Disjunctive Justifications for Mathematics

7 11 2005

One of Penelope Maddy’s main reasons for objecting to the indispensability argument in her book Naturalism in Mathematics is because it seems to make mathematics too contingent – as she says, if indispensability were our grounds for believing in the axioms, then set theorists arguing about large cardinal axioms should be paying attention to quantum gravity and other cutting-edge physics to see what sorts of math are indispensable for it. And more importantly, she thinks that Quine has shown that indispensability arguments only get us ZFC and not the further large cardinal axioms – and that in fact we are limited to V=L, which is incompatible with most of the larger axioms that set theorists emphatically want us to adopt (and which she thinks we have good mathematical reason to adopt).

However, it seems that a Fieldian nominalist has an easier time justifying our mathematical practice, if the program can ever be made to succeed. The goal is to show that mathematics is actually dispensable (and thus the entities it appears to talk about don’t actually exist) using the Fieldian strategy of giving an attractive nominalistic physical theory that the platonistic theory conservatively extends. If this can be done, it undercuts mathematics in one sense, by saying that it is not literally true. But it supports it in another (perhaps more important?) sense, by showing that it’s a perfectly useful way to talk that (while not itself true) will help us get to the truth more easily in the domains we’re actually concerned with, namely the physical.

Thus, the problem for the Quinean realist justification is that we need to show that the entities quantified over in mathematics are indispensable for our scientific theories in order to justify our mathematical talk. The problem for Fieldian nominalist justification is that we need to show that these same entities are dispensable in our scientific theories. Thus, Maddy rejects both attempts at justification, which are based on reading the indispensability argument in opposite directions, and instead suggests that mathematics needs no external justification, just as most naturalist philosophers think about science as a totality. However, I think that Maddy’s move is unnecessary, and that we may even be able to put together these two justifications to show why!

Combining Platonistic and Nominalistic Justifications

First, note that the important trouble steps in these approaches are nicely complementary. However, they aren’t negations of one another. In one case, we need to show that the only (nice) physical theory that explains all our data is a theory that includes mathematics. In the other case, we need to show that there is a nice theory that explains all the data and includes the mathematics, and there is also a nice theory that explains all the data that doesn’t include the mathematics. So to get my proposed disjunctive justification off the ground, we’ll need to show that there is a nice physical theory including the relevant mathematics that explains all our data. Fortunately, for any theory up to the strength of ZFC, we’ve got such a theory. (In another post I’ll mention why I think going further up isn’t such a problem. For now, I think it’s good enough to observe that as long as we think the large cardinal axioms are at least consistent, then we can extend our theory to one including all of them.)

So let E be the set of all our relevant evidence, let M be the relevant mathematical theory, and T+M be our nice theory that currently explains all of E. At this point there are two possibilities – either every nice theory that explains E includes M, or there is some alternate theory T’ that doesn’t include M that explains E equally well. (For now let’s assume that T’ is a nominalistic theory referring only to entities also referred to in T+M.) In the first case, the indispensability argument suggests to us that M is in fact true, and thus refers to a class of existing entities, and is therefore a justifiable part of our scientific discourse. (Never mind that those entities may well be acausal, atemporal, or whatever – they’re indispensable for our science, so we know about them just as we know about quarks.) In the second case, the indispensability argument suggests that M is in fact false, and there are no entities of the type it refers to. However, following Field, we can still use M in our scientific reasoning, because it is part of T+M, which is just as good a way of making predictions about E as T’ is. In either case, M is justified as part of our scientific reasoning, so Maddy needn’t be concerned.

Some Generalizations

If E is a complete theory, then both T’ and T+M will be conservative extensions of it, so we’ll be in exactly the situation Field takes himself to have given us for Newtonian gravitation. Of course, E is our set of actual observations, so it won’t be complete, but there’s a sense in which this doesn’t matter. Alternative scientific theories don’t have to agree with our current ones in every prediction – they just have to be equally good at explaining our data. (In fact, they don’t even necessarily have to be equally good at all of it – if one theory does a better job of explaining some data, and the other theory does a better job on a different set, then both might be useful theories.) So in a sense, Field might be aiming too high when he aims for conservativity of mathematical theories over nominalistic ones. All he needs is something more like empirical and explanatory adequacy. I think he comes around to a position like this in his 1985 “On Conservativeness and Incompleteness” where he suggests that it might be ok for the nominalistic theory to miss out on some translations of Gödel sentences – these are unlikely to appear in the data, so they aren’t a good reason to decide between two theories that differ severely in their ontological virtues.

Now, let’s note that once we know that T+M exists, we don’t need to know anything about what T’ is, or even whether it exists. In the ideal situation (which Field approximately gives us for Newtonian gravitation and calculus, and also sketches for a very general theory of counting medium-sized dry goods and the natural numbers) we know exactly what T’ is, and that T+M is a conservative extension of it. But even if we don’t know it to be conservative, we’re justified in using M either way. If we even know it not to be conservative, we may then be able to empirically test which theory’s predictions are correct – but until then, they both explain our current data equally well. But even if we don’t know whether such a T’ exists, we’re justified by the existence of T+M in using M.

The only way we can lose this justification (without simultaneously replacing it by a Fieldian one) is by coming up with some other theory U that does a substantially better job of explaining E, and doesn’t contain M. Since this theory is better than T+M, it would undermine our indispensability justification. But if it doesn’t contain M, then we’d need to show that U+M was a conservative extension of U (and a useful one) in order to get a Fieldian justification.

Field (early in Science Without Numbers) claims to have an argument that mathematical theories are conservative extensions of any physical theory (though not necessarily useful extensions). But even ignoring this claim, I find it hard to imagine that we will find a useful theory to explain the world that neither includes (some substantial fragement of) ZFC nor has a useful conservative extension including it. This is even less plausible if we’re talking about the theory of real-valued functions. And I venture to say that it’ll be impossible to describe the world in a way that wouldn’t be usefully and conservatively supplemented by Peano arithmetic. So whatever the status of our current theories, I think this disjunctive justification will let us use ZFC in good conscience, or at least the theory of real-valued functions, and certainly PA. And once we’ve got these axioms, I think we can get all the way up to where Maddy wants us to be, as I’ll show in a later post.

So Maddy really has no reason to be concerned about indispensability arguments depriving us of mathematics. Field has shown how to convert indispensability refutations into alternative justifications for mathematics, showing why the minor amount of empiricism the indispensability argument brings to mathematics is so utterly invisible to us.




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