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The free newsletter unpacks important and thought-provoking news stories every month, and helps you to think about them critically. Subscribe here.

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Download a HUGE amount of free lessons, newsletters, TED talks docs, assessment guidance, and other goodies on this page.

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You can use the stories to develop a clearer understanding of this element of TOK, see how TOK concepts manifest in the real world, and gather examples to support arguments in the essay and exhibition.

We've arranged these stories in a way that relates to our 5 Big Question framework for TOK, which is our favoured way of structuring the course. There are other ways to structure TOK, such as the Classic structure.

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A Conversation article, which offers readers additional mathematical constants to Pi that deserve to be celebrated.

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A Conversation article, which offers readers additional mathematical constants to Pi that deserve to be celebrated.

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UNPACKING THE SOURCE To what extent is mathematics ‘hardwired’ into our reality?

KEY TERMS Constant, Abstract subject

LINKS TO Mathematics, Technology, Knowledge & the knower, Human sciences

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Pi gets all the fanfare, but other numbers also deserve their own math holidays

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Pi gets a lot of attention this time of year, but there are plenty of other mathematical constants just as deserving of recognition.

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A Forbes article, arguing that mathematics alone cannot provide us with a complete picture of the nature of the universe.

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A Forbes article, arguing that mathematics alone cannot provide us with a complete picture of the nature of the universe.

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UNPACKING THIS SOURCE This is quite an approachable article that discusses the relationship between mathematics and theoretical physics, and how the former is not sufficient on its own to provide us with a framework to understand the latter. How did mathematics hinder the work of Kepler when he tried to explain the movements of the planets? What was required in order to make the breakthrough (hint - use the phrase ‘observables and measurables’)? Overall, how does mathematics depend on a ‘confrontation with reality’ in order to provide us with useful knowledge about the universe?

KEY TERMS Mathematical framework, Quantitative description, Observables and measurables, Confrontation with reality

LINKS TO Mathematics, Natural sciences

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At the frontiers of theoretical physics, many of the most popular ideas have one thing in common: they begin from a mathematical framework that seeks to explain more things than our currently prevailing theories do. Our current frameworks for General Relativity and Quantum Field Theory are great for what they do, but they don't do everything. They're fundamentally incompatible with one another, and cannot sufficiently explain dark matter, dark energy, or the reason why our Universe is filled with matter and not antimatter, among other puzzles.

It's true that mathematics enables us to quantitatively describe the Universe, it's an incredibly useful tool when applied properly. But the Universe is a physical, not mathematical entity, and there's a big difference between the two. Here's why mathematics alone will always be insufficient to reach a fundamental theory of everything.

About 400 years ago, a battle was unfolding about the nature of the Universe. For millennia, astronomers had accurately described the orbits of the planets using a geocentric model, where the Earth was stationary and all the other objects orbited around it. Armed with the mathematics of geometry and precise astronomical observations - including tools like circles, equants, deferents, and epicycles - the precise mathematical description of the heavenly bodies' orbits matched what we saw spectacularly.

The match wasn't perfect, however, and attempts to improve upon it either led to more epicycles or, in the 16th century, Copernicus' heliocentrism. By placing the Sun at the center, explanations of retrograde motion became simpler, but the fits to the data were worse. When Johannes Kepler came along, he had a brilliant idea that sought to solve everything.

He noticed that there were six planets total, if you included Earth but not Earth's Moon. He also noticed that mathematically, there were only five Platonic solids: five mathematical objects whose faces are all equal-sided polygons. By drawing a sphere inside and outside each one, he could "nest" them in a way that fit the planetary orbits extremely well: better than anything Copernicus had done. It was a brilliant, beautiful mathematical model, and arguably the first attempt at constructing what we might call "an elegant Universe" today.

But observationally, it failed. It failed to even be as good as the ancient Ptolemaic model with its epicycles, equants and deferents. It was a brilliant idea, and the first attempt to argue - from pure mathematics alone - how the Universe ought to be. But it just didn't work.

What came next was a stroke of genius that would define Kepler's legacy.

He took his beautiful, elegant, compelling model that disagreed with observations, and threw it away. Instead, he went and dove into the data to find what types of orbits would match how the planets actually moved, and came away with a set of scientific (not mathematical) conclusions.

1. Planets didn't move in circles around the centrally located Sun, but rather in ellipses with the Sun at one focus, with a different set of parameters describing the ellipse of each planet.

2. Planets didn't move at a constant speed, but rather moved at a speed that varied with the planet's distance from the Sun, in such a way that planets sweep out equal areas in equal times.

3. And finally, planets exhibited orbital periods that were directly proportional to the long axis (the major axis) of each planet's ellipse, raised to a specific power (determined to be 3/2).

This was a revolutionary moment in the history of science. Mathematics wasn't at the root of the physical laws governing nature; it was a tool that described how the physical laws of nature manifested themselves. The key advance that happened is that science needed to be based in observables and measurables, and that any theory needed to confront itself with those notions. Without it, progress would be impossible.

This idea came up again and again throughout history, as new mathematical inventions and discoveries empowered us with new tools to attempt to describe physical systems. But each time, it wasn't simply that new mathematics told us how the Universe worked. Instead, new observations told us that something beyond our currently understood physics was required, and pure mathematics alone was insufficient to get us there.

By the early 1900s, it was clear that Newtonian mechanics was in trouble. It could not explain how objects moved near the speed of light, leading to Einstein's special theory of relativity. Newton's theory of universal gravitation was in similarly hot water, as it could not explain the motion of Mercury around the Sun. Concepts like spacetime were just being formulated, but the idea of non-Euclidean geometry (where space itself could be curved, rather than flat like a 3D grid) had been floating around for decades among mathematicians.

Unfortunately, developing a mathematical framework to describe spacetime (and gravitation) required more than pure mathematics, but the application of mathematics in a particular, tweaked way that would agree with observations of the Universe. It's the reason why we all know the name "Albert Einstein," but very few people know the name "David Hilbert."

Both men had theories that linked spacetime curvature to gravity and the presence of matter and energy. Both of them had similar mathematical formalisms; today an important equation in General Relativity is known as the Einstein-Hilbert action. But Hilbert, who had come up with his own, independent theory of gravity from Einstein, pursued bigger ambitions than Einstein: his theory applied to both matter and electromagnetism as well as gravity.

And that simply didn't agree with nature. Hilbert was constructing a mathematical theory as he thought it ought to apply to nature, and could never get out successful equations that predicted the quantitative effects of gravity. Einstein did, and that's why the field equations are known as the Einstein field equations, with no mention of Hilbert. Without a confrontation with reality, we don't have physics at all.

This almost identical situation came up again just a few years later in the context of quantum physics. You couldn't simply fire an electron through a double slit and know, based on all the initial conditions, where it would wind up. A new type of mathematics - one rooted in wave mechanics and a set of probabilistic outcomes - was required. Today, we use the mathematics of vector spaces and operators, and physics students hear a term that might ring a bell: Hilbert space.

The same mathematician, David Hilbert, had discovered a set of mathematical vector spaces that was enormously promising for quantum physics. Only, once again, its predictions didn't quite make sense when confronted with physical reality. For that, some tweaks needed to be made to the math, creating what some call a rigged Hilbert space or a physical Hilbert space. The mathematical rules needed to be applied with certain specific caveats, or the results of our physical Universe would never be recoverable.

Today, it's grown very fashionable in theoretical physics to appeal to mathematics as a potential way forward to an even more fundamental theory of reality. A number of mathematical-based approaches have been tried over the years:

imposing additional symmetries,

adding extra dimensions,

adding new fields into General Relativity,

adding new fields into quantum theory,

using larger groups (from mathematical group theory) to extend the Standard Model,

along with many others. These mathematical explorations are interesting and potentially relevant for physics: they may hold clues as to what secrets the Universe might have in store beyond what's presently known. But mathematics alone cannot teach us how the Universe works. We will obtain no definitive answers without confronting its predictions with the physical Universe itself.

