# 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. Dec 2, 2019 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. Dec 2, 2019 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. Dec 2, 2019 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). Dec 2, 2019 at 22:05
• @NoahSchweber This is a question about Frege's logicism, how is first-order logic introduced decades later relevant at all? Dec 2, 2019 at 22:07