UPDATE: added paragraph explaining the meaning of the commutative diagram more thoroughly.
I think I’ll start in on some more fundamentals. Today: natural numbers.
The natural numbers are such a common thing that everyone has an intuitive idea what they are. Still, we need to write down specific rules in order to work with them. Back at the end of the 19th century Giuseppe Peano did just that. For our purposes I’ll streamline them a bit.
- There is a natural number .
- There is a function from the natural numbers to themselves, called the “successor”.
- If and are natural numbers, then implies .
- If is a natural number, then .
- For every set , if is in and the successor of each natural number in is also in , then every natural number is in .
This is the most common list to be found in most texts. It gives a list of basic properties for manipulating logical statements about the natural numbers. However, I find that this list tends to obscure the real meaning and structure of the natural number system. Here’s what the axioms really mean.
The natural numbers form a set . The first axiom picks out a special element of , called . Now, think of a set containing exactly one element: . A function from this set to any other set picks out an element of that set: the image of . So the first axiom really says that there is a function .
The second axiom plainly states that there is a function . The third axiom says that this function is injective: any two distinct natural numbers have distinct successors. The fourth says that the image of the successor function doesn’t contain the image of the zero function.
The fifth axiom is where things get really interesting. So far we have a diagram . What the fifth axiom is really saying is that this is the universal such diagram of sets! That is, we have the following diagram:
with the property that if is any set and and are any functions as in the diagram, then there exists a unique function making the whole diagram commute. In fact, at this point the third and fourth Peano axioms are extraneous, since they follow from the universal property!
Remember, all a commutative diagram means is that if you have any two paths between vertices of the diagram, they give the same function. The triangle on the left here says that . That is, since has a special element, has to send to that element. The square on the right says that . If I know where sends one natural number and I know the function , then I know where sends the successor of . The universal property means just that has nothing in it but what we need: and all its successors, and is not the successor of any of them.
Of course, by the exact same sort of argument I gave when discussing direct products of groups, once we have a universal property any two things satisfying that property are isomorphic. This is what justifies talking about “the” natural number system, since any two models of the system are essentially the same.
This is a point that bears stressing: there is no one correct version of the natural numbers. Anything satisfying the axioms will do, and they all behave the same way.
The Bourbaki school like to say that the natural numbers are the following system: The empty set is zero, and the successor function is . But this just provides one model of the system. We could just as well replace the successor function by , and get another perfectly valid model of the natural numbers.
In the video of Serre that I linked to, he asks at one point “What is the cardinality of 3?” This betrays his membership in Bourbaki, since he clearly is thinking of 3 as some particular set or another, when it’s really just a slot in the system of natural numbers. The Peano axioms don’t talk about “cardinality”, and we can’t build a definition of such a purely set-theoretical concept out of what properties it does discuss. The answer to the question is “無!” (“mu”). The Bourbaki definition doesn’t define the natural numbers, but merely shows that within the confines of set theory one can construct a model satisfying the given abstract axioms.
This is how mathematics works at its core. We define a system, including basic pieces and relations between them. We can use those pieces to build more complicated relations, but we can only make sense of those properties inside the system itself. We can build models of systems inside of other systems, but we should never confuse the model with the structure — the map is not the territory.
This point of view seems to fetishize abstraction at first, but it’s really very freeing. I don’t need to know — or even care — what particular set and functions define a given model of the natural numbers. Anything I can say about one model works for any other model. As long as I use the properties as I’ve defined them everything will work out fine, and whether I use Bourbaki’s model or not.