Beauty, intuition and the knower · Perspectives

Why do some correct mathematical results feel wrong, even to experts?

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.

Check dates and figures in a reliable source before you use them, and cite that source.

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.

Link it to the prescribed title or the exhibition prompt you are working on, in your own words. See using maths examples in your essay and choosing exhibition objects.

Themes and study heading

Knowledge and the knower Perspectives

Different views of what mathematics is and whose it is.