TOK knowledge questions about mathematics

Proof and certainty: TOK knowledge questions

Proof is what makes mathematics look different from every other area of knowledge. These questions ask how much certainty a proof really gives, who checks it, and what happens when a proof is too long for any person to read.

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 a mathematical proof a stronger kind of justification than the evidence used in the natural sciences?

Knowledge and the knower Methods and tools

Mathematicians say a result is known once it is proved, however many examples already support it. Scientists, by contrast, accept claims on the strength of evidence that could, in principle, be overturned. This question asks whether proof really is a different and better kind of justification, or only a different one.

Claims

  • A valid proof shows a result must hold in every case, including the infinitely many no one will ever check, so it gives a certainty that no amount of observation can.
  • Once proved, mathematical results tend to stay proved: Euclid's theorem that there are infinitely many primes is as secure today as it was over two thousand years ago, while many accepted scientific theories have been replaced.

Counterclaims

  • A proof is only as certain as its axioms and the people checking it. Published proofs have contained errors that went unnoticed for years, so in practice mathematical knowledge also rests on trust and testing.
  • Mathematicians use evidence too: they believe conjectures because of numerical checks long before anyone proves them, and much research is guided by that kind of experimental confidence.

Real-life situations from mathematics

Goldbach's conjecture. The claim that every even number greater than 2 is the sum of two primes has been checked by computer for every even number up to about 4 × 10¹⁸, yet it is still called a conjecture. To a scientist that much evidence would settle the matter; to a mathematician it does not.

The Pólya conjecture. Proposed in 1919, this claim about how many whole numbers have an odd number of prime factors held for every number anyone checked. In 1958 Haselgrove showed it was false, and the smallest counterexample, found later, is 906,150,257. Evidence from the first few hundred million cases had been misleading.

Use this in your TOK work

Essay. Useful for titles that compare how areas of knowledge justify claims, or that ask whether certainty is possible. Contrast mathematics with a natural science, and make sure your counterclaim shows that proof also depends on people.

Exhibition. A printout of a long computer check of a conjecture, beside a short handwritten proof, makes a strong object for a prompt about certainty or about what counts as evidence.

The mathematics behind it

Can we know a result is true if no human can check every step of its proof?

Knowledge and technology Knowledge and the knower Methods and tools

Some modern proofs depend on computers checking thousands of cases, or run to more pages than any one person could read. This question asks whether knowledge must be something a human mind can survey, or whether trusted machines and communities can know on our behalf.

Claims

  • A computer follows its instructions exactly and does not tire, so for long case-by-case checks it is more reliable than a person, and a correct program gives knowledge just as a correct proof does.
  • Mathematicians already rely on results they have not checked themselves; trusting a well-tested program is not very different from trusting a published theorem.

Counterclaims

  • A proof is meant to explain why a result is true. A computer check can tell us that something is true without giving any insight, which some mathematicians feel is not full mathematical knowledge.
  • Programs, compilers and hardware can all contain errors, so a computer-assisted proof adds new places where things can go wrong, which human readers cannot easily inspect.

Real-life situations from mathematics

The four colour theorem. In 1976 Kenneth Appel and Wolfgang Haken proved that any map can be coloured with four colours so that neighbouring regions differ, by having a computer check a very large number of configurations. Many mathematicians were uneasy. In 2005 Georges Gonthier and colleagues checked the whole proof with a formal proof assistant, which reassured many of the doubters.

The Kepler conjecture. Thomas Hales announced a proof in 1998 that the familiar greengrocer's stacking of oranges is the densest way to pack equal spheres. The referees reported that they could not be completely certain of every computer step. Hales led a project to verify the proof formally by computer, which was completed in 2014.

Use this in your TOK work

Essay. Fits titles about technology, trust in experts or whether knowledge needs understanding. Weigh 'knowing that' against 'understanding why'.

Exhibition. A map coloured with four colours, with a note on how the theorem was proved, is an accessible object for a prompt about technology or about the limits of human knowledge.

The mathematics behind it

How far does mathematical knowledge depend on the agreement of a community of experts?

Knowledge and the knower Knowledge and politics Perspectives

We often imagine a proof as something anyone could check alone. In reality, new results are accepted when the experts in a field have read them, questioned them and agreed. This question asks what happens when that community is small, disagrees or makes mistakes.

Claims

  • Peer review spreads the work of checking across many experts, so the community's acceptance is good evidence that a proof is correct.
  • The history of famous proofs shows that the community does catch mistakes: errors are found, discussed openly and either repaired or the claim is withdrawn.

Counterclaims

  • When a proof is so specialised that only a handful of people understand it, 'the community agrees' may mean only that a few people are persuaded, which is a weaker kind of knowledge.
  • Reputation can influence whether a proof is read carefully at all, so acceptance may depend partly on who the author is rather than only on the mathematics.

Real-life situations from mathematics

Fermat's Last Theorem. Andrew Wiles announced a proof in June 1993. During checking, a gap was found in one part of the argument. Working with Richard Taylor, Wiles repaired it in 1994 and the proof was published in 1995. The community's checking process found the problem and then accepted the fix.

The abc conjecture dispute. In 2012 Shinichi Mochizuki released a long proof of the abc conjecture built on new theory of his own. In 2018 Peter Scholze and Jakob Stix said one key step did not work. The proof was published in 2021 in a journal Mochizuki helps to edit, yet most number theorists still do not regard the conjecture as proved.

Use this in your TOK work

Essay. Good for titles about the role of experts, consensus or disagreement. Contrast mathematics with a human science where disagreement is expected.

