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#4 - Do mathematical symbols have meaning in the same way that words have meaning?

The role of natural language when learning the symbolic language of mathematics

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Where do math symbols come from? - John David Walters

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4. Do mathematical symbols have meaning in the same way that words have meaning?

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4. Do mathematical symbols have meaning in the same way that words have meaning?

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Matthew + Ben

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Math isn't hard, it's a language | Randy Palisoc | TEDxManhattanBeach

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3.How is an axiomatic system of knowledge different from, or similar to, other systems of knowledge?

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3.How is an axiomatic system of knowledge different from, or similar to, other systems of knowledge?

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Matthew and Ben

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Science and maths skills are widely celebrated as keys to economic and technological progress, but abstract mathematics may seem bafflingly far from industrial optimisation or medical imaging. Pure mathematics often yields unanticipated applications, but without a time machine to look into the future, how do mathematicians like me choose what to study?

Over Thai noodles, I asked some colleagues what makes a problem interesting, and they offered a slew of suggestions: surprises, contradictions, patterns, exceptions, special cases, connections. These answers might sound quite different, but they all support a view of the mathematical universe as a structure to explore.

In this view, mathematicians are like anatomists learning how a body works, or navigators charting new waters. The questions we ask take many forms, but the most interesting ones are those that help us see the big picture more clearly.

Making maps

Mathematical objects come in many forms. Some of them are probably quite familiar, like numbers and shapes. Others might seem more exotic, like equations, functions and symmetries.

Instead of just naming objects, a mathematicians might ask how some class of objects is organised. Take prime numbers: we know there are infinitely many of them, but we need a structural understanding to work out how frequently they occur or to identify them in an efficient way.

Other good questions explore relationships between apparently different objects. For example, shapes have symmetry, but so do the solutions to some equations.

Classifying objects and finding connections between them help us assemble a coherent map of the mathematical world. Along the way, we sometimes encounter surprising examples that defy the patterns we've inferred.

Such apparent contradictions reveal where our understanding is still lacking, and resolving them provides valuable insight.

Consider the triangle

The humble triangle provides a famous example of an apparent contradiction. Most people think of a triangle as the shape formed by three connecting line segments, and this works well for the geometry we can draw on a sheet of paper.

However, this notion of triangle is limited. On a surface with no straight lines, like a sphere or a curly kale leaf, we need a more flexible definition.

Read more: Pythagoras' revenge: humans didn't invent mathematics, it's what the world is made of

So, to extend geometry to surfaces that aren't flat, an open-minded mathematician might propose a new definition of a triangle: pick three points and connect each pair by the shortest path between them.

This is a great generalisation because it matches the familiar definition in the familiar setting, but it also opens up new terrain. When mathematicians first studied these generalised triangles in the 19th century, they solved a millennia-old mystery and revolutionised mathematics.

The parallel postulate problem

Around 300 BC, the Greek mathematician Euclid wrote a treatise on planar geometry called The Elements. This work presented both fundamental principles and results that were logically derived from them.

One of his principles, called the parallel postulate, is equivalent to the statement that the sum of the angles in any triangle is 180°. This is exactly what you'll measure in every flat triangle, but later mathematicians debated whether the parallel postulate should be a foundational principle or just a consequence of the other fundamental assumptions.

This puzzle persisted until the 1800s, when mathematicians realised why a proof had remained so elusive: the parallel postulate is false on some surfaces.

On a sphere, the sides of a triangle bend away from each other and the angles add up to more than 180°. On a rippled kale leaf, the sides bow in towards each other and the angle sum is less than 180°.

Triangles where the angle sum breaks the apparent rule led to the revelation that there are kinds of geometry Euclid never imagined. This is a deep truth, with applications in physics, computer graphics, fast algorithms, and beyond.

Salad days

People sometimes debate whether mathematics is discovered or invented, but both points of view feel real to those of us who study mathematics for a living. Triangles on a piece of kale are skinny whether or not we notice them, but selecting which questions to study is a creative enterprise.

Interesting questions arise from the friction between patterns we understand and the exceptions that challenge them. Progress comes when we reconcile apparent contradictions that pave the way to identify new ones.

Today we understand the geometry of two-dimensional surfaces well, so we're equipped to test ourselves against similar questions about higher-dimensional objects.

Read more: Corals, crochet and the cosmos: how hyperbolic geometry pervades the universe

In the past few decades we've learned that three-dimensional spaces also have their own innate geometries. The most interesting one is called hyperbolic geometry, and it turns out to act like a three-dimensional version of curly kale. We know this geometry exists, but it remains mysterious: in my own research field, there are lots of questions we can answer for any three-dimensional space ... except the hyperbolic ones.

In higher dimensions we still have more questions than answers, but it's safe to say that study of four-dimensional geometry is entering its salad days.

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Exploring the mathematical universe - connections, contradictions, and kale

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Science and maths skills are widely celebrated as keys to economic and technological progress, but abstract mathematics may seem bafflingly far from industrial optimisation or medical imaging. Pure mathematics often yields unanticipated applications, but without a time machine to look into the future, how do mathematicians like me choose what to study?

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https://www.newworldencyclopedia.org/entry/Axiomatic_systems

How is an axiomatic system of knowledge different from, or similar to, other systems of knowledge?

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North Carolina School of Science and Mathematics

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How many ways are there to prove the Pythagorean theorem? - Betty Fei

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Many different ways to prove Pythagorean Theorem (Hao and Taeyoung)

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Many different ways to prove Pythagorean Theorem (Hao and Taeyoung)

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This following article is an real life example of the "proof" in mathematics. It helps us to answer the first knowledge question as it gives us an idea of what the proof in mathematics look like and how it's differed from the proof in other AOKs. According to this following article, there are more than 300 proofs to pythagorean theorem.

https://link.springer.com/article/10.1057/jt.2009.16#:~:text=There%20are%20well%20over%20371,the%20United%20States%20James%20A.