In some ways, it's a lesson that every physics student learns the first time they calculate the trajectory of an object thrown into the air. How far does it go? Where does it land? How long does it spend in the air? When you solve the mathematical equations - Newton's equations of motion - that govern these objects, you don't get "the answer." You get two answers; that's what the mathematics gives you.

But in reality, there's only one object. It only follows one trajectory, landing in one location at one specific time. Which answer corresponds to reality? Mathematics won't tell you. For that, you need to understand the particulars of the physics problem in question, as only that will tell you which answer has a physical meaning behind it. Mathematics will get you very far in this world, but it won't get you everything. Without a confrontation with reality, you cannot hope to understand the physical Universe.

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No, The Universe Is Not Purely Mathematical In Nature

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At the frontiers of theoretical physics, many of the most popular ideas have one thing in common: they begin from a mathematical framework that seeks to explain more things than our currently prevailing theories do. Our current frameworks for General Relativity and Quantum Field Theory are great for what they do, but they don't do everything.

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A Conversation article, looking at how research in mathematics is a deeply imaginative and intuitive process - aided by AI. 

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A Conversation article, looking at how research in mathematics is a deeply imaginative and intuitive process - aided by AI. 

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UNPACKING THE SOURCE This article provides a great introduction to the way mathematicians draw on intuition, and whether AI can help us to develop this mathematical skill. Begin by making sure you know what mathematical conjectures and theorems are - and how you move from one to the other. Why does the article refer to intuition as a “miraculous something”, and why does this present problems to anyone hoping to use AI to support the discovery of mathematical knowledge? Why does the article assert that “intelligence is best thought of as having many dimensions” - and can this be applied to other AOKs?

KEY TERMS Imaginative and intuitive, Conjectures, Theorems, “Miraculous something”, Discoveries, Another dimension

LINKS TO Mathematics, Technology

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Research in mathematics is a deeply imaginative and intuitive process. This might come as a surprise for those who are still recovering from high-school algebra.

What does the world look like at the quantum scale? What shape would our universe take if we were as large as a galaxy? What would it be like to live in six or even 60 dimensions? These are the problems that mathematicians and physicists are grappling with every day.

To find the answers, mathematicians like me try to find patterns that relate complicated mathematical objects by making conjectures (ideas about how those patterns might work), which are promoted to theorems if we can prove they are true. This process relies on our intuition as much as our knowledge.

Over the past few years I've been working with experts at artificial intelligence (AI) company DeepMind to find out whether their programs can help with the creative or intuitive aspects of mathematical research. In a new paper published in Nature, we show they can: recent techniques in AI have been essential to the discovery of a new conjecture and a new theorem in two fields called "knot theory" and "representation theory".

Machine intuition

Where does the intuition of a mathematician come from? One can ask the same question in any field of human endeavour. How does a chess grandmaster know their opponent is in trouble? How does a surfer know where to wait for a wave?

The short answer is we don't know. Something miraculous seems to happen in the human brain. Moreover, this "miraculous something" takes thousands of hours to develop and is not easily taught.

The past decade has seen computers display the first hints of something like human intuition. The most striking example of this occurred in 2016, in a Go match between DeepMind's AlphaGo program and Lee Sedol, one of the world's best players.

AlphaGo won 4-1, and experts observed that some of AlphaGo's moves displayed human-level intuition. One particular move ( "move 37") is now famous as a new discovery in the game.

Read more: AI has beaten us at Go. So what next for humanity?

How do computers learn?

Behind these breakthroughs lies a technique called deep learning. On a computer one builds a neural network - essentially a crude mathematical model of a brain, with many interconnected neurons.

At first, the network's output is useless. But over time (from hours to even weeks or months), the network is trained, essentially by adjusting the firing rates of the neurons.

Such ideas were tried in the 1970s with unconvincing results. Around 2010, however, a revolution occurred when researchers drastically increased the number of neurons in the model (from hundreds in the 1970s to billions today).

Traditional computer programs struggle with many tasks humans find easy, such as natural language processing (reading and interpreting text), and speech and image recognition.

With the deep learning revolution of the 2010s, computers began performing well on these tasks. AI has essentially brought vision and speech to machines.

Training neural nets requires huge amounts of data. What's more, trained deep learning models often function as "black boxes". We know they often give the right answer, but we usually don't know (and can't ascertain) why.

A lucky encounter

My involvement with AI began in 2018, when I was elected a Fellow of the Royal Society. At the induction ceremony in London I met Demis Hassabis, chief executive of DeepMind.

Over a coffee break we discussed deep learning, and possible applications in mathematics. Could machine learning lead to discoveries in mathematics, like it had in Go?

This fortuitous conversation led to my collaboration with the team at DeepMind.

Mathematicians like myself often use computers to check or perform long computations. However, computers usually cannot help me develop intuition or suggest a possible line of attack. So we asked ourselves: can deep learning help mathematicians build intuition?

With the team from DeepMind, we trained models to predict certain quantities called Kazhdan-Lusztig polynomials, which I have spent most of my mathematical life studying.

In my field we study representations, which you can think of as being like molecules in chemistry. In much the same way that molecules are made of atoms, the make up of representations is governed by Kazhdan-Lusztig polynomials.

Amazingly, the computer was able to predict these Kazhdan-Lusztig polynomials with incredible accuracy. The model seemed to be onto something, but we couldn't tell what.

However, by "peeking under the hood" of the model, we were able to find a clue which led us to a new conjecture: that Kazhdan-Lusztig polynomials can be distilled from a much simpler object (a mathematical graph).

This conjecture suggests a way forward on a problem that has stumped mathematicians for more than 40 years. Remarkably, for me, the model was providing intuition!

Read more: How explainable artificial intelligence can help humans innovate

In parallel work with DeepMind, mathematicians Andras Juhasz and Marc Lackenby at the University of Oxford used similar techniques to discover a new theorem in the mathematical field of knot theory. The theorem gives a relation between traits (or "invariants") of knots that arise from different areas of the mathematical universe.

Our paper reminds us that intelligence is not a single variable, like the result of an IQ test. Intelligence is best thought of as having many dimensions.

My hope is that AI can provide another dimension, deepening our understanding of the mathematical world, as well as the world in which we live.

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Mathematical discoveries take intuition and creativity - and now a little help from AI

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Machine learning systems can now aid human intuition to create mathematical conjectures and proofs.

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Mordechai Levy-Eichel in Aeon considers various aspects of mathematics for – its “joyful, beautiful” nature, its role in shaping modern society, and the way in which it is often neglected when people consider the history of ideas.

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Mordechai Levy-Eichel in Aeon considers various aspects of mathematics for – its “joyful, beautiful” nature, its role in shaping modern society, and the way in which it is often neglected when people consider the history of ideas.

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UNPACKING THIS SOURCE Provide your students with some anchors to the article, and pose them questions to explore mathematics (and the development of ideas). For example, why does Levy-Eichel say that “Perhaps more than any other subject, mathematics is about the study of ideas.” Do you agree that mathematics is taught in a way that “obscures the actual insights and reasoning, the grandeur and insight, the excitement and frustration, that drive mathematicians.”? And do you agree that mathematics “like any literature, is created by human beings for their own amusement.” How should we envisage the nature of mathematics in order to properly understand it? How does mathematical knowledge underpin other areas of knowledge? Should we always challenge our assumptions about the nature of knowledge?

KEY TERMS Aesthetics, Beauty, Neo-Platonic

LINKS TO Mathematics, Knowledge & the knower

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'These foundations [of mathematics] are, moreover, a perpetual source of reflection and discovery for the masters of the science. Even in number system you will find material for long reflection. Remember that Leibniz did not disdain to occupy himself with it.' - Paul Valéry, letter to Pierre Honnorat, 1942

There are almost too many examples of the power and pervasiveness of mathematical ideas. For instance, this essay was written on a computer. The software of the computer, its mind and spirit, if you will, is a compilation of code that is based on the ideas of Claude Shannon, the father of information theory, and his article 'A Mathematical Theory of Communication' (1948). But perhaps this is both too obvious and too slight an example. The personal computer is hardly an essential part of human existence, even if most of us have structured our lives around it today. Let's take something more basic and widespread, something most of us probably already apprehend, even if only dimly - like the idea of regression to the mean.

