principles yields no verdict in the case of (e) and (f), and so does not yield a complete set of principles.

More generally, it is hard to codify rules for choosing among infinite utility streams. Indeed, there are some impossibility results in the economics literature which suggest that there is no fully satisfactory theory that countenances them.

For further discussion of infinite utility streams, see, for example: Segerberg (1976), Jeffrey (1983), Nelson (1991), Vallentyne (1993, 1994, 1995), Cain (1995), Ng (1995), Van Liedekerke (1995), Lauwers (1997a, 1997b, 1997c, 1997d), Vallentyne and Kagan (1997), Basu and Mitra (2003), Crespo, Nuñez, and Rincou-Zapatero (2009), Bartha, Barker and Hájek (2014), and Jonsson and Voorneveld (2015).

Each of these decisions problems wears its infinitude on its sleeve: it is obvious that there are infinitely many possible actions, or infinitely many states, or infinite utility, or infinite streams of utility. However, in some problems, such infinitude is not foregrounded, but it is tacitly there nonetheless. The two-envelope paradox is such a problem. See the

Supplement on Tacitly Infinite Decision Problems: Two Envelopes.

There are various other paradoxes of infinity in decision theory-the interested reader may follow these references:

"An Infinite Decision Puzzle": Barrett and Arntzenius (1999)

"Trumped": Arntzenius and McCarthy (1997)

"Rouble Trouble": Arntzenius and Barrett (1999)

"The Airtight Dutch Book": McGee (1999)

Several paradoxes in Arntzenius, Elga and Hawthorne (2004)

"The Cable Guy": Hájek (2005).

8. Space and time

Considering whether space and time are infinite in extent and divisibility has led to many famous puzzles, paradoxes and antimonies. It was on account of such paradoxes that Kant was led to the claim that whether space is finite or infinite escapes any possible empirical determination. Kant's presentation of the antinomies rested on a number of assumptions (such as the distinction between infinite and unbounded) that were undermined by later results in mathematics or were simply found to be philosophically questionable. Another interesting paradox relates to divisibility. In this section we discuss Kant on the antinomies of space and time and a measure-theoretic solution to this paradox of divisibility. This is followed by a quick overview of some developments in non-Euclidean geometries and relativistic cosmology. In the final part, we mention some recent developments in cosmic topology, an area of cosmology that attempts to determine whether space is finite or infinite by a combination of empirical observation and mathematical theorizing. The emphasis will be on the latter aspect.

8.1 Antinomies of space and time

Many philosophers have devised paradoxes and even putative 'antinomies' that exploit the structural features of space and time in a way that essentially involves the infinite. Among the ancients, Zeno is renowned for his paradoxes of space, time and motion. They involve infinitely many spatial or temporal subdivisions or processes that are putatively impossible-see the entry on Zeno's paradoxes. Among the moderns, Kant is particularly notable for his treatment of the extent of space and time in his 'First Antinomy of Pure Reason'. We turn to it now.

8.1.1 Kant

In the Critique of Pure Reason, A426-A434, B454-B462-Kant gives 'proofs' of conflicting theses-'thesis' and 'antithesis'-about the extent of space and time. The 'thesis' says that:

1. the world has a beginning in time; and

2. the world has a finite extension.

The 'antithesis' says:

1. the world has no beginning in time; and

2. the world has infinite extension.

To a reasonable approximation, the 'proofs' run as follows:

1. If the world has no beginning in time, then, up to any given moment, an eternity has elapsed: there has passed away an infinite series of successive states of things. But the infinity of a series consists in the fact that it can never be completed through successive synthesis. So it is impossible for an infinite series of successive states of things to have passed away: the world has a beginning in time.

2. Since infinite extension cannot be thought in a single completed act of thought, the world can only be thought to have infinite extension through an act of synthesis in which completion is achieved via the addition of units. But an act of synthesis that achieved completion via the addition of units requires the lapse of an infinite amount of time-and we have already seen, in (a), that this is impossible. Consequently, it cannot be thought that the world has infinite extension. So the world does not have infinite extension.

3. Something begins to exist only if there is a prior time at which it does not exist. Hence, if the world has a beginning in time, there must be an earlier time at which it does not exist: an empty time. But nothing can come into existence in an empty time, since there is no sufficient reason for the thing to come into existence in one rather than another part of empty time. So the world does not have a beginning in time.

4. If the world has finite extension, then the world is contained in an unlimited empty space. Consequently, the objects in the world are not only related in space but also related to space. In particular, the relation of the world to empty space is a relation of the world to no object, i.e. to nothing. But there can be no such relation. So the world has infinite extension.

Much in Kant's discussion of the antinomies of space and time is marred by his conflation of the modern definition of infinity as lack of finitude with the Aristotelian conception of the impossibility of completion. In addition, at A487/B515 we have confirmation that Kant uses "infinite" and "unbounded", as well as "finite" and "bounded", synonymously: "For if it [the magnitude of the world in space] is infinite and unbounded, then it is too big for every possible empirical concept. If it is finite and bounded, then you can rightfully ask: What determines this boundary?" It was only with the work of Bernhard Riemann in the nineteenth century that geometrical concepts of space were introduced that allowed the decoupling of unboundedness and infinity (and correspondingly of bounded and finite). See section 8.2.

You can find further-sometimes sympathetic-discussion of these arguments in Bennett (1966), Huby (1971), Whitrow (1978), Craig (1979), Moore (1990/2019), Oppy (2006), Huemer (2016), and the entry on Kant's Critique of Metaphysics.

8.1.2 Measure

Here is a Zeno-style argument:

Suppose for reductio that a finite line segment of non-zero length is composed of infinitely many disjoint parts of equal, real-valued length.

1. Either the parts all have zero length or they all have the same non-zero length.

2. The length of the whole line segment is the sum of the lengths of the parts.

3. If the parts all have zero length, then the line segment has zero length, contradicting our assumption that it has non-zero length.

4. If the parts all have non-zero length, then the line segment has infinite length contradicting our assumption that it is finite.

Conclusion 1: A finite line segment cannot be composed of infinitely many disjoint parts of equal real-valued length.

Therefore,

Conclusion 2: It cannot be composed of points.

Premise 1 is beyond reproach. However, premises 2, 3, and 4 require us to be careful about how lengths add. Recall that in section 3.2 we discussed how to add a countable sequence of numbers - but the method described there depends on the order, and requires a countable, well-ordered sequence. Although there are techniques for summing uncountable sets of non-negative numbers, most mathematicians deny that lengths or other measures can be added in these ways. This is parallel to what Kolmogorov said about probability (see section 6.3). Probability and length are two paradigms of the more general mathematical field of "measure theory", which includes all such countably additive real-valued functions. For a more detailed discussion of this problem, including approaches involving infinitesimal length, see Skyrms (1983).

For more about measure theory, see Bartle (1995) and Tao (2011).

8.2. Non-Euclidean geometries, relativistic space-time, and cosmic topology

8.2.1 Non-Euclidean geometries

In section 1 we anticipated that Archytas' argument for the infinitude of the cosmos, and Kant's treatment of the antinomies, conflated the notions of finiteness and boundedness.

We now need to introduce another aspect of 19th century mathematics that brought that crucial distinction into focus. The distinction between finiteness and boundedness (and consequently that between infinitude and unboundedness) greatly improved our understanding of issues concerning the structure of space, and what shape a finite, or an infinite space, might take. Recall that for two centuries after Newton, cosmology was developed within the framework of Euclidean infinite space. Such a space is infinite in all directions, it is homogeneous and isotropic-that is, it is the same at all locations and in all directions.

In the middle of the 19th century, alternative conceptions of geometrical space were developed, so-called non-Euclidean geometries. Gauss, Bolyai, Lobachevski and Riemann, showed that one can develop geometries that falsify Euclid's parallel axiom while preserving all the other Euclidean axioms. The axiom states (in a later but equivalent version to the one given by Euclid) that for any line and a point external to that line, there is one and only one parallel to the given line passing through the point. The statement contains a claim of existence and a claim of uniqueness. It can be thus falsified by denying existence or by accepting the existence but denying the uniqueness. Both alternatives have been developed, with some of the earliest interpretations using surfaces. The first alternative, where no parallels exist to any given line that pass through a given point external to the given line, is known as elliptic geometry. An instance of elliptic geometry is spherical geometry, so called because it can be modeled on the surface of a sphere. The second alternative is known as hyperbolic geometry and in it for every line in the model and any point outside of the line there are infinitely many parallels to that line passing through the point. A portion of the surface of a horse saddle can be used to model hyperbolic geometry. (The pictures below are based on those in Luminet 2008: 49.)

The curvature of a surface at a point \(p\) on the surface measures how much the surface bends away from its tangent plane at \(p\). The curvature of a surface is constant if at every point \(p\) of the surface the surface bends away from the tangent plane by the same quantity. Examples of surfaces which can be used to model various geometries are the surface of the cylinder (Euclidean; constant curvature 0), the sphere (spherical; positive curvature), and the horse saddle (hyperbolic; negative curvature). They are all homogenous and isotropic but they have different constant curvature.

Such geometries on surfaces spurred the development of three-dimensional and higher dimensional spaces with different curvatures: positive, null, and negative. An example of a space of positive constant curvature is the 3-sphere (also called hypersphere) used by Bernhard Riemann in his 1854 dissertation (see Riemann 1868; for an English translation see Riemann 2016). A 3-sphere is a surface in a four-dimensional space that is obtained as a generalization of the 2- sphere as visualized in three dimensions: in both cases we define the relevant notion as a locus of points that have a constant distance from a point (its center). For instance, the unit 2-sphere centered at the origin is the set of triples of real numbers \((x, y, z)\) that are one unit away from (0, 0, 0), i.e. that satisfy \(x^2 +y^2 +z^2 =1\), and the 3-sphere with distance 1 from the origin (0, 0, 0, 0) is the set of quadruples \((x, y, z, w)\) of real numbers satisfying \(x^2 +y^2 +z^2 +w^2 =1\). It is a model of physical space that is finite but unbounded, in explicit opposition to Newton's conception of space.