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BRIEF BIOGRAPHY OF PYTHAGORAS

The Greek mathematician Pythagoras has high name recognition, not only in the history of mathematics. Everyone who has studied geometry can recall, well after the high school years, some aspect of the Pythagorean Theorem. However, the story of Pythagoras and his famous theorem is not well known. Some story plot points are: the famous theorem goes by several names grounded in the behavior of the day (discussed later in the text), including the Pythagorean Theorem, Pythagoras' Theorem and notably Euclid I 47. The Pythagorean Theorem is arguably the most famous statement in mathematics, and the fourth most beautiful equation.1, 2 There are well over 371 Pythagorean Theorem proofs originally collected by an eccentric mathematics teacher, who put them in a 1927 book, which includes those by a 12-year-old Einstein, Leonardo da Vinci (a master of all disciplines) and President of the United States James A. Garfield.3, 4, 5

Pythagoras' likeness in pictures and sculptures, as shown in Figure 1, appears in all geometry textbooks, and books about the history of mathematics. However, ironically, not much is really known about him - not even his likeness. What is known about Pythagoras is generally considered more fiction than fact, as historians who lived hundreds of years later provided the facts about his life. There are definite details of Pythagoras' life from early biographies that use original sources, yet are written by authors who attribute divine powers to him, and present him as a deity figure. Consequently, most historians treat this information as legend. Few historians view the information with any degree of historical importance because it is obtained from rare original sources.6

Historians generally agree that Pythagoras of Samos (born circa 569 BC in Samos, Ionia and died circa 475 BC) was the first mathematician. He is an extremely important figure in the development of mathematics, yet relatively little is known about his mathematical achievements. Unlike many later Greek mathematicians, who wrote a number of books, there are no writings by Pythagoras. However, there is evidence that Pythagoras founded a school (in what is now Crotone, to the east of the heel of southern Italy) named the Semicircle of Pythagoras - half-religious and half-scientific, which followed a code of secrecy. Because secrecy is often controversial, Pythagoras is a mysterious figure. The members of the Semicircle of Pythagoras - the Pythagoreans - were bound by an allegiance that was strictly enforced. However, the Semicircle was more than just a school that studied intellectual disciplines, including in particular philosophy, mathematics and astronomy.7 The scientific dimension of the school treated numbers in ways similar to the Jewish mysticism of Kaballah, where each number has divine meaning and combined numbers reveal the mystical worth of life. The latter is reflected in the Pythagorean motto: Number Rules the Universe.6 The religious dimension of the school included diverse lectures held by Pythagoras attended by men and women, even though the law in those days forbade women from being in the company of men. Pythagoreans consumed vegetarian dried and condensed food and unleavened bread (as matzos, used by the Biblical Jewish priestly class (the Kohanim), and used today during the Jewish holiday of Passover).8

One reason for the rarity of Pythagoras original sources was that Pythagorean knowledge was passed on from one generation to the next by word of mouth, as writing material was scarce. Moreover, out of respect for their leader, many of the discoveries made by the Pythagoreans were attributed to Pythagoras himself; this would account for the term 'Pythagoras' Theorem'. Consequently, of Pythagoras' actual work nothing is known. On the other hand, his school practiced collectivism, making it hard to distinguish between the work of Pythagoras and that of his followers; this would account for the term 'Pythagorean Theorem'. Therefore, the true discovery of a particular Pythagorean result may never be known. Regardless of the uncertainty of Pythagoras' actual contributions, however, his school made outstanding contributions to mathematics.

Is seems that Pythagoras was the first person to define the consonant acoustic relationships between strings of proportional lengths. Specifically, strings of equal tension of proportional lengths create tones of proportional frequencies when plucked. For example, a string that is 2 feet long will vibrate x times per second (that is, hertz, a unit of frequency equal to one cycle per second), while a string that is 1 foot long will vibrate twice as fast: 2 x. Furthermore, those two frequencies create a perfect octave.9

The most important discovery of Pythagoras' school was the fact that the diagonal of a square is not a rational multiple of its side.10 This result proved the existence of irrational numbers.11 This finding greatly disturbed the Pythagoreans, as it was inconsistent with their divine belief in numbers: whole numbers and their ratios, which account for geometrical properties, were challenged by their own result. The Pythagoreans were so troubled over the finding of irrational numbers that they swore each other to secrecy about its existence. It is known that one Pythagorean did tell someone outside the school, and he was never to be found thereafter, that is, he was murdered, as Pythagoras himself was murdered by oppressors of the Semicircle of Pythagoras.12

THE BABYLONIANS

Babylonia was situated in an area known as Mesopotamia (Greek for 'between the rivers'). Mesopotamia (arrow 1 in Figure 2) was in the Near East in roughly the same geographical position as modern Iraq.13 Two great rivers flowed through this land: the Tigris and the Euphrates (arrows 2 and 3, respectively, in Figure 2). Mesopotamia was one of the great civilizations of antiquity, rising to prominence 4000 years ago. Thousands of clay tablets, found over the past two centuries, confirm a people who kept accurate records of astronomical events, and who excelled in the arts and literature. Only a small fraction of this vast archeological treasure trove has been studied by scholars. The great majority of tablets lie in the basements of museums around the world, awaiting their turn to be deciphered and to provide a glimpse into the daily life of ancient Babylon.14

Ancient Near East Empires around 1350 BC(Egypt, Babylon, Mitanni, Hittite, Assyria, plus Route of Abraham).

Among the tablets that have received special scrutiny is that with the identification 'YBC 7289', shown in Figure 3, which represents the tablet numbered 7289 in the Babylonian Collection of Yale University.15 The tablet dates from the Old Babylonian period, roughly 1800-1600 BCE, and shows a tilted square and its two diagonals, with some marks engraved along one side and under the horizontal diagonal. The marks are in wedge-shaped characters, carved with a stylus into a piece of soft clay that was then dried in the sun or baked in an oven. They turn out to be numbers, written in the Babylonian numeration system that used the base 60. In this sexagestimal system, numbers up to 59 were written in essentially the modern base-10 numeration system, but without a zero. Units were written as vertical Y-shaped notches, while tens were marked with similar notches written horizontally.15

A fortuitous event: the find of tablet YBC 7289 was translated by Dennis Ramsey and dating to YBC 7289, circa 1900 BC:

4 is the length and 5 is the diagonal. What is the breadth? Its size is not known. 4 times 4 is 16. And 5 times 5 is 25. You take 16 from 25 and there remains 9. What times what shall I take in order to get 9? 3 times 3 is 9. 3 is the breadth.15

The number along the upper left side is easily recognized as 30. The number immediately under the horizontal diagonal is 1; 24, 51, 10 (this is the modern notation for writing Babylonian numbers, in which the commas separate the sexagesition 'digits', and a semicolon separates the integral part of a number from its fractional part). Writing this number in the base-10 system, one gets 1+24/60+51/60 ²+10/60 ³=1.414213, which is nothing other than the decimal value of the square root of 2, accurate to the nearest one hundred thousandth.