As the Oxford English Dictionary explains, regression to the mean is 'the tendency for the values of any distributed variable to move towards the mean over repeated independent trials'. In other words, the more trials, the less random or mistaken the measure. For instance, say you're in a race at school. You do surprisingly well and beat most of your classmates. All things being equal, the next time around, you're actually not likely to do as well, relative to the other runners. Obviously, one's actual rank depends on skill and talent - if you did well the first time, you probably are pretty fast - but each result also depends on luck as well as a host of other circumstances. Therefore, in order to mitigate against any selection effect, one has to run the experiment multiple times. In order to be able to see just where you actually place or rank, you have to be able to know the shape or form of the distribution of outcomes. The notion of regression to the mean informs how we think about a wide range of things, from the design of clinical trials, to gambling, to, well, the prosaic pep-talks we give ourselves after coming up short by saying: 'OK, next time will be better.' Actually, it probably will be.

The Victorian polymath (and eugenicist) Sir Francis Galton first discovered and formulated the idea of regression to the mean in his studies of heredity. The 'regression toward mediocrity', in Galton's memorable phrase, grew out of his studies of why genius seemed to run through families (think of the Bachs), but then also how and why it dissipated over time. Genius and talent themselves being impossible to measure, Galton studied other more quantifiable phenomena, for instance, the height differences between parents and children, and the spatial distribution of falling objects. He devised a contraption called the Quincunx, and found that when uniformly dropped objects encountered small deviations (think of a pinball machine with lots of stoppers), they nevertheless distributed themselves at the bottom along a normal distribution - what we generally call a bell curve. The bell curve, of course, is just one example of the various forms used to plot the relationship among independent and dependent variables. Galton's conceptual innovation, his idea of regression to the mean, made certain features of the world newly visible. By doing so, he helped us count and describe other things we can't see, or test, but that shape our lives.

The modern separation among scholars between intellectual history and the history of mathematics is untenable as mathematics might be the ultimate intellectual endeavour. In the words of the 19th-century German mathematician Carl Friedrich Gauss: 'mathematics is the queen of the sciences'; like literacy, widespread numeracy is one of the defining features of modernity. In fact, one of the great shifts of modernity has been how mathematicians changed their view of mathematics, transforming the focus of their work from the study of the natural world to the study of ideas and concepts. Perhaps more than any other subject, mathematics is about the study of ideas. Yet, when people invoke the history of ideas, you are unlikely to hear about Dedekind's cut (that is, the technique by which the real numbers are rigorously defined from the rational numbers), or L E J Brouwer's rejection of Aristotle's 'law of excluded middle', which states that any proposition is either true or that its negation is (put technically: for all propositions p, either p or not p). Nor are you likely to hear about the contested history of these ideas. Generally, when they talk about ideas, intellectual historians today mean political thought, cultural analysis, and maybe a sprinkling of economic and religious concepts, too.

Nor has this divide been one-sided. As the historian John Tresch recently noted: 'For most historians of science trained in the past 30 years, doing history of science has meant avoiding the history of ideas.' Even though most historians of science would surely consider the history of mathematics part of their field, in fact the subject is more often conspicuous by its absence than its presence. However, the larger problem among historians of science is that, while the consolidation of the field over the previous century has been an institutional and economic success, it has segregated the field. Today, it primarily takes place within separate departments or committees, with separate training. This reinforces, in practice and effect, a division between the study of science and the study of society - something its own literature repeatedly criticises.

'Mathematics, like any literature, is created by human beings for their own amusement'

One of the many damaging results of this intellectual division is that most of us - that is, those of us who are not mathematicians, physicists or engineers - adopt a view of mathematics that is primarily the product of our encounter with it in grade school. For most people, unfortunately, mathematics is a set of confusing, repetitive, formalistic and abstract techniques. Yet this is exactly the opposite of how mathematicians see their own work. Rather, what attracts them, in the words of The Mathematics Lover's Companion (2017) by the graph theorist Ed Scheinerman, are 'joyful, beautiful' theorems and proofs, which they arrive at through the sweat of intellectual play. Akin to the best poems, they contain truths about the world perfectly expressed.

Perhaps the most impassioned remonstration against our bifurcated view of mathematics is the pamphlet A Mathematician's Lament (2009) by the American private-school teacher Paul Lockhart. Written in 2002 and circulated for years among mathematicians and educators, Lockhart's essay ruthlessly criticised the simplistic nature of most mandatory mathematics education. Mathematics, Lockhart wrote, is almost always taught in a way to obscure the actual insights and reasoning, the grandeur and insight, the excitement and frustration, that drive mathematicians. 'At no time are students let in on the secret that mathematics, like any literature, is created by human beings for their own amusement,' Lockhart writes, attempting to humanise the subject.

Continuing with his literature analogy, Lockhart emphasises that: 'A piece of mathematics is like a poem, and we can ask if it satisfies our aesthetic criteria: is this argument sound? Does it make sense? Is it simple and elegant? Does it get me closer to the heart of the matter?' And like most pieces of art, truly appreciating it is not a rote exercise: 'works of mathematics are subject to critical appraisal; that one can have and develop mathematical taste'. By teaching mathematics as a series of techniques, the appeal and nature of mathematics is masked: 'Of course there's no criticism going on in school - there's no art being done to criticise!' The notable English mathematician G H Hardy also compared mathematics to poetry in his own work A Mathematician's Apology (1940): 'A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.'

What then might we all learn from better integrating the two? What might be gained if, to quote Egon Spengler in the film Ghostbusters (1984), 'we cross the streams' of the history of mathematics and the history of ideas?

We should not understand the history of ideas as one besotted by speculations about politics. With that, hopefully, we can gain a wider sense of the diverse ideas and visions that, deep down (for better and for worse), motivate and inform any of our endeavours. Intellectual history today is dominated by elite political ideas. There are many factors that have contributed to the shortchanging of the history of mathematics and its ideas. High among them is how narrowly political the work of leading intellectual historians has been.

The British scholar Quentin Skinner and the French philosopher Michel Foucault, for example, are arguably the most influential intellectual historians of the past 50 years. Superficially, they are unlike each other in almost every way, from their prose style, to their subjects. Yet both portray intellectual history mostly in terms of political languages and expressions of power. The discourses and mentalities of individuals, for both of them, describe much of the social and political world, and account for when and why change happens. But one of the most important reasons for the rise of the modern state is the increasing possible reach of politicians and bureaucrats, changes powerfully facilitated by improvements in mathematics and technology, and that are absent in both of their accounts.

Recognising the role of mathematics in the history of how we think would actually make us more sophisticated in our political judgment. Those wishing to lend certainty and authority to their ideologies use mathematics, but they also abuse it. As the epitome of formal knowledge and the premier example of abstract reason, the spectre of mathematics has haunted European thought for centuries. From theologians to politicians and economists, many have naively - even manipulatively - adopted mathematical methods to develop, and too often implement, overly simplistic theories of the world. Despite the repeated problems with this approach, the impulse has been seminal, formative and comparatively neglected.

The discipline of economics offers a recent example, but this is also true for areas as widely separate as architecture and philosophy, as the political scientist and anthropologist James C Scott shows in his book Seeing Like a State: How Certain Schemes to Improve the Human Condition Have Failed (1999). The allure of formal schemes has turned people's attention to the results of mathematics, rather than to the creative ways that mathematicians work and think. Commenting on the aftermath of the late 19th- and early 20th-century controversy over the foundations of his subject, the influential mathematician John von Neumann wrote in the essay The Mathematician (1947) that '[T]he very concept of "absolute" mathematical rigour is not immutable. The variability of the concept of rigour shows that something else besides mathematical abstraction must enter into the makeup of mathematics.' While the reason that mathematics is authoritative is due of its supposed rigour and certainty, what we mean by rigour has changed throughout history, based on novel ideas and realisations. How mathematics works rests on no absolute timeless standard, despite what many assume today, given its precision and efficacy.

