# Did logicists use mathematical entities (in their attempt) to reduce mathematics to logic?

Two concepts F,G are equinumerous if there exists a one-to-one correspondence between the objects that fall under F and G.

Equinumerosity is one the most fundamental building blocks of Gottlob Frege's attempt to reduce mathematics to logic. Reading the definition, though, this struck me:

Isn't it problematic that the concept of one-to-one correspondence is used prior to defining the notion of maps-functions? Or is it preassumed that is a concept familiar through empirical means?

If we assume the first, then that would imply we use mathematical entities (not rigorously defined) to explore the nature of mathematics and this seems to me like circular reasoning.

On the other hand, if we assume the last, this would imply that the whole construction of mathematics has some empirical basis, which seem even more counterintuitive.

Could you help me clarify this confusion?

• Although 1-1 correspondence sounds like it uses maps it can be expressed without them. Namely, for every x under F there is a y under G, for every y under G there is an x under F, and different x-s have different y-s. You only need logical connectives, quantifiers and the equality symbol to express all of that in addition to "x falls under F". But we do use non-rigorously defined words to explain what symbols mean and how to manipulate them anyway, so there is no escaping that. – Conifold Dec 2 '19 at 21:11
• @Conifold I strongly disagree. "and different x-s have different y-s." presupposes a specific way of choosing ys for xs - that is, a map. "You only need logical connectives, quantifiers and the equality symbol to express all of that" that's not true - bringing dependence into things without a function symbol pushes you into past first-order logic. – Noah Schweber Dec 2 '19 at 21:56
• @NoahSchweber Except Frege did not care which order, nor was the notion extant at the time. See e.g. Heck, The Julius Caesar Objection, p.14 for a symbolic version of his Hume's principle. – Conifold Dec 2 '19 at 22:00
• @Conifold That seems a minor point here - I don't buy that symmetric relations are ontologically simpler than functions - and regardless doesn't address the logical issue I mentioned (the existence of a bijection between two sets can't be expressed in a first-order way). – Noah Schweber Dec 2 '19 at 22:05
• @NoahSchweber This is a question about Frege's logicism, how is first-order logic introduced decades later relevant at all? – Conifold Dec 2 '19 at 22:07

Frege considered the notion of functions to be logically primitive and so to be undefinable. He tried to give some elucidations of his idea of functions by saying that functions comprise all and only the unsaturated or incomplete things. So Frege's logicism simply takes functions for granted as it does regarding objects (the complement of functions).

As a remark it should be noted that while many Frege scholars think that the central logical entities for Frege's logicism are extensions of concepts it should be noted that Frege's Axiom V in the Grundgesetze is actually about functions. It says that functions have the same Werthverlauf or course of value iff they yield the same value for every argument. So functions appear to be of much greater importance to Frege's project than extensions of concepts. This is confirmed by a posthumous piece of Frege's titled 'Was kann ich als Ergebnis meiner Arbeit ansehen?', published after Russell's letter apparently shattered Frege's project. There Frege still accepts a distinction between function and objects but is silent on extensions of concepts.

Reference : Russell, Introduction to Mathematical Philosophy.

As to what belongs to "logic" : See Alonzo Church's article " logic , formal - " in Rune's Dictionary Of Philosophy . ( at Archive.org). You will see that the algebra of classes ( set algebra) , the algebra of relations, and even " set theory" is considered as a part of logic.

Brief answer : there would be circularity in the logicist attempt only in case logicists would use concepts pertaining to arithmetics.

When logicists talked about " mathematics" they thought of arithmetics.

For they considered it had been shown that all mathematics can be reduced to arithmetics: (1) geometry had been reduced to analysis (2) analysis had been reduced to arithmetics (Kronecker, Weierstrass). This is what is called " arithmetization" of mathematics.

Peano had proposed an axiomatized version of arithmetics.

This axiomatization left the term " number" ( more precisely, natural number) undefined.

The logicists project is to define this concept of number via logical means.

Their " tour de force" is to use the concept of " equinumericity" as a concept logically prior to the concept of number.

That is, you do not need to have defined any particular number, you do not even need to have defined the general concept of number, to define the relation " having the same number of elements".

To the contrary, you need the concept of equinumericity to define what is a number , that is " an equivalence class of equinumerous sets".