David Foster Wallace. Everything and More: A Compact History of Infinity. New York: Norton, 2003. Print.
Having enjoyed and admired Infinite Jest, I had to read what its author would say about the concept of infinity. Everything and More keeps the reader’s interest. It is not what I expected, but that is OK. By the way, I spell out the word infinity in most cases in this review, except for a couple of equations. Wallace frequently uses the symbol for infinity, the lemniscate. Even the book’s subtitle is written A Compact History of ∞.
Everything and More is truly a mathematics-based history of the concept of infinity, starting with the ancient Greeks. This work is a combination of mathematics and philosophy. Wallace frequently quotes a high school AP math teacher of his. One of my high school math teachers actually majored in Philosophy and Mathematics at Harvard. If Mr. Galvin is still around, I am sure he would enjoy this book—if he has not already read it.
I will be honest. I did take math in college up to theoretical calculus, but I have not used much of the higher math I studied since taking those classes, so I did skim over some of the more formulaic parts. For any serious mathematicians, there is an erratum at http://www.thehowlingfantods.com/dfw/images/enmerrata.pdf. However, the philosophical parts were fascinating, and I think I got the gist of the main arguments.
Wallace notes that it was the Greeks who, as far as we know, were the first people to look at numbers a potential abstractions. For example, the Egyptians and Babylonians knew about a 3-4-5 right triangle and used them to form and measure right angles. However, it was the Greek Pythagoras who showed the relationship among the sides by the theorem that still bears his name. Prior to some of those Greeks, numbers always stood for something—as Wallace puts it, five meant “five of something” like, say, five oranges. I guess one could say that numbers were adjectives that the Greeks began to see as nouns.
So they began looking at numbers as numbers. Indeed, Wallace asserts that Platonists and Aristotelians had a different view of number. The Platonists would see a number as a form or ideal. Aristotelians saw them as representing or describing something in the physical world. Aristotle dismissed the concept of infinity because nothing in the material world is infinite. Wallace has some fun with some ridiculously small and large numbers to illustrate that even the smallest measurable division of time or the number of electrons in the universe may be numbers so large or tiny as to be unimaginable, but they are still not infinite.
The Greeks were made aware of infinity largely through the infinitesimal, namely Zeno’s Paradox. Most students who have finished junior high math have heard of it. If you keep going halfway, you will never cross the street, yet how can two halves make a whole? What Zeno was suggesting was that between any two integers there are an infinite number of numbers.
As with so many things, the teachings of Aristotle held sway for about a millennium and a half (some still do). Yet people were aware not only of Zeno’s Paradox but others as well. For example, the invention of calculus brought a kind of corollary to Zeno—that in any given position or moment of time an object is at rest, so how does one account for motion?
Even the ancient Greeks had an idea that rational numbers (i.e., numbers that can be expressed as a ratio) do not account for all the numbers. Thanks to Pythagoras, they saw that the diagonal of a square is the square root of two [√ 2 ], which cannot be expressed as a ratio. Neither can pi [π]. So to have continuity on the number line—or even to account for motion mathematically—one has to account for every point, and there are an infinite number of points between any two rational points on a line.
Galileo came up with his own paradox: Even though there are many more numbers that are not perfect squares, when dealing with all the integers (an infinite number of integers) the number of integers and perfect squares are the same because every integer can be squared. Wallace explains this very clearly.
Everything and More focuses on Georg Cantor, the nineteenth century mathematician who developed much of modern set theory and, in doing so, was able to answer many of the questions people had about infinity such as the two paradoxes mentioned here. Unlike some of the calculations in the book, Wallace explains very clearly why there are different infinities. Although there are an infinite number of integers or squares or rational numbers, there are many more irrational numbers. This means that the set of real numbers is a degree of infinity greater than the set of integers or rational numbers.
Cantor used the Hebrew letter aleph [א]to designate an infinite set. A greater infinite set would then be designated by an aleph with a numerical subscript, with the first level aleph being aleph sub zero [א0] corresponding to the set of rational numbers. So aleph sub one [א1] would describe the set of real numbers, which corresponds to a line of one dimension. Since then, people have shown that there are sets of infinities based on dimensions. Yes, there are an infinite number of points on a plane, but that is a degree of infinity greater so that becomes aleph sub two [א2]. Three dimensional space has aleph sub three [א3], and so on. Other mathematicians have made a case that these different alephs can be expressed as two to the aleph power of the aleph that precedes it, so א1=2א0, א2=2א1, and so on. I think I presented this correctly.
Going back to Zeno, this “solves” the problems of continuity and motion mathematically from the perspective of number theory. Aristotle may have been correct saying that in the physical universe there are not an infinite number of anything, but when we observe motion and measurements, we take such an idea into account. This also becomes a topic of discussion in this book and among math scholars: Is induction a valid technique for finding what is true in mathematics or should we, like Euclid, stick to deduction?
Because Infinite Jest was a wonderfully funny work of speculative fiction, I expected Everything and More to be more speculative than it was. Of course, because its topic is infinity, a certain amount of speculation is unavoidable. Still, it was mostly history. And Wallace desires, though cannot quite bring himself, to agree with the professor in the Narnia stories who explains, “It’s all in Plato. What do they teach in the schools these days?”
Ironically, for someone who is contemplating infinity, Wallace takes a narrow view of things. He asserts sadly, “That our thoughts and feelings are really just chemical transfers in 2.8 pounds of electrical pate.” (22) Well, as our last review notes, the mind and the brain are not identical. Similarly, he trivializes love as “a function of natural selection.” (23) How drearily mechanistic. How much like Roger Chillingworth!
Everything and More dismisses, for example, Aquinas’s speculation (and the old saint’s only disagreement with Aristotle) about infinity—that God is infinite and that eternity is infinite in time.
I confess that if I were writing about infinity, that is what I would be speculating about. Take one simple example I share with my students when we study Tom Stoppard. We know from the Second Law of Thermodynamics that the universe is winding down. Stoppard tells us “the future is disorder” and some day “heat is gone from the earth.” However, eternity is different. Eternity means unlimited energy forever. I can prove it mathematically.
We learned about a hundred years ago that the inherent energy of something is its mass times the square of the speed of light, or E=mc2. Eternity is timeless. That means that speed, which is distance divided by time (d/t in math class) is infinite in eternity. Instead of c=186,000 miles/second, in a timeless environment the speed of anything, light and everything else, is the distance over zero because there is no time in eternity. Any number divided by zero is either nonsense or infinity. Wallace makes a case that any real number divided by infinity is zero [e.g., 1/0=∞]. If eternity exists, that means E equals m times infinity [E=m·∞], so E=infinity [E=∞], so eternity has infinite energy. That helps explain creation, miracles, and other things we might consider supernatural. It is a thought.
Everything and More is very well written. Wallace is first and foremost a story teller. Though nonfiction, this book tells a story in a pretty effective way even it its speculation is materialistic, something Infinite Jest starts with but does not end with. Indeed, the last three sentences of Everything and More suggest such things as I speculated on in the last paragraph, but instead of seeing eternity, Wallace saw the “Void.” (305, his capital) Alas.
P.S. The copy of this book that I obtained is the original edition. In 2013 an edition of Everything and More came out with a preface by Neal Stephenson. Stephenson is also a very clever speculative writer. It would be interesting to see what he had to say about this book, but that will have to wait.