The conclusion is inescapable. The Babylonians knew the relation between the length of the diagonal of a square and its side: d=square root of 2. This was probably the first number known to be irrational. However, this in turn means that they were familiar with the Pythagorean Theorem - or, at the very least, with its special case for the diagonal of a square ( d ²= a ²+ a ²=2 a ²) - more than a thousand years before the great sage for whom it was named. The square root of 2, known as Pythagoras' constant, is the positive real number that, when multiplied by itself, gives the number 2 (see Figures 3 and 4).16, 17

Two factors with regard to this tablet are particularly significant. First, it proves that the Babylonians knew how to compute the square root of a number with remarkable accuracy. The unknown scribe who carved these numbers into a clay tablet nearly 4000 years ago showed a simple method of computing: multiply the side of the square by the square root of 2. But there remains one unanswered question: Why did the scribe choose a side of 30 for his example? Probably, 30 was used for convenience, as it was part of the Babylonian system of sexagesimal, a base-60 numeral system. From this one derives the modern day usage of 60 seconds in a minute, 60 min in an hour and 360 (60 × 6) degrees in a circle.18

Today, the Pythagorean Theorem is thought of as an algebraic equation, a ²+ b ²= c ²; but this is not how Pythagoras viewed it. To Pythagoras it was a geometric statement about areas. It was with the rise of modern algebra, circa 1600 CE, that the theorem assumed its familiar algebraic form.

Formally, the Pythagorean Theorem is stated in terms of area:

In any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle). An area interpretation of this statement is shown in Figure 5.19

Area version of Pythagoras' Theorem.

The theorem is usually summarized as follows:

The square of the hypotenuse of a right triangle is equal to the sum of the squares on the other two sides.

A PEOPLE WHO USED THE PYTHAGOREAN THEOREM?

Ancient Egyptians (arrow 4, in Figure 2), concentrated along the middle to lower reaches of the Nile River (arrow 5, in Figure 2), were a people in Northeastern Africa. The ancient civilization of the Egyptians thrived 500 miles to the southwest of Mesopotamia. The two nations coexisted in relative peace for over 3000 years, from circa 3500 BCE to the time of the Greeks. As to the claim that the Egyptians knew and used the Pythagorean Theorem in building the great pyramids, there is no evidence to support this claim.

Egypt has over 100 pyramids, most built as tombs for their country's Pharaohs. I would be remiss if I did not include an image of the iconic Egyptian Pharaoh Tutankhamen, aka King Tut ( Figure 6). Egypt (arrow 4, in Figure 2) and its pyramids are as immortally linked to King Tut as are Pythagoras and his famous theorem. King Tut ruled from the age of 8 for 9 years, 1333-1324 BC. He was born in 1341 BC and died (some believe he was murdered) in 1323 BC at the age of 18.20

THE TEACHER WHO COLLECTED PYTHAGOREAN THEOREM PROOFS

Elisha Scott Loomis (1852-1940) ( Figure 7), an eccentric mathematics teacher from Ohio, spent a lifetime collecting all known proofs of the Pythagorean Theorem and writing them up in The Pythagorean Proposition, a compendium of 371 proofs. The manuscript was prepared in 1907 and published in 1927. A revised second edition appeared in 1940, and this was reprinted by the National Council of Teachers of Mathematics in 1968 as part of its 'Classics in Mathematics Education' series. Loomis received literally hundreds of new proofs from after his book was released up until his death, but he could not keep up with his compendium. As for the exact number of proofs, no one is sure how many there are. Surprisingly, geometricians often find it quite difficult to determine whether certain proofs are in fact distinct proofs. Befitting of someone who collects solutions of the Pythagorean Theorem (I belittle neither the effort nor its value), Loomis, known for living an orderly life, extended his writing to his own obituary in 1934, which he left in a letter headed 'For the Berea Enterprise immediately following my death'. He died on 11 December 1940, and the obituary was published as he had written it, except for the date of his death and the addresses of some of his survivors.21

A 12-YEAR-OLD EINSTEIN 'PROVES' THE PYTHAGOREAN THEOREM

According to his autobiography, a preteen Albert Einstein ( Figure 8)

devised a new 'proof' (he was careful to put the word in quotation marks, evidently not wishing to take credit for it) of the Pythagorean Theorem based on the properties of similar triangles. Many known proofs use similarity arguments, but this one is notable for its elegance, simplicity and the sense that it reveals the connection between length and area that is at the heart of the theorem.22

About his 'holy geometry book', Einstein in his autobiography says:

At the age of 12, I experienced a second wonder of a totally different nature: in a little book dealing with Euclidean plane geometry, which came into my hands at the beginning of a school year. Here were assertions, as for example the intersection of the three altitudes of a triangle in one point, which - though by no means evident - could nevertheless be proved with such certainty that any doubt appeared to be out of the question. This lucidity and certainty made an indescribable impression upon me. For example I remember that an uncle told me the Pythagorean Theorem before the holy geometry booklet had come into my hands. After much effort I succeeded in 'proving' this theorem on the basis of the similarity of triangles ... for anyone who experiences [these feelings] for the first time, it is marvelous enough that man is capable at all to reach such a degree of certainty and purity in pure thinking as the Greeks showed us for the first time to be possible in geometry.23

EINSTEIN'S CHILDHOOD FASCINATION WITH THE PYTHAGOREAN THEOREM BEARS FRUIT

Einstein ( Figure 9) used the Pythagorean Theorem in the Special Theory of Relativity (in a four-dimensional form), and in a vastly expanded form in the General Theory of Relatively. The following excerpts are worthy of inclusion.

Pythagorean Theorem in the Special Theory of Relativity (1905)

Albert Einstein's Metric equation is simply Pythagoras' Theorem applied to the three spatial co-ordinates and equating them to the displacement of a ray of light.

Special relativity is still based directly on an empirical law, that of the constancy of the velocity of light.

d x ²+d y ²+d z ²=( c d t) ² where c d t is the distance traveled by light c in time d t. The fact that such a metric is called Euclidean is connected with the following. The postulation of such a metric in a three-dimensional continuum is fully equivalent to the postulation of the axioms of Euclidean Geometry. The defining equation of the metric is then nothing but the Pythagorean Theorem applied to the differentials of the co-ordinates.

In the special theory of relativity those co-ordinate changes (by transformation) are permitted for which also in the new co-ordinate system the quantity ( c d t) ² (fundamental invariant d S ²) equals the sum of the squares of the co-ordinate differentials. Such transformations are called Lorentz transformations.24

Pythagorean Theorem in the General Theory of Relativity (1915)

From the latest results of the theory of relativity, it is probable that our three-dimensional space is also approximately spherical, that is, that the laws of disposition of rigid bodies in it are not given by Euclidean geometry, but approximately by spherical geometry. According to the general theory of relativity, the geometrical properties of space are not independent, but they are determined by matter. I wished to show that space time is not necessarily something to which one can ascribe to a separate existence, independently of the actual objects of physical reality. Physical objects are not in space, but these objects are spatially extended. In this way the concept 'empty space' loses its meaning. The fact that such a metric is called Euclidean is connected with the following. The postulation of such a metric in a three-dimensional continuum is fully equivalent to the postulation of the axioms of Euclidean Geometry. The defining equation of the metric is then nothing but the Pythagorean Theorem applied to the differentials of the co-ordinates.25

The Theory of Relativity and the Pythagorean Theorem

The above excerpts - from the genius himself - precede any other person's narrative of the Theory of Relativity and the Pythagorean Theorem. Accordingly, I now provide a less demanding excerpt, albeit one that addresses the effects of the Special and General theories of relativity.

