Here is a quotation from G.K. Chesterton: “Poets do not go mad; but chess players do. Mathematicians go mad, and cashiers; but creative artists very seldom. I am not attacking logic: I only say that this danger does lie in logic, not in imagination.” Here also is a snippet from the flap copy for a recent pop bio of G.F.L.P. Cantor: “In the late nineteenth century, an extraordinary mathematician languished in an asylum. . . . The closer he came to the answers he sought, the further away they seemed. Eventually it drove him mad, as it had mathematicians before him.”
The cases of great mathematicians with mental illness have enormous resonance for modern pop writers and filmmakers. This has to do mostly with the writers’/directors’ own prejudices and receptivities, which in turn are functions of what you could call our era’s particular archetypal template. It goes without saying that these templates change over time. The Mentally Ill Mathematician seems now in some ways to be what the Knight Errant, Mortified Saint, Tortured Artist, and Mad Scientist have been for other eras: sort of our Prometheus, the one who goes to forbidden places and returns with gifts we all can use but he alone pays for. That’s probably a bit overblown, at least in most cases.¹ Cantor fits the template better than most. And the reasons for this are a lot more interesting than whatever his problems and symptoms were.²
Merely knowing about Cantor’s accomplishments is different from appreciating them, which latter is the general project here and involves seeing transfinite math as kind of like a tree, one with its roots in the ancient Greek paradoxes of continuity and incommensurability and its branches entwined in the modern crises over math’s foundations—Brouwer and Hilbert and Russell and Frege and Zermelo and Gödel and Cohen et al. The names right now are less important than the tree thing, which is the main sort of overview-trope you’ll be asked to keep in mind.
Chesterton above is wrong in one respect. Or at least imprecise. The danger he’s trying to name is not logic. Logic is just a method, and methods can’t unhinge people. What Chesterton’s really trying to talk about is one of logic’s main characteristics—and mathematics’. Abstractness. Abstraction.
It is worth getting straight on the meaning of abstraction. It’s maybe the single most important word for appreciating Cantor’s work and the contexts that made it possible. Grammatically, the root form is the adjectival, from the L. abstractus = ‘drawn away.’ The O.E.D. has nine major definitions of the adjective, of which the most apposite is 4.a.: “Withdrawn or separated from matter, from material embodiment, from practice, or from particular examples. Opposed to concrete.” Also of interest are the O.E.D.’s 4.b., “Ideal, distilled to its essence,” and 4.c., “Abstruse.”
Here is a quotation from Carl B. Boyer, who is more or less the Gibbon of math history³ : “But what, after all, are the integers? Everyone thinks that he or she knows, for example, what the number three is—until he or she tries to define or explain it.” W/r/t which it is instructive to talk to 1st- and 2nd-grade math teachers and find out how children are actually taught about the integers. About what, for example, the number five is. First they are given, say, five oranges. Something they can touch or hold. Are asked to count them. Then they are given a picture of five oranges. Then a picture that combines the five oranges with the numeral ‘5,’ so they associate the two. Then a picture of just the numeral ‘5’ with the oranges removed. The children are then engaged in verbal exercises in which they start talking about the integer 5 per se, as an object in itself, apart from five oranges. In other words they are systematically fooled, or awakened, into treating numbers as things instead of as symbols for things. Then they can be taught arithmetic, which comprises elementary relations between numbers. (You will note how this parallels the ways we are taught to use language. We learn early on that the noun ‘five’ means, symbolizes, the integer 5. And so on.)
Sometimes a kid will have trouble, the teachers say. Some children understand that the word ‘five’ stands for 5, but they keep wanting to know 5 what? 5 oranges, 5 pennies, 5 points? These children, who have no problem adding or subtracting oranges or coins, will nevertheless perform poorly on arithmetic tests. They cannot treat 5 as an object per se. They are often then remanded to Special Ed Math, where everything is taught in terms of groups or sets of actual objects rather than as numbers “withdrawn from particular examples.”4
The point: ‘Abstract’’s basic def. for our purposes is going to be the somewhat concatenated ‘removed from or transcending concrete particularity, sensuous experience.’ Used in just this way, ‘abstract’ is a term from metaphysics. Implicit in all mathematical theories, in fact, is some sort of metaphysical position. The father of abstraction in mathematics: Pythagoras. The father of abstraction in metaphysics: Plato.
