John Psmith does his level best to make mathematics interesting to layfolk like you’n’me:
In our end of year post I threatened to write more math reviews, and multiple people in the comments egged me on. So now, with Jane laid up in the final stages of pregnancy, I have seized control of the Substack for a very special lightning round of math textbooks I recently enjoyed. No, wait! Don’t close the tab! I promise that some of these will be fun for non-mathematicians as well.
- Counterexamples in Topology, by Lynn Arthur Steen and J. Arthur Seebach Jr.
The mathematicians I have known included some eccentric characters. In fact when one considers research mathematicians as a class, it’s usually the normal people who are the exception. But there are degrees of weirdness. One of the most delightful things about the world is how fractal it is, and this extends to human hierarchies. Take any unusual group of people — frequent-flyers, monastics, the ultra-wealthy, members of genealogical societies — and zoom in on them, and it turns out there are even stranger or more elite subgroups buried within. This is true of mathematicians too, each subfield has its reputation, some of them regarded with awe, others with disdain. But ask any mathematician, “Who are the real weirdos? Who are the ones who are truly cracked?” The answer will be unanimous: it’s the topologists.
Topology is the study of spaces in the most abstract sense, so abstract that they may not even support a well-defined notion of distance (if your spaces are guaranteed to have distances, then you are now doing geometry rather than topology). Topology takes a coarser view of space: forget about curvature, distance, or really anything involving numbers at all. To a topologist, two points can be “near” each other or not, “connected” or not, and beyond that it doesn’t matter. This is the source of all the jokes about topologists mistaking donuts for coffee cups,1 but the kind of topology that studies multi-holed donuts, algebraic topology, is actually comparatively tame and normal. Also relatively normal is differential topology, which is the next neighborhood over from differential geometry, and which produces cool videos like this.
No, the scary part of town, the place where the true freaks and degenerates hang out, is general topology. General topology is where we go to figure out the basic definitions and frameworks that underlie the rest of topology. It’s about exploring what nearness and connectedness even are, and when mathematicians are trying to figure out what things are, that usually means probing the outer limits of what they can be. So general topology turns into the study of the most bizarre and deformed and disturbing spaces accessible to human cognition. No wonder its practitioners are a little weird.
Which brings me to this book, whose perversity is laid out right there in the title. It’s a big book of counterexamples to statements which seem obviously, intuitively correct. In general topology, things that seem intuitively correct are usually wrong:2 the field is notorious for proofs that almost work but twist out of your grasp at the last moment. A big book of counterexamples is exactly what you want for understanding why your proof that “all Xs are Ys and all Ys are Xs” falls flat. Seeing the logic fail is one thing, but seeing a concrete example of an X that is not a Y (or vice versa) brings it home with a satisfying finality.
But the real reason I love this book is the names, oh, the names. Let me flip through the table of contents with you: are “the Infinite Cage” and “the Wheel Without Its Hub” examples of topological spaces, or planes of the underworld? Are “Cantor’s Leaky Tent” and “Tychonoff’s Corkscrew” important counterexamples, or Level 2 wizard spells? I could spend hours idly leafing through this book, pondering these twisted and prosperous spaces, imagining them as worlds in themselves, imagining the bizarre sorts of creatures that might live there. Is this a math textbook or an RPG sourcebook? Trick question, they’re the same thing.
1. One of the proudest moments of my mathematical career was when I attended a faculty tea and a distinguished topologist asked me for a donut and I handed him a cup of coffee instead. Everybody lost it. Alas, I turned out to be much worse at math than I am at improvisational comedy, and my mathematical career ended shortly afterwards.
2. This is why we have the “separation axioms“. Every rung on that latter is the “well, actually …” to something that seems self-evident but isn’t.