... the most important effects of special and general theory of relativity can be understood in a simple and straightforward way. The system of units in which the speed of light c is the unit of velocity allows to cast all formulas in a very simple form. The Pythagorean Theorem graphically relates energy, momentum and mass.26

EUCLID'S ELEMENTS

Euclid of Alexandria was a Greek mathematician ( Figure 10), and is often referred to as the Father of Geometry. The date and place of Euclid's birth, and the date and circumstances of his death, are unknown, but it is thought that he lived circa 300 BCE. His work Elements, which includes books and propositions, is the most successful textbook in the history of mathematics. In it, the principles of what is now called Euclidean Geometry were deduced from a small set of axioms.

When Euclid wrote his Elements around 300 BCE, he gave two proofs of the Pythagorean Theorem: The first, Proposition 47 of Book I, relies entirely on the area relations and is quite sophisticated; the second, Proposition 31 of Book VI, is based on the concept of proportion and is much simpler. The first could not be Pythagoras' own proof because geometry was simply not advanced enough at that time. He may have used Book VI Proposition 31, but, if so, his proof was deficient, because the complete theory of Proportions was only developed by Eudoxus, who lived almost two centuries after Pythagoras.27

Euclid's Elements furnishes the first and, later, the standard reference in geometry. It is a mathematical and geometric treatise consisting of 13 books. It comprises a collection of definitions, postulates (axioms), propositions (theorems and constructions) and mathematical proofs of the propositions. Elements' table of contents is shown in Figure 11.28 One of the oldest surviving fragments of Euclid's Elements is shown in Figure 12.29

One of the oldest surviving fragments of Euclid's elements.

Euclid provided two very different proofs, stated below, of the Pythagorean Theorem. Euclid was the first to mention and prove Book I, Proposition 47, also known as I 47 or Euclid I 47. This is probably the most famous of all the proofs of the Pythagorean proposition.

Book VI, Proposition 31:

Book I, Proposition 47:

Euclid I 47 is often called the Pythagorean Theorem, called so by Proclus - a Greek philosopher who became head of Plato's Academy and is important mathematically for his commentaries on the work of other mathematicians - and others centuries after Pythagoras and even centuries after Euclid.30

Although many of the results in Elements originated with earlier mathematicians, one of Euclid's accomplishments was to present them in a single, logically coherent framework, making them easy to use and easy to reference, including a system of rigorous mathematical proofs that remains the basis of mathematics twenty-three centuries later.

Although best known for its geometric results, Elements also includes number theory. It considers the connection between perfect numbers and Mersenne primes, the infinitude of prime numbers and the Euclidean algorithm for finding the greatest common divisor of two numbers.31

The geometrical system described in the Elements was long known simply as geometry, and was considered to be the only geometry possible. Today, however, this system is often referred to as Euclidean Geometry to distinguish it from other so-called Non-Euclidean geometries that mathematicians discovered in the nineteenth century.32

A simple proof of the Pythagorean Theorem

At this point in my plotting of the 4000-year-old story of Pythagoras, I feel it is fitting to present one proof of the famous theorem. For me, the simplest proof among the dozens of proofs that I read in preparing this article is that shown in Figure 13.33, 34, 35

1. 1
Start with four copies of the same triangle. Three of these have been rotated 90°, 180° and 270°, respectively. Each has area ab/2. (See upper part of Figure 13.)

2. 2
Combine the four triangles to form an upright square with the side ( a+ b), and a tilted square-hole with the side c. (See lower part of Figure 13.)

3. 3
Compute the area of the big square in two ways:

4. 4
( a+ b) ²=4( ab/2)+ c ²

QED ( abbreviation, Latin, Quod Erat Demonstrandum: that which was to be demonstrated.)

A GENERALIZED VERSION OF THE PYTHAGOREAN THEOREM

In the seventeenth century, Pierre de Fermat (1601-1665) ( Figure 14) investigated the following problem: for which values of n are there integer solutions to the equation

It is known that when n=2 then an integer solution exists from the Pythagorean Theorem. Fermat conjectured that there were no non-zero integer solutions for x and y and z when n was greater than 2. He did not leave a proof, though. Instead, in the margin of a textbook, he wrote that he knew that this relationship was not possible, but he did not have enough room on the page to write it down. His conjecture became known as Fermat's Last Theorem. This may appear to be a simple problem on the surface, but it was not until 1993 when Andrew Wiles of Princeton University finally proved the 350-year-old marginalized theorem, which appeared on the front page of the New York Times.36

Today, Fermat is thought of as a number theorist, in fact perhaps the most famous number theorist who ever lived. It is therefore surprising to find that Fermat was a lawyer, and only an amateur mathematician. Also surprising is the fact that he published only one mathematical paper in his life, and that was an anonymous paper written as an appendix to a colleague's book.

Because Fermat refused to publish his work, his friends feared that it would soon be forgotten unless something was done about it. His son Samuel undertook the task of collecting Fermat's letters and other mathematical papers, comments written in books and so on with the goal of publishing his father's mathematical ideas. Samuel found the marginal note ( the proof could not fit on the page) in his father's copy of Diophantus's Arithmetica. In this way the famous Last Theorem came to be published.37

FERMAT'S LAST THEOREM: SOLVED

Sir Andrew John Wiles, KBE (Knight Commander of the Order of the British Empire), mathematician and professor at Princeton University, specializing in number theory, is forever famous for proving Fermat's Last Theorem ( Figure 15).

Andrew Wiles was born in Cambridge, England in 1953, and attended King's College School, Cambridge (where his mathematics teacher David Higginbottom first introduced him to Fermat's Last Theorem). He earned his BA in 1974 after study at Merton College, Oxford, and a PhD in 1980 after research at Clare College, Cambridge. His graduate research was guided by John Coates beginning in the summer of 1975. Together they worked on the arithmetic of elliptic curves with complex multiplication using the methods of Iwasawa theory.

He further worked with Barry Mazur on the main conjecture of Iwasawa theory over Q and soon afterwards generalized this result to totally real fields. Taking approximately 7 years to complete the work, Wiles was the first person to prove Fermat's Last Theorem, earning him a place in history. Andrew Wiles' most famous mathematical result is that all rational semi-stable elliptic curves are modular, which, in particular, implies Fermat's Last Theorem.

