Everything & More & More

August 26, 2021

EVERYTHING AND MORE: A SHORT HISTORY OF INFINITY by DAVID FOSTER WALLACE

a review by BALCKWELL

For a man who refused to learn anything in high school mathematics classes, I sure have become someone who enjoys learning about mathematics. I don't enjoy learning mathematics itself - every time I begin an online algebra course I get stuck as soon as I have to remember how to divide fractions - but I enjoy learning about the process of mathematical discovery, and the moments when mathematics eats itself and explodes. In the last two hundred years, with the rise of number theory, set theory, analytics and etc., there have been plenty of such moments.

What separates mathematicians from the common person is, in a word, rigor. You really can't go halfway with this stuff. Every step has to accord with the last, and follow a strict set of rules. There is room for creativity and fanciness, but it all has to check out, in the end. This slow, methodical approach flies in the face of the styles of thinking common to the rest of us, which can be described essentially as flying by the seat of our pants.

According to the book I am currently reviewing, during the advent of what we now call physics circa the 18th c., mathematicians got a little wrapped up in the worldly world and abandoned their rigorous manipulation of abstract conceptions, instead discovering mechanical laws and being content to say: "Well, it seems to work." (Note that this is an oversimplification of what is already an oversimplification in the text.) This eventually led a contraction period, during which people such as Georg Cantor stepped in to actually prove what everyone else had merely conjectured.

Cantor is the primary protagonist of this book. The point of the book is to guide you toward the proofs that he invented/discovered regarding the place of infinities within mathematics. Infinitely large or small numbers, it turns out, are actually quite useful, as evidenced by the fact that people had been surreptitiously using them to do things like invent/discover calculus for a while before Cantor even showed up.

Everything & More is not a "popular-mathematics" book, if such a thing can be said to exist. It is not non-technical. This is because you can't talk about what he is talking about without at least getting semi-technical. You can skim and simplify as much as you like; in the end, we are still talking about infinities here, and infinities not just in the sense of "really big things," but actually grappling with and performing mathematical operations using infinities. Which, as is seen in the book, is an activity that is likely to heat up your brain and then bend it like a metal beam.

The book makes several assumptions about its readers, mostly regarding their level of mathematical education, all of which are untrue for me except perhaps the most important one: The assumption that the reader will be interested enough to struggle through dense technical explanations in order to reach "the real potatoes.". Even the simplest proof requires at least a second read to penetrate my brain's anti-math defenses, which made parts of this book slow-goings indeed. And in the end, it's hard to say how much of the mathematics itself will actually stick with me.

I suppose I read these books not to learn about math so much as to learn about mathematicians. They are truly a different breed. They approach abstract philosophical questions with their heads screwed on tight, tackling them with a rationality that is hard for me to even conceptualize. They're out there in the long grass, but unlike me, they actually brought a scythe. They've got a job to do, is why they're out there. I'm just here on a picnic. Once I finish my tea and sandwich, I'm out of there. Heading back home. I'm just visiting, really. I think the long grass is sorta neat, is all. It never even occurred to me to cut it.

A concept that is brought up regularly throughout the book is one of Zeno's Paradoxes, specifically the one where he divides a certain distance into an infinite amount of tiny distances, and says that to cross any one of these infinitesimal distances, you would have to cross an infinite number of other, even "more" infinitesimal distances, thus rendering moving anywhere, in any direction at all, impossible. The reason that it is a paradox is because we can and do move places. But to say that it takes a finite time to move an infinite amount of distances is circumspect, to say the least.

I've encountered this paradox before. Henri Bergson uses it in his book "Matter and Memory," his conclusion being that such paradoxes should deter us from considering time and space as mathematical continuums, thus freeing himself to posit (if I remember correctly) that memory is not, in fact, contained within the organ known as our brains, but exists outside of space in some sort of time-space (not space-time) that we reach by expanding our consciousness beyond the present moment. (Bergson, by the way, totally rules.) Suffice to say, actual mathematicians see the matter slightly differently.

This example perhaps highlights the difference between philosophy and mathematics. Henri Bergson is a perfect example of a philosopher who couches emotional appeals in the trappings of logic. It's convincing, at the time of reading, and even somewhat rigorous, but it is much more inclined to fanciful leaps than mathematics.

You may then wonder at the point of philosophy, with its reputation for abritrary answers to unanswerable questions. There are many responses to such a wonder, but for me, I would say the main benefit of reading such texts is a brain-explosion.

