Questions on the Culture and Philosophy of Mathematics

- How are we affected by the difficulty communicating meaningfully between sub-fields? Is the problem comparable for other areas of science or humanities?
- In what sense are mathematical objects real? Do mathematicians discover or invent these objects?
- Is there an objective way to decide whether a proof is rigorous?
- Why do we place more value on proofs that are simple and aesthetic?
- How is our teaching of mathematics affected by our philosophy of mathematics? Do formalists teach differently than those who view mathematics as an imperfect and exploratory science?