Wiles was introduced to Fermat's Last Theorem at the age of 10. He tried to prove the theorem using textbook methods and later studied the work of mathematicians who had tried to prove it. When he began his graduate studies, he stopped trying to prove the theorem and began studying elliptic curves under the supervision of John Coates.

In the 1950s and 1960s, a connection between elliptic curves and modular forms was conjectured by the Japanese mathematician Goro Shimura based on some ideas that Yutaka Taniyama posed. In the West, this conjecture became well known through a paper by André Weil. With Weil giving conceptual evidence for it, it is sometimes called the Shimura-Taniyama-Weil conjecture. It states that every rational elliptic curve is modular. The full conjecture was proven by Christophe Breuil, Brian Conrad, Fred Diamond and Richard Taylor in 1998 using many of the methods that Andrew Wiles used in his 1995 published papers.38

CONCLUSION

I provide the story of Pythagoras and his famous theorem by discussing the major plot points of a 4000-year-old fascinating story in the history of mathematics, worthy of recounting even for the math-phobic reader. It is more than a math story, as it tells a history of two great civilizations of antiquity rising to prominence 4000 years ago, along with historic and legendary characters, who not only define the period, but whose life stories individually are quite engaging.

The intriguing plot points of the story are:

1. 1
Pythagoras is immortally linked to the discovery and proof of a theorem, which bears his name - even though there is no evidence of his discovering and/or proving the theorem.

2. 2
Greek mathematician Euclid, referred to as the Father of Geometry, lived during the period of time about 300 BCE, when he was most active. His work Elements is the most successful textbook in the history of mathematics. Euclid I 47 is often called the Pythagorean Theorem, called so by Proclus, a Greek philosopher who became head of Plato's Academy and is important mathematically for his commentaries on the work of other mathematicians centuries after Pythagoras and even centuries after Euclid.

3. 3
There is concrete (not Portland cement, but a clay tablet) evidence that indisputably indicates that the Pythagorean Theorem was discovered and proven by Babylonian mathematicians 1000 years before Pythagoras was born.

4. 4
So many people, young and old, famous and not famous, have touched the Pythagorean Theorem. The eccentric mathematics teacher Elisha Scott Loomis spent a lifetime collecting all known proofs and writing them up in The Pythagorean Proposition, a compendium of 371 proofs. The manuscript was published in 1927, and a revised, second edition appeared in 1940. As for the exact number of proofs, no one is sure how many there are. Surprisingly, geometricians often find it quite difficult to determine whether some proofs are in fact distinct proofs. Moreover, the theorem seemingly has no ending, as every year students, academicians and problem solvers with a mathematical bent tackle the theorem in an attempt to add new and innovative proofs.

5. 5
In addition, many people's lives have been touched by the Pythagorean Theorem. A 12-year-old Albert Einstein was touched by the earthbound spirit of the Pythagorean Theorem. The wunderkind provided a proof that was notable for its elegance and simplicity. However, the spirit of the Pythagoras' Theorem was not finished with young Einstein: two decades later he used the Pythagorean Theorem in the Special Theory of Relativity (in a four-dimensional form), and in a vastly expanded form in the General Theory of Relativity. That Einstein used Pythagorean Theorem for his Relativity would be enough to show Pythagorean Theorem's value, or importance to the world. But, people continued to find value in the Pythagorean Theorem, namely, Wiles.

6. 6
The theorem's spirit also visited another youngster, a 10-year-old British Andrew Wiles, and returned two decades later to an unknown Professor Wiles. Young Wiles tried to prove the theorem using textbook methods, and later studied the work of mathematicians who had tried to prove it. When he began his graduate studies, he stopped trying to prove the theorem and began studying elliptic curves, which provided the path for proving Fermat's Theorem, the news of which made to the front page of the New York Times in 1993. Sir Andrew Wiles will forever be famous for his generalized version of the Pythagoras Theorem.

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Pythagoras: Everyone knows his famous theorem, but not who discovered it 1000 years before him - Journal of Targeting, Measurement and Analysis for Marketing

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Everyone who has studied geometry can recall, well after the high school years, some aspect of the Pythagorean Theorem. However, the story of Pythagoras and his famous theorem is not well known. Some of the plot points of the story are presented in this article.

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Look at this guy's unique perspective on the real world implementation of mathematical knowledge through the use of the methods and tools of glass blowing (a simultaneous science and art). Does the ability to create a physical object "prove" something about a mathematical concept? (Question 1)

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Your Klein Bottle is in the Post - Numberphile

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1. Do mathematical symbols have meaning in the same way that words have meaning?

https://owlcation.com/stem/Mathematical-Language-and-Symbols#:~:text=It%20is%20distinct%20and%20unique,to%20communicate%20abstract%2C%20logical%20ideas.

https://youtu.be/S-F5Xhrolts

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Mathematical Language and Mathematical Symbols

Mathematical language is a system used in the field of mathematics to communicate mathematical ideas, concepts, and theories among people. It is distinct and unique from the usual language most people are used to and is used to communicate abstract, logical ideas.

Burns (n.d) mentioned three characteristics of mathematical language. These are being precise, being concise, and being powerful. Precision of mathematical language means the language is able to make very fine distinctions about and between things. Conciseness is the ability to say things briefly. Mathematical language being powerful is its ability to express complex thoughts with relative ease so that most people can understand them.

Mathematical symbols are used to refer to certain quantities, concepts, and ideas, among other things. The most commonly used symbols in basic mathematics are the numbers (0, 1, 2, 3, 4, 5, 6, 7, 8, 9), the four fundamental symbols of operation (+, -, ×, ÷), the inequality symbols (≠, ≤, ≥, < >), and the equal sign (=).

Mathematical Expressions vs. Mathematical Sentences

A mathematical expression is the mathematical analogue of an English noun. That is, it is a correct arrangement of mathematical symbols used to represent a mathematical object of interest (Burns, n.d). It does not state a complete thought and it does not make sense to ask if an expression is true or false. In mathematics areas, such as Algebra, the most common expressions are numbers, sets, and functions.

A mathematical sentence is the mathematical analogue of an English sentence. That is, it is a correct arrangement of mathematical symbols that state a complete thought. Hence, it makes sense to ask if a sentence is true, false, sometimes true, or sometimes false.

Take a look at the table below to visualize the difference between expression and sentence in mathematics.

The Language of Algebra

Mathematical sentences and expressions are commonly used in Algebra. In fact, these concepts are generally taught to you during your 7th grade academic year. Let's take a look back at the Language of Algebra where symbols, expressions, and mathematical sentences are first introduced.

Common Words in Algebra

1. Variable - A symbol that represents a value or a number. Usually, small letters are used as variables. For instance, a, x, and y.

2. Constant - A number on its own, meaning the value is never definite and will never change. For instance, 4, 9, and 35.

