What of mathematics? Numbers seem to be the paradigm of changeless objects. The number 7 is not somewhere; it does not move; it does not decay; it does not depend on anything for its existence; it stands in eternal mathematical relations with the other numbers and with the structures it figures in. If anything is a counterexample to the present account, it is the realm of mathematical objects: abstract, atemporal, aspatial, immaterial, and indubitably real (or at least real enough that mathematical practice across cultures and millennia has converged on remarkably stable conclusions about them). The Platonist about mathematics has the most natural case for objects exempt from the structural conditions of [§2.6](<2.6 Triconditionality of Nature.md>).

The move is venerable. Plato's Forms, the original Platonist proposal, are paradigmatic of changeless reality: the Form of the Triangle, of Justice, of the Good, each eternal and unchanging, the temporal world of becoming a degraded copy of the timeless world of being. Frege's *Grundlagen der Arithmetik* -(Frege, Gottlob, 1884) revives mathematical Platonism in modern dress: numbers are objective abstract objects, and arithmetic is the science of their structure. Gödel -(Gödel, Kurt, 1944); -(Gödel, Kurt, 1964) defends a robust Platonism on which mathematical objects are perceived through a faculty analogous to sense perception. Quine and Putnam's indispensability arguments -(Putnam, Hilary, 1971); -(Putnam, Hilary, 1979) make mathematical realism respectable to naturalists who would otherwise have shied away from abstract objects. Shapiro's *Philosophy of Mathematics: Structure and Ontology* -(Shapiro, Stewart, 1997) develops structuralism, on which mathematics is the science of structures rather than of particular objects. Maddy's *Defending the Axioms* -(Maddy, Penelope, 2011) and *Naturalism in Mathematics* -(Maddy, Penelope, 1997) defend a naturalized Platonism. Balaguer's *Platonism and Anti-Platonism in Mathematics* -(Balaguer, Mark, 1998) gives the comprehensive survey. The classical theist sometimes adopts a Platonism-plus theology: God's mind contains the abstract objects, and the realm of Forms is the divine intellect.

The target of the present reply is the structural claim: that mathematical objects are abstract entities exempt from the structural conditions of [§2.6](<2.6 Triconditionality of Nature.md>), and therefore that the Modal Identity of [§2.6.6](<2.6.6 Modal Identity.md>) (the result that the necessary possibility of change entails the necessary co-instantiation of Time, Space, and Substance) holds at most for the concrete world while a wider reality contains entities that violate it. The reply is not that Platonism is incoherent. One available response is nominalist or modal-structuralist: mathematical objects can be treated as descriptions of what structured reality, actual and possible, affords. But that is a substantive philosophy of mathematics, not something established merely by the structural argument. A robust Platonist may instead retain abstract objects while denying that the book's concrete change-realizing types apply to them.

Four forms of the move develop the position at four depths: the classical-Platonist form, the structuralist form, the naturalized form, and the plenitudinous form. Each fails for the same structural reason at increasing levels of refinement.

A note on the shape of the reply before the four forms. The reply does *not* claim that mathematical objects are abstract entities possessing their own time-like, space-like, and substance-like structure in some rarefied mode. That move would only relocate the Platonist's exempt object and quietly stretch Time, Space, and Substance into a second, analogical sense, trading the determinacy of the concrete bearers for a metaphor. The reply runs the other way. There is no number 7 standing apart from things, awaiting structural analysis; there are pluralities that can be counted, structures that can be instantiated, and *7* is the description of a feature they have. Mathematical objects are descriptions of what structured reality affords, not residents of a separate realm that must be shown to satisfy the conditions after all. Because the conditions govern the *possible* and not merely the actual ([§2.6.6](<2.6.6 Modal Identity.md>)), the descriptions reach as far as the possible reaches (to collections larger than anything the actual world contains, and to the infinite) without positing a single abstract object.

*First.* Classical Platonism (the Plato–Frege–Gödel line) holds that mathematical objects are particular abstract entities, each numerically distinct from the others, standing in eternal relations of mathematical structure. The number 7 is a particular object, distinct from the number 6 and from the number 8; the natural numbers stand in the successor relation; the relation is eternal and changeless.

