Everything and More - Book Review

This is a pop-science book on math, a genre I quite enjoy as someone deeply interested in these topics, but without a formal grounding in them. Usually, the author is an eminent physicist, mathematician, or occasionally a very well-informed journalist. To read a book on math by a well-known fiction writer and essayist was a departure, albeit one I enjoyed.

I will admit upfront that I didn’t know a lot about David Foster Wallace going into this. Yes, I’d heard Infinite Jest was a book many people loved, but it also sounded like a lot of work, a post-modern Ulysess, and I haven’t had a lot of time for that sort of thing the last two decades. But I do enjoy good writing, and increasingly I am seeking out real human voices amidst the AI slop drowning our society.

I’ve read past essays by DFW and found them interesting. And now that I’ve read more of him I can wrap my head around his particular style: witty, lots of tangents often stuffed into pages of footnotes, a mind that races around and has lots of facts and things to say, albeit one that is clearly unquiet. He also rose to my radar back in 2018 around the #MeToo movement, which I guess we’ve all moved on from, given our current political climate, and so that was a further reason to not dive into his head for fiction.

But, but, a good friend read the book and recommended it. My father-in-law, who has a master’s in mathematics, read it and found things to enjoy. And so I decided to dive in. (I realize in re-reading this piece that I’ve done the most DFW-like of intros here, touching upon the main point while dropping a lot of words and loosely-connected points along the way, without really getting to the review until now.)¹

So on to the book itself. I will give DFW credit for writing what is a surprisingly hardcore book on math. He does not shy away from details, equations, or the like. This is decidedly not a book on theoretical physics that fails to feature a single equation. Instead, it is DFW sharing the wonders of modern math, its rich history, and offering many insights, mixed with some truly deep dives into the underlying maths.

I am fond of marking up my books, usually in pencil, to highlight thoughts or facts I want to revisit later while skimming. I found myself writing a ton in this book, which is a compliment of sorts, because DFW is unrelenting with the connections and absurdities he points out in modern math, of which there are many.²

Take, for example, this given page, which I thumbed to at random. It has, of course, a footnote in the “If You’re Interested (IFI)” abbreviation that he uses throughout. This is a useful technique for threading the needle with a book intended for a wide audience. A lay reader with little formal math training will find such tangents overwhelming and perhaps less interesting, whereas someone with more mathematical background can enjoy them, as indeed DFW himself does.

I found myself reading all the footnotes and enjoying most of them. Perhaps I’m flattering myself here, as I don’t have a strong formal math background, but I’ve always enjoyed the subject, and in the alternate-timeline of things, if I could redo college or even go back as an older adult and just be a student again, I suspect math and physics are what I would study.

So why does infinity deserve an entire book to itself, and why does it matter? Isn’t infinity, by definition, something that we can’t really define? It turns out that infinity was the canary in the coalmine for much of modern mathematics, something touched upon and noted by the Greeks and others, but viewed as a source of paradox and absurdity, rather than an absolute to be integrated into the whole.

DFW does an admirable job of highlighting the various means of abstraction implicit in mathematical thought. The first stage of math is counting, as anyone with children can attest, but how do you make the leap to proofs and more abstract ideas about why something works rather than how?

There was quite a bit of math worked out by the Babylonians and early Greeks, culminating in Euclid’s Elements from around 300 BC, that has definitions, axioms, theorems, and proofs. But math as we know it was largely stagnant until the late 1700s, and one factor in this was an inability to resolve the question of infinity.

DFW spends quite a while on Zeno and Bertrand Russell, quoting Russell at some length on the “founder of the philosophy of infinity [which is to say, Zeno].” For even 2,000 years ago, Zeno was winning philosophical debates with his various postulates. The most famous is the idea that any distance, no matter how small, can be divided into two smaller distances forever. For example, if you want to cross the road but can only get there by navigating half the distance at a time, in one sense, you’ll never arrive. And yet, in the real world, we know it’s quite possible to do so.

Calculus addressed these assumptions by formalizing the concept of “limits” and allowing mathematicians to calculate and verify solutions to problems in engineering, physics, astrology, and other fields. Calculus emerged in the late 17th century through the efforts of Newton and Leibniz (DFW wisely avoids delving into the deeper history of who was first), but there were roughly another 200 years before Georg Cantor emerged, the father of modern infinity and set theory.

Here is what DFW has to say:

“The situation of mathematics after 1700 is intensely weird […] mathematical discoveries enabled scientific advances, which themselves motivated further math discoveries. [This] created for math a situation that resembled a tree with great lush proliferant systems of branches but no real roots. There were still no grounded, rigorous definitions of the differential, derivative, integral, limit, or convergent/divergent series. Not even of the function. There was constant controversy, and yet at the same time nobody seemed to care.” David Foster Wallace - Everything and More, §5a

Unless you take college-level math, it is unlikely you’ve even heard of Cantor, even though his ideas around infinity and set theory underpin much of modern mathematics, including real analysis, topology, functional analysis, probability, and more.

His revolution/revelation was that there are different sizes of infinity. Infinity is not one single thing, but rather a stack of infinites. Another way to think of this is a measure of size without counting, which leads to set theory.

Consider one set of infinity to be the set of all integers, which means every whole number, positive, negative, or zero stretching in both directions forever. In mathematical notation that looks like this: Z={…,−5,−4,−3,−2,−1,0,1,2,3,4,5,…} where Z is just a symbol for the set of all integers.

Now consider a different set which is just positive integers, often represented by N for natural numbers: N={1,2,3,4,5,…}.

Which infinity is bigger do you think? N or Z? On first glance, you might say, well infinity is infinity, so they are the same. Or you might posit, well N has more numbers in it since it includes zero and negative integers, therefore it must be bigger.

So what’s the answer? It turns out that N and Z are the same size of infinity because they are both countably infinite, meaning it is possible to have a one-to-one pairing between them. Even though Z looks bigger because it contains negative numbers and zero, you can match them all up like this: 1↔0,2↔1,3↔−1,4↔2,5↔−2,6↔3,7↔−3,…

Every natural number matches exactly one integer. Therefore, using vertical bars to indicate the size of a set: ∣𝑁∣ = ∣𝑍∣ = ℵ₀

In words, the size of set N is the same as the size of set Z which is known as ℵ₀, countable infinity, or Aleph-zero. They are the same size–known as cardinality with sets–even though they are not equal.³

You might be wondering about Aleph numbers. Is there an Aleph-One (N₁)? It turns out there is and it has a strict definition: ℵ₁ (“aleph-one”) is the smallest infinite size that is strictly larger than the natural numbers.

Aleph-Two (ℵ₂) is the next smallest cardinal number larger than ℵ₁, then Aleph-Three, Aleph-Four and so on.

One more wild thing to consider: the size/cardinality of real numbers between 0 and 1 (such as 0.1, 0.001, 0.000001, and onwards, you get the idea) is a bigger kind of infinity than all integers counting from 0, 1, 2…. In fact, this type of infinity has a strict definition, c (continuum).

Cantor proved way back in 1874 that: ∣(0,1)∣>∣N∣. This small interval on the number line actually contains more numbers than all integers put together and represents a fundamentally larger infinity.

Even though we can’t count infinities, we can compare their sizes. There isn’t one type of infinity, but rather a never-ending tower of infinities.

I can’t help but think of modern parallels here, as computing is rife with such inconsistencies, illogical areas, and now with LLMs, a whole universe of things we use but don’t fully understand.