On our maths ToK-day we learned to question what mathematics is and where its limitations lie, rather than treating it as a set of fixed rules. 

The day began by asking the question: What is mathematics? Through group work and discussion, we examined whether mathematics is discovered or invented and how far it can be considered a universal language. 
The second part of the day focused on axioms, proofs, and paradoxes. We learned that axioms are statements accepted without proof and form the foundation of mathematical systems. From there, the idea of proof was examined, including examples that appear logical but lead to absurd conclusions such as “0 = 1”. These examples illustrate how small errors can undermine an argument, reinforcing the importance of proof checking. Paradoxes like “Achilles and the Tortoise” further challenged intuitive thinking and showed how logic can conflict with common sense.
After lunch, we watched the film “The Proof”, which connected mathematical reasoning to real-world problem-solving and creativity. This was followed by deeper discussion of whether mathematics is truly complete or whether it contains unavoidable “holes.”

The final part of TOK Day addressed one of the most important ideas in modern mathematics: incompleteness. We were introduced to the incompleteness theorems of Kurt Gödel, which show that any sufficiently powerful and consistent formal mathematical system cannot prove all truths about itself. This demonstrated that even mathematics has limits. This, however does not mean the end of mathematics, but rather the end of the dream of a perfectly complete and self contained system.

Overall, the ToK day aimed to build skepticism, understanding of proof and reasoning, and show that mathematics, relies on assumptions and is not free from paradoxes. 

Alessandra Pilotto Stocker and Marko Susic, 5i