1. On one influential interpretation, Hilbert is taken to have thought that any mathematical theory can be added to finitist mathematics so long as additions are consistent with the finistist base theory. Thus, we are justified in using transfinite mathematics because/if it is consistent with finite mathematics. Moreover, on this interpretation of Hilbert, the additional (transfinite) theory needn't be true to be used (or useful). Even if transfinite numerals fail to refer, they can be used in the service of finite mathematics given that the theories in which they occur are consistent with it. The underlying intuition here seems to be that the additions aren't really adding anything over and above the base theory because they always agree with it, and hence we can use the additional apparatus in the service of the base theory.

I think a similar intuition motivates Field's argument that the convervativeness of mathematics undermines reasons for thinking mathematical objects exist. Field claims a good mathematical theory is conservative, that is, it is consistent with every internally consistent physical theory. Again, I think the intuition here is that (pure) mathematics doesn't add anything over and above the physical things, it is merely a tool for thinking about the physical world. Field's view differs from Hilbert's (thus interpreted) in that he requires not only consistency with the rest of mathematics, but also with any way the physical world could be (as indicated by our theories) when applied to it. In other words, Hilbert required only that the union of finitist mathematics and additional mathematical theories (e.g. transfinite theories) be consistent while Field require that the union of physical theories and pure mathematical thoeries not only be consistent, but also not have any physical consequence beyond those of the physical theories alone. Both views argue that consistency (of one form or another) can replace the realists' truth as a justification for the use of pure mathematics. We needn't be commited to transfinite numbers (Hilbert) or numbers at all (Field) to use theories that purport to refer them because/if we know that they are consistent with everything else we know. All we need to know about those mathematical theories we take an antirealist attitude toward is that they work, and a guarantee of consistency with our accepted theories (which we take a realist attitude toward) seems to get us that without any mention of thier truth.

2. One argument against the kind of fictionalist approach to pure mathematics Field adopts claims that regarding a mathematical theory as good is sufficient for (or constituitive of) believing it. In other words, it follows from what we mean when we say a mathematical theory is good that we believe it (to be true). This means that fictionalism, the claim that pure mathematics is a useful fiction, as a distinct position from realism, is untenable. For Field, however, a mathematical need only be conservative to be good, not true. The only way we can justifiably make the leap from mere convervativeness to truth is when inference to best explanation (or direct observation) requires the entities mentioned in the theory to exist. In the case of pure mathematics, there is reason to think this requirement cannot be met. To the extent that it's possible to do science without pure mathematics, we don't need it's entities to exist to explain anything. -- GlicK - 28 Nov 2006