This is an extremely elementary post about numbers. (“Numbers” means the natural numbers 0,1,2 and so forth.) It is a sort of sequel to my three recent posts on basic arithmetic, which are here, here and here. But it can be read separately from those posts.
Today’s question is: Do numbers exist? The answer is: Of course, and I don’t believe there’s much in the way of serious doubt about this. You were familiar with numbers when you were five years old, and you’ve been discovering their properties ever since. Extreme skepticism on this point is almost unheard of among mathematicians or philosophers, though it seems to be fairly common among denizens of the Internet who have gotten it into their head that extreme skepticism makes them look sophisticated.
Let me be clear that I am not (yet) asking in what sense the natural numbers exist — whether they have existed since the beginning of time, or whether they exist outside of time, or whether they exist only in our minds. Those are questions that reasonable people disagree about (and that other reasonable people find more or less meaningless.) We can — and will — come back to those questions in future posts. For now, the only question: Do the natural numbers exist? And the answer is yes. Or better yet — if you believe the answer is no, then there’s obviously no point in thinking about them, so why are you reading this post?
“Existence” here is used in the ordinary everyday sense of the word, according to which rocks and trees exist, you and I exist, your hopes and dreams exist, and the idea of a unicorn exists. Unicorns themselves do not exist and therefore it makes no sense to study their properties. (Though you can have fun inventing some properties for them.) By contrast, it makes perfect sense for geologists to study the properties of rocks, for botanists to study the properties of trees, for folklorists to study the properties of the idea of a unicorn, and for mathematicians to study the properties of the natural numbers.
An extreme skeptic might deny the existence of rocks. The only possible answers are: a) It’s hard to believe you’re serious, since you’ve been encountering rocks — just like you’ve been encountering numbers — your entire life. b) If you really are serious, I suppose your best strategy is to stop thinking about rocks, and leave them to those of us who find geology interesting. And c) Do not fool yourself into believing that your position is anywhere close to any mainstream school of thought.
Another extreme skeptic might deny the existence of numbers. I’ll leave it to my readers to replace rocks with numbers in the above retorts.
What else might one say to an extreme skeptic? Answer: One might attempt to acquaint him with Godel’s Completeness Theorem. (This is not the same as the far more famous Godel’s Incompleteness Theorem.) Here is (part of) what the Completeness Theorem says: First, without making any assumptions about existence, write down a list of axioms for the natural numbers. For example, write down the Peano Axioms. Then the Completeness Theorem tells you that as long as those axioms are consistent, there must be some mathematical structure that obeys those axioms. (Note that “be” is a synonym for “exist”.) The smallest of those structures (known as “models”) is our good old friend the natural numbers.
In other words, Godel’s Theorem tells you that if the Peano axioms are consistent, then the natural numbers must exist. (Don’t confuse the map with the territory! “Consistency” applies to the axioms; “existence” applies to the natural numbers themselves.)
On the other hand, we can also argue in the opposite direction: If the natural numbers exist, then the Peano axioms, being true statements about existing objects, must be consistent. An accurate map of an existing territory cannot contradict itself.
So — We know that the natural numbers exist because we know the Peano axioms are consistent. And we know that the Peano axioms are consistent because we know that the natural numbers exist. Does that sound circular? It’s not. Here’s the point: We have extremely good reasons for believing in the existence of the natural numbers (beginning with intuition, lifelong familiarity, and the fact that we seem to be able to discover their properties). We have (partly) separate extremely good reasons for believing in the consistency of the Peano axioms (beginning with intuition and the fact that they’ve never yet led us to a contradiction). The fact that our two beliefs reinforce each other — that if either is true, then so must be the other — should build up our confidence that the whole picture hangs together.
Now let’s get back to our extreme skeptic. He denies the existence of the natural numbers. We respond that Godel’s Completeness Theorem proves the existence of the natural numbers, as a consequence of the consistency of the Peano axioms. He now has only two recourses (other than to concede defeat). One is to deny the consistency of the Peano axioms, and the other is to deny the accuracy of Godel’s Completeness Theorem. Let’s see how those strategies are likely to work out for him.
Should he doubt the consistency of the axioms? The Peano Axioms lay out the rules of arithmetic that you’ve used your whole life; they say things like “Every number has exactly one immediate successor” and “x + (y+1) = (x+y) + 1”. People (and to some extent animals) have been applying these axioms, explicitly or implicitly, since long before the dawn of history and no contradiction has ever arisen; moreover, for what it’s worth, the consistency of these simple axioms is instantly clear to most people’s intuitions. If we were to jettison our belief that these axioms are consistent, then we’d pretty much have to give up all quantitative reasoning.
Well, then, should our skeptic doubt Godel’s Completeness Theorem? The theorem is proved using elementary notions about sets — the idea that it’s possible to talk about sets of things and about membership in a set, that it’s possible to form the union of two sets, and so on. This has nothing to do with the more esoteric subject of “axiomatic set theory”; instead, it uses only the most fundamental notions associated with forming collections of things. (These notions, in fact, are prerequisite for axiomatic set theory and therefore cannot depend on it.) Once again, if you were to abandon this sort of reasoning, you’d pretty much have to abandon reasoning altogether.
For anyone who accepts the simplest sorts of combinatorial reasoning, there is no longer an out. The natural numbers are real. Again, this says nothing about where they came from — be it Plato’s heaven, the minds of humans or the mind of God. We’ll get back to that in the next installment of this occasional series.