Nominalist Programs

The indispensability argument puts the nominalist about abstracta in a specific position: it is not enough to find platonism epistemically uncomfortable, since science apparently quantifies over numbers and we believe science. The nominalist must therefore do something — either show that the mathematics can be removed, or show that its use does not commit us. This page surveys the programs, which divide into a hard road and an easy one, and closes with the position that accepts abstracta but makes them cheap.

Field's hard road

Science Without Numbers (1980) is the most ambitious attempt and remains the benchmark. Field accepts the whole Quinean framework — take science at face value, read the commitments off the quantifiers — and tries to show that premise 2 of the indispensability argument is false: mathematics is dispensable.

The strategy has two halves.

Nominalise the physics. Field reformulates Newtonian gravitational theory quantifying only over spacetime points and regions, using primitive comparative predicates ("$x$ is between $y$ and $z$", "$xy$ is congruent to $zw$") in place of numerical functions. The model is Hilbert's axiomatisation of geometry, where synthetic axioms about points and lines are shown by a representation theorem to be equivalent to the numerical theory: any model of the synthetic axioms admits a coordinate assignment, unique up to the relevant transformations. The numbers become a convenient description of a structure that is itself nominalistically specified.

Show mathematics is conservative. Even dispensable, mathematics is enormously useful, and the nominalist must explain why using it is legitimate. Field's answer: mathematics is conservative over nominalistic theories — if $N$ is a nominalistic theory and $M$ is mathematics, then any nominalistic consequence of $N+M$ is already a consequence of $N$. Mathematics adds no content about the physical world; it is a device for shortening derivations. Note the shape of this claim: it explains why mathematics is safe, not why it is useful, and the difference matters.

The programme's difficulties are substantial and instructive:

• Spacetime points are a large concession. Field's ontology contains continuum-many points with a rich structure. Whether this is genuinely more austere than accepting the real numbers, or a relabelling of them, is disputed — the nominalist has arguably kept the mathematics and renamed it "physics." The substantivalism debate then becomes load-bearing for the philosophy of mathematics, which is an unexpected dependency.
• Modality in the metatheory. Conservativeness is a modal claim about all models, and proving it uses set theory. The nominalist appears to need the very apparatus being eliminated, at one level up.
• Quantum mechanics. The programme was carried out for a classical theory whose structure is unusually congenial. Extending it to quantum mechanics, where the state space is not the manifold of physical events and where the relevant structure is a Hilbert space, has not been achieved and is widely doubted to be achievable.
• Not all applications are representational. Some scientific uses of mathematics — explanations that appeal to a number's being prime, or to a topological impossibility — do not look like shorthand for a nominalistic structure at all. These "explanatory indispensability" cases are the current front line.

The easy road

An alternative strategy accepts that mathematics cannot be removed but denies that using it commits one to its objects. It is called the easy road because it skips the nominalisation.

• Figuralism (Yablo): mathematical language is figurative, on a par with "the average family has 2.3 children" or "he did it for her sake." We do not believe in averages or sakes; the figure lets us express something about real things more efficiently. The mathematics is a representational aid whose apparent commitments are no more serious than those of a metaphor.
• Weaseling (Melia): one may assert a theory and then take back part of what it says — "there are numbers... but of course there aren't." What is communicated is the nominalistic content, and the retraction is intelligible even though the theory as stated implies otherwise.
• Modal structuralism (Hellman): mathematical claims are claims about what would hold in any possible structure of the appropriate type. Nothing actual need exist; the ontology is traded for primitive modality. Whether that is a saving depends on one's view of possible worlds, and if one is a Lewisian realist about them it is emphatically not.
• Constructibility quantifiers (Chihara): replace assertions that mathematical objects exist with assertions that certain open sentences are constructible. The domain becomes possible linguistic constructions rather than abstract objects.

Colyvan's objection to the whole family is that the easy road cannot be taken while granting that mathematics is genuinely explanatory in science. If the primeness of a cicada's life-cycle length explains its evolutionary success, an account on which the number is a figure of speech owes a story about how a figure of speech explains. The easy-roader replies that the explanatory work is done by the physical structure the mathematics represents.

Neo-Fregeanism: platonism made cheap

A quite different response accepts abstract objects and denies that they are expensive. The neo-Fregean programme of Wright and Hale builds arithmetic from Hume's Principle:

$$\#F=\#G\leftrightarrow F\approx G$$

— the number of $F$s equals the number of $G$s if and only if the $F$s and $G$s can be put in one-to-one correspondence. The right-hand side quantifies only over the $F$s and $G$s; the left introduces numbers as objects. Frege's Theorem is the striking technical result: second-order Peano arithmetic is derivable from Hume's Principle in second-order logic alone, without Frege's inconsistent Basic Law V.

The philosophical claim is that abstraction principles are implicit definitions — they stipulate the meaning of a new term-forming operator, and their truth is available a priori. If so, we get numbers without the access problem: knowing about them requires no causal contact, because the principle that introduces them is analytic of the concept. Platonism, but with the epistemology of stipulation.

Two objections dominate:

• The Bad Company problem. Abstraction principles of the same form can be inconsistent — Basic Law V is one — or individually consistent but jointly unsatisfiable, or satisfiable only in domains of a particular size. Since form alone does not distinguish the good from the bad, the neo-Fregean needs a criterion of acceptability, and any criterion strong enough to exclude the bad company seems to presuppose knowledge about what exists, which was to be the output rather than the input.
• The Caesar problem. Hume's Principle fixes when the number of $F$s equals the number of $G$s, but says nothing about whether the number of $F$s is Julius Caesar. An implicit definition that leaves identity conditions incomplete has not fully determined its objects.

The interest of the position for this section is that it is a deflationary platonism — it accepts the entities and denies that the acceptance is a substantial commitment. That places it next to the deflationary meta-ontology treated later, where the same move is made about existence questions generally.

God and abstracta

A final pressure on the debate comes from theology, and it is a genuine ontological argument rather than a piece of piety. Classical theism holds that God is a se — dependent on nothing — and the sovereignty-aseity intuition extends this to a claim that everything other than God depends on God. But abstract objects are standardly held to exist necessarily and independently, which contradicts it.

The options are the same ones this section has been cataloguing: nominalism (there are no abstracta, so nothing competes with divine aseity), theistic activism (abstracta exist but are constituted by divine thoughts — which runs into a bootstrapping problem, since God's having properties presupposes properties), or the concession that some things exist independently of God. The theistic case is developed on the goodness, simplicity, and aseity page.

Where the programs sit

These programs are the inventory question worked out in the hardest case, and they are also the clearest demonstration of the method the criterion of commitment licenses: given an unwanted commitment, either find the paraphrase or accept the entity. Field's is the most systematic paraphrase ever attempted in ontology, and its partial success and partial failure are more informative than either outcome alone would be.

They also illustrate the discipline the commitment page demands of paraphrase — that it not covertly reintroduce what it eliminates. Every program here is charged with violating it somewhere: Field with spacetime points and modal metatheory, Hellman with primitive modality, Chihara with possible constructions, the neo-Fregean with second-order logic. Whether any of these is a genuine reintroduction or an acceptable primitive is the live question, and it is not one the criterion can settle.