the Ramanujan machine

Nature of 4 Feb. 2021 offers a rather long (Nature-like) paper on creating Ramanujan-like expressions using an automated process. Associated with a cover in the first pages. The purpose of the AI is to generate conjectures of Ramanujan-like formulas linking famous constants like π or e and algebraic formulas like the novel polynomial continued fraction of 8/π²:

which currently remains unproven. The authors of the “machine” provide Python code that one can run to try uncover new conjectures, possibly named after the discoverer! The article is spending a large proportion of its contents to justify the appeal of generating such conjectures, with several unsuspected formulas later proven for real, but I remain unconvinced of the deeper appeal of the machine (as well as unhappy about the association of Ramanujan and machine, since S. Ramanujan had a mystical and unexplained relation to numbers, defeating Hardy’s logic,  “a mathematician of the highest quality, a man of altogether exceptional originality and power”). The difficulty is in separating worthwhile from anecdotal (true) conjectures, not to mention wrng conjectures. This is certainly of much deeper interest than separating chihuahua faces from blueberry muffins, but does it really “help to create mathematical knowledge”?

A wealth of numbers [book review]

Another general audience book sent to me by Princeton University Press! This Wealth of Numbers is a compilation of one hundred texts on mathematics for the general audience, à la Martin Gardner but starting in 1481! Very few well-known authors in this compilation, apart from Voltaire, Euler, Carroll, Pólya, van der Waerden, Shaw, Rademacher, Toeplitz. and Feynman… I must acknowledge I did not read each entry in detail over breakfast, either by laziness about the old English style or because the topic was not of direct interest to me. This leads me to wonder who would appreciate the book. The styles and contents are quite mixed, from puzzles to historical entries, to older and newer ways of introducing basic notions, to science-fiction (for the very last entry) [if not Anathem!]… A linear reader, going from page 1 to page 365, must thus be quite open-minded if this reader does not want to skip anything. The book can however be seen as a terrific source for short illustrations in talks and classes.

A few gems I appreciated: the wrong resolution of a probability problem by (the highly obscure or even imaginary) L. Despiau in 1801 (page 19) [which makes me regret the absence of a critical postface to the texts, so that they could be replaced into a larger context and straightened out if necessary!];  from a contemporary of Bayes, Banson’s 1760 way of extracting square roots (page 46); Wells’ 1714 limpid introduction to trigonometry (page 94) that reminded me very much of the way my daughter was taught the same a few weeks ago; Ball’s 1892 reproduction of Kempe’s false proof of the four-colour theorem (page 118); a 1561 entry on maritime maps  by Martin Cortés, son of the conquistador Hernán Cortés (pages 153-154); Patridge’s 1648 description of Napier’s “speaking rods” (also known as “Napier’s bones”, page 157) that reminded me of my slide rule in high school (that I learned to use the year before the pocket calculator was allowed at exams, just like the pinched cards I had to handle the year before terminals got accessible in my statistics graduate school!); Voltaire’s amazing 1733 eulogy of Newton, against Leibniz and Bernoulli (page 178); Eicholz’ and O’Daffer’s 1964 explanation of set theory axioms within the “New Math” pedagogy, just a few years before I learned them in primary school (pages 278-281); LOGO programming on the Spectrum 48K (!)  by Gascoigne in 1985 (pages 282-289), quite in tune with the LISP and ADA programing languages my wife was learning at the time, while I stuck to Pascal…; Playfair’s 1798 chart of exchange balance between England and Ireland (page 306) and the only place in the book where statistics is mentioned; Richard Feynman’s very honest acknowledgement of the primacy of mathematics, even though he wished it could be different (pages 320-321). I am sure other readers would find at least as much entries, if not necessarily the same ones, to their taste.

As you can judge from the above, the book also has a very nice cover, by Eugen Jost, relating to Hardy’s taxi number, 1729. (And a nice picture of the author in the back flap, taken in a place reminiscent of Scotland, even though it could as well be the Yorkshire dales or the Lake District.)