Unlike AI, why don’t human mathematicians explain how they proved something?

I quote results by Artificial Intelligence, but I’m asking my question for human mathematicians (NOT AI). I quote https://www.anthropic.com/research/riemann-zeta

Claude’s explanation of how it arrived at its result is available in a separate Appendix here.

I quote https://vibemathed.com/problem/sphere-packing-upper-bounds-cohn-elkies

Transcript Reasoning walkthroughs (PDF)

These notes were written by an AI model that read the original chains of thought together with the resulting mathematical papers and writeups. For each problem, the model reconstructs how the proof came together: which ideas first suggested a path forward, which substantial approaches encountered genuine obstacles, what changes of perspective revealed the underlying structure [emphasis mine], and how the decisive insights developed into the final argument.

The aim is to collect and explain the mathematical thinking behind the proofs in a readable, high-level narrative. Rather than simply reproducing their finished presentations, the chapters emphasize the connections, intermediate discoveries, and sustained detours that clarify why the successful arguments work [emphasis mine].

Every year for the past four years, on the math departmental student surveys for the past 4 years, graduate and undergraduate students gripe that most of their professors and textbooks merely pull “out ready-made solutions” “that are some kind of black magic incantation outside the ken of mortal humans.” Students bemoan that proofs reproduced in their finished perfect presentations “lack deep motivated intuition, and are raw algebraic manipulations for the most part. They prove things well, but they don't explain things well.” Unlike the AI above, lecturers and textbooks fail to expound “which ideas first suggested a path forward, which substantial approaches encountered genuine obstacles, what changes of perspective revealed the underlying structure [emphasis mine]” along with “the connections, intermediate discoveries, and sustained detours that clarify why the successful arguments work”. Students bellyache about “the experience of reading a proof and being convinced by it, and yet having no idea how someone would have thought of the proof or why someone would have expected this result to be true. Then, after thinking about the theorem a lot and reading other sources, I eventually find a way to look at it that makes it seem easy and obvious. That's what I mean by "intuition" -- an explanation for why the result is secretly easy or obvious, even though it might appear daunting upon first sight.

However, students find invaluably helpful, and hanker to peruse ““which ideas first suggested a path forward, which substantial approaches encountered genuine obstacles, what changes of perspective revealed the underlying structure [emphasis mine]” along with “””the connections, intermediate discoveries, and sustained detours that clarify why the successful arguments work”.

For purely human proofs, why don’t humans expound “the mathematical thinking behind the proofs in a readable, high-level narrative”? To wit, why isn’t it usual standard practice to explain “to motivate and explain definitions, calculations, concepts, counterexamples, proofs, disproofs, results to students in such a way that they feel like they could have come up with it themselves, naturally, given a decade or two?

posted 22 days ago

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Mark V‭

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