These are specifically things which mathematics has taught me which have applications outside the field of mathematics... there's a whole lot of much more boring stuff.

1. That there are such things as unanswerable questions - indeed, provably unanswerable questions
2. That Occam's Razor is fallible
3. To assume nothing
4. That the word "obvious" means "a proof springs immediately to mind"
5. The importance of proof
6. How to reason logically
7. To reason logically
8. How to recognise faulty reasoning and find counterexamples
9. That intuition counts for zilch
10. Point nine recurring equals one