Conservative Extensions Are Permissible, But Non-Conservative Ones Can Be Mandatory

18 03 2005

On the face of it, Hartry Field’s insistence that the conservativity of mathematics (together with the existence of acceptable nominalistic scientific theories) means that we shouldn’t believe in numbers flies in the face of mathematical history. It seems that historically, it’s precisely when complex numbers were shown to be conservative over the reals that they were first fully accepted. However, I think that this can be explained in a Fieldian way. The mathematician who believes in complex numbers believes in them exactly the way she believes in real numbers – which I think a hermeneutic reading of Field’s fictionalism suggests is just fictional. The established conservativity of the complex numbers means that they are a consistent extension that is conservative over the nominalistic physical theory we already have, and therefore it is just as acceptable to talk about complex numbers as it is to talk about real numbers. This sort of acceptability doesn’t mean that the claims should be literally read as true (on Field’s account), but they are still legitimate to talk about as fictional entities.

This legitimacy is meant to contrast with the non-acceptability of the talk of a set of all sets, or Frege’s Basic Law V. Once they were established to be conservative, complex numbers became just as much a permissible subject of mathematical discourse as the real numbers. The Russell set or Fregean extensions have never become permissible, because of the particular form of their non-conservativity (ie, contradictoriness).

However, non-conservativity of an extension sometimes makes the extension mandatory rather than impermissible. The parts of our physical theories that talk about subatomic particles are not conservative over the nicely formulated parts that don’t, but we must accept them and their implied entities anyway. This is because the extra statements that get proven by the non-conservative extension have explanatory power for facts that were observable at the macroscopic level. Similarly, Donald Martin (in “Mathematical Explanation”, in eds. Dales and Oliveri, Truth in Mathematics, 1998, Oxford University Press) suggests that once one countenances the entities whose existence can be derived from ZFC, one should countenance large cardinals as well, because they are a non-conservative addition in precisely a way that explains certain facts about the existence of Turing cones and Wadge degrees in all known sets of a certain complexity. So in whatever sense one accepts ZFC, one must accept certain large cardinal axioms, and in whatever sense one accepts one’s observations of macrosopic objects one must accept subatomic particles.

However, just by accepting macroscopic objects and subatomic particles, one is not forced to accept sets. They can be taken (fictionally, according to Field) or left.

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