**H**ow-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.

**W**hile 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.

**I**n 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?

**T**here 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.

**A**nother 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.

**I**n 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.

**L**et 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"But this condenses together three separate assumptions:

P ~ Q means "there is a one-to-one correspondence between P and Q"

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"

**T**he 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.

**F**rege, 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.

**B**ut 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.

**B**ut 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.

**B**ut 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.

**I**n 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.

**C**ounting, 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".

**C**antor 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.