Archytas's argument (in section 1 above), which conflated unboundedness with infinitude, could finally be put to rest. Riemann's model allows for the universe to be finite and unbounded at the same time. In 1854 he wrote: "The unboundedness of space possesses in this way a greater empirical certainty than any external experience. But its infinite extent by no means follows from this; on the other hand if we assume independence of bodies from position, and therefore ascribe to space constant curvature, it must necessarily be finite provided this curvature has ever so small a positive value. If we prolong all the geodesics starting in a given surface-element, we should obtain an unbounded surface of constant curvature, i.e., a surface which in a flat manifoldness of three dimensions would take the form of a sphere, and consequently be finite." (Riemann 2016: 39)

The distinction between infinite and unbounded is an integral part of the conceptual leap that leads to the idea that physical space need not be Euclidean. In the following section we will briefly describe how issues of curvature and topology play a role in addressing the question of whether the world is spatially finite or infinite in cosmology.

On non-Euclidean geometries the reader will find useful Greenberg (2007) and Gray (2010). On the philosophical relevance of curvature and Riemannian geometry see the classic Torretti (1984).

8.2.2 Relativistic space-time and cosmic topology

In 1915 Einstein introduced general relativity, and our conception of the universe is based on it. General relativity rests on a conception of space and time-or better, space-time matter-that stands in opposition to the Newtonian one we have described above. In Einstein's theory, space-time is deformable and its shape depends on the presence of matter. Space-time is, in technical terms, a four-dimensional manifold. We may think of an \(n\)-dimensional manifold as a set of \(n\)-tuples of real numbers. The spatial section of a four-dimensional manifold of space-time is a three-dimensional manifold (one can think of it as a set of triples of real numbers), and when cosmologists ask about the shape of the universe they try to characterize this three-dimensional manifold. The curvature of space-time corresponds to gravitation, and light rays and other material particles follow the geodesics (shortest paths) in the manifold. In general, the geodesics differ depending on the matter-energy content of the space being considered. The geodesics of the surface of the sphere (i.e. a two-dimensional surface) are portions of great circles. In the Euclidean plane, they are segments of straight lines. There are analogous notions for four-dimensional manifolds. Einstein's equations for general relativity describe how the matter-energy content of the universe determines the geometry of space-time. The equations also yield cosmological models, which must be tested by empirical observation. The equations allow for multiple solutions and, as Alexander Friedmann (1924) observed, "in the absence of additional hypotheses, Einstein's equations for the universe do not allow to definitely answer the question of the finiteness of the universe". Let us briefly explain what is at stake in this comment, by pointing out the role of curvature and topology with respect to the issue of finiteness vs. infiniteness of our universe in relativistic cosmology. Topology is the branch of geometry that classifies spaces according to whether they can be transformed into one another "continuously", that is with transformations that do not lead to cuts or tears.

In 1917, Einstein posited a static finite universe. The finiteness was given by the choice of the 3-sphere (see section 8.2.1) and the static nature of the universe by the fact that the radius of the hypersphere did not change with time. With Friedmann (1922-24) and Lemaître (1927), Einstein's static model would be replaced by dynamical models (to account for the empirical evidence that by 1930 led to the postulation that the universe is expanding, i.e., most galaxies, galaxy clusters, etc. are growing further apart, just as spots on an uninflated balloon grow further apart when the balloon is blown up). Such models are also among the possible solutions for Einstein's equations and they were the source for the so-called "Big Bang" theories. But what about finiteness? The finiteness or infiniteness of the universe are not determined by Einstein's equations, which allow for both possibilities. In his choice of the 3-sphere, Einstein was motivated by considerations that had to do with preserving a hypothesis by Mach on inertial mass and inertial motion. Friedmann and Lemaître also opted for the finiteness of the universe (we need not get into why they did so). Their dynamical models assumed a uniform distribution of matter in the universe and that space is homogeneous and isotropic. But the Friedmann-Lemaître dynamical solutions still allow for a great variety of mathematical solutions and do not settle the issue of finiteness. Our observations in what follows are restricted to such models.

Space, in this context, is characterized by its curvature (taken to be constant) and its topology. Let us consider curvature first. In these models, there are three possible types of spaces depending on whether the curvature of the space is negative, null, or positive. The spaces corresponding to such curvatures are called hyperbolic, Euclidean, and elliptical. A spherical space (constant positive curvature) is always of finite extension, no matter what its topology is. This explains, at least in part, why many early cosmologists (including Einstein, de Sitter, Friedmann, Lemaître and others) opted for this solution. Indeed, for a long time issues concerning the topology of space were not brought to the fore due to the implicit assumption that the topology of space was a simply connected topology (in a simply connected topology every loop on the surface can be continuously contracted to a point). Under that assumption, spaces of positive constant curvature are finite and those of null and negative constant curvature are infinite. The issue then of the finiteness vs. infiniteness of the universe rests on the mean density of matter and energy and on the value of a parameter \(\lambda\) introduced by Einstein in 1917, called the cosmological constant (measuring a sort of anti-gravitational force). With most cosmologists (but not Einstein in 1917) assuming \(\lambda = 0\), and with the assumption that space is simply connected, determining curvature (and hence resolving the finiteness vs infinity issue) depends only on a critical value for the mean density of matter-equivalently, on a density parameter \(\Omega\). Thus, under those assumptions, it would be in principle possible to determine the curvature of space experimentally.

Different values of \(\lambda\) lead to different scenarios for the evolution of the universe. With \(\lambda = 0\), if the curvature of space is negative or null we end up with a constantly expanding universe; if the curvature of space is positive, a phase of expansion would be followed by a contraction leading to a "big crunch". Other values of the cosmological constant are possible and if \(\lambda \lt 0\) a "big crunch" will occur no matter what the curvature of space is. By contrast, if \(\lambda \gt 0\), no "big crunch" will occur. New experimental evidence (coming from type 1A supernovae and fossil radiation) seems to indicate a positive mean density of matter and a value of \(\lambda \gt 0\). In this case the universe would be finite while still remaining in perpetual accelerated expansion.

Moreover, recent work has pointed out the importance of considering multiply connected topologies. Unlike what happens for simply connected topologies, curvature does not immediately determine the finiteness or infinity of the space. Indeed, there are spaces of null or negative curvature that can be finite or infinite depending on the multiply connected topology associated to them. This leads into cosmic topology, which investigates the global shape of space and how it can be determined experimentally. If space has positive curvature, then the universe is finite independently of the specific topology associated with it. But if the curvature is negative or null, whether the universe is finite or not will depend on the topology. Thus, determining whether the universe is finite or infinite requires not only determining the mean density of matter (which determines the curvature of space) but also the topology of space. Two major techniques that are employed experimentally to determine the topology of space are cosmic crystallography and the circles in the sky method (based on the cosmic microwave background).

For further information on cosmic topology see Luminet, Starkmann, and Weeks, (1999), Luminet and Lachièze-Rey (2005), Luminet (2005) (English 2008). See also Aguirre (2011) and Luminet (2015). For more technical treatment see Thurston (1997), Weeks (2001), and Hitchmann 2018.

9. Conclusion

We are well aware that our discussion of infinity is incomplete-but then, so is any such discussion. We take some comfort from the fact that it is impossible to give balanced coverage to a boundless set of issues in finite space.

There are many more philosophically significant paradoxes and puzzles that involve infinity in one way or another; we have only given a small sample. And new paradoxes involving infinity seem to appear at an ever-increasing rate (doubtless yet another one can be fashioned out of this very fact!). However, so too are our tools for understanding infinity. Of course, we cannot give a definitive assessment of the state of play, but the theoretical developments that we have sketched and references that we have cited make us sanguine that overall, the prospects for our relationship with infinity are good: we can indeed live with it.

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# data[].attributes.attachment_link.title
Infinity (Stanford Encyclopedia of Philosophy)

# data[].attributes.attachment_link.description
for each real number ϵ>0ϵ>0, there exists a natural number
NN, such that for every natural number n>Nn>N,
|an−l|<ϵ|an−l|<ϵ’ that
appears in this definition later, but for now it just indicates the
behavior of the sequence beyond any particular finite point.

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Any five-year-old has no trouble turning an old blanket and a couple of chairs into an impenetrable fort. But as we get older, knowledge and experience increasingly displace imagination and our ability to see an object for anything other than its original purpose. This is called Functional Fixedness and while you probably won't need to build a fort during your professional career, chances are you do suffer from it and it is impacting your work. How can I be so sure? Well ... you're human, right?

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The Cognitive Bias Keeping Us from Innovating

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"Functional fixedness" is the hobgoblin of uncreative minds.

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Scientific Method (Stanford Encyclopedia of Philosophy)

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This entry could have been given the title Scientific Methods and gone
on to fill volumes, or it could have been extremely short, consisting
of a brief summary rejection of the idea that there is any such thing
as a unique Scientific Method at all. Both unhappy prospects are due
to the fact that scientific activity varies so much across
disciplines, times, places, and scientists that any account which
manages to unify it all will either consist of overwhelming
descriptive detail, or trivial generalizations.

