TOK knowledge questions about mathematics

Discovered or invented?: TOK knowledge questions

Is mathematics out there waiting to be found, or something people make? These questions look at what numbers are, why abstract results keep turning out to be useful, and how new ideas such as negative and imaginary numbers came to be accepted.

Questions on this page

Each question has claims, counterclaims, real-life situations from mathematics and pointers for your essay and exhibition. Check dates and figures in a reliable source before you use them, and cite that source. Link a question to the prescribed title or exhibition prompt you are working on, in your own words: see using maths examples in your essay and choosing exhibition objects. The study headings (scope, perspectives, methods and tools, ethics) are our own.

Is mathematics discovered or invented?

Knowledge and the knower Knowledge and language Perspectives

Do mathematicians find truths that were already there, like explorers mapping a coastline, or create them, like composers writing music? Your answer affects what you think a proof shows and why mathematics works. The best discussions avoid a simple either-or.

Claims

  • Mathematical results are not up to us: once the definitions are fixed, no one can choose whether there are infinitely many primes. That suggests they are discovered.
  • Different cultures working independently have found the same results, such as the relationship between the sides of a right-angled triangle, which is what you would expect of discoveries.

Counterclaims

  • Notation, definitions and axioms are human choices, and different choices lead to different mathematics, which suggests invention.
  • Some objects, such as imaginary numbers, began as tools people were reluctant to believe in, and became accepted because they were useful, not because anyone saw them.

Real-life situations from mathematics

Imaginary numbers. In 1545 Gerolamo Cardano published a method for solving cubic equations that sometimes required square roots of negative numbers. Rafael Bombelli worked out rules for them in 1572. They were long treated with suspicion, yet today they are essential in electrical engineering and quantum physics.

The Mandelbrot set. This intricate shape comes from repeating one simple rule on complex numbers. Benoit Mandelbrot made it famous in 1980 using computer plots, but its detail was fixed by the rule before anyone drew it. Was it invented with the rule, or discovered by the pictures?

Use this in your TOK work

Essay. A classic, so make it fresh: use one specific example in depth, and argue that the answer may differ for definitions and for results.

Exhibition. A printed zoom of the Mandelbrot set works well for prompts about whether knowledge is created or found.

The mathematics behind it

Why does mathematics developed for its own sake so often turn out to be useful?

Knowledge and technology Scope

In 1960 the physicist Eugene Wigner wrote about the 'unreasonable effectiveness' of mathematics in the natural sciences. Ideas pursued for curiosity have repeatedly become the language of later science and technology. This question asks whether that is luck, design, selection or something deeper.

Claims

  • Mathematics studies patterns and structure in general, so it is not surprising that the patterns it describes reappear in nature and technology.
  • Usefulness is evidence that mathematics captures something true about the world rather than being a human game.

Counterclaims

  • We remember the pure mathematics that became useful and forget the large amount that did not, so the effect may be exaggerated by selection.
  • Scientists often choose the mathematics that fits, adjusting models until they work, so the effectiveness is partly built in by the people applying it.

Real-life situations from mathematics

Number theory and online security. In A Mathematician's Apology (1940) G. H. Hardy took pride in number theory having no practical use. In 1977 Rivest, Shamir and Adleman published RSA encryption, based on prime numbers, which now helps protect online payments.

Curved geometry and gravity. Bernhard Riemann developed the geometry of curved spaces in 1854 as pure mathematics. About sixty years later Einstein used it as the language of general relativity (1915), his theory of gravity.

Use this in your TOK work

Essay. Fits titles about the value of knowledge, or about the relationship between areas of knowledge. Compare with the natural sciences.

Exhibition. A padlock icon from a web browser, with a note on RSA, makes an everyday object for a prompt about the value or usefulness of knowledge.

The mathematics behind it

Can mathematics give us knowledge about the world before we observe it?

Knowledge and the knower Methods and tools

Several famous discoveries began on paper: a calculation said something should exist, and observers then found it. This question asks whether mathematics itself produced that knowledge, or whether the observation was what turned a prediction into knowledge.

Claims

  • A successful mathematical prediction shows that reasoning alone can reach truths about the physical world.
  • Predictions make knowledge testable: by saying exactly where to look, mathematics guides observation and saves effort.

Counterclaims

  • The mathematics only works because it was built on earlier observations and physical theory, so the knowledge was never independent of experience.
  • Mathematics also predicts things that do not exist; equations need physical interpretation, and many predicted objects are never found.

Real-life situations from mathematics

The discovery of Neptune. Irregularities in the orbit of Uranus led Urbain Le Verrier (and independently John Couch Adams) to calculate where an unseen planet should be. In September 1846 Johann Galle found Neptune close to Le Verrier's predicted position.

The positron. In 1928 Paul Dirac's equation for the electron had solutions that seemed to describe a particle like the electron but with positive charge. In 1932 Carl Anderson observed such a particle, the positron.

