En Passant V

(warning: today’s persiflage comes with possible extra snark due to sleep deprivation)

The Ramanujan Machine: I learnt from John Baez on twitter about The Ramanujan Machine, a project designed to “help reveal [the] underlying structure” of the “fundamental constants” of mathematics. It seems that more effort has been spent on hype rather than on learning anything about continued fractions, and there is nothing there that would be surprising to Gauss let alone Ramanujan. Despite the overblown rhetoric (sample nonsense from the website: Suggest a proof to any of the conjectures that were discovered by the Ramanujan Machine. Have a formula named after you!), the chances of finding anything novel by these methods seem slim. While I’m all for the use of computers in mathematics, at this point (for these type of identities) we are very much in a world where the guiding hand of human mathematical intuition is very much required. As for the “new conjectures” that are animated on their flashy website (screenshot below):

well, the less said the better. It’s harder to say which one is sillier to pretend is original. The latter is a specialization of a specialization of a specialization of a specialization of Gauss’ 2F1 continued fraction identity, sending the four parameters \(a,b,c,z\) to \(a = 0, b = 1/2, c = 5/2, z = -1.\) (Hint: when your result is a specialization of a formula on Wikipedia it does not count as original.) The former is even more elementary and proves itself by induction:

The rest of the paper is littered with mathematical trivialities and grandiose bombast. (All the other “new” identities are similar trivializations of Gauss’ hypergeometric continued fraction or just known infinite (or finite) sums in disguise via Euler’s continued fraction.) What would be a much more interesting project is to find a way of taking a continued fraction and recognizing it as a specialization of one of the (very many) known results. Lest I be considered a luddite, I should note that when it comes to infinite sums and integrals, this is something that Mathematica does amazingly well, so respect to the people who worked on that.

3Blue1Brown: I don’t really get the appeal, to be honest (yes, I know, I’m not the target audience). The presenter has a geometric perspective which gets shoehorned into everything whether it is appropriate or not. I watched two videos, both of which seemed to miss (or at least elide) at least one key underlying mathematical point. The first was a video on quaternions. The fundamental property of \(\mathbf{H}\) is that it is (the unique!) non-commutative division algebra over the real numbers. But the video really only talked about the multiplicative structure, in which case you may as well talk about \(\mathrm{SU}(2).\) Are you really “visualizing quaternions” when you only think about the multiplicative structure? Then there was this video on the Riemann hypothesis. The video does a reasonably good job of explaining analytic continuation in terms of conformal maps (not that I think of it that way, but this is a perfectly reasonable way to think about it). However, the entire video ignores (once again) the key point that what is amazing is that the zeta function has an analytic continuation at all rather than it is unique (time stamp 16m 45s):

https://www.youtube.com/watch?v=sD0NjbwqlYw?#t=16m45s

The closest the video comes to acknowledging this is the quote “which through more abstract derivation we know much exist” which is somewhere between wrong (suggesting that the extension exists for formal reasons) and misleading (using “abstract” to mean something like “beyond the scope of this video” or something). How can you start to think about the Riemann Hypothesis without appreciating why the analytic continuation exists? I guess the video helps bridge the gap between mathematics and physics; popular accounts of physics have long since enabled people to think that they understand something about physics while actually not really having any real idea what is going on, now people can think that way about the Riemann zeta function as well! (The number of views of these videos is in the millions.)

Hatcher on Class Groups: I learnt that Allen Hatcher, author of a wonderful and free textbook on algebraic topology, is writing a textbook on the geometry of binary quadratic forms. I’m sure it will be a great read, and I don’t quite mean to lump it together with the two examples above, although I do notice that it does not appear to mention (in any way) class field theory. That seems to be a strange omission: couldn’t the book at least have a sentence or two indicating that two centuries of algebraic number theory has been built on generalizing Gauss’ work on the class group?