An Innocent and Impossible Question
Every Labor Day weekend, my mother's siblings get together for an annual family reunion in the beautiful mountains of Wheeling, West Virginia. I went for the first time in four years last September, in 2015. In those four years I was an undergrad at Hamilton College, a private liberal arts college in western upstate NY that doesn't believe in three-day weekends, so while I was at Hamilton, I was never able to attend this reunion. When I went last September, that was the first time I had seen many of my relatives since I’d decided to take on mathematics as a career. Now, being a good graduate student, I decided to take a couple notebooks with me so I could study during our down-time.
I have one cousin in particular. Her name's Olivia. Olivia joined our family around when I started school at Hamilton, so this weekend was the first time I'd ever met her in person. (At least some of my friends just read that and went, "Oh God, Dom. What did you do?") So I'm being my friendly self and trying to make a good impression, when she asked me at one point if she could take a look at one of my notebooks. I gave her my linear algebra notebook to leaf through. Seconds later, with an expression of shock, she showed it to one of our aunts and cried out,
"This doesn't have any numbers in it! And look, this entire paragraph doesn't have any numbers OR words in it!"
I use a ton of symbolic logic shorthand in my notes. This was a while ago, but I think this was the paragraph she was referring to:
(I dunno, there are a couple words... and some 1s and 0s...)
As far as first impressions go, this was a weird one for me. So, Olivia, I feel like I owe you an apology, because you probably thought I was out of my mind. And... well... you were probably right.
Naturally, Olivia’s exclamation prompted my aunt to ask me a very specific question. It’s the same question that everyone asks once they learn that after a certain point in mathematics, you stop working with numbers. Without knowing what can of worms she was about to open, my aunt asked me the most difficult question for any mathematician to answer, regardless of age or experience.
“So, Dom… what exactly is mathematics?”
This is a dangerous question to ask a mathematician, because everyone thinks they know the answer but there’s no telling what answer they’ll give you. Surprisingly, there’s no universally accepted satisfactory answer to this seemingly simple question; in fact several mathematicians bitterly disagree about this. (Read: There’s a nonzero chance that this post is going to offend some mathematicians out there. I apologize in advance.)
“But Dom,” you may be asking, “if this is such a complex and controversial issue, why take it on now?” Well, because if I don’t, I’m worried that future posts won’t make sense. Many of the arguments I’d like to make on this blog stem from a premise of my beliefs that requires justification. The premise is this:
I believe that mathematics is, at its core, a naturally recurring entity, and would still exist in some respect even if there weren’t any humans around to do mathematics. (Whatever it means to “do mathematics;” there’s no satisfactory definition for that, either.)
Among others, there are at least two noteworthy views on the nature of mathematics. One is Platonism, and the other is Formalism. (There are many others, but I want to focus on those two.) Platonists believe that mathematics exists, on its own, independent of human thought, in a sense removed from physical reality. This comes from Plato’s belief in a second universe independent of our own, a perfect universe of ideals. Plato conjectured that a circle really does “exist;” when you draw a circle, you’re really only using your drawing to represent an ideal circle, which exists outside of our physical reality. That’s how several mathematicians (including me) think of numbers. In fact, that’s where the title of this blog comes from. This may sound a tad melodramatic, but I think it’s pretty accurate.
I believe in numbers in the same way that some people believe in angels.
Try defining the Number 2. You may be tempted to use the word “number” or a variation on the phrase “to have two of something.” Okay, but then what’s a “number,” and what does it mean to “have two of something?” Circular definitions aren’t allowed; you can’t say “to have two of something” means “to have a thing, and in particular, to have two of that thing.”
Go on. Try it. I’ll wait.
It’s not so easy, is it? Now, you can point to the symbol “2” drawn on a blackboard, but deep down, you know you’re not pointing to the Number 2; you’re pointing to an Arabic glyph representing the Number 2. When I visit a church and I see a painting of an angel, I know I’m not actually looking at a real angel; I’m looking at a representation of one. This is the heart of Platonism: I don’t know what a number is, but I believe that numbers exist, in some metaphysical sense. A similar argument can be made about geometric and topological objects like circles, spheres, tori, Möbius bands, and others.
Formalists, however, find this line of reasoning severely dissatisfactory. Formalism is the idea that mathematics can be reduced to a set of axioms, or basic laws, from which all other theorems may be proven. In the late 19th and early 20th century, the German mathematician David Hilbert (pictured right) initiated an astronomical project, titled the “Hilbert Program,” which was an effort by mathematicians around the world to ground all of mathematics in axiomatized logic—to “formalize” mathematics. Their objective was to prove all of the most elementary results, like additive commutativity (a.k.a. the rule that x + y = y + x no matter what numbers x and y you choose). Here’s an example: in 1889, an Italian logician named Giuseppe Peano (pronounced “piano”) published a list of nine axioms of natural number arithmetic, from which one can prove several critical and fundamental results, including additive commutativity. In other words, Peano found a way to “formalize” number theory. For those interested, Peano’s axioms—forming the basis of what’s now known as “Peano arithmetic”—are listed in simplified language below (only for the curious reader; the rest of this post will not refer to the specific axioms of Peano):
0 is a number.
Given any number x, it is always true that x = x.
Given any two numbers x and y, if x = y, then y = x.
Given any three numbers x, y, and z, if x = y and y = z, then x = z.
Given any number x, and an object y, if x = y, then y is also a number.
Given any number n, there is a successor to n, which is also a number. (In particular, the “successor” to the number n is n+1, but Peano wrote this as S(n).)
Given any numbers n and m, it is true that n = m if and only if n+1 = m+1 (or in Peano’s notation, if and only if S(n) = S(m)).
Given any number n, it is false that its successor S(n) is 0. (That is, we never have n+1 = 0; Peano was interested only in natural number arithmetic, so negative numbers were not allowed in this system.)
If K is a set of positive whole numbers for which the following two properties hold, then K is the set of all positive whole numbers:
0 is in K, and
whenever n is a number in K, the successor S(n), or n+1, is also a number in K.
For decades, Hilbert’s Program was the most ambitious collective undertaking of the international mathematical community in history. These researchers were working around the clock to axiomatize everything in mathematics. They wanted to create a finite list of axioms, like Peano arithmetic, from which all of mathematics can be based, and which could prove its own consistency—that is, one could prove that these axioms never contradict each other, using those same axioms. And for a while, it seemed like they had a good thing going. For a while, they thought they were getting closer and closer.
Well… for a while.
What the Formalists didn’t understand was that the Hilbert Program was absolutely, positively, unquestionably, dead on arrival. The project was doomed from the start. To be fair, there was no way they could have known that when they had begun. Even a Platonist like me can appreciate the appeal of having a perfectly axiomatized, well-grounded system on which mathematics can be based. Unlike Platonism, which answers the question in a heady, abstract, esoteric way, Formalism offers a very clean, concrete answer to the question, “What is mathematics?”
But all it took was one mathematician. One journal. Inside the journal was a paper. Inside the paper were two theorems. And those two theorems were all it took to bring the perfect, pristine tower of Formalism, on which all of mathematics was to be based, crumbling to the ground.
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