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How does Frege's definition of number solve the Julius Caesar problem?
Frege's definition of number in the end of Foundation is such: the number belonging to the concept F is the extension of the concept equal to the concept F such that F and G are equal if they can put in one to one correspondence.
I cannot wrap my head around the problem as to how to apply this definition to conclude that JC problem (about the nature of numbers) is overcome
Help would be much appreciated!
Thanks in advance!
Here is some historical context. In Grundlagen der Arithmetik (1884) Frege introduced his ill-fated Axiom V, now known as the axiom of unrestricted comprehension: every predicate defines a class of objects that satisfy it, called its extension (Frege's own formulation is more technical). This led to the set of all sets and then to the Russell's paradox in 1901 (apparently discovered by Zermelo before Russell, see How did Russell arrive at the paradox demonstrating the inconsistency of naive set theory?). But already at the time of the writing Frege already knew that the full force of Axiom V was not needed for his proofs, or for deriving the philosophical consequences he wanted. A weaker claim sufficed, the so-called Hume's principle (it was considered even before Hume, but adopted only after Cantor, see Mancosu's Measuring the Size of Infinite):
The number of Fs is the same as the number of Gs when there is a one-one correspondence between the Fs and the Gs.
Of course it was harder to sell this as a "law of thought", and the whole point of Frege's logicism was to derive mathematics from logic, the laws of thought. But even after Russell pointed out the problem with the set of all sets not belonging to themselves Frege did not try to save his system by switching to the Hume's principle. He was apparently deterred by the "Julius Caesar problem" laid out as the "third doubt" in Grundlagen §66:
"In the proposition [“the number of Fs is the same as the number of Gs”] [the number of Fs] plays the part of an object, and our definition affords us a means of recognizing this object as the same again, in case it should happen to crop up in some other guise, say as [the number of Gs]. But this means does not provide for all cases. It will not, for instance, decide for us whether [Julius Caesar] is the same as [the number zero] — if I may be forgiven an example which looks nonsensical. Naturally, no-one is going to confuse [Julius Caesar] with [the number zero]; but that is no thanks to our definition of [number]. That says nothing as to whether the proposition [“the number of Fs is identical with q”] is to be affirmed or denied, except for the one case where q is given in the form of [“the number of Gs”]. What we lack is the concept of [number]; for if we had that, then we could lay it down that, if q is not a [number], our proposition is to be denied, while if it is a [number], our original definition will decide whether it is to be affirmed or denied." [boldface mine]
What Frege is saying here is that mere Hume's principle does not allow us to distinguish between numbers and non-numbers (like Julius Caesar), because the concept of number is lacking. We need a definition of numbers, not just a rule for establishing their equality in a special case. Axiom V provides for such a definition (for example, number zero is the extension of a concept under which nothing falls, i.e. the null set), while the Hume's principle does not. A good review of related issues is Heck's The Julius Caesar Objection, who makes an interesting comment:
"All of this having been said, the question arises why, upon receiving Russell’s famous letter, Frege did not simply drop Axiom V, install Hume’s Principle as an axiom, and claim himself to have established logicism anyway. The question is not only of historical interest. Though Frege did not himself adopt it, this position has seemed to some a worthy heir to Frege’s logicism: On one version of it, Hume’s Principle is thought of as embodying an explanation of the concept of number, whence, even though it is not a principle of logic, perhaps it has a similarly privileged epistemological position... The historical question is made pressing by the fact that, in a letter to Russell, Frege explicitly considers adopting Hume’s Principle as an axiom, remarking only that the “difficulties here” are not the same as those plaguing Axiom V."
In a nutshell, the issue arises from the definition of number of as a second-order concept (i.e. a numerical quantifier) in Die Grundlagen der Arithmetik (1884).
Consider e.g. 0xϕ(x)=df Card[xy] (y ≠ y) [ϕx], that reads:
To assert 0xϕ(x) is to say that the objects that are ϕ are in one-to-one correlation with the objects that are not self-identical,
i.e. there are no objects that are ϕ.
Here we have defined the "second-order" concept (a predicate 0x(ξ) of a first-order predicate ϕ(x)). But we have not defined what is the number 0.
With that definition, how can we answer the question :
"A sentence of the form ‘The number of F's = q’ is false when q is not a number"
(where the paradigmatic example is with Julius Caesar in place of the "name" q) ?
In the "logically perfect" language considered by Frege, every singular term ("name") must have a denotation (Beduetung) and every function must be defined for every possible argument (every concept must be "sharply defined").
For instance, if ‘q’ is a singular term, its semantic interpretation must fix the truth-conditions of ‘q = y’ for any given object y.
In the case of Frege’s attempt to define the cardinal numbers, the problem is that the provided criterion for being the number belonging to the concept ϕ does not determine the Bedeutung of the number words completely, and thus it does not justify the definite article ‘the’.
The "mature" Fregean solution in Grundgesetze der Arithmetik (1893/1903) is based on the introduction of value-courses (Wertverläufe):
The Bedeutung of the singular term ‘ε´ φ(ε)’ is the value-course of the function φ(ξ),
which rules out that a sentence of the form ‘ε´ ϕ(ε) = q’ is true when q is not a value-course.
It is highly debate in the literature whether the value-course theory really solves the julius Caesar problem.
Caezar can be defined as number or not? As Frege thought the numbers can not be applied to uncertain World. The numbers are not conceived from sensations. The numbers are not our thinkings because the thoughts are varied from person to person. Therefore we can not define Caezar as number. The question is for Frege, which Caezar we will put the number such as 1. Everything changes in outer World.
