IB Theory of Knowledge
TOK and Mathematics
Mathematics is the area of knowledge where proof seems to give certainty, and where that certainty is most worth questioning. Here are 49 original knowledge questions, 29 exhibition object ideas and essay scaffolds for maths-minded IB students, free.
What mathematics brings to TOK
Mathematics is a test case for almost every big TOK question. It claims certainty through proof, yet rests on axioms. Its abstract results describe the physical world uncannily well. Its statistics decide medical, legal and political questions, and its algorithms shape what we see online. And as a maths student you know things about it that others do not: what it feels like to understand a proof, trust a calculator, or doubt a result that turned out to be true.
Mathematics as an area of knowledge
To compare areas of knowledge, it helps to ask the same questions of each. We use four study headings of our own; they are a revision aid, not taken from the IB guide, so follow your TOK teacher if your class uses different ones. Here is mathematics under each, in our words.
Scope
What mathematics is about, and where its edges lie. Mathematics studies quantity, structure, space, change and pattern through abstract objects such as numbers, sets, functions and shapes. Its scope reaches from pure questions asked for their own sake to applied work in science, finance and technology. TOK asks where the edges are: is statistics mathematics or a science, are computer checks proofs, and can practical skills such as navigation count as mathematical knowledge?
Questions: Gödel and the limits of proof · Pure maths that became useful · Do numbers exist?
Perspectives
Different views of what mathematics is and whose it is. Mathematicians disagree about whether mathematical objects exist independently of us, whether infinity is real, and how much proof should rely on intuition or computers. Mathematics also has a history across many cultures, and who gets credit has changed over time. Your own perspective as a student, someone who learns results before proving them, is a valid starting point too.
Questions: Proof and the community · Discovered or invented? · When new numbers were resisted
Methods and tools
How mathematical knowledge is produced and justified. The central method is proof: deducing results from definitions and axioms by steps that anyone can check. Around it sit other methods and tools: conjecture and experiment, counterexamples, models, statistical inference, notation, calculators, computer searches and, increasingly, proof assistants and AI. TOK asks how far each can be trusted and what each adds.
Questions: Proof versus evidence · Proofs too long for humans · Axioms and certainty
Ethics
Responsibility in making and using mathematics. Mathematics can look ethically neutral, yet its applications decide loans, grades, sentences and public health policy, and mathematical work has served codebreaking and weapons. TOK asks who is responsible for the uses of mathematical knowledge, whether mathematical fairness is the same as real fairness, and what honesty means in statistics and modelling.
Questions: Statistics in the courtroom · Algorithms and bias · Credit and priority disputes
How mathematics links to the themes
| Theme | The maths angle | Start with |
|---|---|---|
| Knowledge and the knower (core) | How you come to trust a proof, a formula or a teacher; intuition, beauty and confidence in mathematics. | Proof versus evidence Proofs too long for humans |
| Knowledge and technology | Calculators, computer-assisted proofs, proof assistants, algorithms and AI in mathematics. | Proofs too long for humans Proof assistants and certainty |
| Knowledge and language | Notation, symbols, definitions and whether mathematics is itself a language. | Gödel and the limits of proof Discovered or invented? |
| Knowledge and politics | Statistics in public life, voting systems, census categories, maps and models behind policy. | Proof and the community When new numbers were resisted |
| Knowledge and religion | Infinity, certainty and faith; historical links between mathematics and religious thought. | Knowing the infinite Calculus before rigour |
| Knowledge and indigenous societies | Counting systems, navigation, weaving and building practices, and what counts as mathematics. | Do numbers exist? Intuition and proof |
Students study the core theme and 2 optional themes.
A maths student's TOK journey
Start with these questions
- Is a mathematical proof a stronger kind of justification than the evidence used in the natural sciences?ProofMethods and tools
- Is mathematics discovered or invented?Nature of mathsPerspectives
- What does the replication crisis reveal about how statistics turns data into knowledge?StatisticsMethods and tools
- Can an algorithm be biased if it only follows mathematics?TechnologyEthics
- If an AI system produces a correct proof, does anyone know the result?TechnologyMethods and tools
- Can a representation be accurate and misleading at the same time?ModelsPerspectives
How TOK is assessed
| Exhibition | Essay | |
|---|---|---|
| What | 3 objects with commentaries, on one of 35 IB prompts | One of 6 prescribed titles for your session |
| Length | Up to 950 words | Up to 1,600 words |
| Marked by | Internal: marked by the school and moderated by the IB | External: marked by IB examiners |
| Weight | one third | two thirds |
Our summary, not the IB's wording. Each task is marked out of 10. TOK and the Extended Essay together can add up to 3 points to your Diploma. Checked on the IB's own pages on 5 October 2026: TOK subject brief, EE subject brief (Diploma points).
Guides
Frequently asked questions
Is mathematics an area of knowledge in TOK?
Yes. Mathematics is one of the 5 areas of knowledge in the current TOK course, alongside history, the human sciences, the natural sciences and the arts.
What is a knowledge question about mathematics?
A question about knowledge itself rather than about a mathematical topic: for example, how proof justifies a claim, whether a model can be trusted, or what makes a statistical result reliable. It is open to more than one reasonable answer and can be discussed using examples from mathematics and beyond.
Can I use my maths IA or EE in TOK?
Yes, as an example. Your own experience of modelling data, choosing assumptions or checking a proof is a strong real-life situation in TOK, as long as your TOK work makes its own argument and you follow your school's rules on using the same work in more than one component.
Do these pages contain the IB's exhibition prompts or essay titles?
No. The prompts and prescribed titles are the IB's, and the titles change every session. Your TOK teacher or your school's IB coordinator has them. Our pages help you think about mathematics in any prompt or title you choose.
Can your tools write my TOK essay or exhibition commentary?
No. Everything here is coaching: ideas, examples, scaffolds and checklists. Your TOK work must be your own, and any tool you use must be acknowledged as your school requires.
IB Math Revision is independent and not affiliated with or endorsed by the IB. These pages are our own writing. They do not reproduce the IB's exhibition prompts or prescribed essay titles: your TOK teacher or IB coordinator has the current ones.