TOK knowledge questions about mathematics
Beauty, intuition and the knower: TOK knowledge questions
Mathematicians talk about beautiful proofs and results that feel wrong. These questions are about the knower: the part intuition, taste and understanding play in what we accept as mathematical knowledge.
Questions on this page
- 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
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 beauty a reliable guide to mathematical truth?
Knowledge and the knower Perspectives
Mathematicians often call proofs beautiful or ugly, and some physicists have trusted beautiful equations as signs of truth. This question asks whether a sense of beauty is evidence, a useful guide, or only a feeling.
Claims
- Beautiful results usually reveal deep connections and simple structure, which are signs of having understood something true.
- Many mathematicians report that aesthetic judgement guides their research successfully, choosing which problems and approaches to pursue.
Counterclaims
- Some true results have long, ugly proofs, and some elegant ideas are false; beauty cannot replace proof.
- Judgements of beauty vary with training and culture, so they are a perspective rather than evidence.
Real-life situations from mathematics
Euler's identity. The equation e^(iπ) + 1 = 0 has repeatedly topped polls of mathematicians' most beautiful equations, including a reader poll in The Mathematical Intelligencer published in 1990. In a 2014 brain-imaging study led by Semir Zeki, mathematicians' ratings of beautiful equations were linked to activity in the same brain region as other experiences of beauty.
Hardy on ugly mathematics. In A Mathematician's Apology (1940) G. H. Hardy argued that beauty is the first test of mathematics and that ugly mathematics has no lasting place.
Use this in your TOK work
Essay. Fits titles linking the arts and mathematics, or about the role of emotion in knowledge.
Exhibition. A poster of Euler's identity, or a page with a proof you find beautiful, is a personal object for a prompt about the knower.
The mathematics behind it
What role does intuition play in producing mathematical knowledge?
Knowledge and the knower Knowledge and indigenous societies Methods and tools
Mathematics is famous for proof, yet many results are first guessed, sometimes long before anyone can prove them. This question asks whether intuition is part of mathematical knowledge or only a step before it.
Claims
- Intuition guides mathematicians to true results and promising methods; without it, proof would have nothing to aim at.
- Experienced mathematicians' intuitions are trained by years of practice, so they are a reliable form of expertise.
Counterclaims
- Intuition is often wrong, especially with probability and infinity, so it can never count as knowledge on its own.
- Insisting on proof is exactly how mathematics protects itself against the errors of intuition.
Real-life situations from mathematics
Ramanujan's notebooks. In early 1913 Srinivasa Ramanujan, largely self-taught, sent G. H. Hardy pages of remarkable formulas, many without proofs; he arrived in Cambridge in April 1914. Most turned out to be correct and some were new to the experts; a few were wrong. Mathematicians have spent decades proving results from his notebooks.
Conjectures before proofs. Fermat's Last Theorem was stated in the 1630s and proved in the 1990s. For over 350 years many mathematicians believed it on the strength of intuition and partial evidence.
Use this in your TOK work
Essay. Fits titles about intuition, imagination or the role of the individual knower.
Exhibition. A page from your own rough working, with a guess you later proved or disproved, is a genuinely personal object.
The mathematics behind it
Why do some correct mathematical results feel wrong, even to experts?
Knowledge and the knower Perspectives
Some correct answers provoke strong disbelief, from students and professionals alike. This question asks what that tells us about the relationship between understanding, intuition and knowledge.
Claims
- When intuition and proof disagree, proof should win, and learning to accept that is part of gaining mathematical knowledge.
- Counter-intuitive results show that mathematics can correct our everyday thinking, which is a reason to value it.
Counterclaims
- If a result still feels wrong after a proof, the knower may know that it is true without understanding why, which is a weaker kind of knowledge.
- Intuitions can be retrained: what feels wrong to one generation may feel obvious to the next.
Real-life situations from mathematics
The Monty Hall problem. In 1990 Marilyn vos Savant answered a reader's question in Parade magazine: on a game show, switching doors doubles your chance of winning the car. Thousands of readers wrote in to say she was wrong, many of them with doctorates. Simulations show she was right.
The birthday problem. In a group of only 23 people, the chance that at least two share a birthday is just over 50%. Most people guess a much larger group is needed.
Use this in your TOK work
Essay. Fits titles about intuition, persuasion or the difference between knowing and understanding.
Exhibition. Three paper cups and a coin (a Monty Hall set-up) make a simple, memorable object.
The mathematics behind it
- Probability (AA SL)
- Conditional probability and Bayes' theorem (AA HL)
- Conditional probability with tree diagrams (AI SL)
- Maths IA idea: The Monty Hall problem when the host is forgetful
- Maths IA idea: The birthday problem, extended: what about birthdays within a day?
- Maths EE idea: Monty Hall and its variants
- Maths EE idea: Birthday problems beyond birthdays
Can you know a mathematical result without understanding its proof?
Knowledge and the knower Scope
Most people who know that Fermat's Last Theorem is true could not follow its proof. Students use the quadratic formula long before they can derive it. This question asks whether this is knowledge or trust, and whether it matters.
Claims
- Knowledge can be gained from reliable testimony, so accepting a proved result from experts is genuine knowledge.
- Using results we have not proved is unavoidable in a large subject; no one could prove everything they use.
Counterclaims
- Mathematics claims to be the area where each person can check everything; relying on authority goes against what makes mathematical knowledge distinctive.
- Without understanding, people cannot tell when a result applies, so their knowledge is fragile.
Real-life situations from mathematics
The classification of finite simple groups. The proof that all finite simple groups fall into known families is spread across many hundreds of journal articles by around a hundred authors. Few people, if any, have read all of it.
Formula booklets. IB maths students are given a formula booklet in exams. Using a formula from the booklet correctly shows some knowledge, but not the same knowledge as deriving it.
Use this in your TOK work
Essay. Fits titles about testimony, expertise or the difference between knowing and understanding.
Exhibition. Your IB formula booklet is an ideal personal object for this question.
The mathematics behind it
Other areas: Proof · Nature of maths · Infinity · Models · Statistics · Technology · Culture · Ethics · all 49 knowledge questions