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But isn't math strucutured in the same way subdivided in lemmas, theorems, corollarys with limited depth.
by looofooo0 2y ago
But isn't math strucutured in the same way subdivided in lemmas, theorems, corollarys with limited depth.
- llm_trw 2y agoNo. It's pretty common to have depth 20+ on even relatively simple proofs when you start looking at decomposing proofs to their axioms. The old joke about 1 + 1 = 2 being on page 360 of Principia Mathematica still largely holds.
- auggierose 2y agoThat depends on the granularity of your proof steps. That old joke doesn't hold for a very long time now (take a look at Isabelle/Isar proofs), yet people keep bringing it up.
- llm_trw 2y agoDoesn't matter what the granularity of your proof step is you still need to understand what that proof step does to be a mathematician. You seem to mistake syntactic brevity for semantic meaning.
- auggierose 2y agoMathematicians do very large proof steps, and so can the computer these days. If you do something like (proof by auto: thm_1 thm_2 thm_3) and it goes through, then I know why the proof works (because of these theorems), and this is similar to how a mathematician understands something. Syntactic brevity is indeed an indicator that you understand the proof. The better you understand something, the more concise you can formulate and prove it.