3. Term - A constant, or a variable, or constant(s) and variable(s) multiplied together. For instance,4, z, 5xy, 87xy, and xyz. The literal coefficient is the variable in a particular term. The numerical coefficient is the constant in a particular term.

4. Expression - Also called an algebraic expression, it is a group of terms separated by the "plus" or "minus" symbols. For instance, 3x-2 and x+3.

As mentioned, mathematics uses sentences. In Algebra, these sentences can be written mathematically or verbally. Mathematical sentences are written using variables, constants, and an equal sign or inequality sign. On the other hand, verbal sentences are written in words using phrases.

Commonly Used Verbal Phrases in Algebra

1. Addition ("plus," "increased by," "added to," "the sum of," and "more than")

2. Subtraction ("decreased by," "subtracted from," "the difference of," "less than," and "diminished by")

3. Multiplication ("multiplied by," "of," "the product of," and "times")

4. Division ("ratio of," "the quotient of," and "divided by")

5. Inequality ("is greater than," "is less than," "is at least," "is less than or equal," "is greater than or equal to," "is at most," and "is not equal to")

6. Equality ("the same as," "is equal to," and "equals")

Translating English to Math: Further Help With the Language of Algebra
This content is accurate and true to the best of the author's knowledge and is not meant to substitute for formal and individualized advice from a qualified professional. © 2020 Ryan Bernido Network
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Mathematical language is a system used in the field of mathematics to communicate mathematical ideas, concepts, and theories among people. It is distinct and unique from the usual language most people are used to and is used to communicate abstract, logical ideas. Burns (n.d) mentioned three characteristics of mathematical language.

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Real World Example - Phillip and Zakaria

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Real World Example - Phillip and Zakaria

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https://theconversation.com/cancer-evolution-is-mathematical-how-random-processes-and-epigenetics-can-explain-why-tumor-cells-shape-shift-metastasize-and-resist-treatments-199398

Would be useful for question number three as an example of the use of statistics.

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Cancer is often seen as a disease that arises from genetic mutations causing cells to divide uncontrollably and invade other parts of the body. But the spread of cells away from their origins is actually a normal process in some cases. The embryo burrows into the uterus during early pregnancy. Immune cells spread from lymph nodes to sites of infection to attack the invading bacteria. And germ cells migrate to where the gonad will be during early human development.

Cancer is not a uniform disease. Rather, cancer is a disease of phenotypic plasticity, meaning tumor cells can change from one form or function to another. This includes reverting to less mature states and losing their normal function, which can result in treatment resistance, or changing their cell type altogether, which facilitates metastasis.

In addition to direct changes in your DNA in cancer, a key driver of cancer progression is where and when your DNA is activated. If your DNA contains the "words" that spell out individual genes, then epigenetics is the "grammar" of your genome, telling those genes whether they should be turned on or off in a given tissue. Even though all tissues in the body have almost exactly the same DNA sequence, they can all carry out different functions because of chemical and structural modifications that change which genes are activated and how. This " epigenome " can be influenced by environmental exposures such as diet, adding a dimension to how researchers understand drivers of health beyond the DNA code inherited from your parents.

I'm a cancer researcher, and my laboratory at Johns Hopkins University studies how the differences among normal tissues are controlled by an epigenetic code, and how this code is disrupted in cancer. In our recently published review, colleague Andre Levchenko at Yale University and I describe a new approach to understanding cancer plasticity by combining epigenetics with mathematics. Specifically, we propose how the concept of stochasticity can shed light on why cancers metastasize and become resistant to treatments.

What is stochasticity?

Stochasticity is a mathematical concept that refers to the idea that the randomness of the steps in a process affects the predictability of its outcome. Albert Einstein famously studied this concept applied to the movement of particles suspended in a liquid or gas. Researchers can apply stochasticity to study the spread, resistance and evolution of COVID-19, the behavior of the stock market and almost any game inside a casino.

A key way to measure the stochasticity of a process is entropy, which quantifies the degree of uncertainty in an outcome. For example, a fair coin toss has an entropy of one, or low information, because there is no way to predict whether the coin toss will be heads or tails. But a weighted coin toss has an entropy of zero, or high information, as the outcome is already known and no new information will be gained by tossing the coin.

Researchers can use entropy to measure the amount of informational noise in telecommunications. Entropy can also help players beat the game Wordle. The word with highest entropy and thus the largest expected new information after each guess would be your best bet.

Epigenetics ties stochasticity and cancer together

Incorporating stochasticity and entropy into biology allows researchers to better understand phenotypic plasticity in cancer. This could even reconceptualize development by including reversibility, or moving against "time's arrow." This departs from a more classic perspective of embryonic development that views tissues as becoming progressively and irreversibly differentiated into their final state as they develop.

Experimental and computational biologists are using entropy to understand the underlying randomness in how cells are internally organized, respond to environmental signals and mature and form tissues.

Stochasticity in epigenetics is critical to how cancer evolves. For example, a condition called Barrett's esophagus occurs when healthy cells in the esophagus develop features more like what cells lining the intestine normally have, which can ultimately lead to esophageal cancer. This is caused by progressive random changes in the epigenetic code, and this shift occurs more rapidly once it reaches a certain threshold. The stochastic nature of these epigenetic changes also leads to increased entropy in the function of those genes, and progression toward cancer.

By measuring the gene activity and epigenetic changes of individual cells, biologists and mathematicians can compare the entropy in cancer cells with the normal cells surrounding them. Scientists are now beginning to identify regions of the genome that mediate stochasticity in cancer. A study that has not yet been peer-reviewed found that entropy is related to how chromosomes are physically compacted in the nucleus, another key epigenetic mechanism for controlling gene activity in cancer.

There is a connection between entropy and aging as well. My colleagues and I found that human aging is associated with an increase in epigenetic entropy in sun-damaged skin. Parts of the genome that have high entropy experience further loss of epigenetic information in sun-exposed skin, which can lead to cancer. Recently, researchers have identified DNA damage as a cause for this age-associated entropy in mice. Thus, if epigenetic entropy increases in aging and is related to DNA damage, it might help explain why cancer risk starkly increases with age.

By identifying how epigenetic entropy triggers cancer, scientists might be able to better detect cancer at its early stages, and design drugs that reduce entropy and so decrease the risk of tumors spreading and becoming resistant to treatment.

And perhaps most importantly, epigenetic entropy shows that you can't fully understand cancer without mathematics. Biology is catching up with other hard sciences in incorporating mathematical methods with biological experimentation.

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Cancer evolution is mathematical - how random processes and epigenetics can explain why tumor cells shape-shift, metastasize and resist treatments

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Cancer is often seen as a disease that arises from genetic mutations causing cells to divide uncontrollably and invade other parts of the body. But the spread of cells away from their origins is actually a normal process in some cases. The embryo burrows into the uterus during early pregnancy.

