For our 6th ToK day, this time about mathematics, we surprisingly did not start off with book presentations. Instead, we first engaged in a discussion of the question: "What is math?". Although Wikipedia defines math as "an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes", we came to the conclusion that math is not something which can be easily defined.

After this we debated the pros and cons of several statements such as "my parents did not understand math - as a consequence, I will never understand math as well" as opposed to "everyone can understand math no matter what his or her background is."

Lastly, we learned about the various technical terms which build up math including axioms, definitions, propositions, proofs, theorems and lemmas before trying our own hands at mathematically proving a statement and then breaking for lunch.

We started off the afternoon by watching the movie “The Proof”. It shows the story of Andrew Wiles and how he proved Fermat’s last theorem. It took him seven years and when he eventually managed it was a big achievement in the world of mathematics. The film shows how complex certain mathematical problems are and how only a few of us can solve them. On the other hand, it showed how fascinating math can be. Even though there might be no use for the formula, it’s still a great thing to know how it works.

In our last lesson we first fell back on paradoxes which we discussed earlier this day and previously touched upon in lessons. Paradoxes can be very exhausting to discuss since they often play with our minds. How can Achilles catch up with the tortoise even if he never reaches it? (Zeno’s Paradox). Mathematics explains how it is possible. Even though it is a paradox, math has a way of proving that it is possible to solve the problem. This shows the power of math.

The second part of this lesson was mainly on the Incompleteness Theorem and how Kurt Gödel proved it. “Any consistent axiomatic system for arithmetic […] must of necessity be incomplete”. This is very advanced math but again shows its complexity and power.

This TOK day gave us a very interesting insight into the structure of mathematics. Thanks to Mr. Benz for his teaching and organization of the day!