You may recall that in our last episode we discussed the results of 19th-century attempts to rigorously define the concept of whole numbers. These attempts culminated in the Peano axioms, a set of simple properties that defined numbers on the basis of the primitive concepts of zero and succession, plus a few related rules. While this definition has its merits, Bertrand Russell pointed out that it also has some major flaws: sets that we don’t think of as whole numbers, such as all numbers above 100, all even numbers, or all inverted powers of two, could also satisfy these axioms. So, how did Russell propose to define numbers?
Here’s Russell’s definition: “A number is anything which is the number of some class”. Great, problem solved, we can end the podcast early today! Or… maybe not. Let’s explore Russell’s concepts a bit, to figure out why this definition isn’t as circular as it seems.
The basic concept here is that we think of a whole number as a description of a class of sets, all sets which contain that number of elements. Let’s take a look at the number 2. The set of major US political parties, the set of Mars’s moons, and the set of my daughter’s cats are all described by the number two. But how do we know this? You might say we could just count the elements in each one to see they have the same number— but Russell points out that that would be cheating, since the concept of counting can only exist if we already know about whole numbers. So what do we do?
The concept of 1-1 correspondence between sets comes to the rescue. While we can’t count the elements of a set before we define whole numbers, we can describe similar sets: a pair of sets are similar if their elements can be put into direct 1-1 correspondence, without any left out. So despite lacking the intellectual power to count to 2, I can figure out that the number of my daughter’s kittens and the number of moons of Mars are the same: I’ll map Mars’s moon Phobos to Harvey, and Mars’s moon Deimos to Freya, and see that there are no moons or cats left over.
Thus, we are now able to look at two sets and figure out if they belong in the same class of similar sets. Russell defines the number of a class of sets as the class of all sets that are similar to it. Personally, I think this would have been a bit clearer if Russell hadn’t chosen to overload the term ‘number’ here, using it twice with slightly different definitions. So let’s call the class of similar sets a numerical grouping for clarity. Then the definition we started with, “A number is anything which is the number of some class”, becomes “A number is anything which is the numerical grouping of some class”, which at least doesn’t sound quite as circular. The wording gets a little tricky here, and I’m sure some Russell scholars might be offended at my attempt to clarify it, but I think the key concept is this: A number is defined as a class of sets, all of which can be put into 1-1 correspondence with each other, and which, if we conventionally count them (not allowed in the definition, but used here for clarification), have that number of elements.
Is this more satisfying than the Peano axioms? Well, if we identify zero with the empty set and the succession operation with adding an element to a set and finding its new number, we can see that those axioms are still satisfied. Furthermore, this interpretation does seem to rule out the pathological examples Russell mentions: the numbers greater than 100, even numbers, and inverted powers of two all fail to meet this set-based definition. And Russell successfully used this definition as the basis for numerous further significant mathematical works. On the other hand, Russell’s method was not the final word on the matter: philosophers of mathematics continue to propose and debate alternate definitions of whole numbers to this day.
Personally, I know a whole number when I see it, and maybe that’s all the definition a non-philosopher needs on a normal day. But it’s nice to know there are people out there somewhere thinking hard about why this is true.
And this has been your math mutation for today.