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https://www.youtube.com/watch?v=P8oEutoqSmo

How is an axiomatic system of knowledge different from, or similar to, other systems of knowledge

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In mathematics and set theory, an axiomatic system is any set of specified axioms from which some or all of those axioms can be used, in conjunction along with derivation rules or procedures, to logically derive theorems. A mathematical theory or set theory consists of an axiomatic system and all its derived theorems. An axiomatic system that is completely described is a special kind of formal system; usually, however, the effort towards complete formalization brings diminishing returns in certainty and a lack of readability for humans. Therefore discussion of axiomatic systems is normally only semi-formal. A formal theory typically means an axiomatic system, for example formulated within model theory. A formal proof is a complete rendition of a mathematical or set-theoritical proof within a formal system.

Properties

An axiomatic system is said to be consistent if it lacks contradiction (i.e. it is not possible to derive both a statement and its negation from the system's axioms).

In an axiomatic system, an axiom is called independent if it is not a theorem that can be derived from other axioms in the system. A system will be called independent if each of its underlying axioms is independent.

The most important criterion for assessment of an axiomatic system is that particular system's consistency. Inconsistency in an axiomatic system is universally regarded as being a fatal flaw for that system.

Independence is also a desireable property, but its lack is not a fatal flaw. Lack of independence means that the system has redundancy in its axioms, meaning that one or more of its axioms is not needed. This is usually considered to be a flaw because reducing the number of axioms of a system to the minimum necessary for deriving all the needed or desired theorems of that system is considered to be a virtue, because axioms are unproved and unprovable; having as little of that as possible means that as few unproved assumptions as possible are being made in that system.

An axiomatic system will be called complete if for every statement, either itself or its negation, is derivable in that system. This is very difficult to achieve, however, and as shown by the combined works of Gödel and Coen, impossible for axiomatic systems involving infinite sets. So, along with consistency, relative consistency is also the mark of a worthwhile axiom system. This occurs when the undefined terms of a first axiom system are given definitions from a second, such that the axioms of the first are theorems of the second.

A good example is the relative consistency of neutral geometry, or absolute geometry, with respect to the theory of the real number system. Lines and points are undefined terms in absolute geometry, but assigned meanings in the theory of real numbers in a way that is consistent with both axiom systems.

A model for an axiomatic system is a well-defined set, which assigns meaning for the undefined terms presented in the system, in a manner that is correct with the relations defined in the system. The existence of a concrete model proves the consistency of a system.

A model is called concrete if the meanings assigned are objects and relations from the real world, as opposed to an abstract model which is based on other axiomatic systems. The first axiomatic system was Euclidean geometry.

Models can also be used to show the independence of an axiom in the system. By constructing a valid model for a subsystem without a specific axiom, we show that the omitted axiom is independent if its correctness does not necessarily follow from the subsystem.

Two models are said to be isomorphic if a one-to-one correspondence can be found between their elements, in a manner that preserves their relationship. An axiomatic system for which every model is isomorphic to another is called categorial (sometimes categorical), and the property of categoriality (categoricity) ensures the completeness of a system.

Axiomatic method

The axiomatic method is often discussed as if it were a unitary approach, or uniform procedure. With the example of Euclid to appeal to, it was indeed treated that way for many centuries. Up until the beginning of the nineteenth century it was generally assumed in European mathematics and philosophy (for example in Spinoza's work) that the heritage of Greek mathematics represented the highest standard of intellectual finish (development more geometrico, in the style of the geometers).

That approach, in which axioms were supposed to be self-evident and thus indisputable, was swept away during the course of the nineteenth century. One important episode in this was the development of Non-Euclidean geometry, based on denial of Euclid's parallel postulate (or axiom). It was found that consistent geometries can be constructed by denying that postulate, taking as an axiom that more than one parallel to a given line can be drawn through a point outside that line, or a different axiom that no parallel can be drawn-both of those result in different and consistent geometric systems that may or may not be applicable to an experienced world.

Other challenges to the supposed self-evidence of axioms came from the foundations of real analysis, from Georg Cantor's set theory, and from the failure of Frege's work on foundations. Russell was able to derive a paradox-a kind of contradiction-from Frege's axioms for set theory, thus showing that Frege's axiomatic system was not consistent, and this showed that the supposed self-evidence of Frege's axioms was mistaken.

Another challenge came from David Hilbert's 'new' use of axiomatic method as a research tool. For example, group theory was first put on an axiomatic basis towards the end of that century. Once the axioms were clarified (that inverse elements should be required, for example), the subject could proceed autonomously without reference to the transformation group's origins of those studies.

Therefore, there are at least three 'modes' of axiomatic method current in mathematics, and in the fields it influences. In caricature, possible attitudes are:

1. Accept my axioms and you must accept their consequences;

2. I reject one of your axioms and accept extra models;

3. My set of axioms defines a research program.

The first case is the classic deductive method. The second goes by the slogan be wise, generalise; it may go along with the assumption that concepts can or should be expressed at some intrinsic 'natural level of generality'. The third was very prominent in the mathematics of the twentieth century, in particular in subjects based around homological algebra.

It is easy to see that the axiomatic method has limitations outside mathematics and set theory. For example, in political philosophy axioms that lead to unacceptable conclusions are likely to be rejected wholesale; so that no one really assents to version 1 above.

Axiomatic systems in science

Historically, the most important purpose of an axiom system was to reach an overview of some science or part of science. Euclid succeeded in doing that for geometry. Beyond its successful use in set theory and mathematics, there have been attempts to use the axiomatic method in physics (by Ludwig Boltzmann, Heinrich Hertz, and some members of the Vienna Circle), biology (by J. H. Woodger), quantum mechanics (by Günther Ludwig), and possibly other sciences.

Those attempts have been at best only partly successful. If such efforts at axiomatization for sciences were to be successful, this would make it possible to study these sciences simply by drawing logical inferences from the axioms, without needing any new empirical input. When conclusions are drawn from general scientific laws or principles that method is in fact employed, empirical testing of such theoretical derivations is still always needed. Thus, the axiomatization and formalization of the system is incomplete and does not solve the problem of whether the system yields actual scientific knowledge.

Axiomatization

Model theory

Gödel's incompleteness theorem

References

Bernays, Paul. Axiomatic Set Theory. With a hist. introd. by Abraham A. Fraenkel, Amsterdam: North-Holland Pub. Co., 1958.

Euclid. 2002. Euclid's Elements: All Thirteen Books Complete in One Volume. trans. Thomas L. Heath, Dana Densmore, ed. Santa Fe, N.M.: Green Lion Press. ISBN 1888009187 ISBN 1888009195

Henkin, Leon. 1959. The Axiomatic Method With Special Reference to Geometry and Physics. Amsterdam: North-Holland Pub. Co.