I almost said brain-expansion, but that makes it sound like I'm getting quantitatively smarter, which I assure you is definitely not the case. Instead, I receive an explosion in my head, which destroys my ability to think about the world properly, and throws me helpless and screaming into a void. Philosophy is not about learning; it is about forgetting. It is akin to taking a psychedelic drug and losing your ego. The reference points that carry you through everyday life are taken away, and you are forced to see the world at its bare components, which vary depending on the individual philosopher. The reason why philosophical question sound so stupid is because they tend to have simple, logical answers; answers which you have to consciously avoid thinking about, because we're going past the realm of simplicity and logic, and entering directly into the bizarre brain-space of the weirdest people you could possibly imagine.

Philosophy, in a certain respect, is an attempt to use language (either natural or logical) in order to reach conclusions that are unintuitive and borderline crazy. A good philosophical work will carry you from the world of sense into a fantasy world of abstract concepts, and anything you learn inside is almost impossible to carry back. The quest (for the writer) is to guide you, step by step, into a trap. Once you are caught, you can look back and wonder, "Where did I lose my way?" but (and this depends on the skill of the philosopher) you likely won't be able to figure it out. And now you are stuck with this thought that, "logically," space and time do not exist. And that doesn't feel right, because a few seconds just went by, and you just walked from the bathroom to the kitchen.

Obviously, you can't really live your life under the impression that space and time don't truly exist. It's not exactly possible. But, for a brief moment, or moments, you were brought into a mind-space where, as far as your logical mind was concerned, space and time didn't exist. And that's a beautiful, weird place to be, a place that you probably couldn't get to on your own. Just knowing that such a place (I'm using the word "place" metaphorically here) exists leaves a certain imprint on your mind. The shrapnel from the explosion never really goes away.

Philosophy is better for the imagination than any work of fiction. Fiction is second-hand imagination; it can, but often doesn't, expand your view of the world itself. It presents to you a situation, and then tells you what happens. The most you may learn is, "Ah, such a thing could happen" or "such a person could exist." A philosophy book on the other hand takes you into a depraved weird person's mind, teaches you how they think, forces you to think that way, and then leaves you there to figure out what to do next. These books leave me reeling for days, or weeks.

Reading books is often associated, especially in modern times, with the accumulation of information. You read because you want to learn things, and therefore, the more you read, the more things you know. This association is not altogether inaccurate; books do, in fact, contain information that you can learn. Many people see reading as adding data to a hard drive, that can later be accessed for maximal gain and profits. I prefer to think that reading expands my RAM, allowing me to perform more interesting "lateral" brain movements.

...Okay, I don't know how computers work. For some reason, whenever I attempt to concoct a metaphor, the first comparison that comes to mind is often hard drives and RAM, despite the fact that I don't particularly know what RAM does, other than that I need a lot of it to play the hottest new computer games.

Let's say this: Reading is fun. I like the way my brain feels when I am reading. That's why I do it so much. I'm not trying to become smart; at least, not in any quantitative sense. I am not trying to build up a wellspring of information that I can use for purposes. Most of what I read just falls straight out of my head at the nearest opportunity, and even that which sticks with me is not often useful in any practical sense. I can't even use it to impress anyone anymore. (Everyone I talk to knows me too well.)

When I read about mathematics, I do so in the same way I might read about, say, medieval Europe. I'm not going to go out and do math, in the same way I'm not about to run off to medieval Europe. I'm just trying to get a different angle on the world. These people (mathematicians) exist, and they use the same tools I do (a brain/nervous system) to look at this world, and yet our interpretations are wildly different. I am averse to systematization and formalization; every thought of mine exists in a fundamentally indescribable relation to all others.

When I started this book, I was inclined to look at it in my usual way, and take it as a given that certain conclusions would for all intents and purposes come out of nowhere. I did not expect to be led, kicking and screaming, from axiom to theorem to theorem as concepts are literally proved to me to be correct (not in a strict mathematical sense, I suppose, but in the baby-math way that is required to speak to people like me.) I wanted to yell, I wanted to protest, but in the end it all comes off as fairly logical. Maybe I've just been duped harder than ever before, but I would downright say it makes sense that transfinite numbers exist. And I can even say that I understand why Cantor felt it necessary to prove their existence.

I will never understand the depths and complexities of number theory, or set theory, or even game theory. I used to think that this was due to the fact that all my forays into such topics occurred when I was drunk. I now understand that there is no adding or taking away of drugs that can change my physiology enough to make these things make sense to me. That's alright. I also don't know much about birds; but hey, I can at least tell you the names of some of them.

I am a man of jumping. I am a man of lateral movements. I do not dig deep into the Earth, for I am afraid of its core. It's hot down there. Sometimes, I feel that I belong among the stars, although I suppose up there I'd have the opposite problem. Perhaps I belong just slightly above where I currently am:

Thus, the journey continues:

BALCKWELL RISING!