The O.E.D.’s other defs. are not irrelevant, though. Not just because modern math is abstract in the sense of being extremely abstruse and arcane and often hard to even look at on the page. Also essential to math is the sense in which abstracting something can mean reducing it to its absolute skeletal essence, as in the abstract of an article or book. As such, it can mean thinking hard about things that for the most part people can’t think hard about—because it drives them crazy.
All this is just sort of warming up; the whole thing won’t be like this. Here are two more quotations from towering figures. M. Kline: “One of the great Greek contributions to the very concept of mathematics was the conscious recognition and emphasis of the fact that mathematical entities are abstractions, ideas entertained by the mind and sharply distinguished from physical objects or pictures.” F.d.l. Saussure: “What has escaped philosophers and logicians is that from the moment a system of symbols becomes independent of the objects designated it is itself subject to undergoing displacements that are incalculable for the logician.”
Abstraction has all kinds of problems and headaches built in, we all know. Part of the hazard is how we use nouns. We think of nouns’ meanings in terms of denotations. Nouns stand for things—man, desk, pen, David, head, aspirin. A special kind of comedy results when there’s confusion about what’s a real noun, as in ‘Who’s on first?’ or those Alice in Wonderland routines—‘What can you see on the road?’ ‘Nothing.’ ‘What great eyesight! What does nothing look like?’ The comedy tends to vanish, though, when the nouns denote abstractions, meaning general concepts divorced from particular instances. Many of these abstraction-nouns come from root verbs. ‘Motion’ is a noun, and ‘existence’; we use words like this all the time. The confusion comes when we try to consider what exactly they mean. It’s like Boyer’s point about integers above. What exactly do ‘motion’ and ‘existence’ denote? We know that concrete particular things exist, and that sometimes they move. Does motion per se exist? In what way? In what way do abstractions exist?
Of course, that last question is itself very abstract. Now you can probably feel the headache starting. There’s a special sort of unease or impatience with stuff like this. Like ‘What exactly is existence?’ or ‘What exactly do we mean when we talk about motion?’ The unease is very distinctive and sets in only at a certain level in the abstraction process—because abstraction proceeds in levels, rather like exponents or dimensions. Let’s say ‘man’ meaning some particular man is Level One. ‘Man’ meaning the species is Two. Something like ‘humanity’ or ‘humanness’ is Level Three; now we’re talking about the abstract criteria for something qualifying as human. And so forth. Thinking this way can be dangerous, weird. Thinking abstractly enough about anything . . . surely we’ve all had the experience of thinking about a word—‘pen,’ say—and of sort of saying the word over and over until it ceases to denote; the very strangeness of calling something a pen begins to obtrude on the consciousness in a creepy way, like an epileptic aura.
As you probably know, much of what we now call analytic philosophy is concerned with Level Three– or even Four–grade questions like this. As in epistemology = ‘What exactly is knowledge?’; metaphysics = ‘What exactly are the relations between mental constructs and real-world objects?’; etc.’5 It might be that philosophers and mathematicians, who spend a lot of time thinking (a) abstractly or (b) about abstractions or (c) both, are eo ipso rendered prone to mental illness. Or it might just be that people who are susceptible to mental illness are more prone to think about these sorts of things. It’s a chicken-and-egg question. One thing is certain, though. It is a total myth that man is by nature curious and truth-hungry and wants, above all things, to know.6 Given certain recognized senses of ‘to know,’ there is in fact a great deal of stuff we do not want to know. Evidence for this is the enormous number of very basic questions and issues we do not want to think about abstractly.