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The idea that science is a collective enterprise of researchers in successive generations is characteristic of the Modern Age (Nisbet 1980). Classical empiricists (Francis Bacon) and rationalists (René Descartes) of the seventeenth century urged that the use of proper methods of inquiry guarantees the discovery and justification of new truths. This cumulative view of scientific progress was an important ingredient in the optimism of the eighteenth century Enlightenment, and it was incorporated in the 1830s in Auguste Comte's program of positivism: by accumulating empirically certified truths science also promotes progress in society. Other influential trends in the nineteenth century were the Romantic vision of organic growth in culture, Hegel's dynamic account of historical change, and the theory of evolution. They all inspired epistemological views (e.g., among Marxists and pragmatists) which regarded human knowledge as a process. Philosopher-scientists with an interest in the history of science (William Whewell, Charles Peirce, Ernst Mach, Pierre Duhem) gave interesting analyses of some aspects of scientific change.

In the early twentieth century, analytic philosophers of science started to apply modern logic to the study of science. Their main focus was the structure of scientific theories and patterns of inference (Suppe 1977). This "synchronic" investigation of the "finished products" of scientific activities was questioned by philosophers who wished to pay serious attention to the "diachronic" study of scientific change. Among these contributions one can mention N.R. Hanson's Patterns of Discovery (1958), Karl Popper's The Logic of Scientific Discovery (1959) and Conjectures and Refutations (1963), Thomas Kuhn's The Structure of Scientific Revolutions (1962), Paul Feyerabend's incommensurability thesis (Feyerabend 1962), Imre Lakatos' methodology of scientific research programmes (Lakatos and Musgrave 1970), and Larry Laudan's Progress and Its Problems (1977). Darwinist models of evolutionary epistemology were advocated by Popper's Objective Knowledge: An Evolutionary Approach (1972) and Stephen Toulmin's Human Understanding (1972). These works challenged the received view about the development of scientific knowledge and rationality. Popper's falsificationism, Kuhn's account of scientific revolutions, and Feyerabend's thesis of meaning variance shared the view that science does not grow simply by accumulating new established truths upon old ones. Except perhaps during periods of Kuhnian normal science, theory change is not cumulative or continuous: the earlier results of science will be rejected, replaced, and reinterpreted by new theories and conceptual frameworks. Popper and Kuhn differed, however, in their definitions of progress: the former appealed to the idea that successive theories may approach towards the truth, while the latter characterized progress in terms of the problem-solving capacity of theories.

Since the mid-1970s, a great number of philosophical works have been published on the topics of change, development, and progress in science (Harré 1975; Stegmüller 1976; Howson 1976; Rescher 1978; Radnitzky and Andersson 1978, 1979; Niiniluoto and Tuomela 1979; Dilworth 1981; Smith 1981; Hacking 1981; Schäfer 1983; Niiniluoto 1984; Laudan 1984a; Rescher 1984; Pitt 1985; Radnitzky and Bartley 1987; Callebaut and Pinxten 1987; Balzer et al. 1987; Hull 1988; Gavroglu et al. 1989; Kitcher 1993; Pera 1994; Chang 2004; Maxwell 2017). These studies have also led to many important novelties being added to the toolbox of philosophers of science. One of them is the systematic study of inter-theory relations, such as reduction (Balzer et al. 1984; Pearce 1987; Balzer 2000; Jonkisz 2000; Hoyningen-Huene and Sankey 2001), correspondence (Krajewski 1977; Nowak 1980; Pearce and Rantala 1984; Nowakowa and Nowak 2000; Rantala 2002), and belief revision (Gärdenfors, 1988; Aliseda, 2006). Another was the recognition that, besides individual statements and theories, there is also a need to consider temporally developing units of scientific activity and achievement: Kuhn's paradigm-directed normal science, Lakatos' research programme, Laudan's research tradition, Wolfgang Stegmüller's (1976) dynamic theory evolution, Philip Kitcher's (1993) consensus practice. A new tool that is employed in many defenses of realist views of scientific progress (Niiniluoto 1980, 2014; Aronson, Harré, and Way 1994; Kuipers 2000, 2019) is the notion of truthlikeness or verisimilitude (Popper 1963, 1970).

Lively interest about the development of science promoted close co-operation between historians and philosophers of science. For example, case studies of historical examples (e.g., the replacement of Newton's classical mechanics by quantum theory and theory of relativity) have inspired many philosophical treatments of scientific revolutions. Historical case studies were important for philosophers who started to study scientific discovery (Hanson 1958; Nickles 1980). Historically oriented philosophers have shown how instruments and measurements have promoted the progress of physics and chemistry (Chang 2004). Experimental psychologists have argued that the strive for broad and simple explanations shapes learning and inference (Lombrozo 2016). Further interesting material for philosophical discussions about scientific progress is provided by quantitative approaches in the study of the growth of scientific publications (de Solla Price 1963; Rescher 1978) and science indicators (Elkana et al. 1978). Sociologists of science have studied the dynamic interaction between the scientific community and other social institutions. With their influence, philosophers have analyzed the role social and cultural values in the development of science (Longino 2002). One of the favorite topics of sociologists has been the emergence of new scientific specialties (Mulkay 1975; Niiniluoto 1995b). Sociologists are also concerned with the pragmatic problem of progress: what is the best way of organizing research activities in order to promote scientific advance. In this way, models of scientific change turn out to be relevant to issues of science policy (Böhme 1977; Schäfer 1983).

Science is a multi-layered complex system involving a community of scientists engaged in research using scientific methods in order to produce new knowledge. Thus, the notion of science may refer to a social institution, the researchers, the research process, the method of inquiry, and scientific knowledge. The concept of progress can be defined relative to each of these aspects of science. Hence, different types of progress can be distinguished relative to science: economical (the increased funding of scientific research), professional (the rising status of the scientists and their academic institutions in the society), educational (the increased skill and expertise of the scientists), methodical (the invention of new methods of research, the refinement of scientific instruments), and cognitive (increase or advancement of scientific knowledge). These types of progress have to be conceptually distinguished from advances in other human activities, even though it may turn out that scientific progress has at least some factual connections with technological progress (increased effectiveness of tools and techniques) and social progress (economic prosperity, quality of life, justice in society).

All of these aspects of scientific progress may involve different considerations, so that there is no single concept that would cover all of them. For our purposes, it is appropriate here to concentrate only on cognitive progress, i.e., to give an account of advances of science in terms of its success in knowledge-seeking or truth-seeking.

"Progress" is an axiological or a normative concept, which should be distinguished from such neutral descriptive terms as "change" and "development" (Niiniluoto 1995a). In general, to say that a step from stage \(A\) to stage \(B\) constitutes progress means that \(B\) is an improvement over \(A\) in some respect, i.e., \(B\) is better than \(A\) relative to some standards or criteria. In science, it is a normative demand that all contributions to research should yield some cognitive profit, and their success in this respect can be assessed before publication by referees (peer review) and after publication by colleagues. Hence, the theory of scientific progress is not merely a descriptive account of the patterns of developments that science has in fact followed. Rather, it should give a specification of the values or aims that can be used as the constitutive criteria for "good science."

The "naturalist" program in science studies suggests that normative questions in the philosophy of science can be reduced to historical and sociological investigations of the actual practice of science. In this spirit, Laudan has defended the project of testing philosophical models of scientific change by the history of science: such models, which are "often couched in normative language," can be recast "into declarative statements about how science does behave" (Laudan et al. 1986; Donovan et al. 1988). It may be the case that most scientific work, at least the best science of each age, is also good science. But it is also evident that scientists often have different opinions about the criteria of good science, and rival researchers and schools make different choices in their preference of theories and research programs. Therefore, it can be argued against the naturalists that progress should not be defined by the actual developments of science: the definition of progress should give us a normative standard for appraising the choices that the scientific communities have made, could have made, are just now making, and will make in the future. The task of finding and defending such standards is a genuinely philosophical one which can be enlightened by history and sociology but which cannot be reduced to empirical studies of science. For the same reason, Mizrahi's (2013) empirical observation that scientists talk about the aim of science in terms of knowledge rather than merely truth cannot settle the philosophical debate about scientific progress (cf. Bird, 2007, Niiniluoto, 2014).

For many goal-directed activities it is important to distinguish between quality and progress. Quality is primarily an activity-oriented concept, concerning the skill and competence in the performance of some task. Progress is a result-oriented concept, concerning the success of a product relative to some goal. All acceptable work in science has to fulfill certain standards of quality. But it seems that there are no necessary connections between quality and progress in science. Sometimes very well-qualified research projects fail to produce important new results, while less competent but more lucky works lead to success. Nevertheless, the skillful use of the methods of science will make progress highly probable. Hence, the best practical strategy in promoting scientific progress is to support high-quality research.

The failure of science indicators to function as definitions of scientific progress is due to the fact that they do not take into account the semantic content of scientific publications. To determine whether a work \(W\) gives a contribution to scientific progress, we have to specify what \(W\) says (alternatively: what problems \(W\) solves) and then relate this content of \(W\) to the knowledge situation of the scientific community at the time of the publication of \(W\). For the same reason, research assessment exercises may use science indicators as tools, but ultimately they have to rely on the judgment of peers who have substantial knowledge in the field.

The classical sceptic argument against science, repeated by Laudan (1984a), is that knowing the truth is a utopian task. Kant's answer to this argument was to regard truth as a regulative principle for science. Charles S. Peirce, the founder of American pragmatism, argued that the access to the truth as the ideal limit of scientific inquiry is "destined" or guaranteed in an "indefinite" community of investigators. Almeder's (1983) interpretation of Peirce's view of scientific progress is that there is only a finite number of scientific problems and they will all be solved in a finite time. However, there does not seem to be any reason to think that truth is generally accessible in this strong sense. Therefore, the crucial question is whether it is possible to make rational appraisals that we have made progress in the direction of the truth (see Section 3.4).