Platonic Forms presuppose form. The word *form* (English *form*, Latin *forma*, Greek εἶδος) carries both the verbal sense (to take shape, to come into being) and the nominal sense (the eternal pattern, the abstract universal). εἶδος descends from ἰδεῖν, *to see*: it named originally what something looks like as it takes shape, the seen aspect of something undergoing process. Plato's move is to read the seen-aspect as an eternal pattern that prescribes the shape the process takes. We observe trees taking shape, mammals developing, planets accreting; the Platonist infers Forms of Tree, Mammal, and Planet prescribing what those processes converge on. The inference trades on the equivocation between the verbal sense (formation) and the nominal sense (the eternal Form): what follows from the observation is that formation-processes occur, and that the conditions of reality are such as to permit them, not that there is an eternal pattern prior to them prescribing what they must produce.

Three pressures on the classical-Platonist line are worth naming before the section turns to the math-Platonist refinements. *The natural-kind pressure.* Trees vary across species, climates, and life-stages; mammals span the spectrum from blue whale to shrew; the Form of Tree or Form of Mammal that would be at the same explanatory level as the Form of the Triangle becomes harder to specify the closer the inspection goes. Plato himself hesitates on this point in the *Parmenides* (130c–d), where Socrates flinches over whether there are Forms of hair, mud, and dirt. The modern reactivation runs through Wittgenstein's family-resemblance critique (Wittgenstein, Ludwig, 1953), §§65–71, Dupré's species pluralism (Dupré, John, 1993), and Boyd's homeostatic-property-cluster account of natural kinds (Boyd, Richard, 1991), each pressing the point that the kinds the world contains are not the kinds the Platonist's Form-talk requires.

*The process pressure.* A tree is not a static form: it is seed, sapling, mature, senescent, snag, and across seasons evergreen or deciduous. The Form of Tree would have to be either the form of a static cross-section (which is not what a tree is) or the form of the developmental sequence (which makes the Form temporal, conceding the present account). Aristotle presses this against his teacher in *Metaphysics* A.9, 991a8–b9 (where the Forms are shown useless for change and coming-to-be) and develops the immanent, hylomorphic alternative in *De Anima* II.1, 412a19–28; the process-philosophical tradition from Heraclitus through Whitehead -(Whitehead, A. N., 1929) and Bergson -(Bergson, Henri, 1907) develops the point systematically.

*The reflexive-substrate pressure.* The Platonist standing on Earth is held by what they describe (a planet) and owes an account of the Form of Earth that supports them. Retreat to property-universals (Armstrong, Tooley) abandons Forms of individuals; retreat to the *Timaeus* elemental geometries (53c–55c) confuses earth-the-element with Earth-the-planet; retreat to Plotinian emanation (*Enneads* V.9, VI.9) substitutes a Neoplatonist theology for the Platonist analysis; retreat to Lewisian modal realism (Lewis, David, 1986) changes the ontology under discussion. Each retreat concedes that the Form of the individual thing supporting the Platonist was not available.

The pressures share a structural diagnosis. The Platonist treats Forms as prescriptive: the Form of Tree explains why trees take the shape they do; the Form of Triangle explains why triangles have the angle-sum they have; the Form of Justice explains what just acts have in common. The structural account inverts the direction. That trees take the shape they do follows from the structural conditions of [§2.6](<2.6 Triconditionality of Nature.md>) being such as to permit trees: the necessary co-instantiation of Time, Space, and Substance yields a Reality in which tree-formation is among the possible processes. The descriptive content the Platonist reads as the Form of Tree is the description of what those conditions afford for that kind of configuration. The 180° angle-sum of a Euclidean triangle is not prescribed by a Form of Triangle; it is a structural consequence of the metric specification of the relevant space ([§2.4.7](<2.4.7 Distinction Requires Spatiality.md>) develops the metric-as-contingent-specification point) and spherical and hyperbolic geometries produce different angle-sums under different metric specifications, all licensed within the structural conditions. What the Platonist calls a Form runs downstream of actuality plus structural conditions, never upstream. This is the polarity inversion: what looks like a prior pattern prescribing actuality is a description of what actuality, under the structural conditions, produces.

