Against Angles and the Fregean-Cantorian Theory of Number
by Andrew Boucher
v1.00 Created: 28 Jan 2002 (as a posting to sci.logic and sci.math)
Please send your comments to abo

How-many numbers, such as 2 and 1000, relate or are capable of expressing the size of a group or set. Both Cantor and Frege analyzed how-many number in terms of one-to-one correspondence between two sets. That is to say, one arrived at numbers by either abstracting from the concept of correspondence, in the case of Cantor, or by using it to provide an out-and-out definition, in the case of Frege.

While CFT (the "Cantorian-Fregean Theory") can be traced back to Hume and perhaps further, nonetheless it was only in the latter part of the nineteenth century that it "gained widespread acceptance" (in the words of Frege). It produced a revolution, because it contradicted what was, apparently anyway, the obvious theory of how-many numbers: that they arise from and should be based on counting. In this view a set has a number precisely when it can be counted, and its number is the last number which has been used in the counting. Such a prosaic philosophy evidently guarantees only the finite numbers. Freed from these strictures, and using correspondence, Cantor was able to introduce numbers beyond the finite. Among other consequences, set theory has largely arrived at its present axiomatization to provide a systematic basis for these transfinite cardinals.

In spite of, indeed because of, its widespread influence, it is important to take another look at CFT . Did mathematics make the right turn? Or should we just go back to the common-sense view? In brief, is a counter-revolution in order?

There are, after all, well-known problems with CFT. While agreeing with our intuition in the case of finite numbers, it stretches and even contradicts it in the case of the infinite. The set of even numbers, according to CFT, has the same number as the set of natural numbers. Yet, evidently, there are more natural than even numbers. One would even insist that there are obviously many, many more. Superficially anyway, it is bizarre to say that the two sets have the same size.

Another problem is that numbers are no longer comparable, in the sense of Trichotomy, which says that for any numbers n and m, either n is less than or equal to m, or m is less than or equal to n. This would seem to be a sine qua non for any theory of number. Yet, the Theorem can only be proven, in the general case, if one assumes the Axiom of Choice. Most people take this as a good reason to accept the Axiom. Nonetheless, the domino-effect should not pass uncommented. First, we are made to believe that the CFT provides a useful generalization of number. Then, in order for it to be as useful as advertized, we are forced, almost at gunpoint, to accept a new principle, which makes large if not extraordinary ontological claims.

In brief, CFT can be criticized as cherry-picking which properties of numbers actually hold. It extends the concept of number from the finite to the infinite, but makes, so to speak, a subjective valuation of which properties of finite number are the important ones. Not all are carried over. The more die-hard supporters of CRT will even boast this to be one of its advantages, by exposing the failure of our intution and then improving on it. Nonetheless, any theory will claims to be providing an analysis of an intuitive concept but then renounces intuition, is surely only making a confession of its own inadequacy.

Let us therefore take a closer look at the precise assumptions that CFT makes. It is often written (under the influence of what is called "Frege Arithmetic") as:

(P)(Q) (#P = #Q <=> P ~ Q),
which we will call "Hume's Principle" or HP for short.

Here:

#P means "the number of P"
P ~ Q means "there is a one-to-one correspondence between P and Q"
But this condenses together three separate assumptions:

1) One-to-one correspondence is equivalent to equinumerosity, that is:

(P)(Q)(n)(Mn,P => (Mn,Q <=> P ~ Q))

2) Existence, that is:

(P)[n] Mn,P

3) Uniqueness, that is:

(P)(n)(m)(Mn,P & Mm,P => n = m)
Here:
Mn,P means "P numbers n"
P ~ Q means that "there is a one-to-one correspondence between P and Q"
(P) means "for all P"
[n] means "there exists an n"

The separation is revealing. By bundling all three together, HP presents a take-it-or-leave-it package. Splitting it into three, we see that the first axiom, the philosophical heart of the theory, which says that correspondence and equinumerosity are equivalent, is independent of the other two assumptions. The first in no way implies the second and third. Indeed, there does not even seem any possible way to motivate the assertion of existence or uniqueness from considerations of correspondence alone.

Frege, of course, tried. He defined the number of P to be the set {Q | Q ~ P}. This produces uniqueness (because of existensionality) and, in Frege's system, existence. Unhappily for Frege, its existence leads to contradiction. So Frege's "way out" can be put to one side.

But even forgetting about Russell's Paradox, Frege's move is illegitimate. He is pretending it to be a stipulation, which cannot therefore be disputed. However, a stipulation is the assignment of a meaning to a word or word phrase, any of whose real meanings must be ignored and are therefore inessential. So, in a stipulation, the defiendum (in this case "the number of P') can be replaced by any other word, including nonsense ones like 'plok', without impacting the nature of the definition. For instance, the definition of 'prime number' to mean "a natural number different than 1 divisible only by itself and 1" is indeed a stipulation, because no one cares if instead one had defined 'plok' to be such and then proven that there are an infinite number of ploks. The defiendum --be it 'plok' or 'prime number'--is inessential.

But clearly in his case, Frege needs the defiendum to be 'the number of P' and not something else--precisely because he is trying to provide a theory of number. His is therefore not a stipulation, but an analysis, and it can be disputed. But this opens the floodgates, and as an analysis, it is evidently wrong. Who would say that "the number of people in this room" is a subset of the universal set? Or that (since this number happens to be 1, since I am alone) it contains the singleton set of the moon? Whatever a theory of something should be, it should not be ridiculous; but Frege's theory is precisely that.

But even grant Frege the correctness of his definition. He still cannot provide it any justification, other than that it "works", in the sense that HP and (allowing for even more so-called stipulations) the Peano Axioms fall out as consequences. Such a theory is a non-theory, and it does not explain anything. Frege is a long way from doing what he set out to do: namely, showing what supports the truths of arithmetic.

In brief, while there may be a way to motivate the existence and uniqueness of numbers using the idea of one-to-one correspondence, no one, to my knowledge anyway, has produced one.

Counting, on the contrary, provides a basis for both existence and uniqueness, although of course restricted to the finite. Knowing how to count, means one understands, if one has counted P, and if there is a thing t which is not P, that one can then also count P and t. That is, there is always a next number to be used in counting, hence existence. Moreover, if one has counted up to a certain point, and one has a new thing to count, there can only be one number to do so, hence uniqueness. Finally, with axioms about counting, one can prove (rather than assume) that one-to-correspondence is equivalent to equinumerosity. This can, of course, be formalized; please see (in pdf format) "Systems for a Foundation of Arithmetic".

Cantor and Frege can therefore be accused of putting the cart before the horse, of mistaking a theorem for a definition. Counting, not one-to-one correspondence, is the basis of number. The plain, boring philosophy may not be as intriguing or fantastic as its competitor, but it has the one great advantage of being correct. Numbers are finite, and there is no reason to believe in the transfinite. They say that in the Middle Ages, the greatest thinkers disputed how many angels could fit on the head of a pin. Today, the best logicians engage in learned discourse about inaccessible cardinals. So to speak, transfinite numbers are the angels of our day. Let the counter-revolution begin.