A goal is effectively recognizable if there are routine or mechanical tests for showing that the goal has been reached or approached. If the defining criteria of progress are not recognizable in this strong sense, we have to distinguish true or real progress from our perceptions or estimations of progress. In other words, claims of the form 'The step from stage \(A\) to stage \(B\) is progressive' have to be distinguished from our appraisals of the form 'The step from stage \(A\) to stage \(B\) seems progressive on the available evidence'. The latter appraisals, as our own judgments, are recognizable, but the former claims may be correct without our knowing it. Characteristics and measures that help us to make such appraisals are then indicators of progress.

Laudan requires that a rational goal for science should be accessible and effectively recognizable (Laudan 1977, 1984a). This requirement, which he uses to rule out truth as a goal of science, is very strong. The demands of rationality cannot dictate that a goal has to be given up, if there are reasonable indicators of progress towards it.

A goal may be backward-looking or forward-looking: it may refer to the starting point or to the destination point of an activity. If my aim is to travel as far from home as possible, my success is measured by my distance from Helsinki. If I wish to become ever better and better piano player, my improvement can be assessed relative to my earlier stages, not to any ideal Perfect Pianist. But if I want to travel to San Francisco, my progress is a function of my distance from the destination. Only in the special case, where there is only one way from \(A\) to \(B\), the backward-looking and the forward-looking criteria (i.e., distance from \(A\) and distance to \(B)\) determine each other.

Kuhn and Stegmüller were advocating backward-looking criteria of progress. In arguing against the view that "the proper measure of scientific achievement is the extent to which it brings us closer to" the ultimate goal of "one full, objective true account of nature," Kuhn suggested that we should "learn to substitute evolution-from-what-we-know for evolution-toward-what-we-wish-to-know" (Kuhn 1970, p. 171). In the same spirit, Stegmüller (1976) argued that we should reject all variants of "a teleological metaphysics" defining progress in terms of "coming closer and closer to the truth."

A compromise between forward-looking and backward-looking criteria can be proposed in the following way. If science is viewed as a knowledge-seeking activity, it is natural to define real progress in forward-looking terms: the cognitive aim of science is to know something that is still unknown, and our real progress depends on our distance from this destination. But, as this goal is unknown to us, our estimates or perceptions of progress have to be based on backward-looking evidential considerations. This kind of view of the aims of science does not presuppose the existence of one unique ultimate goal. To use Levi's words, our goals may be "myopic" rather than "messianic" (Levi 1985): the particular target that we wish to hit in the course of our inquiry has to be redefined "locally," relative to each cognitive problem situation. Furthermore, in addition to the multiplicity of the possible targets, there may be several roads that lead to the same destination. The forward-looking character of the goals of inquiry does not exclude what Stegmüller calls "progress branching." This is analogous to the simple fact that we may approach San Francisco from New York along two different ways-via Chicago or St Louis.

Some philosophers use the concepts of progress and rationality as synonyms: progressive steps in science are precisely those that are based upon the scientists' rational choices. One possible objection is that scientific discoveries are progressive when they introduce novel ideas, even though they cannot be fully explained in rational terms (Popper 1959; cf. Hanson 1958; Kleiner 1993). However, another problem is more relevant here: By whose lights should such steps be evaluated? This question is urgent especially if we acknowledge that standards of good science have changed in history (Laudan 1984a).

A major controversy among philosophers of science is between instrumentalist and realist views of scientific theories (Leplin 1984; Psillos 1999; Niiniluoto 1999a; Saatsi 2018). The instrumentalists follow Duhem in thinking that theories are merely conceptual tools for classifying, systematizing and predicting observational statements, so that the genuine content of science is not to be found on the level of theories (Duhem 1954). Scientific realists, by contrast, regard theories as attempts to describe reality even beyond the realm of observable things and regularities, so that theories can be regarded as statements having a truth value. Excluding naive realists, most scientists are fallibilists in Peirce's sense: scientific theories are hypothetical and always corrigible in principle. They may happen to be true, but we cannot know this for certain in any particular case. But even when theories are false, they can be cognitively valuable if they are closer to the truth than their rivals (Popper 1963). Theories should be testable by observational evidence, and success in empirical tests gives inductive confirmation (Hintikka 1968; Kuipers 2000) or non-inductive corroboration to the theory (Popper 1959).

It might seem natural to expect that the main rival accounts of scientific progress would be based upon the positions of instrumentalism and realism. But this is only partly true. To be sure, naive realists as a rule hold the accumulation-of-truths view of progress, and many philosophers combine the realist view of theories with the axiological thesis that truth is an important goal of scientific inquiry. A non-cumulative version of the realist view of progress can be formulated by using the notion of truthlikeness. But there are also philosophers who accept the possibility of a realist treatment of theories, but still deny that truth is a relevant value of science which could have a function in the characterization of scientific progress. Bas van Fraassen's (1980) constructive empiricism takes the desideratum of science to be empirical adequacy: what a theory says about the observable should be true. The acceptance of a theory involves only the claim that it is empirically adequate, not its truth on the theoretical level. Van Fraassen has not developed an account of scientific progress in terms of his constructive empiricism, but presumably such an account would be close to empiricist notions of reduction and Laudan's account of problem-solving ability (see Section 3.2).

An instrumentalist who denies that theories have truth values usually defines scientific progress by referring to other virtues theories may have, such as their increasing empirical success. In 1906 Duhem expressed this idea by a simile: scientific progress is like a mounting tide, where waves rise and withdraw, but under this to-and-fro motion there is a slow and constant progress. However, he gave a realist twist to his view by assuming that theories classify experimental laws, and progress means that the proposed classifications approach a "natural classification" (Duhem 1954).

Evolutionary epistemology is open to instrumentalist (Toulmin 1972) and realist (Popper 1972) interpretations (Callebaut and Pinxten 1987; Radnitzky and Bartley 1987). A biological approach to human knowledge naturally gives emphasis to the pragmatist view that theories function as instruments of survival. Darwinist evolution in biology is not goal-directed with a fixed forward-looking goal; rather, species adapt themselves to an ever changing environment. In applying this account to the problem of knowledge-seeking, the fitness of a theory can be taken to mean that the theory is accepted by members of the scientific community. But a realist can reinterpret the evolutionary model by taking fitness to mean the truth or truthlikeness of a theory (Niiniluoto 1984).

For a constructive empiricist, it would be natural to think that among empirically adequate theories one theory \(T_{2}\) is better than another theory \(T_{1}\) if \(T_{2}\) entails more true observational statements than \(T_{1}\). Such a comparison makes sense at least if the observation statements entailed by \(T_{1}\) are a proper subset of those entailed by \(T_{2}\). Kemeny and Oppenheim (1956) gave a similar condition in their definition of reduction: \(T_{1}\) is reducible to \(T_{2}\) if and only if \(T_{2}\) is at least as well systematized as \(T_{1}\) and \(T_{2}\) is observationally stronger than \(T_{1}\), i.e., all observational statements explained by \(T_{1}\) are also consequences of \(T_{2}\). Variants of such an empirical reduction relation has been given by the structuralist school in terms of set-theoretical structures (Stegmüller 1976; Scheibe 1986; Balzer et al. 1987; Moulines 2000). A similar idea, but applied to cases where the first theory \(T_{1}\) has been falsified by some observational evidence, was used by Lakatos in his definition of empirically progressive research programmes: the new superseding theory \(T_{2}\) should have corroborated excess content relative to \(T_{1}\) and \(T_{2}\) should contain all the unrefuted content of \(T_{1}\) (Lakatos and Musgrave 1970). The definition of Kuipers (2000) allows that even the new theory \(T_{2}\) is empirically refuted: \(T_{2}\) should have (in the sense of set-theoretical inclusion) more empirical successes, but fewer empirical counter-examples than \(T_{1}\).

Against these cumulative definitions it has been argued that definitions of empirical progress have to take into account an important complication. A new theory often corrects the empirical consequences of the previous one, i.e., \(T_{2}\) entails observational statements \(e_{2}\) which are in some sense close to the corresponding consequences \(e_{1}\) of \(T_{1}\). Various models of approximate explanation and approximate reduction have been introduced to handle these situations. An important special case is the limiting correspondence relation: theory \(T_{2}\) approaches theory \(T_{1}\) (or the observational consequences of \(T_{2}\) approach those of \(T_{1})\) when some parameter in its laws approaches a limit value (e.g., theory of relativity approaches classical mechanics when the velocity of light c grows without limit). Here \(T_{2}\) is said to be a concretization of the idealized theory \(T_{1}\) (Nowak 1980; Nowakowa and Nowak 2000). However, these models do not automatically guarantee that the step from an old theory to a new one is progressive. For example, classical mechanics can be related by the correspondence condition to an infinite number of alternative and mutually incompatible theories, and some additional criteria are needed to pick out the best among them.

Kuhn's (1962) strategy was to avoid the notion of truth and to understand science as an activity of making accurate predictions and solving problems or "puzzles". Paradigm-based normal science is cumulative in terms of the problems solved, and even paradigm-changes or revolutions are progressive in the sense that "a relatively large part" of the problem-solving capacity of the old theory is preserved in the new paradigm. But, as Kuhn argued, it may happen that some problems solved by the old theory are no longer relevant or meaningful for the new theory. These cases are called "Kuhn-losses." A more systematic account of these ideas is given by Laudan (1977): the problem-solving effectiveness of a theory is defined by the number and importance of solved empirical problems minus the number and importance of the anomalies and conceptual problems that the theory generates. Here the concept of anomaly refers to a problem that a theory fails to solve, but is solved by some of its rivals. For Laudan the solution of a problem by a theory \(T\) means that the "statement of the problem" is deduced from \(T\). A good theory is thus empirically adequate, strong in its empirical content, and-Laudan adds-avoids conceptual problems.

