Abstract Entities, Universals, and Properties

Abstract

Distinct things share properties — three ripe tomatoes are each red — and the one-over-many argument takes this to require a single repeatable entity, a universal, that the many instances have in common. Realists (Plato's transcendent Forms, Armstrong's immanent universals) accept such entities; nominalists reject them and must account for resemblance by other means (predicate nominalism, class nominalism, resemblance nominalism); trope theorists occupy a middle position, admitting particular property-instances that resemble but denying any shared entity. Behind the dispute stands the abstract/concrete distinction and the special case of mathematical objects, whose apparent indispensability to science is the strongest argument for Platonism. This paper gives the Force of Time's account, which resolves the realism–nominalism quarrel by placing the universals inside the world. The {2,3,5,π} lattice — the fixed structure of Τ that fixes the fine-structure ratio, the spectral wavelengths and the rest — is a system of real, repeatable, mind-independent universals; but it is not an abstract realm apart from concrete reality. It is the concrete structure of the one substance. The theory is thus a realism about universals (they exist and are shared) and an immanentism (they are in things, not beyond them), and it makes the truthmakers of property-attributions the configurations of Τ on the lattice. Mathematical objects are real for the same reason — the lattice numbers are not describers of nature but constituents of it. We give the position as numbered propositions.

1. The one over many

The problem of universals begins with an argument of great simplicity (Fig. 1). Many distinct things are, say, red; they are alike in this respect; their likeness is a fact needing explanation; and the natural explanation is that there is some one thing — redness — that each of the many has. If we accept it, we have admitted a universal: an entity that is wholly present in each of its many instances and repeatable across them, unlike the particulars, which are each in one place at a time. The question of universals is whether such entities exist, and if not, what does the explanatory work they were introduced to do [4,6].

2. Realism, nominalism, and tropes

Three families of answer divide the field (Fig. 2). Realism affirms universals: for Plato they are transcendent Forms existing apart from their instances; for Armstrong [4] they are immanent, existing only in the things that instantiate them, and discoverable a posteriori. Realism explains resemblance directly — things resemble by sharing a universal — but pays in ontology, admitting a category of repeatable, abstract entities. Nominalism refuses that category: there are only particulars, and resemblance must be handled by predicates we apply, or classes things fall into, or primitive resemblance among the particulars themselves. It is economical but owes an account of why the predicates apply as they do. Trope theory splits the difference: it admits properties but as particulars — this tomato's redness is numerically distinct from that one's — and grounds talk of shared properties in the resemblance of tropes, not in a shared entity.

3. The abstract, the concrete, and mathematics

Universals, if real, are usually classed as abstract — outside space and time, causally inert — and this raises the general problem of abstract entities, of which mathematical objects are the hardest case. The indispensability argument observes that our best science quantifies ineliminably over numbers, and concludes by Quine's own criterion that we are committed to them; the Platonist takes the commitment at face value, the nominalist labours to paraphrase it away. The status of numbers is thus not a side-issue but the sharp end of the problem of universals: if there are abstract objects at all, the numbers are they.

4. The Force of Time: universals in the substance

The Force of Time cuts across the realism–nominalism dispute by relocating the universals (Fig. 3). It agrees with the realist that there are real, repeatable, mind-independent structures that distinct things share: the {2,3,5,π} lattice is exactly such a system. The fine-structure ratio 9/125π², the carrier values, the geometry that fixes the water angle — these are not features we project but structures the world instantiates wherever the relevant configuration of Τ occurs, wholly and repeatably. That is realism about universals. But the theory denies what makes realism costly: that these universals form an abstract realm apart from the concrete world. The lattice is not elsewhere. It is the structure of Τ itself — the concrete substance of which everything is made. The universals are therefore immanent in the strongest possible sense, not merely present in their instances but identical with the structure of the one stuff those instances are configurations of.

5. Numbers as constituents, not describers

This dissolves the special problem of mathematics. On the Force of Time the numbers of the lattice are not abstract objects that our theories describe from outside; they are constituents of the world's fabric. To say that the fine-structure ratio is 9/125π² is not to relate a physical magnitude to a Platonic number in a separate realm; it is to state the actual lattice structure that Τ has at that address. The indispensability of mathematics to physics is then unmysterious and expected: mathematics is indispensable because the world is arithmetical in its substance, built on {2,3,5,π}. The numbers are real, as the Platonist insists, and concrete, as the nominalist wants — because they are the structure of the concrete substance.

6. Properties, truthmakers, and what this claims

A property-attribution is made true, on this account, by a configuration of Τ: 'this is red' is true in virtue of Τ being arranged, at the relevant address, in the lattice pattern that constitutes redness. Properties are neither transcendent Forms nor free-floating tropes but modes of the one substance, individuated by their lattice structure; their sharing is the recurrence of the same structure at many addresses. Truthmakers, the concern of the companion literature [7], are thus supplied uniformly: the truthmaker of any property-claim is the configuration of Τ that satisfies it. The claim of the paper is that this immanent realism keeps the explanatory advantage of realism (resemblance is the sharing of real structure) without its ontological cost (no realm of abstracta), and keeps the concreteness the nominalist prizes without its explanatory debt (the structure is really there to be shared). Its adequacy turns on whether the lattice can in fact furnish the properties the world displays — the running burden of the series.

7. Conclusion

The problem of universals is the problem of the one in the many, and the Force of Time answers it by finding the one within the many rather than above them. The shared structures are real and repeatable — the realist's insight — but they are the structure of the single concrete substance — the nominalist's scruple honoured. Redness, and the number three, are equally in the world, because the world is made of a substance whose fabric is arithmetical.

Figures

The position, in full

References