|God created the integers.
All else is the work of man.
The problem with knowledge is that you have to start somewhere. Once you know something, you can start deducing other things. But how do you know the first thing?
Descartes’s famous solution was “I think, therefore I am”. I have direct knowledge of my own thoughts, and this in turn tells me that I exist. Now we can go on.
Mathematics, like all other knowledge, needs a starting point. Most of our mathematical knowledge is deduced from prior mathematical knowledge. I know that every positive integer is the sum of four squares because I know how to deduce this fact from other things I know. But where do I start?
There seems to be a widespread misconception (widespread, that is, among non-mathematicians — mathematicians know better) that all we know is what we can derive from axioms. This is wrong for several reasons. Two of these I’ve blogged about repeatedly in the past (e.g. here, here, and here and here), but the third is even more fundamental:
- To say all we know about mathematics is what we can derive from the axioms is like saying that everything we know about Nebraska is what we can derive from maps of Nebraska. It confuses a partial description of reality with reality itself.
- Godel’s Incompleteness Theorem (among other theorems) assures us that no system of axioms can serve as a complete description of the natural numbers. Any set of axioms that can describe the natural numbers also describes many other mathematical structures — just as a sufficiently undetailed map of Nebraska might serve equally well as a map of part of Montana. If the map fits both territories equally well, then it can’t be said to specify either territory.
- Any axiomatic system presupposes that you know something about numbers and therefore cannot be the basis for your knowledge about numbers. In an axiomatic system, a proof is a list of statements, each of which is either an axiom or a logical consequence of earlier statements on the list. You can’t make sense of that notion until you know what a list is, and you can’t know what a list is unless you know what it means to come first, second, third and fourth in a series.
Since I’ve blogged before about points 1 and 2, here I’ll emphasize point 3: To employ an axiomatic description of the natural numbers, you must already know something about natural numbers. That knowledge must come first.
In fact, any sort of mathematical reasoning at all seems to rely on at least some prior familiarity with both numbers and sets. We write down axioms to formalize some of that knowledge, but numbers and sets are the starting points. The basic properties of numbers and sets are the things we have to “just know”, in the same sense that Decartes “just knew” that he was conscious.
Some people find it unsettling that all knowledge must have a starting point. But nobody’s ever found a way around that problem, and in math, the starting points are numbers, sets, and their basic properties.
But there is a key difference between our knowledge of numbers and our knowledge of sets. The difference is important and profound.
When it comes to numbers, we write down (say) the Peano axioms to formalize some of what we already know. Then we discover (thanks to Godel and others) that the Peano Axioms have multiple models — that is, there are alternative mathematical territories that they describe equally well. Of the many structures that obey the Peano Axioms, we are particularly interested in the standard model — that is, the ordinary natural numbers that you’ve known about since age three.
When it comes to sets, we write down (say) the Zermelo-Frankel axioms to formalize some of what we already know. Then we discover that the Zermelo-Frankel axioms have multiple models — that is, alternative mathematical territories that obey the axioms for a “universe of sets”. Of the many structures that obey the Zermelo-Frankel axioms, we’d like to pick one out and call it the standard model — the ordinary universe of sets that we’ve all been studying since fourth grade.
But when you meditate on that notion, you discover that “the universe of sets” is a much fuzzier notion than “the natural numbers” — sufficiently fuzzy that it’s not clear there is a standard model of the Zermelo-Frankel axioms. Roughly this means the following: When we talk about the natural numbers, we are quite confident that we’re all talking about the same thing. When we talk about the universe of sets, it’s not so clear.
This in turn makes the real numbers into a fuzzier notion than you might expect — because you can’t describe the real numbers without talking about sets, and sets are fuzzy. The rational numbers are fine — a rational number consists of a numerator and a denominator (together with some simple rules for deciding when two rational numbers are equal, and for adding them etc.), and numerators and denominators are natural numbers, which we understand. But to describe an arbitarary real number, you’ve got to talk about sets. The square root of two, for example, is the real number that separates the set of rational numbers whose squares are less than two from the set of rational numbers whose squares exceed two. Other real numbers have more complicated descriptions in terms of sets. So if there is no preferred “universe of sets” — and it’s not at all obvious that there is — then there is no preferred system of real numbers.
Of course we all know how to specify a real number — it’s a (possibly infinite) string of digits, punctuated by a decimal point. But which infinite strings are permitted? Well, all of them course. But what are “all of them”? What infinite strings are there? Ultimately, infinite strings are defined in terms of sets, which brings us full circle back to the fundamental ambiguity.
The real numbers, then, are a little fuzzy. The natural numbers are sharp. And their sharpness derives, ultimately, from our direct intuition of them, from which all mathematical knowledge flows.