One difficulty for the problem-solving account is to find a proper framework for identifying and counting problems (Rescher 1984; Kleiner 1993). When Newton's mechanics is applied to determine the orbit of the planet Mars, this can be counted as one problem. But, given an initial position of Mars, the same theory entails a solution to an infinite number of questions concerning the position of Mars at time \(t\). Perhaps the most important philosophical issue is whether one may consistently hold that the notion of problem-solving may be entirely divorced from truth and falsity: the realist may admit that science is a problem-solving activity, if this means the attempt to find true solutions to predictive and explanatory questions (Popper, 1972; Niiniluoto 1984). Bird's (2007) main criticism against the "functional account" of Kuhn and Laudan is its consequence that the cumulation of false solutions from an entirely false theory counts as scientific progress (e.g. Oresme in the fourteenth century believed that hot goat's blood could split diamonds).

According to Shan (2019), "science progresses if more useful research problems and their corresponding solutions are proposed". This definition involves both problem-defining and problem-solving, as illustrated by the development of early genetics from Darwin to Bateson. Shan gives up the typical Kuhn-Laudan assumption that the scientific community is able to know whether it makes progress or not, and is open to the introduction of the notions of know-how and perspectival truth, so that his "new functional approach" is a compromise with what Bird (2007) calls the "epistemic view" of progress.

A different view of problem-solving is involved in those theories which discuss problems of decision and action. A radical pragmatist view treats science as a systematic method of solving such decision problems relative to various kinds of practical utilities. According to the view called behavioralism by the statistician L J. Savage, science does not produce knowledge, but rather recommendations for actions: to accept a hypothesis is always a decision to act as if that hypothesis were true. Progress in science can then be measured by the achievement of the practical utilities of the decision maker. An alternative methodological version of pragmatism is defended by Rescher (1977) who accepts the realist view of theories with some qualifications, but argues that the progress of science has to be understood as "the increasing success of applications in problem-solving and control." Similarly, Douglas (2014), after suggesting that the distinction between pure and applied science should be relinquished, defines progress "in terms of the increased capacity to predict, control, manipulate, and intervene in various contexts." In this view, the notion of scientific progress is in effect reduced to science-based technological progress (cf. Niiniluoto 1984).

Already the ancient philosophers regarded explanation as an important function of science. The status of explanatory theories was interpreted either in an instrumentalist or realist way: Plato's school started the tradition of "saving the appearances" in astronomy, while Aristotle took theories to be necessary truths. Both parties can take explanatory power to be a criterion of a good theory, as shown by van Fraassen's (1980) constructive empiricism and Wilfrid Sellars' scientific realism (Pitt 1981; Tuomela 1985). When it is added that a good theory should also yield true empirical predictions, the notions of explanatory and predictive power can be combined within the notion of systematic power (Hempel 1965). If the demand of systematic power simply means that a theory has many true deductive consequences in the observational language, this concept is essentially equivalent to the notion of empirical success and empirical problem-solving ability discussed in Section 3.2, but normally explanation is taken to include additional structural conditions besides mere deduction (Aliseda 2006). Inductive systematization should also be taken into account (Hempel 1965; Niiniluoto and Tuomela 1973).

One important idea regarding systematization is that a good theory should unify empirical data and laws from different domains (Kitcher 1993; Schurz 2015). For Whewell, the paradigm case of such "consilience" was the successful unification of Kepler's laws and Galileo's laws by means of Newton's theory.

On the other hand, instead of requiring consensus on a single unifying theory, many philosophers have defended pluralist approaches by arguing that scientific progress needs a variety of conceptual classifications (Dupré 1993; Kitcher 2001), a non-fundamentalist patchwork of laws for "a dappled world" (Cartwright 1999), and different perspectives and values (Longino 2002).

After Hempel's pioneering work in 1948, various probabilistic measures of explanatory power have been proposed (Hempel 1965; Hintikka 1968). Most of them demand that the explanatory theory \(h\) should be positively relevant to the empirical data \(e\). This is the case also with the particular proposal \[ \frac{P(h\mid e) - P(h\mid\neg e)}{P(h\mid e) + P(h\mid\neg e)} \] defended by Schupbach and Sprenger (2011) as the unique measure which satisfies seven intuitively plausible adequacy conditions.

Realist theories of scientific progress take truth to be an important goal of inquiry. This view is built into the classical definition of knowledge as justified true belief: if science is a knowledge-seeking activity, then it is also a truth-seeking activity. However, truth cannot be the only relevant epistemic utility of inquiry. This is shown in a clear way by cognitive decision theory (Levi 1967; Niiniluoto 1987).

Let us denote by \(B = \{h_{1}, \ldots ,h_{n}\}\) a set of mutually exclusive and jointly exhaustive hypotheses. Here the hypotheses in \(B\) may be the most informative descriptions of alternative states of affairs or possible worlds within a conceptual framework \(L\). For example, they may be complete theories expressible in a finite first-order language. If \(L\) is interpreted on a domain \(U\), so that each sentence of \(L\) has a truth value (true or false), it follows that there is one and only one true hypothesis (say \(h^*\)) in \(B\). Our cognitive problem is to identify the target \(h^*\) in \(B\). The elements \(h_{i}\) of \(B\) are the (potential) complete answers to the problem. The set \(D(B)\) of partial answers consists of all non-empty disjunctions of complete answers. The trivial partial answer in \(D(B)\), corresponding to 'I don't know', is represented by a tautology, i.e., the disjunction of all complete answers.

For any \(g\) in \(D(B)\), we let \(u(g, h_{j})\) be the epistemic utility of accepting \(g\) if \(h_{j}\) is true. We also assume that a rational probability measure \(P\) is associated with language \(L\), so that each \(h_{j}\) can be assigned with its epistemic probability \(P(h_{j}\mid e)\) given evidence \(e\). Then the best hypothesis in \(D(B)\) is the one \(g\) which maximizes the expected epistemic utility

\[\tag{1} U(g\mid e) = \sum_{i=1}^{n} P(h_j \mid e)u(g, h_j) \]

For comparative purposes, we may say that one hypothesis is better than another if it has a higher expected utility than the other by formula (1).

If truth is the only relevant epistemic utility, all true answers are equally good and all false answers are equally bad. Then we may take \(u(g, h_{j})\) simply to be the truth value of \(g\) relative to \(h_{j}\):

\[ u(g, h_j) = \begin{cases} 1 \text{ if } h_j \text { is in } g \\ 0 \text{ otherwise.} \end{cases} \]

Hence, \(u(g, h^*)\) is the real truth value \(tv(g)\) of \(g\) relative to the domain \(U\). It follows from (1) that the expected utility \(U(g\mid e)\) equals the posterior probability \(P(g\mid e)\) of \(g\) on \(e\). In this sense, we may say that posterior probability equals expected truth value. The rule of maximizing expected utility leads now to an extremely conservative policy: the best hypotheses \(g\) on \(e\) are those that satisfy \(P(g\mid e) = 1\), i.e., are completely certain on \(e\) (e.g. \(e\) itself, logical consequences of \(e\), and tautologies). On this account, if we are not certain of the truth, then it is always progressive to change an uncertain answer to a logically weaker one.

The argument against using high probability as a criterion of theory choice was made already by Popper in 1934 (see Popper 1959). He proposed that good theories should be bold or improbable. This idea has been made precise in the theory of semantic information.

Levi (1967) measures the information content \(I(g)\) of a partial answer \(g\) in \(D(B)\) by the number of complete answers it excludes. With a suitable normalization, \(I(g) = 1\) if and only if \(g\) is one of the complete answers \(h_{j}\) in \(B\), and \(I(g) = 0\) for a tautology. If we now choose \(u(g, h_{j}) = I(g)\), then \(U(g\mid e) = I(g)\), so that all the complete answers in B have the same maximal expected utility 1. This measure favors strong hypotheses, but it is unable to discriminate between the strongest ones. For example, the step from a false complete answer to the true one does not count as progress. Therefore, information cannot be the only relevant epistemic utility.

Another measure of information content is \(cont(g) = 1 - P(g)\) (Hintikka 1968). If we choose \(u(g, h_{j}) = cont(g)\), then the expected utility \(U(g\mid e) = 1 - P(g)\) is maximized by a contradiction, as the probability of a contradictory sentence is zero. Any false theory can be improved by adding new falsities to it. Again we see that information content alone does not give a good definition of scientific progress. The same remark can be made about explanatory and systematic power.

Levi's (1967) proposal for epistemic utility is the weighted combination of the truth value \(tv(g)\) of \(g\) and the information content \(I(g)\) of \(g\):

\[\tag{2} aI(g) + (1 - a)tv(g), \]

where \(0 \lt a \lt \bfrac{1}{2}\) is an "index of boldness," indicating how much the scientist is willing to risk error, or to "gamble with truth," in her attempt to be relieved from agnosticism. The expected epistemic utility of \(g\) is then

\[\tag{3} aI(g) + (1 - a)P(g\mid e). \]

A comparative notion of progress '\(g_{1}\) is better than \(g_{2}\)' could be defined by requiring that both \(I(g_{1}) \gt I(g_{2})\) and \(P(g_{1}\mid e) \gt P(g_{2}\mid e)\), but most hypotheses would be incomparable by this requirement. By using the weight \(a\), formula (3) expresses a balance between two mutually conflicting goals of inquiry. It has the virtue that all partial answers \(g\) in \(D(B)\) are comparable with each other: \(g\) is better than \(g'\) if and only if the value of (3) is larger for \(g\) than for \(g'\).

If epistemic utility is defined by information content cont(g) in a truth-dependent way, so that

\[ U(g,e) = \begin{cases} cont(g) \text{ if } g \text{ is true}\\ -cont(\neg g) \text{ if } g \text{ is false}, \end{cases} \]

(i,e., in accepting hypothesis \(g\), we gain the content of \(g\) if \(g\) is true, but we lose the content of the true hypothesis \(\neg g\) if \(g\) is false), then the expected utility \(U(g\mid e)\) is equal to

\[\tag{4} P(g\mid e) - P(g) \]

This measure combines the criteria of boldness (small prior probability \(P(g))\) and high posterior probability \(P(g\mid e)\). Similar results can be obtained if \(cont(g)\) is replaced by Hempel's (1965) measure of systematic power \(syst(g, e) = P(\neg g\mid \neg e)\).