Use this in your TOK work

Essay. Fits titles about prediction, the relationship between reason and observation, or areas of knowledge working together.

Exhibition. A star chart showing where Neptune was found could anchor a prompt about how we gain knowledge of things we cannot yet see.

The mathematics behind it

How do new mathematical ideas come to be accepted as knowledge?

Knowledge and the knower Knowledge and politics Perspectives

Negative numbers, irrational numbers, imaginary numbers and infinite sets were all resisted at first, sometimes for centuries. This question asks what changed people's minds: proof, usefulness, familiarity or the authority of respected mathematicians.

Claims

  • Ideas are accepted when they are shown to be consistent and fruitful: once rules for a new kind of number are clear and lead to correct results, the case for it is strong.
  • Resistance is part of how knowledge is checked; it forces new ideas to be justified carefully.

Counterclaims

  • Acceptance often follows familiarity and education rather than argument: each generation takes for granted what the previous one doubted.
  • Respected figures can delay or speed acceptance, so the history of mathematics is partly a social history, not only a logical one.

Real-life situations from mathematics

Negative numbers. Brahmagupta stated rules for calculating with zero and negative numbers in 628 CE. In Europe, many mathematicians treated negative solutions as false or absurd well into the seventeenth and eighteenth centuries.

Cantor's infinite sets. Georg Cantor's work on different sizes of infinity, from the 1870s, was strongly opposed by some leading mathematicians, notably Leopold Kronecker. A generation later David Hilbert defended Cantor's set theory as something mathematics should not give up.

Use this in your TOK work

Essay. Fits titles about how knowledge changes over time, or the role of authority. Compare with paradigm shifts in the natural sciences.

Exhibition. A thermometer reading below zero is a simple object for a prompt about how ideas once rejected become everyday knowledge.

The mathematics behind it

Is mathematics a language, and if so, what does it describe?

Knowledge and language Perspectives

Galileo described the book of nature as written in mathematics. Mathematics has symbols, grammar and conventions, and people talk of 'reading' an equation. This question asks whether the comparison with language helps or misleads, and what mathematical sentences are about.

Claims

  • Mathematics behaves like a language: it has a vocabulary of symbols and rules for combining them, and it lets people communicate precisely across national languages.
  • Good notation, like good language, makes some thoughts easier to have, which is why changes in notation have changed what mathematicians could do.

Counterclaims

  • Unlike a language, mathematics has results that are true or false regardless of how they are written; changing notation does not change the theorem.
  • Mathematics also involves pictures, intuition and physical models that are not language-like at all.

Real-life situations from mathematics

The equals sign. Robert Recorde introduced the symbol = in The Whetstone of Witte (1557), choosing two parallel lines because no two things could be more equal. Before that, equality was written out in words.

Newton's dots and Leibniz's d. Newton and Leibniz developed calculus with different notations. British mathematicians largely kept Newton's notation, and some historians argue this slowed their progress until the Analytical Society promoted Leibniz's dy/dx in Cambridge from 1812.

Use this in your TOK work

Essay. Fits titles about language and knowledge, or about how representations shape knowledge.

Exhibition. A page of mathematics in an unfamiliar script that you can still partly read because of its symbols is a strong object for a language prompt.

The mathematics behind it

Do mathematical objects such as numbers exist independently of human minds?

Knowledge and the knower Knowledge and language Knowledge and indigenous societies Scope

You can point to three apples but not to the number three. Yet statements about numbers seem objectively true. This question asks what mathematical objects are: abstract things that exist outside space and time, patterns in the world, or useful fictions.

Claims

  • Mathematical statements are true whether or not anyone believes them, and true statements are about something, so numbers must exist in some sense.
  • Facts about numbers, such as the digits of π, can be discovered, which suggests they were already fixed independently of us.

Counterclaims

  • We never interact with numbers in space and time, so it is hard to explain how we could know about them if they exist independently of us.
  • Numbers can be understood as tools we built for counting and measuring; the truths follow from how we defined them.

Real-life situations from mathematics

Computing π. In 2022 a team at Google Cloud led by Emma Haruka Iwao computed π to 100 trillion digits. NASA's Jet Propulsion Laboratory has explained that about 15 decimal places are enough for its interplanetary navigation. The extra digits were fixed before anyone computed them, but did they 'exist'?

Counting without number words. Researchers working with the Pirahã people of the Amazon in the 2000s reported that their language has no words for exact numbers beyond roughly 'one' and 'two'. The findings are debated, but they raise the question of whether exact number is universal or learned.

Use this in your TOK work

Essay. Use with care in titles about the nature of knowledge; make sure you connect the philosophy to how mathematicians actually work.

Exhibition. A printout of a thousand digits of π invites questions about what we know when we know a number.

The mathematics behind it

Other areas: Proof · Infinity · Models · Statistics · Technology · The knower · Culture · Ethics · all 49 knowledge questions