Logicism and the Ontological Commitments of Arithmetic

Once this much is accepted, further familiar metaphysical theses become close to irresistible: (3) Numbers (what numerical terms designate, what numerical predicates apply to, what number-theoretic quantifiers range over) exist outside of space and time; they are in no sense created by us, by thought, or by "mathematical activity" (whatever that may be).This is all prima facie plausible.After all, sciences have subject matters: ichthyology is abotit fish, ecology about eco-systems, etc.These subject matters are kinds or types of objects of a reasonably non-ad-hoc sort.The mathematical sciences seem to be no exception.It would sound silly to deny that a book on number theory is about numbers.This is how mathematicians and nonmathematicians alike speak.So once we have learned some logic and start seeing the logical form of statements about people, planets, positrons, and nations through the goggles of the first-order predicate calculus, and all looks clear and basically right, and we come across syntactically similar discourse in Mathematics departments and mathematics texts, it's only natural, for generalizing creatures like ourselves, to try on these goggles in this land, too.Within certain limits, to be discussed momentarily, things look fine: mathematical discourse can be regimented into first-order languages, and mathematical arguments can be formalized in any complete formalization of first-order logic.^All this is built into doctrine (2), and it all seems right.When other considerations brought Frege's research project to grief, he seems to have been quite sensible to hold to his slogan, "Numbers are objects," as a fixed point, a beacon of light in the confusing darkness:The prime problem of arithmetic may be taken to be the problem: How do we apprehend logical objects, in particular numbers?What justifies us in recognizing numbers as objects?Even if this problem is not yet solved to the extent that I believed it was . . ., nonetheless I do not doubt that the way to a solution has been found. Several passages in the Nachlasssuggest that late in his life Frege was inclined to abandon this fixed point.I contend that this is the move that should be made.It is natural, harmless-at least when one is not doing philosophy-and even helpful, to think and speak ^ Perhaps Category theory is an exception.