For Levi, the best hypothesis in \(D(B)\) is the complete true answer. But his utility assignment also makes assumptions that may seem problematic: all false hypotheses (even those that make a very small error) are worse than all truths (even the uninformative tautology); all false complete answers have the same utility (see, however, the modified definition in Levi, 1980); among false hypotheses utility covaries with logical strength (i.e. if \(h\) and \(h'\) are false and \(h\) entails \(h'\), then \(h\) has greater utility than \(h')\). These features are motivated by Levi's project of using epistemic utility as a basis of acceptance rules. But if such utilities are used for ordering rival theories, then the theory of truthlikeness suggests other kinds of principles.

Popper's notion of truthlikeness is also a combination of truth and information (Popper 1963, 1972). For him, verisimilitude represents the idea of "approaching comprehensive truth." Popper's explication used the cumulative idea that the more truthlike theory should have (in the sense of set-theoretical inclusion) more true consequences and less false consequences, but it turned out that this comparison is not applicable to pairs of false theories. An alternative method of defining verisimilitude, initiated in 1974 by Pavel Tichy and Risto Hilpinen, relies essentially on the concept of similarity.

In the similarity approach, as developed in Niiniluoto (1987), closeness to the truth is explicated "locally" by means of the distances of partial answers \(g\) in \(D(B)\) to the target \(h^*\) in a cognitive problem \(B\). For this purpose, we need a function \(d\) which expresses the distance \(d(h_{i}, h_{j}) =: d_{ij}\) between two arbitrary elements of \(B\). By normalization, we may choose \(0 \le d_{ij} \le 1\). The choice of \(d\) depends on the cognitive problem \(B\), and makes use of the metric structure of \(B\) (e.g., if \(B\) is a subspace of the real numbers \(\Re)\) or the syntactic similarity between the statements in \(B\). Then, for a partial answer \(g\), we let \(D_{\min}(h_{i}, g)\) be the minimum distance of the disjuncts in \(g\) from \(h_{i}\), and \(D_{\rmsum}(h_{i}, g)\) the normalized sum of the distances of the disjuncts of \(g\) from \(h_{i}\). Then \(D_{\min}(h_{i}, g)\) tells how close to \(h_{i}\) hypothesis \(g\) is, so that the degree of approximate truth of \(g\) (relative to the target \(h^*\)) is \(1 - D_{\min}(h^*, g)\). On the other hand, \(D_{\rmsum}(h_{i}, g)\) includes a penalty for all the mistakes that \(g\) allows relative to \(h_{i}\). The min-sum measure

\[\tag{5} D_{\rmms}(h_{i},g) = aD_{\min}(h_{i},g) + bD_{\rmsum}(h_{i},g), \]

where \(a \gt 0\) and \(b \gt 0\), combines these two aspects. Then the degree of truthlikeness of \(g\) is

\[\tag{6} Tr(g, h^*) = 1 - D_{\rmms}(h^*, g). \]

Thus, parameter \(a\) indicates our cognitive interest in hitting close to the truth, and parameter \(b\) indicates our interest in excluding falsities that are distant from the truth. In many applications, choosing \(a\) to be equal to \(2b\) gives intuitively reasonable results.

If the distance function \(d\) on \(B\) is trivial, i.e., \(d_{ij} = 1\) if and only if \(i = j\), and otherwise 0, then \(Tr(g, h^*)\) reduces to the variant (2) of Levi's definition of epistemic utility.

Obviously \(Tr(g, h^*)\) takes its maximum value 1 if and only if \(g\) is equivalent to \(h^*\). If \(g\) is a tautology, i.e., the disjunction of all elements \(h_{i}\) of \(B\), then \(Tr(g,h^*) = 1 - b\). If \(Tr(g, h^*) \lt 1 - b\), \(g\) is misleading in the strong sense that its cognitive value is smaller than that of complete ignorance.

Oddie (1986) has continued to favor the average function instead of the min-sum measure. An alternative account of truth approximation is given by Kuipers (2019).

When \(h^*\) is unknown, the degree of truthlikeness (6) cannot be calculated. But the expected degree of verisimilitude of a partial answer \(g\) given evidence \(e\) is given by

\[\tag{7} ver(g\mid e) = \sum_{i=1}^n P(h_i \mid e) Tr(g, h_i) \]

If evidence \(e\) entails some \(h_{j}\) in \(B\), or makes \(h_{j}\) completely certain (i.e., \(P(h_{j}\mid e) = 1)\), then \(ver(g\mid e)\) reduces to \(Tr(g,h_{j})\). If all the complete answers \(h_{i}\) in \(B\) are equally probable on \(e\), then \(ver(h_{i}\mid e)\) is also constant for all \(h_{i}\).

The truthlikeness function \(Tr\) allows us to define an absolute concept of real progress:

(RP) Step from \(g\) to \(g'\) is progressive if and only if \(Tr(g, h^*) \lt Tr(g', h^*)\),

and the expected truthlikeness function \(ver\) gives the relative concept of estimated progress:

(EP) Step from \(g\) to \(g'\) seems progressive on evidence \(e\) if and only if \(ver(g\mid e) \lt ver(g'\mid e)\).

(Cf. Niiniluoto 1980.) According to definition RP, it is meaningful to say that one theory \(g'\) satisfies better the cognitive goal of answering problem \(B\) than another theory \(g\). This is an absolute standard of scientific progress in the sense of Section 2.5. Definition EP shows how claims of progress can be fallibly evaluated on the basis of evidence: if \(ver(g\mid e) \lt ver(g'\mid e)\), it is rational to claim on evidence \(e\) that the step from \(g\) to \(g'\) in fact is progressive. This claim may of course be mistaken, since estimation of progress is relative to two factors: the available evidence \(e\) and the probability measure \(P\) employed in the definition of \(ver\). Both evidence \(e\) and the epistemic probabilities \(P(h_{i}\mid e)\) may mislead us. In this sense, the problem of estimating verisimilitude is as difficult as the problem of induction.

Rowbottom (2015) argues against RP and EP that scientific progress is possible in the absence of increasing verisimilitude. He asks us to imagine that the scientists in a specific area of physics have found the maximally truthlike theory C*. Yet this general true theory could be used for further predictions and applications. This is indeed the case if we do not make the idealized assumption that the scientists know all the logical consequences of their theories. Then the explanations and predictions from C* constitute new cognitive problems. Moreover, in Rowbottom's thought experiment further progress is possible by expanding the conceptual framework in order to consider as a target a deeper truth than C* (Niiniluoto 2017).

The measure of expected truthlikeness can be used for retrospective comparisons of past theories \(g\), if evidence \(e\) is taken to include our currently accepted theory \(T\), i.e., the truthlikeness of \(g\) is estimated by \(ver(g\mid e \amp T)\) (Niiniluoto, 1984, 171). In the same spirit, Barrett (2008) has proposed that-assuming that science makes progress toward the truth through the elimination of descriptive error-the "probable approximate truth" of Newtonian gravitation can be warranted by its "nesting relations" to the General Theory of Relativity.

The definition of progress by RP can be contrasted with the model of belief revision (Gärdenfors, 1988). The simplest case of revision is expansion: a theory \(T\) is conjoined by an input statement \(A\), so that the new theory is \(T \amp A\). According to the min-sum measure, if \(T\) and \(A\) are true, then the expansion \(T \amp A\) is at least as truthlike as \(T\). But if \(T\) is false and \(A\) is true, then \(T \amp A\) may be less truthlike than \(T\). For example, let the false theory \(T\) state that the number of planets is 9 or 20, and let \(A\) be the true sentence that this number is 8 or 20. Then \(T \amp A\) states that the number of planets is 20, but this is clearly less truthlike than \(T\) itself. Similar examples show that the AGM revision of a false theory by true input need not increase truthlikeness (Niiniluoto 2011).

Bird (2007) has defended the epistemic definition of progress (accumulation of knowledge) against the semantic conception (accumulation of true beliefs or succession of theories with increasing verisimilitude). Here knowledge is not defined as justified true belief, but still it is taken to entail truth and justification, so that Bird's epistemic view in fact returns to the old cumulative model of progress. According to Bird, an accidentally true or truthlike belief reached by irrational methods without any justification does not constitute progress. This kind of thought experiment may seem artificial, since there is always some sort of justification for any hypothetical theory which is accepted or at least seriously considered by the scientific community. But Bird's argument raises the important question whether justification is merely instrumental for progress (Rowbottom, 2008) or necessary for progress (Bird, 2008). Another interesting question is whether the rejection of unfounded but accidentally true beliefs is regressive. The truthlikeness approach replies to these problems by distinguishing real progress RP and estimated progress EP: justification is not constitutive of progress in the sense of RP, but claims of real progress can be justified by appealing to expected verisimilitude (Cevolani and Tambolo, 2013). On the other hand, the notion of progress explicated by EP (or by the combination of RP and EP) is relative to evidence and justification but at the same time non-cumulative.

Bird (2015) can reformulate his initial example by assuming that an accidentally true or truthlike theory \(H\) has been obtained by scientific but yet unreliable means, perhaps by derivation from an accepted theory which turns out to be false. Does such application of mistaken reasoning constitute progress? The interplay of RP and EP allows several possibilities here. Later evidence might show that the initial estimate \(ver(H\mid e)\) was too high. Or the Tr-value was in fact high but initially the ver-value was low (e.g. Aristarchus on heliocentric system, Wegener on continental drift) and only later it was increased by new evidence.