Exhibition. A journal article with a published correction, or a referee's report form, could anchor a prompt about who decides what counts as knowledge.

The mathematics behind it

What do Gödel's incompleteness theorems tell us about the limits of mathematical knowledge?

Knowledge and the knower Knowledge and language Scope

In 1931 Kurt Gödel showed that any consistent formal system rich enough to describe ordinary arithmetic contains true statements it cannot prove, and cannot prove its own consistency. This question asks what that means for the hope that mathematics could be made completely certain.

Claims

  • Gödel's results show that there are limits to what any single set of rules can prove, so mathematical knowledge can never be captured by one complete, mechanical system.
  • Some natural questions have turned out to be undecidable from the usual axioms, which shows the limits are not only theoretical.

Counterclaims

  • The theorems themselves are proved mathematics, so they are an achievement of mathematical knowledge as much as a limit on it.
  • Almost all the mathematics used in schools, science and engineering is unaffected; the limits apply at the edges, not to whether 2 + 2 = 4.

Real-life situations from mathematics

The continuum hypothesis. Georg Cantor asked whether there is a size of infinity between that of the whole numbers and that of the real numbers. Gödel showed in 1940 that the usual axioms of set theory cannot disprove that there is none, and Paul Cohen showed in 1963 that they cannot prove it either. The question cannot be settled from those axioms at all.

Hilbert's programme. In the 1920s David Hilbert hoped to put all of mathematics on axioms that could be shown, by finite methods, never to lead to a contradiction. Gödel's second theorem showed that this goal, as Hilbert framed it, could not be reached.

Use this in your TOK work

Essay. Use it carefully for titles about limits of knowledge or certainty: state the result accurately and do not claim it makes all mathematics uncertain.

Exhibition. A printed self-referential sentence such as 'This sentence cannot be proved' makes a thought-provoking object for a prompt about limits or about language.

The mathematics behind it

If mathematics rests on axioms that cannot themselves be proved, in what sense is it certain?

Knowledge and the knower Methods and tools

Every proof starts from assumptions. In geometry these were once thought to be self-evident truths about space; today mathematicians usually treat them as chosen starting points. This question asks whether certainty in mathematics is certainty about the world, or only certainty that conclusions follow from assumptions.

Claims

  • Mathematical certainty is conditional: if the axioms hold, the theorems must hold. That conditional certainty is still far stronger than anything other areas of knowledge offer.
  • Axioms are not arbitrary: they are chosen because they capture ideas such as number and space that we already understand, so results built on them tell us something real.

Counterclaims

  • Different choices of axioms give different, equally consistent mathematics, which suggests mathematics describes possibilities rather than one certain truth.
  • Some axioms are accepted because they are useful even though their consequences seem strange, so the foundations rest partly on convenience and taste.

Real-life situations from mathematics

Euclid's parallel postulate. For about two thousand years mathematicians tried to prove Euclid's fifth postulate from his others. In the 1820s and 1830s Lobachevsky and Bolyai (and, privately, Gauss) showed that replacing it gives a consistent geometry in which a triangle's angles add up to less than 180°.

The axiom of choice. This widely used axiom of set theory implies the Banach–Tarski result of 1924: a solid ball can, in theory, be split into finitely many pieces and reassembled into two balls of the same size. Most mathematicians accept the axiom because so much useful mathematics depends on it.

Use this in your TOK work

Essay. Fits titles about assumptions, certainty or whether knowledge needs foundations. Compare axioms with the paradigms or assumptions of another area of knowledge.

Exhibition. A triangle drawn on a ball (its angles add up to more than 180°) shows that 'obvious' geometry depends on assumptions.

The mathematics behind it

Why can one counterexample overturn a mathematical claim that thousands of examples could not establish?

Knowledge and the knower Methods and tools

In mathematics a general claim is overturned by a single case that breaks it, while no number of agreeing cases is enough to prove it. This asymmetry between confirming and refuting is also discussed in the philosophy of science. This question asks why it is so sharp in mathematics.

Claims

  • A mathematical claim usually says 'for every case', so one verified failure makes it false outright; there is no room for 'mostly true'.
  • A counterexample in mathematics can be checked completely by anyone, so it gives decisive knowledge in a way that a surprising experiment, which might be faulty, does not.

Counterclaims

  • When a counterexample appears, mathematicians often repair the claim by changing definitions or adding conditions rather than abandoning it, so refutation is less final than it looks.
  • Checking an enormous counterexample may itself need a computer, so even refutation can depend on trust in tools.

Real-life situations from mathematics

Euler's sum of powers conjecture. Euler suggested that you need at least n nth powers to add up to another nth power. In 1966 Lander and Parkin used a computer search to find that 27⁵ + 84⁵ + 110⁵ + 133⁵ = 144⁵, four fifth powers adding to a fifth power, and the conjecture fell.

Lakatos and Euler's polyhedron formula. In Proofs and Refutations (1976) the philosopher Imre Lakatos traced how mathematicians responded to shapes that broke the formula V − E + F = 2: some called them 'monsters' and excluded them, others refined the definition of a polyhedron. Refutation reshaped the concept.

Use this in your TOK work

Essay. Strong for titles comparing how areas of knowledge deal with contrary evidence. Link to falsification in the natural sciences.

Exhibition. A card showing the equation 27⁵ + 84⁵ + 110⁵ + 133⁵ = 144⁵ is a small, striking object about how knowledge is overturned.

The mathematics behind it

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