TOK and mathematics
49 TOK knowledge questions about mathematics
Each question has a short discussion page: claims and counterclaims, real-life situations from mathematics, the maths topics behind it and pointers for your essay and exhibition. They are starting points for your own thinking.
Proof and certainty
- Is a mathematical proof a stronger kind of justification than the evidence used in the natural sciences?ProofMethods and tools
- Can we know a result is true if no human can check every step of its proof?ProofMethods and tools
- How far does mathematical knowledge depend on the agreement of a community of experts?ProofPerspectives
- What do Gödel's incompleteness theorems tell us about the limits of mathematical knowledge?ProofScope
- If mathematics rests on axioms that cannot themselves be proved, in what sense is it certain?ProofMethods and tools
- Why can one counterexample overturn a mathematical claim that thousands of examples could not establish?ProofMethods and tools
Discovered or invented?
- Is mathematics discovered or invented?Nature of mathsPerspectives
- Why does mathematics developed for its own sake so often turn out to be useful?Nature of mathsScope
- Can mathematics give us knowledge about the world before we observe it?Nature of mathsMethods and tools
- How do new mathematical ideas come to be accepted as knowledge?Nature of mathsPerspectives
- Is mathematics a language, and if so, what does it describe?Nature of mathsPerspectives
- Do mathematical objects such as numbers exist independently of human minds?Nature of mathsScope
Infinity and foundations
- How can finite minds have knowledge of the infinite?InfinityScope
- Did the mathematical idea of a limit resolve Zeno's paradoxes, or only give us a way to calculate?InfinityMethods and tools
- Can paradoxes be a source of mathematical knowledge rather than a threat to it?InfinityMethods and tools
- Was calculus knowledge before it was made rigorous?InfinityPerspectives
Models and the real world
- If every mathematical model leaves something out, how can models produce knowledge?ModelsMethods and tools
- Can a representation be accurate and misleading at the same time?ModelsPerspectives
- Does chaos theory set a limit on what mathematics can know about the future?ModelsScope
- Are units of measurement discovered in nature or decided by agreement?ModelsMethods and tools
- How do the assumptions built into a model shape the conclusions it can reach?ModelsMethods and tools
Statistics, data and evidence
- What does the replication crisis reveal about how statistics turns data into knowledge?StatisticsMethods and tools
- When can a statistical correlation justify a claim about cause?StatisticsMethods and tools
- Can a larger set of data ever give less reliable knowledge than a smaller one?StatisticsMethods and tools
- How should we judge a statistical claim reported in the news?StatisticsPerspectives
- Can a single average represent a group fairly?StatisticsPerspectives
- What does a probability mean when an event happens only once?StatisticsScope
- Can the same data support opposite conclusions depending on how it is shown?StatisticsPerspectives
- Should new evidence change what we believe according to a fixed mathematical rule?StatisticsMethods and tools
Technology, algorithms and AI
- Does a proof checked by software give us more certainty than a proof checked by people?TechnologyMethods 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
- How far should we trust tools whose workings we cannot check?TechnologyMethods and tools
- Does big data make mathematical theory less necessary?TechnologyMethods and tools
Beauty, intuition and the knower
- Is beauty a reliable guide to mathematical truth?The knowerPerspectives
- What role does intuition play in producing mathematical knowledge?The knowerMethods and tools
- Why do some correct mathematical results feel wrong, even to experts?The knowerPerspectives
- Can you know a mathematical result without understanding its proof?The knowerScope
Culture, history and language
- Whose contributions are recognised in the history of mathematics, and why does it matter?CulturePerspectives
- Is mathematics the same in every culture?CulturePerspectives
- Does it matter who gets credit for a mathematical discovery?CultureEthics
- Can knowledge built into practices such as weaving, building or navigation count as mathematics?CultureScope
- Does mathematical notation shape what mathematicians are able to think?CultureMethods and tools
Ethics and responsibility
- How can mistakes in mathematical reasoning lead to injustice?EthicsEthics
- Do mathematicians have ethical responsibilities for how their work is used?EthicsEthics
- Can mathematics tell us what a fair voting system is?EthicsEthics
- Should numbers decide questions about health and life?EthicsEthics
- What is gained and lost when a state counts its people?EthicsEthics
- Who is responsible when a trusted mathematical model fails?EthicsEthics
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Next: find exhibition objects or use a question in your essay. Teaching TOK? Try the lesson starters.
Frequently asked questions
What makes a good TOK knowledge question about mathematics?
It is about knowledge rather than about a maths topic, it is open (reasonable people can disagree), and it uses TOK concepts such as evidence, certainty, interpretation or responsibility. 'How do I solve this equation?' is a maths question; 'Why does one counterexample outweigh thousands of examples?' is a knowledge question.
Can I use these questions in my essay?
Use them to think. Your essay must answer the prescribed title you chose, in your own words; these pages give you examples and arguments to test, not text to copy.