In “What is Mathematics About?” (in The Seas of Language), Michael Dummett puts forward a view of logicism that I think is not quite orthodox. Rather than saying that it is an attempt to derive the truth of mathematics purely from logic, he says that mathematics just is the business of working with deductive proofs to produce theorems that can be plugged in later. I took Frege’s program (and Hale and Wright’s revival of it) to be an attempt to justify from logic along quantification over a domain of actually existing mathematical entities, and not merely an attempt to justify mathematical practice.
However, Dummett’s suggestion strikes me as the most reasonable suggestion I’ve heard in all of these debates. He doesn’t require the existence of mathematical entities: numbers don’t need to exist – we just abbreviate long deductive proofs into theorems “about them” that get applied whenever we count things. I think this is also quite close to what mathematicians feel like they’re doing. A group theorist thinks that what she is doing is just writing down results that hold for whatever things happen to satisfy the group axioms, as a topologist is coming up with results that apply for anything that happens to satisfy the topological axioms assumed.
However, in calling this “logicism”, I think Dummett misses some of the point of Hartry Field’s fictionalist program. Dummett says that Field’s program would require a conservativity result for every mathematical theory applied in every scientific theory, and this proof would have to be carried out for each such pair of theories. I think this is basically right. But then Dummett goes on to say that this could never be finished, because there is always the possibility of the discovery or invention of more theories, both mathematical and scientific, and new conservativity proofs would need to be found. Dummett thinks this is a problem, but I think this is also exactly right.
The conservativity proof is just a statement of the applicability of the mathematical theory that is used. After all, although every particular actual scientific application of real numbers might be conservative, it certainly isn’t the case that every application of the real numbers is. One can use real numbers to define physically meaningless quantities of a system and then state that these quantities are conserved, which would result in many many contradictions. (If the physical theory is complete, then of course any non-conservative extention will be inconsistent, as Prior’s addition of “tonk” was to classical logic.) There may be some general patterns to which these conservativity results conform, so that many different applications of real numbers (say to lengths, volumes, times, forces, masses, etc.) can all be seen to be conservative in similar ways. But this makes sense, because it’s basically just showing that the physical objects satisfy the axioms of the real number system, which means that we can take Dummett’s logicist view, that then the application of real numbers is just to give results that were known to be true because of the fact that the system satisfies the axioms. And this is exactly what Field says about conservative extensions – they just shorten the proofs!
Of course, proving the conservativity of the reals over any of these physical systems will be quite hard. Applying reals to two different types of dimensionality sounds confusing! But once we figure out how to do it (maybe Field already has – I should reread Science Without Numbers) it will become much easier to do for all other theories.