Taking mathematics seriously might lead thinking back to questions of aesthetics and beauty

In some ways, mathematics is the hardest, and most realistic, art. The German philosopher Martin Heidegger, for instance, offers an example of an influential thinker who misrepresented mathematics as a kind of thinking divorced from the natural world. In the essay 'Modern Science, Metaphysics, and Mathematics' (1962), he writes: 'In the mathematical project develops an obligation to principles demanded by the mathematical itself.' In Heidegger's view, the embrace of mathematics and technology brings a kind of repetitious, routinised, alienating world in its wake. The paradox, however, is that a desire to bypass repetitious exercises is precisely one of the things that motivates mathematicians. Elementary proofs are nothing if not ways to think about something in all applicable cases so that repetitive testing on each possibility is not necessary. Nothing could be more elegant. Describing technology and mathematical modes of thought as mechanistic and alienating mistakes the result for the process.

Finally, taking mathematical ideas seriously might lead to discovering that technical ideas are as important as political or religious ones. Taking mathematics seriously might also, counter to stereotypes, lead thinking away from current preoccupations with culture and power, and back to questions of aesthetics and beauty. Aesthetics and beauty are ever-present concerns in art and in mathematics, though seemingly small matters to historians and humanists today, preoccupied as they are with power.

Take, for example, the case of the Weils: Simone Weil was an important literary and political writer, regarded as one of the more insightful religious philosophers and social critics of the first half of the 20th century; her brother André, a mathematician, made important contributions to a variety of areas of mathematics, but is much less widely known, unless one is a mathematician. There are many colourful anecdotes about André Weil's life - he escaped war-torn Europe and lived all around the world - but perhaps most importantly, and amusingly, he was a founding member the Bourbaki group. This was a loose association of mostly French mathematicians in the first half of the 20th century who tried to put mathematics on a new and self-contained basis (a recurring goal in the history of mathematics and, more generally, the history of thought). One of the main ways they tried to achieve this goal was by collectively writing textbooks under the name of a fictitious character, one Nicolas Bourbaki. Bourbaki doesn't exist, but he published not a few books, and came up with some of the terms and symbols modern mathematicians use, such as Ø to represent the empty set. André Weil led the first meeting of the Bourbaki group.

And Weil is just the beginning. There's Shannon and his insights into information theory, along with his teacher, the prodigy Norbert Wiener, who coined the term 'cybernetics' (and wrote a scintillating two-volume autobiography). There's the debates between Brouwer and David Hilbert about the basis of mathematics. And there's the extraordinary career of Emmy Noether, who made contributions across many fields of mathematics and physics, and helped to apply abstract algebra to topology (which deals with the preservation of spatial properties during transformations of shape and size, and is one of the main ways topology is done today). In other words, it would mean beginning to be as interested in the creators of our technological civilisation as history has been in those who lament it.

The history of ideas can, should and, historically, has often been precisely about the limitations of politics and the rewards of other ideas; it need not primarily be the study of thinkers who, more often than not, never even directly held power. The ideas that most deeply shape us, and our societies, are just as often not part of the history of political thought, but come from many other areas. This is not an argument for a neo-Platonic worshipful examination of ideas, or for a move away from considerations of politics. It is rather a call to recognise the ways in which humans actually encounter ideas, and the unique vitality of mathematical ideas in the world.

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The secret intellectual history of mathematics - Mordechai Levy-Eichel | Aeon Essays

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'These foundations [of mathematics] are, moreover, a perpetual source of reflection and discovery for the masters of the science. Even in number system you will find material for long reflection. Remember that Leibniz did not disdain to occupy himself with it.'

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A Big Think video and article that considers whether octonions can help us to unravel certain uncertainties about the structure of the universe. 

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A Big Think video and article that considers whether octonions can help us to unravel certain uncertainties about the structure of the universe. 

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UNPACKING THE SOURCE There are some mind-boggling ideas within this real-life situation, and it is very hard for a non-specialist to comprehend. But students should be able to grasp the superficial details of this story, and relate it to the perennial question about whether mathematics is invented or discovered (see the point made about the link between octonions and the force that keeps quarks together in an atomic nucleus). Overall, how can abstract mathematical concepts help us to understand the real world? How ‘real’ are mathematical theories? How do mathematical ideas develop?

KEY TERMS Model, Revolutionary

LINKS TO Mathematics, Natural sciences

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Is our reality, including its forces and particles, based on the strange properties of numbers with eight dimensions called " octonions "? A physicist thinks so, having found a way to expand 40-year-old research to reach surprising new directions.

First, a brief history of numbers.

Regular numbers that we are familiar with in our everyday life can be paired up in a special way to create "complex numbers," which act like coordinates on a two-dimensional plane. This was discovered in 16h-century Italy by the mathematician Gerolamo Cardano. As explains Natalie Wolchover of Quanta Magazine, you can perform operations on complex numbers like adding, subtracting, multiplying and dividing by "translating and rotating positions around the plane."

An Irish mathematician by the name of William Rowan Hamilton discovered in 1843 that if you pair the complex numbers in a certain way, they can form 4-D " quaternions." He was apparently so excited about figuring out that formula, that he immediately carved it into the Broome Bridge in Dublin. Not to be outdone, John Graves, a friend of Hamilton's who was a lawyer and math whiz, showed that quarternions can be paired up to become "octonions" - numbers that can assume coordinates in an abstract 8-dimensional (8-D) space.

Each type of numbers has been utilized extensively in the development of modern physics, with complex numbers used in quantum mechanics and even the quaternions employed in Albert's Einstein's special theory of relativity.

What hasn't been completely understood and put to work - the octonions, usually represented by the capital letter O and whose multiplication rules are encoded in a triangular diagram called the Fano plane (that looks like something the Freemasons would devise).

The mystery of these numbers has led to speculation among researchers that they have a special purpose and can eventually explain the deeper secrets of the universe. In an email interview with Quanta Magazine, the particle physicist Pierre Ramond from the University of Florida explained that "Octonions are to physics what the Sirens were to Ulysses."

In 1973, Murat Günaydin, the then-Yale-graduate student (now professor at Penn State) and his advisor Feza Gürsey, discovered that there is an unexpected link between octonions and the strong force that keeps quarks together in an atomic nucleus. Günaydin continued his research quite outside the mainstream, looking at connecting the numbers to such ideas as string theory and M-theory.

In 2014, Cohl Furey, a graduate student at the University of Waterloo, Canada, built on Günaydin's work by finding a new use for the hard-to-imagine numbers. She devised an octonionic model that includes both the strong and electromagnetic forces. Now a postdoc in UK's University of Cambridge, Furey generated a series of results that link the octonions to the Standard Model of particle physics, in work that has been praised by other scientists. She has "taken significant steps toward solving some really deep physical puzzles," said Shadi Tahvildar-Zadeh, a mathematical physicist at Rutgers University.

Others, like the noted string theorist and Imperial College London professor Michael Duff, are more reserved, excited about her work but saying it's "hard to say" yet if it will become "revolutionary."

Furey is undeterred by working in a currently obscure field, thinking of her research as a "process of collecting clues," as she explained in an interview.

She published a paper in May 2018's The European Physical Journal C, where she consolidated several of her findings, looking to complete the standard model of particle physics and find the rightful place in our understanding of the world for the octonions.

To learn more, watch Furey explain the octonions here:

For those inclined to delve deeper into the math, check out this fascinating graphic:

Lucy Reading-Ikkanda/Quanta Magazine

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Physicists puzzled by strange numbers that could explain reality

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Is our reality, including its forces and particles, based on the strange properties of numbers with eight dimensions called " octonions "? A physicist thinks so, having found a way to expand 40-year-old research to reach surprising new directions. First, a brief history of numbers.