Lightstone, A. H. 1964. The Axiomatic Method; An Introduction to Mathematical Logic. Englewood Cliffs, N.J.: Prentice-Hall.

Ludwig, Günther. 1985. An Axiomatic Basis for Quantum Mechanics. Berlin & New York : Springer-Verlag. ISBN 0387137734

Meyer, Burnett. 1974. An Introduction to Axiomatic Systems. Boston: Prindle, Weber & Schmidt. ISBN 0871501740

Quine, W. V. 1969. Set Theory and Its Logic. Rev. Ed., Cambridge: Belknap Press of Harvard University Press.

Suppes, Patrick. 1972. Axiomatic Set Theory. New York: Dover Publications. ISBN 0486616304

Takeuti, Gaisi, and W. M. Zaring. 1973. Axiomatic Set Theory. New York: Springer-Verlag. ISBN 0387900519

Woodger, Joseph Henry. 1937. The Axiomatic Method in Biology. With appendices by Alfred Tarski and W. F. Floyd. London: Cambridge University Press, Photocopy: Ann Arbor, MI: University Microfilms, 1986.

External links

All links retrieved December 31, 2021.

Stanford Encyclopedia of Philosophy entries:

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# data[].attributes.attachment_link.description
In mathematics and set theory, an axiomatic system is any set of specified axioms from which some or all of those axioms can be used, in conjunction along with derivation rules or procedures, to logically derive theorems. A mathematical theory or set theory consists of an axiomatic system and all its derived theorems. An axiomatic system that is completely described is a special kind of formal system; usually, however, the effort towards complete formalization brings diminishing returns in certainty and a lack of readability for humans.

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Do mathematical symbols have meaning in the same way that words have meaning?

# data[].attributes.subject
Do mathematical symbols have meaning in the same way that words have meaning?

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https://owlcation.com/stem/Mathematical-Language-and-Symbols

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1.5 Translating Words into Mathematical Symbols

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3.How is an axiomatic system of knowledge different from, or similar to, other systems of knowledge?

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3.How is an axiomatic system of knowledge different from, or similar to, other systems of knowledge?

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Matthew and Ben

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Science and maths skills are widely celebrated as keys to economic and technological progress, but abstract mathematics may seem bafflingly far from industrial optimisation or medical imaging. Pure mathematics often yields unanticipated applications, but without a time machine to look into the future, how do mathematicians like me choose what to study?

Over Thai noodles, I asked some colleagues what makes a problem interesting, and they offered a slew of suggestions: surprises, contradictions, patterns, exceptions, special cases, connections. These answers might sound quite different, but they all support a view of the mathematical universe as a structure to explore.

In this view, mathematicians are like anatomists learning how a body works, or navigators charting new waters. The questions we ask take many forms, but the most interesting ones are those that help us see the big picture more clearly.

Making maps

Mathematical objects come in many forms. Some of them are probably quite familiar, like numbers and shapes. Others might seem more exotic, like equations, functions and symmetries.

Instead of just naming objects, a mathematicians might ask how some class of objects is organised. Take prime numbers: we know there are infinitely many of them, but we need a structural understanding to work out how frequently they occur or to identify them in an efficient way.

Other good questions explore relationships between apparently different objects. For example, shapes have symmetry, but so do the solutions to some equations.

Classifying objects and finding connections between them help us assemble a coherent map of the mathematical world. Along the way, we sometimes encounter surprising examples that defy the patterns we've inferred.

Such apparent contradictions reveal where our understanding is still lacking, and resolving them provides valuable insight.

Consider the triangle

The humble triangle provides a famous example of an apparent contradiction. Most people think of a triangle as the shape formed by three connecting line segments, and this works well for the geometry we can draw on a sheet of paper.

However, this notion of triangle is limited. On a surface with no straight lines, like a sphere or a curly kale leaf, we need a more flexible definition.

Read more: Pythagoras' revenge: humans didn't invent mathematics, it's what the world is made of

So, to extend geometry to surfaces that aren't flat, an open-minded mathematician might propose a new definition of a triangle: pick three points and connect each pair by the shortest path between them.

This is a great generalisation because it matches the familiar definition in the familiar setting, but it also opens up new terrain. When mathematicians first studied these generalised triangles in the 19th century, they solved a millennia-old mystery and revolutionised mathematics.

The parallel postulate problem

Around 300 BC, the Greek mathematician Euclid wrote a treatise on planar geometry called The Elements. This work presented both fundamental principles and results that were logically derived from them.

One of his principles, called the parallel postulate, is equivalent to the statement that the sum of the angles in any triangle is 180°. This is exactly what you'll measure in every flat triangle, but later mathematicians debated whether the parallel postulate should be a foundational principle or just a consequence of the other fundamental assumptions.

This puzzle persisted until the 1800s, when mathematicians realised why a proof had remained so elusive: the parallel postulate is false on some surfaces.

On a sphere, the sides of a triangle bend away from each other and the angles add up to more than 180°. On a rippled kale leaf, the sides bow in towards each other and the angle sum is less than 180°.

Triangles where the angle sum breaks the apparent rule led to the revelation that there are kinds of geometry Euclid never imagined. This is a deep truth, with applications in physics, computer graphics, fast algorithms, and beyond.

Salad days

People sometimes debate whether mathematics is discovered or invented, but both points of view feel real to those of us who study mathematics for a living. Triangles on a piece of kale are skinny whether or not we notice them, but selecting which questions to study is a creative enterprise.

Interesting questions arise from the friction between patterns we understand and the exceptions that challenge them. Progress comes when we reconcile apparent contradictions that pave the way to identify new ones.

Today we understand the geometry of two-dimensional surfaces well, so we're equipped to test ourselves against similar questions about higher-dimensional objects.

Read more: Corals, crochet and the cosmos: how hyperbolic geometry pervades the universe

In the past few decades we've learned that three-dimensional spaces also have their own innate geometries. The most interesting one is called hyperbolic geometry, and it turns out to act like a three-dimensional version of curly kale. We know this geometry exists, but it remains mysterious: in my own research field, there are lots of questions we can answer for any three-dimensional space ... except the hyperbolic ones.

In higher dimensions we still have more questions than answers, but it's safe to say that study of four-dimensional geometry is entering its salad days.

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Exploring the mathematical universe - connections, contradictions, and kale

# data[].attributes.attachment_link.description
Science and maths skills are widely celebrated as keys to economic and technological progress, but abstract mathematics may seem bafflingly far from industrial optimisation or medical imaging. Pure mathematics often yields unanticipated applications, but without a time machine to look into the future, how do mathematicians like me choose what to study?