Beauty, intuition and the knower · Scope
Can you know a mathematical result without understanding its proof?
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.
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 testimony, expertise or the difference between knowing and understanding.
Exhibition. Your IB formula booklet is an ideal personal object for this question.
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 Scope
What mathematics is about, and where its edges lie.