Most accounts of truthlikeness satisfy the principle that among true theories truthlikeness covaries with logical strength (for an exception, see Oddie, 1986). So accumulation of knowledge is a special case of increasing verisimilitude, but it does not cover the case of progress by successive false theories. In his attempt to rehabilitate the cumulative knowledge model of scientific progress, Bird admits that there are historical sequences of theories none of which are "fully true" (e.g. Ptolemy-Copernicus-Kepler or Galileo-Newton-Einstein). As knowledge entails truth, Bird tries to save his epistemic account by reformulating past false theories as true ones. He proposes that if \(g\) is approximately true, then the proposition "approximately \(g\)" is true, so that "the improving precision of approximations can be an object of knowledge". One problem with this treatment is that scientists typically formulate their theories as exact statements, and at the time of their proposal it is not known how large margins of errors would be needed to transform them into true theories. With reference to Barrett (2008), Saatsi (2019) argues that the approximate truth of Newtonian mechanics can be assessed only from the vantage point of General Theory of Relativity, so that this knowledge was not epistemically accessible to Newton at his time. Further, many past theories were radically false rather than approximately true or truthlike, but still they could be improved by more truthlike successors. Ptolemy's geocentric theory was rejected in the Copernican revolution, not retained in the form "approximately Ptolemy". Indeed, the progressive steps from Ptolemy to Copernicus or from Newton to Einstein are not only matters of improved precision but involve changes in theoretical postulates and laws. A further problem for Bird's proposal is the question whether his approximation propositions are able to distinguish between progress and regress in science (Niiniluoto, 2014).

In Section 3.5., we made a distinction between real and estimated progress in terms of the truthlikeness measures. A similar distinction can be made in connection with measures of empirical success. For example, one may distinguish two notions of the problem-solving ability of a theory: the number of problems solved so far, and the number of solvable problems. Real progress could be defined by the latter, while the former gives us an estimate of progress.

The scientific realist may continue this line of thought by arguing that all measures of empirical success in fact are at best indicators of real cognitive progress, measured in terms of truth or truthlikeness. For example, if \(T\) explains \(e\), then it can be shown that \(e\) also confirms \(T\), or increases the probability of \(T\) (Niiniluoto 1999b). A similar reasoning can be employed to give the so-called "ultimate argument" or "no miracle argument"for scientific realism: theoretical realism is the only assumption that does not make the empirical success of science a miracle (Putnam, 1978; Psillos 1999; Niiniluoto 2017; Kuipers 2019; cf. criticism in Laudan 1984b). This means that the best explanation of the empirical progress of science is the hypothesis that science is also progressive on the level of theories.

The thesis that science is progressive is an overall claim about scientific activities. It does not imply that each particular step in science has in fact been progressive: individual scientists make mistakes, and even the scientific community is fallible in its collective judgments. For this reason, we should not propose such a definition that the thesis about the progressive nature of science becomes a tautology or an analytic truth. This undesirable consequence follows if we define truth as the limit of scientific inquiry (this is sometimes called the consensus theory of truth), as then it is a mere tautology that the limit of scientific research is the truth (Laudan 1984a). But this "trivialization of the self-corrective thesis" cannot be attributed to Peirce who realized that truth and the limit of inquiry coincide at best with probability one (Niiniluoto 1980). The notion of truthlikeness allows us to make sense of the claim that science converges towards the truth. But the characterization of progress as increasing truthlikeness, given in Section 3.5, does not presuppose "teleological metaphysics" (Stegmüller 1976), "convergent realism" (Laudan 1984), or "scientific eschatology" (Moulines 2000), as it does not rely on any assumption about the future behavior of science.

The claim about scientific progress can still be questioned by the theses that observations and ontologies are relative to theories. If this is true, the comparison of rival theories appears to be impossible on cognitive or rational grounds. Kuhn (1962) compared paradigm-changes to Gestalt switches (Dilworth 1981). Feyerabend (1984) concluded from his methodological anarchism that the development of science and art resemble each other.

Hanson, Popper, Kuhn, and Feyerabend agreed that all observation is theory-laden, so that there is no theory-neutral observational language. Accounts of reduction and progress, which take for granted the preservation of some observational statements within theory-change, thus run into troubles. Even though Laudan's account of progress allows Kuhn-losses, it can be argued that the comparison of the problem-solving capacity of two rival theories presupposes some kind of correlation or translation between the statements of these theories (Pearce 1987). Various replies have been proposed to this issue. One is the movement from language to structures (Stegmüller 1976; Moulines 2000), but it turns out that a reduction on the level structures already guarantees commensurability, since it induces a translation between conceptual frameworks (Pearce 1987). Another has been the point that an evidence statement \(e\) may happen to be neutral with respect to rival theories \(T_{1}\) and \(T_{2}\), even though it is laden with some other theories. The realist may also point that the theory-ladenness of observations concerns at most the estimation of progress (EP), but the definition of real progress (RP) as increasing truthlikeness does not mention the notion of observation at all.

Even though Popper accepted the theory-ladenness of observations, he rejected the more general thesis about incommensurability as "the myth of the framework" (Lakatos and Musgrave 1970). Popper insisted that the growth of knowledge is always revolutionary in the sense that the new theory contradicts the old one by correcting it, but there is still continuity in theory-change, as the new theory should explain why the old theory was successful to some extent. Feyerabend tried to claim that successive theories are both inconsistent and incommensurable with each other, but this combination makes little sense. Kuhn argued against the possibility of finding complete translations between the languages of rival theories, but in his later work he admitted the possibility that a scientist may learn different theoretical languages (Hoyningen-Huene 1993). Kuhn kept insisting that there is "no theory-independent way to reconstruct phrases like 'really there'," i.e., each theory has its own ontology. Convergence to the truth seems to be impossible, if ontologies change with theories. The same idea has been formulated by Putnam (1978) and Laudan (1984a) in the so-called "pessimistic meta-induction": as many past theories in science have turned out to be non-referring, there is all reason to expect that even the future theories fail to refer-and thus also fail to be approximately true or truthlike. But the optimistic reply by comparative realists points out that for all rejected theories in Laudan's list the scientists have been able to find a better, more truthlike alternative (Niiniluoto 2017; Kuipers 2019).

The difficulties for realism seem to be reinforced by the observation that measures of truthlikeness are relative to languages. The choice of conceptual frameworks cannot be decided by means of the notion of truthlikeness, but needs additional criteria. In defense of the truthlikeness approach, one may point to the fact that the comparison of two theories is relevant only in those cases where they are considered (perhaps via a suitable translation) as rival answers to the same cognitive problem. It is interesting to compare Newton's and Einstein's theories for their truthlikeness, but not Newton's and Darwin's theories. When definitions RP and EP are applied to rival theories in different languages, they have to be translated into a common conceptual framework.

Another line is to appeal to theories of reference in order to show that rival theories can after all be regarded as speaking about the same entities (Psillos 1999). For example, Thompson, Bohr, and later physicists are talking about the same electrons, even though their theories of the electron differ from each other. This is not possible on the standard descriptive theory of reference: a theory \(T\) can only refer to entities about which it gives a true description. Kuhn's and Feyerabend's meaning holism, with devastating consequences for realism, presupposes this account of reference. A similar argument is used by Moulines (2000), who denies that progress could be understood as "knowing more about the same," but his own structuralist reconstruction of progress with "partial incommensurability" assumes that rival theories share some intended applications. Causal theories of reference allow that reference is preserved even within changes of theories (Kitcher 1993). The same result is obtained if the descriptive account is modified by introducing a Principle of Charity (Putnam 1975; Smith 1981; Niiniluoto 1999a): a theory refers to those entities about which it gives the most truthlike description. An alternative account, illustrated by the relation of phlogiston theory and oxygen theory, is given by Schurz (2011) by his notion of structural correspondence. This makes it possible that even false theories are referring. Moreover, there can be reference invariance between two successive theories, even though both of them are false; progress means then that the latter theory gives a more truthlike description about their common domain than the old theory.

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The idea that science is a collective enterprise of researchers in successive generations is characteristic of the Modern Age (Nisbet 1980). Classical empiricists (Francis Bacon) and rationalists (René Descartes) of the seventeenth century urged that the use of proper methods of inquiry guarantees the discovery and justification of new truths.

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The great gene editing debate: can it be safe and ethical?

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A UK law allowing gene-edited food has been paused and some British scientists fear being overtaken.

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Self-reporting of psychiatric illness in an online patient registry is a good indicator of the existence of psychiatric illness

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Online registries offer many advantages for research, including the ability to efficiently assess large numbers of individuals and identify potential participants for clinical trials and genetic studies. Of particular interest is the validity and…

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"Sources of Human Psychological Differences: The Minnesota Study of Twins Reared Apart" (1990), by Thomas J. Bouchard Jr, David T. Lykken, Matthew McGue, Nancy L. Segal and Auke Tellegen

In 1990, Thomas J. Bouchard and his colleagues published the paper "Sources of Human Psychological Differences: The Minnesota Study of Twins Reared Apart" in Science Magazine. The paper described the results of a study initiated in 1979 on the development of twins raised in different environments. The scientists conducted their experiment at the University of Minnesota, in Minneapolis, Minnesota. The researchers physiologically and psychologically assessed monozygotic twins or triplets who were reared apart, comparing the similarity of those twins to twins who were reared together. The research team found that identical twins who are reared apart had the same chance of being similar as twins who were raised together. Bouchard and his colleagues concluded that genetic factors have a large influence on behavioral habits demonstrating the influence of the genetics on development.

