The mathematics physics actually needs
A paper on Field's nominalism: if mathematics is only a conservative shortcut, its axioms could be chosen freely. Quantum mechanics needing complex numbers suggests they cannot, and the consistency the argument rests on cannot be proved.
Written for a PHL233 paper assignment, submitted April 5, 2024. The assignment set a passage and asked two questions in 500 words: what is the author’s argument, and what is a reasonable objection to it. Rewritten here from the original, with the citations turned into inline links and a looser register; the argument is unchanged, and the paper as submitted is available as a PDF. The passage is Field, Science Without Numbers, from “This argument isn’t conclusive” on page 13 to “they are not a priori true, for they are not true at all” on page 16. The copy I worked from is here.
Field starts from the presumption that standard mathematics is consistent. It yields no self-contradictions, and if it did we would revise it. From there he argues that standard mathematics is conservative: if you use mathematical machinery to get from nominalistically statable premises to a nominalistically statable conclusion, the same conclusion was already derivable from those premises nominalistically, without the machinery. Mathematics is a shortcut through a deduction that could have been walked.
The consequence is that whether a mathematical system is true does not matter to its use in science, so long as it is conservative. Nobody has to assume mathematics is true before applying it; systems can be picked for how useful they are. That is aimed at the Platonist who argues mathematical entities must be real because mathematics works so well in describing the world. On Field’s account they need only be conservative, and a shortcut is not the sort of thing that is true.
Consistency he cannot prove
The argument is deductively valid, so the premises are where to push.
Field treats the consistency of standard mathematics as something it would be surprising to lose. Mathematics does not run on that kind of expectation. Gödel’s second incompleteness theorem says a formal system meeting certain conditions cannot prove its own consistency, so there is no proving standard mathematics consistent with the tools of standard mathematics. We will not find out. Contradictions could sit inside any axiom system that currently looks consistent, ZFC included, which is presumably what Field has in mind by standard mathematics, and we cannot settle ZFC’s consistency inside ZFC. An inconsistent system is certainly not a conservative one, so the assumption carrying the whole argument is the one part of it nobody can check.
Some mathematics is not optional
Grant the consistency and the conservativeness anyway. Different mathematical systems still differ in what they can reach. A system without real numbers cannot state calculus, so it will struggle to deliver Newtonian physics. A system without complex numbers cannot describe quantum physics.
If mathematical entities were bookkeeping, chosen for convenience, the fact that quantum mechanics leans on a human invention like the complex numbers would be a very large coincidence. It reads more like the entities tracking something about the world.
Conformity, not usefulness
Field’s picture has the axioms as arbitrary shortcuts, bent at the researcher’s discretion. The evidence above says the universe has structures that force particular data types on us, so conservativeness cannot be the only property that matters. Good mathematics has to preserve truth and also answer to what is there, and a system that fails the second is not free to be modified without cost.
That sounds close to Field’s own standard of usefulness, and it is not the same standard. Spacetime geometry and quantum wavefunctions require certain mathematical concepts before they can be studied properly, so our judgment of a system depends on whether it conforms to this universe. That conformity is checked by observation, which makes it mind-independent. Field’s usefulness is decided by the person doing the deducing.
Where the shortcut runs out
Field’s case rests on a consistency claim that Gödel puts out of reach, and on treating the choice of axioms as free. The second is the more interesting failure. If mathematics were only a shortcut, any conservative system would do, and physics would not keep demanding particular ones.
How I first misread Field
I read conservativeness wrong at first, as ordinary truth preservation: use mathematics to get from P to Q, and if P is true then Q is true. That version leaves Platonism standing, since a tool that introduces no falsehoods can still be indispensable. Conservativeness is the stronger claim that the nominalistic conclusion was derivable without the mathematics in the first place. Only then is the mathematics a shortcut, and only a shortcut can be thrown away.
With that straight, the first objection is Gödel: the consistency Field assumes is the one thing his system cannot establish about itself.
The second took longer. I began from the observation that Science Without Numbers rebuilds Newtonian physics without mathematics while quantum theory has resisted the same treatment, and took that to be telling. It is not. It reports what Field managed, not what is necessary. Turning it into a claim about what a system must contain, real numbers before calculus and complex numbers before quantum mechanics, is what made it an argument.
That left one distinction to draw. Judging a mathematical system by its conformity to the universe sounds like Field’s usefulness, and it is a different standard, because conformity can be checked by observation and usefulness is judged by whoever is doing the deducing.