Laconic Proofs: Can Mathematics be Poetry?

Mathematical writing is a narrative form. The development of definitions and theorems are not unlike the story of growth and struggle of a character. A well-written math paper or book has a narrative structure with an initial goal as the introduction, some definitions and lemmas as the rising action, a major theorem or corollary as the climax, and a resolution, which can be surprisingly diverse. This is a topic that I would like to explore in the future (provided I can be coherent enough to talk about it).

But what can we make of the short proofs found in the literature? (No, not “left as an exercise to the reader”.) I mean what do we make of Littlewood’s hypothetical shortest dissertation (pdf)? There is a game of sorts to find who can publish the shortest paper. But beyond ego boosting, there’s a certain aesthetic to these types of laconic proofs that demands extra attention from its reader. It reminds me of laconic poetry that I once read, and similar demands that it made. Now, I confess that very few mathematicians bother writing in verse, but there’s a certain parallel between these two forms. Their brevity betrays their depth. It takes so little time to read, but so much time to unpack. Not all of them have this quality, I would say the counterexample to Euler’s conjecture (pdf) lacks any depth. But a one page proof of the irrationality of pi or a one sentence proof that every prime of the form 4k + 1 can be written as the sum of two squares has a charm and beauty of its own that I dare to say is poetry.

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About minimalrho
Unemployed guy with a PhD in math.

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