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I will always remember when I first got into serious proof-based math in the first semester of college, and I had to work hard to develop my sense of what count
by dcre 3y ago
I will always remember when I first got into serious proof-based math in the first semester of college, and I had to work hard to develop my sense of what counts as a sufficient proof. I would read the proofs in the textbook and hit the QED at the end and not understand why it was enough. Eventually I came to understand something like the framing here, which is that a proof is about persuasion, persuasion is about judgment, and judgment (by definition, maybe) can't be pinned down to clear rules.
The University of Chicago has a special math class format called Inquiry-Based Learning designed around this idea, where you work together in class to put proofs together and work out a shared understanding of what is sufficient. I didn't take it but I wish I had. You can read some people's experiences with it here[0].
[0]: https://www.reddit.com/r/uchicago/comments/i1id9e/what_was_your_experience_like_in_honors_calc_ibl/ https://www.reddit.com/r/uchicago/comments/i1id9e/what_was_y...
- pjacotg 3y agoI recently read this book review [0] where a mathematical proof was described as a dialogue between two semi-adversarial but collaborative actors, the Prover and the Skeptic, who together aim to aquire mathematical insight. I thought it was an interesting perspective. [0] https://jdh.hamkins.org/book-review-catarina-dutilh-novaes-the-dialogical-roots-of-deduction/ https://jdh.hamkins.org/book-review-catarina-dutilh-novaes-t...
- hgsgm 3y agoThis also what philosophical discussion/debate is.
- mikhailfranco 3y agoThis is Game Theoretic Semantics. See Hintikka's Principles of Mathematics Revisited and his framing of Independence Friendly Logic: https://plato.stanford.edu/entries/logic-if/ https://plato.stanford.edu/entries/logic-if/
- agrounds 3y agoI was lucky enough to take two IBL courses from Dr. Michael Starbird at UT Austin. Both were wonderful courses, very engaging and very fun. I never collaborated so much with other math students as I did in those courses. In particular this was my intro to topology and I’ve been hooked on it ever since.
- zerbinxx 3y agoIt definitely takes a second to develop that intuition, I remember being quite confused as well. As I got more into CS, I began to think of proofs similar to how I do unit tests: you externalize or “fix” as many as the inputs as you possibly can, and then abuse the remaining degrees of freedom as much as possible to show that your solution is complete.