To be a Form, at all, is to be a determinate *this* rather than *that*: the Form of Justice, not Beauty; the Form of the Triangle, not the Circle. But that determinacy is borrowed, not native. The Form of the Triangle is determinate because triangles (actual or possible) are determinate; its distinctness from the Form of the Circle is the distinctness of the configurations it describes. The determinacy the Platonist reads as a property of the Form is the determinacy of the structured (possible) reality the Form is a description of. The Form is not a further object that must itself be located in the order of distinctness, placement, and persistence; it is the description that order makes available.

Take the number 7. What is 7 apart from seven of something? The classical Platonist's answer is: a particular abstract object, distinct from 6 and 8, sitting in the eternal successor relation. The present answer is that *7* is the count-structure of any seven-membered plurality there could be, the way any such collection is when it is enumerated. Its distinctness from 6 and 8 is the distinctness of those collections (one fewer thing, one more); the successor ordering is the ordering of the collections under the operation of adding a unit; the stability of "7 is the successor of 6" across every time, person, and culture is the stability of what *any* possible seven-membered collection would be. These can be represented as facts about actual or possible pluralities. Actual concrete pluralities have concrete members, but merely possible, infinite, or abstract pluralities cannot simply be assumed to occupy Space, stand in temporal order, or be made of Substance. The modal-structuralist analysis must explain those cases rather than treating concreteness as automatic. The number needs no private time, space, and substance of its own, because it was never an object over and above the things; it is the description of their count. And because the count is a count of *possible* collections, it reaches numbers larger than all the things that happen to exist, and the transfinite, without any object's being conjured to bear it. Classical Platonism does not exhibit a counterexample to the account developed here; once the number is seen as the description of structured possibility rather than as an object exempt from it, the apparent counterexample dissolves.

Two homely cases make the reading concrete. First, measure. No one is tempted to house "1.5 meters" in a realm of abstracta; it is how long a thing is, a magnitude borne by the rod that has it. Number belongs to the same family: *7* is a *how many* in the way a length is a *how long*, a feature of a plurality rather than an object standing beside it. The Platonist's temptation is to treat the one magnitude we happen to reach by counting as though it were unlike every other magnitude, a thing rather than a measure. Second, matching. A shepherd who cannot count can still keep his flock exactly: he drops a pebble into a sack as each sheep leaves the fold and removes one as each returns, and a pebble left over at dusk is a sheep unaccounted for. He has tracked the number without ever naming it, and the sheep and the pebbles share nothing beyond standing in one-to-one correspondence with none left over. That correspondence is what *7* records, and it is why collections with nothing else in common can be identical in number: sameness of number is sameness under pairing, not joint participation in a shared object. This is the content of Hume's principle (Frege, Gottlob, 1884), §63: the number of *F*s equals the number of *G*s just when the *F*s and *G*s correspond one-to-one; on the present reading the principle fixes a fact about pluralities rather than introducing a name for an abstract particular.

*Second.* Structuralism (Shapiro, Resnik, Hellman) refines the classical position by holding that mathematical objects are not particular entities but positions in structures. The number 7 is not a particular abstract object but the seventh position in any ω-sequence (any infinite sequence ordered like the natural numbers, with a first element and a successor for every element); what mathematics studies is the structure of ω-sequences, not particular sequences. This is meant to handle the multiple-reductions problem (3 can be reduced to {{{{}}}} or to {{},{{}},{{},{{}}}}) by holding that any reduction that preserves the structure preserves what 7 is.

Structuralism makes the structural-conditions reply more direct, not less. The structures mathematics studies are constituted by determinate elements or positions and relations, directly supporting a difference-role. Some structures include asymmetric order; many do not. Their abstract stability does not by itself amount to temporal persistence or physical bearing. Upgrading the mathematical roles to T and Φ therefore requires a realization premise. Shapiro's account of structures as patterns whose positions are individuated by their relations to other positions in the same structure is a description, in the mathematical case, of the distinctness, ordering, and persistence that [§2.6](<2.6 Triconditionality of Nature.md>) names. The reply is not that Shapiro is wrong about mathematical structure; the reply is that Shapiro's account of mathematical structure describes what [§2.6](<2.6 Triconditionality of Nature.md>) names. The structural-conditions account is, on the present reading, structuralism extended to non-mathematical reality, the same insight applied to the wider domain of which mathematics is one structured region.

