Written: Dec. 17, 2024
I recently finished reading Karl Sigmund's The Waltz of Reason, subtitled: "The entanglement of mathematics and philosophy", and I really loved it, and strongly recommend it to professional (mathematicians and philosophers) and laypeople alike. Even though I knew most (but not all!) the material covered, the philosophical angle and the "big picture" were really fascinating, and the style is so engaging, a true page-turner!
But then it occurred to me, that very soon, all the material covered will be only of historical interest, describing the state-of-the-art and the conventional wisdom of current mainstream mathematics, and its philosophical underpinning, at the close of the first quarter of our century. In particular, in the concluding chapter, entitled "Understanding", ten pages before the end (p. 374, line 10 from the bottom), the author states:
"As almost all mathematicians agree, proof is the sine qua non of mathematics."
I am proud to be in the tiny minority that disagrees. I do agree that, mathematics, as practiced today, indeed has this hangup, but sooner or later (hopefully sooner) I am sure that this obsession with fully rigorous, logically impeccable proof, would have to be abandoned, since all the proofs of non-trivial mathematical knowledge would be intractable, i.e. beyond the scope of both humankind and computer-kind, and even our future AI overlords, who would be too wise to keep this "sine qua non" hangup. Mathematics would become again an inductive science, like it was back in Egypt and Babylon, before it got ruined by the Greek mafia of Euclid and his buddies.
Ironically, the last few pages of Sigmund's book follows Polya's description of Euler's beautiful recurrence for the sum of divisors function σ(n) (p. 383)
σ(n)=
σ(n-1)+σ(n-2)
-σ(n-5)-σ(n-7)
+σ(n-12)+σ(n-15)
+...
+(-1)j(σ(n-(3j-1)j/2)+σ(n-(3j+1)j/2))
+...
where the summation goes as long as the argument is non-negative, and if the last argument is zero (i.e. n happens to be a pentagonal number) it ends with n.
As Sigmund (following Polya) tells us, Euler found this formula (obtained from the pentagonal number theorem by logarithmic differentiation) completely "experimentally", by playing around, and then tested it for quite a few special cases and, as quoted by Sigmund (p. 384 there), Euler concluded sardonically :
"I think these examples are sufficient to discourage anyone from imagining that it is by mere chance that my rule is in agreement with the truth."
Euler discovered this beautiful rule in 1751. As it turned out, he did find a proof, ten years later, and the result was (in hindsight) not terribly deep (there are beautiful combinatorial proofs by Fabian Franklin and Bressoud and Zeilberger), but even if there were no fully, iron-tight mathematical proof, it would have been, nevertheless, absolutely certain.
In the future, we would very rarely have the luxury of a fully rigorous proof, and would have to get used to semi-rigorous, and even non-rigorous, experimental and heuristic, proofs. This way mathematica will advance much faster, without the strait-jacket of a "fully rigorous proof".
But future generations, as well as ourselves, from "all walks of life", should read Sigmund's masterpiece. For us contemporaries, it has great mathematical (as well as philosophical and historical) interest. For our children, and especially grand-children and great-grand-children, it would still be of of great historical interest. Enjoy!