The research team, led by Bouchard, included David T. Lykken, Matthew McGue, Nancy L. Segal, and Auke Tellegen. The team conducted the study at the University of Minnesota, where Bouchard directed the Minnesota Center for Twin and Adoption Research. Bouchard, Lykken, McGue, Segal, and Tellegen studied different aspects of psychology and behavioral genetics. Many of those scientists had previously conducted twin studies. Often, twins are a subjet of interest to scientists because identical or monozygotic twins share all of their genes, which allows for the control of genetic differences that is otherwise difficult to achieve with non-identical individuals.

Bouchard and his colleagues conducted their study to determine the origin of individual differences in ability, personality, interest, and social attitudes. As the researchers mention in their article, they studied monozygotic twins who were separated early in life and raised apart, because they provided a way to separate the influence of environmental and genetic factors on human characteristics. Monozygotic or identical twins occur when a single egg is fertilized and forms one zygote that later separates into two embryos. In contrast, dizygotic twins occur when multiple fertilized eggs are implanted in the uterus at the same time and develop into separate embryos. According to the Genetic Science Learning Center of the University of Utah in Salt Lake City, Utah, the strength of twin studies arises from the fact that monozygotic twins share all of their genes, while dizygotic twins on average share about 50 percent of them. Because of that fact, the authors conclude that individual differences between monozygotic twins who are reared apart should only be caused by the influence of the environment they live in. The occurrence of twins being reared apart is rare, but provides an effective way to compare the genetic and environmental influences on human characteristics. The study "Sources of Human Psychological Differences: The Minnesota Study of Twins Reared Apart" is part of the "Minnesota Twin Study," an ongoing study of monozygotic and dizygotic twins since 1979 that involves more than one hundred sets of reared apart twins or triplets.

In the paper "Sources of Human Psychological Differences: the Minnesota Study of Twins Reared Apart," the authors report on the individual differences between monozygotic twins who are reared apart. The authors break the paper into eight distinct sections. In the introduction, the authors start by describing the assessments the participants performed during the study and give a summary of their procedure, results and interpretations. The authors also detail their methods, data, and interpretations in different sections. In the first section, the authors describe the formula they use to correlate the effects of genetic and environmental factors on twins. In the following section, the authors discuss the details and results of multiple tests and assessments that the twins performed during the study. In the third section, the authors explore the effects of similarities in the rearing environment on the IQ results of monozygotic co-twins reared apart. In the fourth section, the authors explore the correlation between monozygotic co-twins reared apart on their IQ results. In the fifth section, the authors discuss various similarities between monozygotic twins. Finally, in the last three sections the researchers discuss what they concluded about the effects of genetic and environmental factors in human development based on their study, and provide different possibilities that could explain the similarity between monozygotic twins reared apart.

In the introduction, the authors describe the twins in their sample and how they were recruited for the study. The authors mention the rarity of reared-apart twin and that the study involved over one hundred sets of twins recruited from many places around the world, including the US, UK, Canada, China, and Germany. The authors state that they recruited the twins in several ways. First, they established a way for friends and relatives of the twins, or the reunited twins themselves, to contact the Minnesota Center for Twin and Adoption Research (MICTAR) upon hearing about the project. In addition, members of the adoption community, social workers, and professionals who encountered reared-apart twins were able to serve as intermediaries between the twins and the researchers. Finally, the scientists were able to recruit individuals who were aware of having a separated twin and who had asked for assistance in locating them from the MICTAR staff.

The authors explain that the study sample consisted of adult twins, separated very early in life, reared apart during their formative years, and reunited in adulthood. The authors mention that they recruited both monozygotic reared apart and dizygotic reared apart twins, and identified their zygosity with extensive serological or body fluid comparisons, fingerprint ridge count, and anthropometric measurements.

The researchers continue the introduction to describe what tasks the participants performed during the experiment. As part of the study the twins underwent approximately fifty hours of medical and psychological assessment. The authors explain that they used multiple test instruments from each major domain of psychological assessment, including personality traits, occupational interests, and mental ability tests to ensure adequate coverage. The authors also explain that an assessment was performed to determine the influence of the rearing environments on the psychological development of the twins involved in the study. The authors state that separate examiners administered IQ tests, life history interviews, psychiatric interviews, and sexual life history interviews with all the twins in the study. The twins also independently completed questionnaires, under supervision from the staff member. The authors do not mention any details about the questions they asked twins during those examinations.

In the first section after the introduction, "Components of Phenotypic Variance," the authors provide a formula they used to correlate between the effects of genetic and environmental factors that can lead to variability between individuals. The authors explain that they used the formula to determine which of the two factors had a larger effect based on the data they collected.

IQ tests that the twins performed. As the authors explain, the twins performed three different intelligence quotient (IQ) measurements, including the Wechsler Adult Intelligence Scale (WAIS), which involved administering a set of six verbal and five performance subtests to the twins at the same time in different rooms. According to the authors, the twins also completed the Raven, Mill-Hill composite measurement, which involved nonverbal measurement of problem-solving ability, vocabulary tests, and multiple-choice word knowledge tests. Finally, the authors administered the first principal component (PC) measurement to measure the cognitive and comprehensive ability of the twins. These tests were done to examine if the twins had significant individual differences in their IQ scores, or if they were mostly similar. Based on the results the authors obtained from the examinations, the authors conclude that the IQ scores of monozygotic co-twins reared apart were similar.

Bouchard and his colleagues continue to analyze the data they collected on the rearing environments of monozygotic twins in the third section, titled "Do Environmental Similarities in Rearing Environments Explain MZA IQ Similarity?" The authors compare the similarity of the rearing environment of the monozygotic twins to the similarity in the twins' IQ scores. Based on that comparison, the authors conclude that rearing environments had no significant effects on the IQ scores of monozygotic twins raised apart.

In the fourth section, titled "Has Pre- and Post-Reunion Contact Contributed to MZA Twin Similarity in IQ?" Bouchard and his colleagues analyze the effects of contact between co-twins on the similarity of the IQ scores. After comparing the frequency of contact between different pairs of co-twins and the IQ scores of those co-twins, the authors concluded that contact has no effect on the similarity between monozygotic twins raised apart.

In the fifth section, titled "Similarity of MZA Twins on a Variety of Dimensions," the authors describe different types of similarities between monozygotic twins. The authors explain that certain similarity measurements were more stable than others. The authors argue that physical similarities like fingerprint ridges are usually more consistent between monozygotic co-twins, while psychophysiological similarities such as blood pressure can vary from time to time. The authors use those measurements to examine if monozygotic reared apart co-twins were more different than monozygotic reared together co-twins.

In the last three sections, the authors move on to explain the results of their study. According to the authors, the monozygotic reared apart twins showed similarities in behavior, interests, religion, and intelligence regardless of their rearing environment. The IQ scores were also consistent between the monozygotic reared apart twins. Bouchard and his colleagues compare the correlation of IQ between monozygotic reared apart twins and the correlation of IQ between monozygotic twins that are reared together, and found that those values were similar. That similarity led them to conclude that rearing monozygotic twins together or apart has no significant effect on IQ. Based on that comparison, Bouchard and his colleagues conclude that monozygotic twins reared apart were just as similar as monozygotic twins reared together.

Finally, the authors state three different possibilities that could cause the similarity between monozygotic twins reared apart. First, the authors state that the genetic factors strongly influence the behavior and general intelligence of individuals and accounts for about 70 percent of the variation in IQ. Second, the authors claim that environmental factors did not have a significant effect on the development of monozygotic twins when they were reared apart which caused them to be similar in social attitude due to the genetic factor. Finally, the authors state that genetic factors control the effects of environmental factors. That means the even if environmental factors had a significant effect, the genetic similarity between the monozygotic reared apart twins might have influenced the twins to focus on, and retain similar environmental influences and, as a result, the twins become similar.

The Minnesota twin studies has been cited over 1500 times. It also raised many questions about the importance of heredity and the environment in shaping human behavior. The evidence that Bouchard and his colleagues provided through their study supported the argument that genetic factors and inheritance play a large role in the development of individuals and the interests and characteristics they show.

Bahjat, Mudhaffar, ""Sources of Human Psychological Differences: The Minnesota Study of Twins Reared Apart" (1990), by Thomas J. Bouchard Jr, David T. Lykken, Matthew McGue, Nancy L. Segal and Auke Tellegen". Embryo Project Encyclopedia (2017-10-19). ISSN: 1940-5030 http://embryo.asu.edu/handle/10776/12997.

Arizona State University. School of Life Sciences. Center for Biology and Society. Embryo Project Encyclopedia.

Copyright Arizona Board of Regents Licensed as Creative Commons Attribution-NonCommercial-Share Alike 3.0 Unported (CC BY-NC-SA 3.0) http://creativecommons.org/licenses/by-nc-sa/3.0/

Twin Study; win Studies; Twins, Monozygotic; Twins, Dizygotic; Diseases in Twins; Minnesota Twin Registry; Minnesota Center for Twin and Adoption Research; Bouchard, T. J. (Thomas J.); Twin; Nature and nurture; Genetics and environment; Nature versus nurture; Genetics; Heredity; Literature

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"Sources of Human Psychological Differences: The Minnesota Study of Twins Reared Apart" (1990), by Thomas J. Bouchard Jr, David T. Lykken, Matthew McGue, Nancy L. Segal and Auke Tellegen

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"Sources of Human Psychological Differences: The Minnesota Study of Twins Reared Apart" (1990), by Thomas J. Bouchard Jr, David T. Lykken, Matthew McGue, Nancy L. Segal and Auke Tellegen In 1990, Thomas J. Bouchard and his colleagues published the paper "Sources of Human Psychological Differences: The Minnesota

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Adoption Study Highlights Impact of Young Children’s Genetic Makeup on Their Parents

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Findings from the ongoing longitudinal Early Growth and Development Study (EGDS), a long-term, prospective adoption study launched by David Reiss, a Yale