Fantastic Numbers and Where to Find Them.

Prof. Antonio Padilla is a theoretical physicist and cosmologist at the University of Nottingham who has also appeared numerous times on the Numberphile Youtube series, including this incredibly popular video where he shows how the sum of all natural numbers (1 + 2 + 3 + 4 + … ?) is actually -1/12. It’s ridiculous – Padilla concedes that it looks like “a bit of mathematical hocus-pocus”, but the pudding is in the proof, or something, and he points out that 1) this only works if you’re adding all of the natural numbers, which means you don’t stop at any point, and 2) this sum appears in physics, where we don’t see infinities (and if we do, it’s a problem).

Padilla describes the interplay between physics and some numbers at both extremes of the mathematical scale, both the very small and the very large, in his book Fantastic Numbers and Where to Find Them: A Journey to the Edge of Physics, an intense but mostly accessible book that runs through nine distinct numbers, from zero to a googolplex to 10^-120 to infinity, and uses them to explain some key concepts or findings about the nature of everything. He waltzes through the history of math – just about every famous figure there makes an appearance at some point, which will make you realize just how many great mathematicians ended up losing their marbles – and just about always finishes up somewhere in the realm of quantum physics, whether it’s things we know or things we think we know, or occasionally things we still don’t know. There’s even a chapter on the cosmological constant, which was in the news just this past week with the revelations that dark energy isn’t as immutable as we believed, which implies that the cosmological constant is, in fact, inconstant.

When Padilla is talking physics and cosmology, at either end of the scale, he’s engaging and by and large easy to follow, other than perhaps near the end of the book when he’s introduced the panoply of particles that populate the quantum world – all the quarks, leptons, and gauge bosons that we know or think exist – where keeping any of them straight was a bit more than I could handle. It doesn’t end up mattering much to the narratives of those chapters, as Padilla’s point is the relevance of the numbers in question, although I ended up a little frustrated that I didn’t entirely know what was going on at some points.

It’s the mathy stuff where Padilla struggles to communicate in a way that a typical reader might follow, and perhaps that’s just a function of the size of the numbers he’s discussing. The chapter on the number TREE(3), which is so large that we can’t even notate it, let alone comprehend it, ultimately lost me not in its prose but in its sea of notation. TREE(3) is much larger than the number of atoms scientists believe exist in the entire universe (around 10⁸⁰, itself a number that we can’t easily envision), a number so big that the universe won’t “allow” it to exist – according to the Poincaré recurrence theorem, at least, which says that the universe will “reset” before TREE(3) happens in any sense of the word. Padilla uses TREE(3) to explain that theorem and the possibility that the universe is a hologram, that we live in two dimensions and only think we perceive the third, but by the end of that chapter I didn’t understand why TREE(3) got us there in the first place. (It doesn’t help that Padilla discusses all of this several chapters before he gets into string theory, which underpins the holographic principle, so we’re walking without a net for a while.)

Padilla is a gifted communicator, clearly, and his enthusiasm for the subject comes through everywhere in the book – it’s just that the topic itself is abstruse and assumes some familiarity with physics and/or with some branches of math like infinite series and set theory. He’s better at explaining concepts like particle spin, which he points out isn’t spin like what we’re talking about in baseball but an innate characteristic of a particle (any more than red or green quarks have those actual colors), than at explaining concepts like the nested powers of TREE(3) or Gödel’s incompleteness proof. It all left me with the sense that I’d enjoyed the book, but that the audience for it might be very narrow – you have to know enough to follow him through his various rambles through math and physics, but not so much that you already know all of this stuff. I was at least lucky enough to mostly be in the first camp, even though I got lost a few times, but that’s just because I love these topics and have read a lot of books about them. It’s not the physics I learned in high school, and not really the math I learned there either.

Next up: Michael Swanwick’s Stations of the Tide, winner of the 1991 Nebula Award for Best Novel.