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A Conversation article, marking the 100th anniversary of the death of John Venn, and discussing why his diagrams are still so important. 

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A Conversation article, marking the 100th anniversary of the death of John Venn, and discussing why his diagrams are still so important. 

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UNPACKING THE SOURCE Do we only understand mathematical and logical concepts when we visualize them?

KEY TERMS Relationships, Visualise, Databases, Syllogism, Logic

LINKS TO Mathematics, Technology

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April 2023 marks the 100th anniversary of the death of mathematician and philosopher John Venn. You may well be familiar with Venn diagrams - the ubiquitous pictures of typically two or three intersecting circles, illustrating the relationships between two or three collections of things.

For example, during the pandemic, Venn diagrams helped to illustrate symptoms of COVID-19 that are distinct from seasonal allergies. They are also often taught to school children and are typically part of the early curriculum for logic and databases in higher education.

Venn was born in Hull, UK, in 1834. His early life in Hull was influenced by his father, an Anglican priest - it was expected John would follow in his footstep. He did initially begin a career in the Anglican church, but later moved into academia at the University of Cambridge.

One of Venn's major achievements was to find a way to visualise a mathematical area called set theory. Set theory is an area of mathematics which can help to formally describe properties of collections of objects.

For example, we could have a set of cars, C. Within this set, there could be subsets such as the set of electric cars, E, the set of petrol based cars, say P, and the set of diesel powered cars, D. Given these, we can operate on them, for example, to apply car charges to the sets P and D, and a discount to the set E.

These sorts of operations form the basis of databases, as well as being used in many fundamental areas of science. Other major works of Venn's include probability theory and symbolic logic. Venn had initially used diagrams developed by the Swiss mathematician Leonard Euler to show some relationships between sets, which he then developed into his famous Venn diagrams.

Venn used the diagrams to prove a form of logical statement known as a categorical syllogism. This can be used to model reasoning. Here's an example: "All computers need power. All AI systems are computers." We can chain these together to the conclusion that "all AI systems need power".

Today, we are familiar with such reasoning to illustrate how different collections relate to each other. For example, the SmartArt tool in Microsoft products lets you create a Venn diagram to illustrate the relationships between different sets. In our earlier car example, we could have a diagram showing electric cars, E, and petrol powered cars, P. The set of hybrid cars that have a petrol engine would be in the intersection of P and E.

Logic and computing

The visualisation of sets (and databases) is helpful, but the importance of Venn's work then - and now - is the way they allowed proof of George Boole's ideas of logic as a formal science.

Venn used his diagrams to illustrate and explore such " symbolic logic " - defending and extending it. Symbolic logic underpins modern computing, and Boolean logic is a key part of the design of modern computer systems - making his work relevant today.

Venn's work was also crucial to the work of philosopher Bertrand Russell, showing that there are problems that are unsolvable. We can express such problems with sets, in which each is an unsolvable problem. One such unsolvable problem can be expressed with the " Barber paradox ". Suppose we had an article in Wikipedia containing all the articles that don't contain themselves - a set. Is this new article itself in that set?

Luckily we can visualise that with a Venn diagram with two circles, where one circle is the set of entries that don't include themselves, A, and the other circle is the set of entries that do include themselves, B.

We can then ask the question: where do we put the article that contains all the articles that don't contain themselves? Have a think about it, then see where you would put it.

The problem is that it cannot be on the left, as it would contain itself, and would therefore be inconsistent. And it cannot be on the right, as then it would be missing, or incomplete. And it can't be in both. It must be in one or the other. This paradox illustrates how unsolvable statements can arise - they are valid in terms of expressing them within the logical system, but ultimately unanswerable. We could possibly extend our system to solve this, but then we would end up with another unanswerable question.

Venn's diagrams were crucial in understanding this. And this area of science is still important, for example when considering the limitations of machine learning and AI, where we may ask questions that cannot be answered.

Venn also had an interest in building mechanical machines - including a bowling machine which proved so effective it was able to bowl out some top Australian batsmen of the day.

Following his abstract work on logic, he developed the concept of a logical-diagram machine with a lot of processing power: though this brilliant idea from 1881 would take many decades to appear as modern computers.

We remember Venn here in Hull, with a bridge close to his birthplace decorated with Venn circle inspired artwork. At the University of Hull's main administration building, there's an intersection of management and academia which is called the Venn building.

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Venn: the man behind the famous diagrams - and why his work still matters today

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April 2023 marks the 100th anniversary of the death of mathematician and philosopher John Venn. You may well be familiar with Venn diagrams - the ubiquitous pictures of typically two or three intersecting circles, illustrating the relationships between two or three collections of things.

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This Conversation article considers the mathematics behind the shape and structure of DNA.

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This Conversation article considers the mathematics behind the shape and structure of DNA.

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UNPACKING THIS SOURCE How does mathematics provide us with an insight into biological structures?

KEY TERMS Computer simulations, Mathematical modeling vs. biological experiments

LINKS TO Mathematics, Natural sciences

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Math shows how DNA twists, turns and unzips

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Mathematical models can describe the many shapes of DNA, as well as cellular processes like DNA replication.

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A Discover Magazine article, evaluating what statistics can reveal about the scale of serial killers who were never caught. 

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A Discover Magazine article, evaluating what statistics can reveal about the scale of serial killers who were never caught. 

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UNPACKING THE SOURCE How can statistical data help us to understand social issues and phenomena?

KEY TERMS Mathematical model

LINKS TO Mathematics, Human sciences

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The most prolific modern serial killer, according to Wikipedia , is probably Harold Shipman, a British doctor who probably killed as many as 250 people.

Shipman's crimes went unnoticed because his victims were mostly elderly and whose deaths were unlikely to raise suspicions. However, researchers have since pointed out that Shipman's murderous tendencies stick out like a sore thumb if they are viewed through the lens of statistics. Too many of his patients died unexpectedly and this statistical signature could have raised the alarm earlier.

Clearly, statistics can play a valuable role in characterizing the behavior of serial killers. Now Mikhail Simkin and Vwani Roychowdhury at the University of California, Los Angeles, say their analysis of data on serial killers reveals how many go uncaught and how many victims these killers must have bagged.

Their analysis begins with the observation that for some serial killers, the time between murders can stretch to decades. So it is reasonable to think that some killers will die during this interval before they can be caught.

Mathematical Model

With this in mind, Simkin and Roychowdhury construct a simple mathematical model that simulates the behavior of these killers. The important parameters in this model are, first, the probability that a killer can commit a murder without being caught and, second, the likelihood of death before he or she commits another murder.

Of course, not all serial killers are equally capable. So the probability of being caught is likely to change from one killer to another. Simkin and Roychowdhury account for this by using a probability distribution.

To calculate the likelihood of death, they use US life tables from 1950 (they are interested in the number uncaught killers in the 20th century).

Finally, the researchers use these probabilities to model the behavior of 1 million killers using a Monte Carlo simulation.

The simulation begins by choosing at random the age of the first killer when he or she strikes first (from a distribution of the actual ages of serial killers when they committed their first crimes).

This killer then commits their first murder and the simulation decides whether or not he or she is caught using the probability distribution described above. The simulation then calculates when the killer will strike next, based on a random choice of interval taken from a distribution of murders by real serial killers.

It next uses the life table to decide whether the killer will still be alive at this time. If not, the killer dies and remains uncaught. If still alive, the simulation repeats the calculations for a second murder. It then starts on the next killer and so on until it has simulated the behavior of a million of them.

Simulated Behavior

The results make for interesting reading. Out of these million killers, 659,684 were caught after the first murder. But 539 died without being caught. Of the rest, 337,729 went on to commit two or more murders and of these 2048 went uncaught.

"The ratio of uncaught to caught killers in the simulated sample was 2,048 divided by 337,729 = 0.006064," say Simkin and Roychowdhury.

That ratio can then be used to calculate the number that went uncaught in real life. They point out that there were 1172 serial killers who were caught in the US during the 20th century which suggests a specific number evaded the law. "The result is that in 20th century there were about seven of such killers," they say.

