I dreamed recently that I opened the Sunday comics to find Scott Adams’s Dilbert strip turned into a somewhat lengthy, weird illustrated diatribe about how all numbers smaller than infinity were essentially the same, with the exception of the privileged number 17, which was the number of kinds of finite groups sharing some interesting property. Before I carry on I should point out that I have no reason to think that Scott Adams has any particularly crankish mathematical views, and no reason to think that he thinks much about infinity, finite groups, or the number 17. Imagining he has some fixation on them is wholly the creation of my unconscious or semiconscious mind, whatever parts of mind and body create dreams. But there are some points I can talk about from that start.
First I should say what a finite group is. It’s not important to the imaginary screed, but, it’s an interesting idea that doesn’t get a lot of pop mathematical attention. Groups, in this context, are creations of abstract algebra. Ordinary algebra, such as you deal with in middle or high school when suddenly arithmetic involves spelling, challenges you to do arithmetic when you don’t know what one or more of the numbers are: suppose that all you know about a number is that three times it minus four is equal to twice it plus two; what can you conclude from that? Abstract algebra looks at the question of can you do the operations of arithmetic, things like addition and subtraction, multiplication, maybe even division, without it having to be numbers you add together? Can you add together other stuff?
A group is a collection of things on which, yes, you can do addition and subtraction. We don’t worry about multiplication, or division (if you can do them, you call the group a ring, or a field, respectively). In some cases this looks obviously like addition and subtraction. Imagine taking an object and rotating it either clockwise or counterclockwise; you can easily say that rotating it some angle clockwise, and then clockwise again, looks like adding two numbers together. Rotating it clockwise, then counterclockwise, looks like adding a number and then subtracting it.
In some cases the structure is less familiar. Imagine that you have a square, and you’re allowed to rotate it, but only 90, 180, 270, or 360 degrees clockwise or counterclockwise; that is, when you set it down, the corners have to be pinned down to the spots the corners were originally in (although the corners themselves can change spots). This addition and subtraction still works, but you only have a couple of possible elements to add or subtract. But that still looks like ordinary arithmetic, just, using the whole numbers 0, 1, 2, and 3, with any number larger than 3 or less than 0 looped back around: 3 + 1 is 0, and a 270 degree rotation plus a 90 degree rotation is a 0 degree rotation; 2 + 3 is 1, and 180 degrees plus 270 degrees is indistinguishable from 90 degrees.
In some cases the correspondence between addition and subtraction is hard to identify. For example, imagine that you have a couple of characters, such as — from today’s Jumble — “PRELUP”. You can think of unscrambling this as swapping any pair of letters, like, the second and fifth, or the third and fourth. These swaps — not the letters, but the idea of swapping the second and fifth letters in a string — done in order are like addition, and you can make a group out of these swapping instructions.
So that’s groups: collections of things that are not necessarily numbers, on which you can do something that works like addition and subtraction. The real definition is more precise than that, but this should make the idea not so intimidating and abstract.
A finite group is just a group — a collection of things on which you can do addition and subtraction — where there’s only finitely many different things to add or subtract. The whole numbers aren’t a finite group, but, the rotations of that square (there’s only four to be done there), or the swaps of letters (for “PRELUP” there’s only — well, that’s a fun little puzzle for after you’ve figured what word that is), are finite groups.
Classifications of groups, well, that’s a little bit like classifying birds or flowers. When you look at their structures you might spot that they look very nearly the same, or they have structures that are the same. For example, it’s possible for a group to have a subgroup, a collection of pieces within it that, by the same rules for addition and subtraction, is itself a group. It’s possible to have subgroups that have really wild properties that take me too far away from my main point to describe here.
So this would just be the background of what my imaginary Scott Adams decided was so all-fired important that it should take over his strip for an imaginary Sunday: the discovery that there was some classification of finite groups which had exactly seventeen different kinds of groups in it, and that this was somehow unique to all numbers less than infinity.
In my dream I only skimmed over the dense scrawling of the argument, partly because that’s what all extremely wordy cartoons get (I don’t disparage cartoonists wanting to write — good writing is always precious — but it can get too much), partly because, well, infinity is one of those things that gets cranky writers a lot, and it’s almost never worth reading. The arguments tend to be loopy or vapid, when they can be made out at all.
It’s too glib to say that every scientific or related field thinks they get the most amateur cranks intruding with what has been so well described as “not even wrong” screeds. I think mathematics is prone to such, since the barriers for entry are quite low: in principle, you just need the ability to reason. I suppose philosophy suffers the same problem, and I have heard that judges are getting fed up with extraordinarily complicated loopy screeds that have to be read carefully just in case the self-representing party has hit on something that’s a coherent legal argument, but I don’t know from experience.
