Most people (except for extreme scientific anti-realists) say that a theory of the form “There are no Xs, but everything observable is just as it would be if there were Xs” is bad. As I mentioned before, Cian Dorr would like to give a fairly novel explanation of why these theories are bad, but here I’m going to try to focus on what I take to be the more traditional account, and its application to the debate about mathematical nominalism. In particular, one objection to fictionalism about mathematical entities that I have seen mentioned in work of Mark Colyvan, Mike Resnik, and possibly Penelope Maddy, is that not only do we need the claims of ZFC to be true for our best applied theories, but we also need them to be true even to use physical theories (like Newton’s) that we take to be false. I will discuss this objection towards the end, but first I will return to the instrumentalist move, saying that things behave just as if there were Xs, even though there aren’t.
The point seems to be one about inference to the best explanation. If things look just as if there were Xs, then one explanation would be if there actually were Xs. However, the “as if” style theory explicitly denies this explanation, without giving another one. Therefore, in one sense it’s not a theory, but rather the data that the theory needs to explain.
However, such data often can be explained. An idealized Adam Smith in an idealized free-market world might have observed that prices generally stay close to levels as if they were set by some “invisible hand” observing the needs of society at large. However, there are decent reasons to believe there is no actual invisible hand, so Adam Smith sought another explanation, and found one in the mechanisms of competition for both supply and demand.
One might try to be purely instrumentalist about the material world, saying there are no material objects, but things appear just as if there were. In particular, I might say “there is no strawberry in front of me, but it looks just as if there were”. However, while the instrumentalist might want to say this always, even the realist says this on certain occasions, when a mirror of a certain type is used. There is no strawberry there, but because there is one three inches below that spot, and the mirror is curved in exactly the right way, and you’re looking at it from an angle that is high enough for your line of sight to intersect the mirror in that place, and low enough not to see the actual strawberry, it looks just as if there were a strawberry there. The advertisement claims the images “defy, yet demand explanation” – it’s true that they demand explanation, but they don’t defy a suitably optically-informed explanation. At any rate, there is a clear contrast between the case of such an illusion and the ordinary cases of seeing an actual strawberry. Realists can make sense of and explain this contrast, but instrumentalists have to be a bit more careful. (I’m sure that it’s possible for them to cash out just what it means to look like there’s a strawberry there without really looking like there’s a strawberry there, or something, but it’ll be more complicated.)
There seems to be no reason why one couldn’t be a global instrumentalist about everything (except maybe sense data, or something of the sort), but at intermediate levels, it seems that one really does need an explanation. Hartry Field, in Science Without Numbers attempts to do something like this for a Newtonian universe – he can explain why everything acts just as if there were the real numbers and continuous functions that Newton talked about, even though all there actually is is just regions of space-time with various three- and four-place betweenness and equidistance properties. He still needs to help himself to some fairly strong logic (a quantifier saying “there are infinitely many Xs such that…”), but it’s a nice development.
More simply, a nominalist can explain why our counting practice works the way it does, just as if there actually were abstract entities known as numbers, even though there aren’t any (according to the nominalist). This explanation would point out the isomorphism between the counting process and the successor operation. It would point out that for any particular application of counting, a non-standard semantics can be given for the numerical terms on which they denote the objects counted rather than numbers. And it would point out that every particular numerical statement can be translated into a statement with numerical quantifiers, which can be translated in terms of existential and universal quantifiers, connectives, and identity.
However, even if this project of nominalizing our best scientific theory succeeds, realists might object that we would still need mathematics to be true to explain our practices of approximation. For instance, say that the population in any year of some herd of bison is given by a difference equation in terms of the populations in the previous few years. In many cases, the behavior of difference equations is hard to predict, but they can be usefully approximated by differential equations, where exact solutions can often be achieved. Thus, we might use a differential equation to model the population of the bison, even though we don’t believe that the population actually increases between breeding seasons, and we believe that the population is always integer-valued rather than real-valued as the differential equation requires. The realist can explain why the differential equation is a good approximation by pointing out all sorts of mathematical theorems about the systems of equations involved. However, even if the nominalist has managed to nominalize away all talk of numbers and equations in using the difference equation, she will have trouble explaining why the differential equation is a good approximation. She can’t appeal to facts about the mathematical structures denoted, because she says there are no such structures. And she presumably can’t nominalize the differential equation in any nice way, because it refers to fractional bison, and bison born at the wrong time of year. Instead, she’ll have to fall back to something like the instrumentalist position and say that the differential equation is not correct, but it makes very good predictions, and she can’t explain why.
The only remotely promising move I see at this point is to say that “according to the fiction of mathematics, the differentical equation and the difference equation will always make very similar predictions”. However, if mathematics isn’t actually true, it will be hard for her to explain why it is still correct about when it says two theories make similar predictions. In a sense, this is the same problem that Field ran into in expressing the claim “mathematical physics is a conservative extension of nominalist physics”, which is what justifies the practice of using mathematical physics to make predictions even though it is not literally true. Except here, we have to deal not only with conservative extensions, but with good approximations that aren’t conservative.
It would be very unappealing to end up in the situation where mathematics was unnecessary to make correct predictions, but necessary for various methods that give approximate predictions. (Some examples other than differential equations involve frictionless planes, infinitely deep oceans, light that travels infinitely fast, and the like.) In that case, the indispensability argument would only apply through our practice of approximation, and not through actual science. At this point, I think more people would be willing to bite the nominalist bullet and say that we’re not really justified in using our approximations, but it would still be an odd situation.
Fortunately, most of science remains un-nominalized, and the people that think we can some day nominalize most of it will probably believe that we can nominalize our approximation methods in some way too. It’s just an extra challenge to meet.