They go on to calculate how many victims these seven killers must have had using the distribution of victim numbers of real killers. These numbers for uncaught killers are sobering. "The most prolific of them likely committed over sixty murders," say Simkin and Roychowdhury.

The researchers point out that their simulation has one obvious weakness. This is that some serial killers would likely be prevented from killing by poor end-of-life health rather than death. So active life span would be a better measure than total life span. "So the fraction of the uncaught killers would be only bigger," they say.

That's interesting work that once again highlights the potential of statistics in the fight against crime. Nevertheless, this will be little comfort to the families of the victims whose murders remain unsolved.

Ref: Estimating The Number Of Serial Killers That Were Never Caught : arxiv.org/abs/2109.11051

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Statisticians Reveal the Number of Serial Killers That Were Never Caught During The 20th Century

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Some prolific serial killers almost certainly die before they can be caught. Now a new statistical analysis reveals just how many evade justice in this way.

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A Guardian article on the ‘hunt’ for a new ninth planet in the solar system. 

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A Guardian article on the ‘hunt’ for a new ninth planet in the solar system. 

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UNPACKING THE SOURCE The article prompts us to think about how knowledge is produced in a field where the line between theory and observation is very blurred. What is the evidence that a ninth planet exists? Will this only become reality when we can see the planet ‘for ourselves’? What will it mean for existing knowledge if the planet’s existence is proven? Does science progress cumulatively, or does new knowledge violently destroy old knowledge?

KEY TERMS Evidence, Eureka moment

LINKS TO Mathematics, Technology

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You'd think that if you found the first evidence that a planet larger than the Earth was lurking unseen in the furthest reaches of our solar system, it would be a big moment. It would make you one of only a small handful of people in all of history to have discovered such a thing.

But for astronomer Scott Sheppard of the Carnegie Institution for Science in Washington DC, it was a much quieter affair. "It wasn't like there was a eureka moment," he says. "The evidence just built up slowly."

He's a master of understatement. Ever since he and his collaborator Chad Trujillo of Northern Arizona University, first published their suspicions about the unseen planet in 2014, the evidence has only continued to grow. Yet when asked how convinced he is that the new world, which he calls Planet X (though many other astronomers call it Planet 9), is really out there, Sheppard will only say: "I think it's more likely than unlikely to exist."

As for the rest of the astronomical community, in most quarters there is a palpable excitement about finding this world. Much of this excitement centres on the opening of a giant new survey telescope named after Vera C Rubin, the astronomer who, in the 1970s, discovered some of the first evidence for dark matter.

Scheduled to begin its full survey of the sky in 2022, the Rubin observatory could find the planet outright or provide the clinching circumstantial evidence that it's there.

Discovery of the planet would be a triumph, but also a disaster for existing theory about how the solar system was created.

"It would change everything we thought we knew about planet formation," says Sheppard, in another characteristic understatement. In truth, no one has a clue how such a large planet could form that far from the sun.

The distant solar system is a place of darkness and mystery. It encompasses an enormous volume of space that begins at the orbit of Neptune, some 30 times further from the sun than Earth, or 30 astronomical units (AU)the , and extends to about 100,000AU. That's almost one-third of the distance from the sun to the next nearest star.

It was in the inner regions of this volume that American astronomer Clyde Tombaugh discovered Pluto in 1930. Although Pluto possessed just two-thirds of the diameter of our moon, it was originally classed as a planet.

Pluto, which was discovered in 1930, was downgraded to a dwarf planet in 2006. Photograph: AFP/Getty Images

By the end of the century, however, telescopes were bigger and astronomers were beginning to find more tiny worlds beyond Neptune. They were all even smaller than Pluto until 2005, when Mike Brown from the California Institute of Technology discovered Eris. It was at least the same size as Pluto and probably bigger, so, if Pluto was a planet, so was Eris. Nasa hastily organised a press conference and announced the discovery of Planet 10.

About a year later, the International Astronomical Union ruled that Pluto and Eris were effectively too small to be called planets and renamed them dwarf planets. So the solar system's roll-call returned to eight: Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune. And a cottage industry of finding distant solar system objects really got going.

The path towards Planet 9 began one night in 2012, when Sheppard and Trujillo were using the Cerro Tololo Inter-American Observatory's telescope in Chile. They were finding more and more distant objects, but one in particular stood out. Catalogued as 2012 VP113, they nicknamed it Biden after the US vice-president at the time (because of the letters VP in the catalogue designation). To their amazement, this far-flung world never came closer to the sun than about 80AU. At its furthest, Biden would reach 440AU into deep space, meaning that it followed a highly elliptical orbit. But that wasn't the most remarkable thing about it.

By some weird coincidence, its orbit appeared to be very similar to that of another distant world known as Sedna. This mini-world had been discovered in 2003 by Brown, Trujillo and David Rabinowitz of Yale University. It immediately stood out because of its highly elliptical orbit, which swings from 76AU to 937AU.

"Objects like Sedna and 2012 VP113 can't form on these eccentric orbits," says Sheppard. Instead, computer simulations suggest that they form much closer and are then ejected by gravitational interactions with the larger planets. The truly odd thing, however, was that the two elongated orbits pointed in roughly the same direction.

And the more Sheppard and Trujillo examined the other objects in their catch, the more they saw that those orbits were aligned, too. It was as if something was corralling those tiny worlds, like a sheepdog manoeuvring its flock. And the only thing they could think of that was capable of doing that was a much larger planet.

An illustration showing some of the more distant solar system objects. Photograph: Illustration by Roberto Molar Candanosa and Scott Sheppard/Carnegie Institution for Science

Curiosity piqued, they did some calculations and discovered that the planet their results were hinting at had to be somewhere between two and 15 times more massive than Earth, on an orbit that lies on average somewhere between 250AU and 1500AU from the sun. Their results were published by the prestigious journal in March 2014 and interest in Planet 9 began to sweep the astronomical world.

The next big leap came in 2015 when Sheppard and Trujillo were among the scientists who discovered 2015 TG387. They nicknamed this one the Goblin. It's the third most extreme object behind Sedna and Biden and it, too, lines up, reducing still further the idea that this alignment is a random coincidence.

In 2016, Brown and his collaborator Konstantin Batygin, also of Caltech, published their own analysis of the data. Agreeing with Sheppard and Trujillo about the size and distance of the planet, they even suggested an area of sky where they thought it might be found.

But not everyone is convinced.

Pedro H Bernardinelli, a PhD candidate at the University of Pennsylvania, realised that Sheppard's data wasn't the only place one could look for distant worldlets. So he turned to some initial data from a cosmological survey that was designed to measure the way in which the universe is expanding by looking at far-away galaxies. He searched the data for the celestial equivalent of a photobomb, looking for distant solar system objects that just happened to get in the way of the camera. He found seven.

At first sight, it looked as if these worldlets were also aligned as expected, but the more rigorously Bernardinelli analysed the data, the weaker he felt the alignment became. "We don't think we see the signal in our data," says Bernardinelli, although he admits the he can't yet definitely rule out the planet and has yet to run the analysis on the full survey data. "Our answer might change the next time we do this," he says.

These days, Sheppard can regularly be found using Japan's Subaru telescope on Mauna Kea, Hawaii, patiently scouring the sky for more evidence of Planet 9, maybe even hoping that he sees the planet itself. The scale of the task is enormous. It really is like looking for the proverbial needle in a haystack. The planet - if it is even there - is very faint and the sky is very large. But help is on its way in the form of the Rubin observatory.

Rubin is a monster that will devour the sky. Whereas most telescopes would take months or years to survey the whole sky, Rubin will do it in just three nights. Then do it again and again and again to see what's changed and so catch the moving objects.

Construction is nearing completion, and the telescope is set to open its giant eye for the first time later this year. Commissioning and tweaking will then take another couple of years.