Theory: The dreads and dangers of abstract thinking are a big reason why we now all like to stay so busy and bombarded with stimuli all the time. Abstract thinking tends most often to strike during moments of quiet repose. As in for example in the early morning, especially if you wake up slightly before your alarm goes off, when it can suddenly and for no reason occur to you that you’ve been getting out of bed every morning without the slightest doubt that the floor would support you. Lying there now considering the matter, it appears at least theoretically possible that some flaw in the floor’s construction or its molecular integrity could make it buckle, or that even some aberrant bit of quantum flux or something could cause you to melt right through. Meaning it doesn’t seem logically impossible or anything. It’s not like you’re actually scared that the floor might give way in a moment when you really do get out of bed. It’s just that certain moods and lines of thinking are more abstract, not just focused on whatever needs or obligations you’re going to get out of bed to attend to. This is just an example. The abstract question you’re lying there considering is whether you are truly justified in your confidence about the floor. The initial answer, which is yes, lies in the fact that you’ve gotten out of bed in the morning thousands—actually well over ten thousand times so far, and each time the floor has supported you. It’s the same way you’re also justified in believing that the sun will come up, that your wife will know your name, that when you feel a certain set of sensations it means you’re getting ready to sneeze, & c. Because they’ve happened over and over before. The principle involved is really the only way we can predict any of the phenomena we just automatically count on without having to think about it. And the vast bulk of daily life is composed of these sorts of phenomena; and without this kind of confidence based on past experience we’d all go insane, or at least we’d be unable to function because we’d have to stop and deliberate about every last little thing. It’s a fact: life as we know it would be impossible without this confidence. Still, though: Is the confidence actually justified, or just highly convenient? This is abstract thinking, with its distinctive staircase-shaped graph, and you’re now several levels up. You’re no longer thinking just about the floor and your weight, or about your confidence re same and how necessary to basic survival this kind of confidence seems to be. You’re now thinking about some more general rule, law, or principle by which this unconsidered confidence in all its myriad forms and intensities is in fact justified instead of just being a series of weird clonic jerks or reflexes that propel you through the day. Another sure sign it’s abstract thinking: You haven’t moved yet. It feels like tremendous energy and effort is being expended and you’re still lying perfectly still. All this is just going on in your mind. It’s extremely weird; no wonder most people don’t like it. It suddenly makes sense why the insane are so often represented as grabbing their head or beating it against things. If you had the right classes in school, however, you might now recall that the rule or principle you want does exist—its official name is the Principle of Induction. It is the fundamental principle of modern science. Without the Principle of Induction, experiments couldn’t confirm a hypothesis, and nothing in the physical universe could be predicted with any confidence at all. There could be no natural laws or scientific truths. The P.I. states that if something x has happened in certain particular circumstances n times in the past, we are justified in believing that the same circumstances will produce x on the (n+1)th occasion. The P.I. is wholly respectable and authoritative, and it seems like a well-lit exit out of the whole problem. Until, that is, it happens to strike you (as can occur only in very abstract moods or when there’s an unusual amount of time before the alarm goes off) that the P.I. is itself merely an abstraction from experience . . . and so now what exactly is it that justifies our confidence in the P.I.? This latest thought may or may not be accompanied by a concrete memory of several weeks spent on a relative’s farm in childhood (long story). There were four chickens in a wire coop off the garage, the brightest of whom was called Mr. Chicken. Every morning, the farm’s hired man’s appearance in the coop area with a certain burlap sack caused Mr. Chicken to get excited and start doing warmup-pecks at the ground, because he knew it was feeding time. It was always around the same time t every morning, and Mr. Chicken had figured out that
t (man + sack) = food, and thus was confidently doing his warmup-pecks on that last Sunday morning when the hired man suddenly reached out and grabbed Mr. Chicken and in one smooth motion wrung his neck and put him in the burlap sack and bore him off to the kitchen. Memories like this tend to be remain quite vivid, if you have any. But with the thrust, lying here, being that Mr. Chicken appears now actually to have been correct—according to the Principle of Induction—in expecting nothing but breakfast from that (n+1)th appearance of man + sack at t. Something about the fact that Mr. Chicken not only didn’t suspect a thing but was apparently wholly justified in not suspecting a thing—this seems concretely creepy and upsetting. Finding some higher-level justification for your confidence in the P.I. seems much more urgent when you realize that, without this justification, our own situation is basically indistinguishable from that of Mr. Chicken. But the conclusion, abstract as it is, seems inescapable: What justifies our confidence in the Principle of Induction is that it has always worked so well in the past, at least up to now. Which would seem to mean that our only real justification for the Principle of Induction is the Principle of Induction, which seems shaky and question-begging in the extreme.