But not all fields of math get crankish submissions. I can’t remember seeing anything about directed graphs and sign nonsingular matrices, for example, probably because you don’t even hear about those until you’ve gotten fairly well into a mathematics education. You might suspect I just made those terms up; I promise I didn’t. Even if someone moderately mathematics-nutty gets to that point, there’s almost nobody it can be communicated to.
In comparison, infinity is a very welcoming subject for the novice mathematician. What’s probably the most remarkable result — that there are different sizes of infinity, and indeed, there are infinitely many different sizes of infinity — is so accessible that its core gimmick, Cantor’s Diagonal Argument, could be understood correctly in elementary school. [Edit: I mean, here, just showing there are more reals than there are integers, the version where you get shown lists of numbers between 0 and 1. The Power Set version is a bigger challenge.] It’s about as easy to understand the thing being talked about as it is to understand the statements of Fermat’s Last Theorem, or the Goldbach Conjecture, or the classic Greek compass-and-straightedge problems of squaring the circle or trisecting an angle. (The other great problem of classic Greek compass-and-straightedge mathematics which can’t be done, the doubling of the cube, gets little attention, possibly because it’s so hard to draw solid geometry if you aren’t a tolerably skilled artist.)
Once you have grasped the idea that the set of real numbers is bigger than the set of integers — and many never get to that point — the next staggering idea, that it’s literally undecidable whether the set of real numbers is the smallest set bigger than the integers, or whether there’s an infinite set of size between the two (or whether there’s many such sets), is easy to hear and still the more staggering. This isn’t a matter of “we don’t know”; it’s a matter of, based on the rules of set theory up to that point, we can build logically consistent mathematics by assuming there’s no infinities of intermediate size, or assuming there’s one, or assuming there’s many. The rules are consistent whatever way, just as the rules of baseball are consistent with there being eight teams in the league, or sixteen, or as few as two or as many as a hundred. (I suppose they’re consistent with one or zero teams, but the pennant races are duller.)
So accessibility is probably important to a field drawing cranks. As best I can tell from the physics side of things the favorite crank magnet is special relativity, the important reasoning behind which is open to anyone who’s got the hang of the Pythagorean Theorem, and maybe who’s comfortable using a letter in place of a number since it’s quite convenient to speak of arbitrary velocities v and to replace the speed of light with the letter c. Quantum mechanics gets a fair amount of crankery too, although since you can’t do most of it without advanced calculus, what really gets built upon is the pop science treatment, which for sad but logical reasons hypes up the “spooky” and “mysterious” and ignores how sound and predictable the subject is. (However, I do recommend Richard Feynman’s QED as it teaches some of the most important stuff using a scheme that looks weirdly arbitrary, but tucks all the trick calculus parts off where they don’t have to bother the reader who doesn’t know calculus. It’s a masterpiece of popular science.)
In biology, evolutionary theory attracts all the attention, possibly in part because — as Adam Gropnik points out in Angels and Ages — Charles Darwin’s The Origin Of Species is pretty near the only important scientific result whose presentation is reliably read for pleasure, or can be. (The only other candidate I can think of is Euclid’s Elements and that’s a distant second. One might fairly quibble that the Elements are more a textbook of mathematics than a collection of fresh results.) I grant there’s some religious influences there too.
Sometimes arguments about infinities attract religious overtones too, since infinity has connotations of being the domain of God. The sense I get is infinity-cranks tend to find the infinity of different infinities an affront to God. I suspect that if I believed there were a God who could make something mathematically sound or not — and I can’t figure any way to do that, since the conclusions follow from deductive logic, with no room for God to rule anything in or out — the idea there could be this incredible structure to the incredible would inspire awe at the work, not horror. But I don’t have any particularly crankish thoughts about infinity, at least not that I’ve noticed.
So all this is why I can imagine someone deciding to write a screed about infinities and how everything which isn’t an infinity is essentially the same thing. Why I should imagine someone getting into finite groups I can’t be quite as sure about, although quite a few neat and accessible mathematics problems, Fermat’s Last Theorem among them, encourage one into the study of how arithmetic works, which takes one to abstract algebra and to groups almost right away, and serve as good places to crash up. Why Scott Adams, I can’t say. Possibly I know Bill Amend wouldn’t do such a thing in FoxTrot. The 17 is probably because it’s the least random of numbers.