A Nasa illustration of its New Horizons spacecraft encountering an object in the Kuiper Belt. Photograph: HO/AFP/Getty Images

"That survey is going to change solar system science as we know it," says Sheppard. And if Planet 9 is out there, Rubin should see it.

"We can detect an Earth-mass planet at around 1000 AU," says Meg Schwamb, of Queen's University Belfast, who co-chairs the Rubin observatory's solar system science collaboration. That puts Sheppard's world easily within its sights. "If others haven't seen Planet 9 before our survey starts then, I think, all eyes are on the Rubin observatory," says Schwamb.

Even if the telescope fails to see the planet directly, it will detect many more distant mini-worlds that can all be use to triangulate the planet's position more precisely, thus helping to narrow the search area. And if Planet 9 really is out there, then the consequences will be huge.

Astronomers think that the solar system formed in a disc of matter surrounding the sun. That matter condensed into smaller bodies, which then collided to form larger ones. At the end of this process, the planets were born. But the matter in this disc thins out further from the sun, meaning there is not enough raw material to form a large planet in the distant solar system.

To rescue the standard theory, some suggest that Planet 9 was once destined to become a gas giant like Jupiter or Saturn and so was forming alongside them. However, a gravitational interaction stunted its growth by hurling it out into the dark.

But Jakub Scholtz of Durham University is sceptical. "It's possible," he says, "but it actually requires quite a lot of coincidences." That's because a single gravitational interaction can't do the job. Instead, a series of interactions is needed to place it in an orbit that never brings it back to where it formed.

Scholtz has a more exotic idea. Together with collaborator James Unwin, of the University of Illinois at Chicago, he has suggested that the object corralling these distant worldlets is not a long-lost planet but a black hole.

If so, not even Rubin will be able to see it, because black holes emit no light whatsoever - they simply swallow light and anything else that happens to cross their path. It is a tantalising possibility because Scholtz's black hole would have to be part of a long-suspected but never-proved population of black holes that were formed shortly after the formation of the universe.

But for the time being, most other astronomers seem more than content with the idea that there's a large planet out there in the darkness, just waiting to come into view in the next few years.

And if Planet 9 really is there, then perhaps the first time Sheppard sees it through a telescope, he will finally experience something akin to a eureka moment.

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Beyond Pluto: the hunt for our solar system's new ninth planet

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Scientists think a planet larger than Earth lurks in the far reaches of the solar system

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A Discover Magazine article about the way most people use the word and concept ‘average’ in a misguided way. 

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A Discover Magazine article about the way most people use the word and concept ‘average’ in a misguided way. 

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UNPACKING THE SOURCE Begin by asking students what they understand the term ‘average’ to be, and how often they refer to it, or think in terms of it. Then ask them to read the article - are they using it correctly? How can the concept of ‘average’ be manipulated in order to skew our perception of reality (see the hypothetical example of ‘Chestnut Street’)? Thinking bigger, what does this tell us about the use and misuse of statistical data, and how the intention behind it can interfere with its objectivity?

KEY TERMS Statistical terms

LINKS TO Mathematics, Human sciences

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On Average, Average May Not Mean What You Think It Does

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The different kinds of averages and what they’re good for.

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This Aeon article examines how we generate figures of probability, pointing out that there are different possible approaches, suggesting that the concept of probability is not as simple as we think. 

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This Aeon article examines how we generate figures of probability, pointing out that there are different possible approaches, suggesting that the concept of probability is not as simple as we think. 

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UNPACKING THE SOURCE Probabilities play a huge role in how we view the world. Get students to look in the news and find examples of how probabilities are used to help us understand the potential outcomes of events and issues. Next, get students to think about how probabilities are generated – from simple ‘coin-tossing’, to more complex calculations of whether someone might be guilty of a crime. How do these link to the ways of knowing? Is there a line where we can’t reasonably talk in terms of probabilities? Finally get students to read through the article. Is probability, ultimately, a philosophical assessment? To what extent do assumptions underpin our understanding of mathematics? Which ways of knowing do we use to calculate probabilities? Is probability a philosophical concept?

KEY TERMS Probabilities, Frequency interpretation, Propensity interpretation, Confidence interpretation, Mathematical rules, Mathematics as philosophy

LINKS TO Mathematics, Knowledge & the knower

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The concept of probability is not as simple as you think | Aeon Ideas

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Working out probabilities is not simple as you might think, whether you’re a gambler, a quantum physicist or a juror

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A Vox article, showing you the mathematics behind Covid-19, and why this is essential to understanding the pandemic.  

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A Vox article, showing you the mathematics behind Covid-19, and why this is essential to understanding the pandemic.  

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UNPACKING THE SOURCE Not only does this present a fantastically clear analysis of Covid-19, it also highlights a really important fact: that part of our perception of the pandemic - perhaps the most important part - has to be based on mathematics. What do we learn about the virus when we understand the statistics behind it (specifically, compared to ordinary flu)? In what way does this allow us to assess the threat and impact of Covid-19? Applying this wider, what is the role of mathematical information in informing us about the world?

KEY TERMS Comparisons

LINKS TO Mathematics, Human sciences, Natural sciences, Knowledge & the knower

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Covid-19, the disease caused by the coronavirus, can look very similar to the flu. They have similar symptoms - a fever, cough, and the possibility of leading to pneumonia - and even spread the same way. So wanting to compare the two is natural and, frankly, understandable.

But Covid-19 is very different from the flu. It's more dangerous in almost every way.

Not only is it twice as contagious, but the time it takes you to realize you are contagious is much longer. With the flu, the average time it takes you to feel sick from the moment you become sick (called the incubation period) is two days.

With Covid-19, most people don't realize they're sick - and possibly contagious - for five days; it can take up to 12 or even 14 days. You can also show no symptoms and still be contagious.

And when you become sick, you have a higher chance of needing to be hospitalized. (Yes, even young people.) While the flu can kill nearly 60,000 Americans in a year, the flu's fatality rate is still just 0.1 percent. Covid-19's fatality rate by some estimates is 10 times that, especially for older people.

Watch the video above to learn more about the differences between coronavirus and the flu, and why we have to take Covid-19 so seriously. Or read more from Vox's senior science reporter Brian Resnick.

You can find this video and all of Vox's videos on YouTube. And if you're interested in supporting our video journalism, you can become a member of the Vox Video Lab on YouTube.
Our goal this month
Now is not the time for paywalls. Now is the time to point out what's hidden in plain sight (for instance, the hundreds of election deniers on ballots across the country), clearly explain the answers to voters' questions, and give people the tools they need to be active participants in America's democracy. Reader gifts help keep our well-sourced, research-driven explanatory journalism free for everyone. By the end of September, we're aiming to add 5,000 new financial contributors to our community of Vox supporters. Will you help us reach our goal by making a gift today?

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Covid-19, the disease caused by the coronavirus, can look very similar to the flu. They have similar symptoms - a fever, cough, and the possibility of leading to pneumonia - and even spread the same way. So wanting to compare the two is natural and, frankly, understandable.

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This Atlantic story looks at a new algorithm-based tool, COMPAS, that is supposed to be able to calculate (ie predict) a defendant’s risk of committing another crime. However, it seems that the tool is no more accurate than “random volunteers recruited from the internet”.

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This Atlantic story looks at a new algorithm-based tool, COMPAS, that is supposed to be able to calculate (ie predict) a defendant’s risk of committing another crime. However, it seems that the tool is no more accurate than “random volunteers recruited from the internet”.

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UNPACKING THIS SOURCE The tone of this article is rather negative in terms of algorithms being used to make valuable ethical decisions. The problems associated with COMPAS seem to indicate one of two possibilities – either this is an issue that is too complex to be predicted accurately in this way, or COMPAS is flawed. Which of these is more likely, and why? Can ethical knowledge be objectively calculated via mathematics?

KEY TERMS Risk-assessment algorithms, Crowdsourcing, Proof

LINKS TO Mathematics, Technology

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The COMPAS tool is widely used to assess a defendant's risk of committing more crimes, but a new study puts its usefulness into perspective.