The only way out of the potentially bedridden-for-life paralysis of this last conclusion is to pursue further abstract side-inquiries into what exactly ‘justification’ means and whether it’s true that the only valid justifications for certain beliefs and principles are rational and noncircular. For instance, we know that in a certain number of cases every year cars suddenly veer across the centerline into oncoming traffic and crash head-on into people who were driving along not expecting to get killed, and thus we also know, on some level, that whatever confidence lets us drive on two-way roads is not 100% rationally justified by the laws of statistical probability. And yet ‘rational justification’ might not apply here. It might be more the fact that, if you cannot believe your car won’t suddenly get crashed into out of nowhere, you just can’t drive, and thus that your need/desire to be able to drive functions as a kind of ‘justification’ of your confidence.7 It would be better not to then start analyzing the various putative ‘justifications’ for your need/desire to be able to drive a car—at some point you realize that the process of abstract justification can, at least in principle, go on forever. The ability to halt a line of abstract thinking once you see it has no end is part of what usually distinguishes sane, functional people—people who when the alarm finally goes off can get hit the floor without trepidation and plunge into the concrete business of the real workaday world—from the unhinged.
1 IYI [If You're Interested] Although so is the other, antipodal, stereotype of mathematicians as nerdy little bowtied fissiparous creatures. In today’s archetypology, the two stereotypes seem to play off each other in important ways.
2 In modern medical terms, it’s fairly clear that G.F.L.P. Cantor suffered from manic-depressive illness at a time when nobody knew what this was, and that his polar cycles were aggravated by professional stresses and disappointments, of which Cantor had more than his share. This is, of course, makes for less interesting flap copy than Genius Driven Mad By Attempts To Grapple With ∞. The truth, though, is that Cantor’s work and its context are so totally interesting and beautiful that there’s no need for breathless Prometheusizing of the poor guy’s life. The real irony is that the view of ∞ as some forbidden zone or road to insanity—which view was very old and powerful and haunted math for 2000+ years—is precisely what Cantor’s own work overturned. Saying that ∞ drove Cantor mad is sort of like mourning St. George’s loss to the dragon: it’s not only wrong but insulting.
3 IYI Boyer is joined at the top of the math-history food chain only by Prof. Morris Kline. Boyer and Kline’s major works are respectively A History of Mathematics and Mathematical Thought from Ancient to Modern Times. Both books are extraordinarily comprehensive and good and will be liberally cribbed from.
4 B. Russell has an interesting ¶ in this regard about high-school math, which is usually the next big jump in abstraction after arithmetic:
In the beginning of algebra, even the most intelligent child finds, as a rule,
very great difficulty. The use of letters is a mystery, which seems to have
no purpose except mystification. It is almost impossible, at first, not to
think that every letter stands for some particular number, if only the teacher
would reveal what number it stands for. The fact is, that in algebra the
mind is first taught to consider general truths, truths which are not asserted
to hold only of this or that particular thing, but of any one of a whole group of
things. It is in the power of understanding and discovering such truths that
the mastery of the intellect over the whole world of things actual and possible
resides; and ability to deal with the general as such is one of the gifts that a
mathematical education should bestow.
5 IYI According to most sources, G.F.L.P. Cantor was not just a mathematician—he had an actual Philosophy of the Infinite. It was weird and quasi-religious and, not surprisingly, abstract. At one point Cantor tried to switch his U. Halle job from the math dept. to philosophy; the request was turned down. Admittedly, this was not one of his stabler periods.
6 IYI The source of this pernicious myth is Aristotle, who is in certain respects the villain of our whole Story.
7 A compelling parallel here is the fact that most of us fly despite knowing that a definite percentage of commercial airliners crash every year. This gets into the various different kinds of knowing v. ‘knowing,’ though. Plus it involves etiquette, since commercial air travel is public and a kind of group confidence comes into play. This is why turning to inform your seatmate of the precise statistical probability of your plane crashing is not false but cruel: you are messing with the delicate psychological infrastructure of her justification for flying.
IYI Depending on mood/time, it might strike you as interesting that people who cannot summon this strange faith in principles that cannot be rationally justified, and so cannot fly, are commonly referred to as having an ‘irrational fear’ of flying.
Excerpted from the forthcoming Everything and More by David Foster Wallace. Copyright (c) 2003 by David Foster Wallace. With permission of Atlas Books-W. W. Norton & Company, Inc. Publication date October 13, 2003.