BRIEFLY NOTED: Mathematical Mayhem

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.

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• 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.

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• Set Theory and the Continuum Problem, by Raymond Smullyan and Melvin Fitting

Imagine you have a set of three fruits — an apple, an orange, and a banana, {A, O, B}. Now imagine all possible subsets of your set. There are eight of them: the three sets each containing a single fruit (they look like: {A}, {O}, and {B}), the three sets that each contain two fruits but are missing the last one ({A, O}, {A, B}, and {O, B}), the set of all three fruits (our original set), and the empty set, which has none of the fruits. Eight is bigger than three, and this is one instance of a very general theorem proved by Georg Cantor in 1891: for any set X, the set of all subsets of X has more elements than X itself.

That sounds incredibly stupid and obvious, because the set of all subsets already has X elements just from each of the 1-element sets of the elements of X, and then you've got lots more stuff besides. And it is kind of obvious in the finite case, but Cantor proved it without any reference to the finitude of the original set X": he proved it in full generality for all sets, and it didn't take long for people to realize that that means there isn't just one kind of infinity, but rather an infinite tower of infinities, all different sizes from one another.

A natural next question to ask would be, “well, is that all of the infinities, or are there others?” Put differently, we know from Cantor that if we have one kind of infinity, and consider the set of all subsets of that kind of infinite set, the number of such subsets is a different, larger kind of infinity. But is it the “next” larger, or could there be a third infinity living between the two?

The Continuum Hypothesis, more or less, is the statement that there is no such third infinity, that the operation of considering the number of all subsets is the smallest “step” you can take from one infinity to another. For the first half of the 20th century, this question was considered the deepest and most difficult one in the foundations of mathematics, until it was decisively settled by Kurt Gödel and Paul Cohen, though not in the way anybody expected. The answer is that the existing axioms of mathematics don't have an answer: you can construct a universe in which there are no intermediate infinities, and you can also construct universes in which there are 1, 5, 17, any finite number, or even an infinity of intermediate infinities. All of these universes “work” mathematically, so there’s no way to say which one we live in without changing the axioms.

I've long considered the proof of the independence of the Continuum Hypothesis to be one of the crown jewels of human knowledge. There’s something dizzying, even Promethean, about mankind’s ability to even consider such questions, let alone settle them. Unfortunately, the proof itself, especially Cohen’s half of it, is widely regarded to be complex and difficult to follow. So while I’ve dreamed since I was young of one day comprehending this proof, I always expected it would take a while before I could make the attempt. But I was intrigued when I learned that one of my favorite authors, Raymond Smullyan, had written a textbook aimed at this precise topic. Smullyan was a stage magician and Zen sage and children’s author with a kind of side hobby in academic mathematics. So I bought the book figuring that even if it was incomprehensible to me, it would at least be an idiosyncratic take on the topic.

It’s certainly idiosyncratic. Totally absent is the drily rigorous style of most modern mathematics textbooks. Smullyan is chatty throughout, and pauses at the top of each foothill to take in the scenery with a few reflections on the philosophical implications of these intermediate results. While his approach to the summit largely follows the contours mapped out by Gödel and Cohen, a lot of the details and supporting lemmas are his own invention. Sometimes this is aggravating, as when he invents his own vocabulary for concepts that the rest of mathematics has already settled on words for. Sometimes it's just perplexing, like when he ignores a very straightforward “canonical” proof of a result, and instead delivers a murky, meandering, indirect proof of the same thing. But Smullyan was spiritually-Chestertonian, and he delighted in nothing more than paradox. So maybe these strange logical contraptions, these twisting passages of logic that criss-cross and double back on themselves, seemed clear and straightforward to him.

I’ll be honest, I skimmed the final quarter of the book. I made it all the way through the preliminaries, and then to the peak of Gödel’s half of the mountain. But the subtle and technical methods that Cohen used were a bit too much for me at this point in my life. So I familiarized myself with the basic plan and outline of the proof, but didn’t actually do the climb. When I revisit it someday, I don’t know that this book will be my first choice of guide. It is very weird. But I’m certain that at some point in that journey I’ll come to an idea that try as I might, I just can’t grasp. And in that moment of frustration I’ll go find this book on my shelf, while muttering to myself, “I wonder how Smullyan explained this, at least it'll be different...”

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• On Numbers and Games, by John Conway

John Conway was feeling ambitious one day, and decided to re-invent math with different numbers. Normal numbers are, y’know, the things you count with. Technically they're equivalence classes of finite sets. That seemed boring to Conway, so his numbers are completely different objects, with their own completely alien rules of addition and multiplication and so on. The numbers can be written in a canonical form, each of them consisting of two different sets each containing more numbers, a left set and a right set. Each of the numbers in each of those sets has its own left set and right set, and so on until they bottom out in the empty set...or don’t. The resulting numbers look like spidery, recursive, fractal, possibly infinite trees. Miraculously, they’re well defined, and his parallel-universe version of arithmetic all just works.

If you search hard within Conway’s numbers, you can find our plain old familiar numbers, like 3 and -17.4, and Pi, and the cube root of twelve, and so on. All of these numbers have their analogues, and again, miraculously, they play identical structural roles in Conway's new system, all the rules of mathematics with the old familiar numbers just keep right on working with his branching and twisting and cross-pollinating recursive trees instead.