In February 2013, Eric Loomis was found driving a car that had been used in a shooting. He was arrested, and pleaded guilty to eluding an officer. In determining his sentence, a judge looked not just to his criminal record, but also to a score assigned by a tool called COMPAS.

Developed by a private company called Equivant (formerly Northpointe), COMPAS-or the Correctional Offender Management Profiling for Alternative Sanctions-purports to predict a defendant's risk of committing another crime. It works through a proprietary algorithm that considers some of the answers to a 137-item questionnaire.

COMPAS is one of several such risk-assessment algorithms being used around the country to predict hot spots of violent crime, determine the types of supervision that inmates might need, or-as in Loomis's case-provide information that might be useful in sentencing. COMPAS classified him as high-risk of re-offending, and Loomis was sentenced to six years.

He appealed the ruling on the grounds that the judge, in considering the outcome of an algorithm whose inner workings were secretive and could not be examined, violated due process. The appeal went up to the Wisconsin Supreme Court, who ruled against Loomis, noting that the sentence would have been the same had COMPAS never been consulted. Their ruling, however, urged caution and skepticism in the algorithm's use.

Caution is indeed warranted, according to Julia Dressel and Hany Farid from Dartmouth College. In a new study, they have shown that COMPAS is no better at predicting an individual's risk of recidivism than random volunteers recruited from the internet.

"Imagine you're a judge and your court has purchased this software; the people behind it say they have big data and algorithms, and their software says the defendant is high-risk," says Farid. "Now imagine I said: Hey, I asked 20 random people online if this person will recidivate and they said yes. How would you weight those two pieces of data? I bet you'd weight them differently. But what we've shown should give the courts some pause." (A spokesperson from Equivant declined a request for an interview.)

COMPAS has attracted controversy before. In 2016, the technology reporter Julia Angwin and colleagues at ProPublica analyzed COMPAS assessments for more than 7,000 arrestees in Broward County, Florida, and published an investigation claiming that the algorithm was biased against African Americans. The problems, they said, lay in the algorithm's mistakes. "Blacks are almost twice as likely as whites to be labeled a higher risk but not actually re-offend," the team wrote. And COMPAS "makes the opposite mistake among whites: They are much more likely than blacks to be labeled lower-risk but go on to commit other crimes."

Northpointe questioned ProPublica's analysis, as did various academics. They noted, among other rebuttals, that the program correctly predicted recidivism in both white and black defendants at similar rates. For any given score on COMPAS's 10-point scale, white and black people are just as likely to re-offend as each other. Others have noted that this debate hinges on one's definition of fairness, and that it's mathematically impossible to satisfy the standards set by both Northpointe and ProPublica-a story at The Washington Post clearly explains why.

The debate continues, but when Dressel read about it, she realized that it masked a different problem. "There was this underlying assumption in the conversation that the algorithm's predictions were inherently better than human ones," she says, "but I couldn't find any research proving that." So she and Farid did their own.

They recruited 400 volunteers through a crowdsourcing site. Each person saw short descriptions of defendants from ProPublica's investigation, highlighting seven pieces of information. Based on that, they had to guess if the defendant would commit another crime within two years.

On average, they got the right answer 63 percent of their time, and the group's accuracy rose to 67 percent if their answers were pooled. COMPAS, by contrast, has an accuracy of 65 percent. It's barely better than individual guessers, and no better than a crowd. "These are nonexperts, responding to an online survey with a fraction of the amount of information that the software has," says Farid. "So what exactly is software like COMPAS doing?"

Only Equivant can say, and they're not revealing the secrets of their algorithm. So the duo developed their own algorithm, and made it as simple as possible -"the kind of thing you teach undergrads in a machine-learning course," says Farid. They found that this training-wheels algorithm could perform just as well as COMPAS, with an accuracy of 67 percent, even when using just two pieces of data-a defendant's age, and their number of previous convictions. "If you are young and have a lot of prior convictions, you are high-risk," says Farid. "It's kind of obvious."

Other teams have found similar results. Last year, a team of researchers led by Cynthia Rudin from Duke University showed that a basic set of rules based on a person's age, sex, and prior convictions-essentially, an algorithm so simple you could write it on a business card-could predict recidivism as well as COMPAS.

The problem isn't necessarily that COMPAS is unsophisticated, says Farid, but that it has hit a ceiling in sophistication. When he and Dressel designed more complicated algorithms, they never improved on the bare-bones version that used just age and prior convictions. "It suggests not that the algorithms aren't sophisticated enough, but that there's no signal," he says. Maybe this is just as good as it gets. Maybe the whole concept of predicting recidivism is going to stall at odds that are not that much better than a coin toss.

Sharad Goel, from Stanford University, sees it a little differently. He notes that judges in the real world have access to far more information than the volunteers in Dressel and Farid's study, including witness testimonies, statements from attorneys, and more. Paradoxically, that informational overload can lead to worse results by allowing human biases to kick in. Simple sets of rules can often lead to better risk assessments -something that Goel found in his own work. Hence the reasonable accuracy of Dressel and Farid's volunteers, based on just seven pieces of information.

"That finding should not be interpreted as meaning that risk-assessment tools add no value," says Goel. Instead, the message is "when you tell people to focus on the right things, even nonexperts can compete with machine-learning algorithms."

Equivant make a similar point in a response to Dressel and Farid's study, published on Wednesday. "The findings of 'virtually equal predictive accuracy' in this study," the statement says, "instead of being a criticism of the COMPAS assessment, actually adds to a growing number of independent studies that have confirmed that COMPAS achieves good predictability and matches the increasingly accepted AUC standard of 0.70 for well-designed risk assessment tools used in criminal justice."

There have been several studies showing that algorithms can be used to positive effect in the criminal-justice system. "We're not saying you shouldn't use them," says Farid. "We're saying you should understand them. You shouldn't need people like us to say: This doesn't work. You should have to prove that something works before hinging people's lives on it."

"Before we even get to fairness, we need to make sure that these tools are accurate to begin with," adds Dressel. "If not, then they're not fair to anyone."

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A Popular Algorithm Is No Better at Predicting Crimes Than Random People

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The COMPAS tool is widely used to assess a defendant's risk of committing more crimes, but a new study puts its usefulness into perspective. In February 2013, Eric Loomis was found driving a car that had been used in a shooting. He was arrested, and pleaded guilty to eluding an officer.

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This short Aeon video by Cathy O’Neil ponders the nature of algorithms, and challenges the idea that they represent “indisputable mathematical truths”.

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This short Aeon video by Cathy O’Neil ponders the nature of algorithms, and challenges the idea that they represent “indisputable mathematical truths”.

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UNPACKING THE SOURCE Get students to think about the type of knowledge produced by algorithms, and how objective it is. Think further, about how this knowledge is applied and used, focusing on the common view that it’s used to generate unbiased decisions. Then play the video, and focus in on what O’Neil (brilliantly, and so simply) contends. Does this change their view of what sort of knowledge algorithms produce? How objective is the knowledge generated by algorithms? Is it possible to approach ethical problems (or any type of problem) completely objectively?

KEY TERMS Algorithms as opinions, Indisputable mathematical truths, Application of ethics

LINKS TO Mathematics, Knowledge & the knower

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It can be easy to simply accept algorithms as indisputable mathematic truths. After all, who wants to spend their spare time deconstructing complex equations? But make no mistake: algorithms are limited tools for understanding the world, frequently as flawed and biased as the humans who create and interpret them. In this brief animation, which was adapted from a 2017 presentation at the Royal Society of Arts (RSA) in London, the US data scientist Cathy O'Neil, author of Weapons of Math Destruction (2016), argues that algorithms can be useful tools when thoughtfully deployed. However, their newfound ubiquity and massive power calls for ethical conduct from modellers, regulation and oversight by policymakers, and a more skeptical, mathematics-literate public.

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Algorithms are opinions, not truth machines, and demand the application of ethics | Aeon Videos

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Algorithms are opinions, not truth machines, and demand the application of ethics