But then you notice that those familiar numbers are just a tiny subset of Conway’s numbers, a sliver of a sliver, and that surrounding them are a vastly larger horde of...other numbers. These are the surreal numbers, and they include infinitesimals, actual infinities, and everything in between. All of them continue to obey all the axioms and rules of mathematics with their own more general sort of arithmetic. In fact these are special numbers: the surreals are provably the largest well-defined and totally-ordered field containing the numbers we know and love.

Now consider games — two player, turn-based, perfect-information games like Chess or Go. In a given position in a game, each player has a set of options they could pick, a set corresponding to all the moves that they could make. Take all the options for the first player and put them in one set called, oh, I don't know...the left set. And all the options for the second player let's call that the right set, and, hey! Where have I heard this before? But just what are these options anyway? If you think in sufficient generality, they’re games themselves. White choosing to move his pawn to E4 on the first turn of a chess game is equivalent to white choosing to play a new game — a game in which black goes first, and white happens to start with a pawn on E4. So a game is just two sets, a left set and a right set, each of which contains more games, which themselves have a left set and a right set, and...wait a minute, I've definitely heard this before.

It turns out that not all games are numbers, but many of them are. And for those games that are just numbers in disguise, all of Conway’s new laws of mathematics, his trippy new methods of addition and multiplication, just keep working. And vice versa, when you play a game, you are actually doing arithmetic in Conway's more general field of numbers. This means you can use numbers to prove which games can be won, and by whom, and what the best strategy is for either side. Conversely, by playing a game, or figuring out how to play it perfectly, you can prove statements in number theory in ways that you would never have imagined.

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• Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, by David Cox

I bought this book for one reason, which is that its author, David Cox, wrote the book on Galois Theory that is my favorite math textbook ever. Have you ever bought and read a book because you loved the author, even though the subject seemed totally boring? It often turns out really well! It sure did this time.

Algebraic geometry is one of the most fearsome branches of modern mathematics (in its contemporary form it was created by that Grothendieck guy). A book about it aimed at undergrads sounds like the setup to a joke, or a newly invented form of sadism. But Cox totally pulls it off, and he does it with two techniques that should be much more widely used. First, he introduces his subject via the historical avenues that its actual development took, rather than dropping the completed modern theory on you in its full generality. This is also what makes his Galois theory book so great, and I talk a lot more about it in my review of that book.

But the second technique is trickier, and even rarer for a math textbook. He reifies every important result he discusses with constructive proofs,³ which he then turns into concrete algorithms that can be compiled and run. In addition to aiding intuition, this provides a Rosetta stone that can turn questions about algebraic geometry into questions about algorithms and vice versa, and he switches back and forth between them as necessary. This turns some terribly abstract definitions and proofs into something delightfully concrete, creating its own little “ladder of abstraction.” It also turns up powerful applications of what is usually a pretty esoteric branch of math,⁴ and teaches you a lot about what really happens in the guts of a computer algebra system.

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• Spacetime and Geometry: An Introduction to General Relativity, by Sean Carroll

Surprise! This one isn’t a math book at all, some dirty physicist slipped this in here! It’s okay though, I think general relativity is a lot like the proof of the independence of the continuum hypothesis, in that they’re among the crowning achievements of the human intellect, and something that every educated renaissance man should try to understand. The trouble with general relativity is that it’s very hard for a layman to find a good textbook — they’re all either much too hard or much too easy, much too formal or much too “string theory is like a taco.” Some of them use mathematical techniques I've never heard of, while others use index-juggling tensorial methods that I understand but which make my eyes fall out of their sockets.

This book, recommended to me by an actual-physicist, is the Goldilocks of general relativity textbooks, just right in every way. It grew out of a set of lecture notes that spread online, and got so popular that the author decided to polish them up into a book. Strangely, the book starts out by looking at electrodynamics in a totally flat spacetime. Huh? I thought this was a book about general relativity. Don’t worry, we’re getting there. Carroll is building up your intuition, and your confidence, by helping you to slay some easier prey. He introduces advanced mathematical machinery like exterior derviatives and Hodge duals, but he does it gently, and then shows how nothing was gratuitous, by using these tools to massively clean up and simplify all of special relativity and relativistic electrodynamics. It’s gratifying in the same way that reaching the end of a story that finally explains the beginning is.

Once that’s out of the way, he takes the step into curved spacetime, but goes easy on the differential geometry. The emphasis throughout is on fully working out solutions for “classic” spacetimes like the Schwarzschild metric for a non-rotating black hole, really stepping in and inhabiting the Einstein field equations and seeing how they shake when you poke them. Pretty soon you’re taking on modern cosmology and the FLRW-metric. I’ll be honest, as with the Smullyan book, I skimmed the end of this one. Mastering general relativity is not compatible with my current lifestyle (too many young children wailing about how they’re bored). But one day I will be old and wrinkled and hopefully not senile, and this treasure of a book will beckon once more.

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.

3. This is math jargon for a proof that ends up with an actual example of the thing you’re proving.

4. Among other things, you learn how to implement completely mechanical ways of solving systems of polynomials, and of implicitizing rational functions.