*Third.* The naturalized form (Maddy, Quine, Putnam) drops the strong realism about abstract objects and holds instead that mathematical objects are whatever the indispensability argument requires, the entities that are quantified over in our best scientific theories. The ontological commitment is to whatever is needed to make the science work; the metaphysics of the entities is left open.

The naturalized form does not aim to exhibit a counterexample to the structural account; it aims to deflate the metaphysical question. As deflation, it is irrelevant to the structural reply: whatever the indispensable mathematical entities turn out to be, they will be whatever the science requires, and the science requires distinctness, ordering, and persistence of mathematical facts. If the naturalist insists on a stronger reading (that mathematical entities have some metaphysical status that exempts them from the structural conditions) then the reply to the classical Platonist applies. If the naturalist remains deflationary, then the position is not in tension with the structural account.

*Fourth.* Plenitudinous Platonism (Balaguer) holds that every consistent mathematical theory describes a real domain of mathematical objects: the universe is mathematically plenitudinous, with every possible mathematical structure instantiated somewhere. This is meant to dissolve the access problem: we do not need a faculty for picking out particular abstract objects, because every consistent description picks out something.

Plenitude is a stronger claim than the classical Platonist makes, and the structural reply applies more straightforwardly. The plenitude of mathematical structures is not itself a further structured object but a description, at maximal extent, of what the structural conditions afford. To speak of the totality of consistent theories is to describe how such theories stand: individuated by their consistency conditions, ordered by relations of inclusion or interpretation, stable across all considerations of which structures exist. Each of those is a description of the distinctness, ordering, and persistence any possible structure would have, not a Time, Space, or Substance instantiated by some super-object 'the totality.' If the present account is right, plenitudinous Platonism is its mathematical correlate, not its refutation.

A kindred move is the divine-intellect Platonism: the abstract objects exist in God's mind, and their atemporality is the atemporality of the divine intellect. The reply combines the present reply with [§4.2.1](<4.2.1 The Changeless Ground.md>) (the changeless-ground reply): the divine intellect that contains the structured realm of abstracts is itself structured, and the structuredness is what [§2.6](<2.6 Triconditionality of Nature.md>) names. The classical theist's Platonism-plus-theology faces the structural reply twice over.

The modal-structuralist position (Hellman) is, on the present reading, not one more Platonism to absorb but the closest mathematical statement of the reply itself: mathematics is not about actual abstract objects but about possible structures. That is precisely the account given here: *7* is the structure of any possible seven-membered collection, and arithmetic is the study of what such possible structures are like. The only adjustment is the ground: the modality is not a primitive realm of its own but is fixed by the structural conditions of [§2.6](<2.6 Triconditionality of Nature.md>), which govern the whole space of the possible ([§2.6.6](<2.6.6 Modal Identity.md>)). Mathematics describes the possible; the possible is structured by Time, Space, and Substance; so mathematics, far from naming exceptions to the conditions, is one of the most general descriptions of what they afford.

*What converges.* Four forms and none exhibits the counterexample the position needs: classical Platonism's particular abstract objects resolve into descriptions of structured possibility, not exceptions to it; structuralism's structures are constituted by exactly the relational facts the account names; the naturalized form is either deflationary (and not in tension) or stronger-than-deflationary (and faces the classical reply); and plenitudinous Platonism is the most extensive description of what structure affords, not a counterexample to it. Across the four forms, no mathematical view by itself supplies a counterexample to the necessity of structure in concrete change-realizing worlds. The stronger nominalist conclusion, that mathematical objects are not objects at all but descriptions of possible structure, remains one proposed interpretation, not a result forced on Platonism by [§2.6](<2.6 Triconditionality of Nature.md>) alone.

Sibling counter-constructions: the changeless-ground objection, on which a timeless being grounds the structure from outside, is [§4.2.1](<4.2.1 The Changeless Ground.md>). The B-theoretic block objection, on which the temporal manifold is a static four-dimensional reality, is [§4.2.6](<4.2.6 Block Universe.md>). The McTaggart objection, on which time itself is unreal, is [§4.2.5](<4.2.5 The Unreality of Time.md>).

What the Platonist tradition genuinely captures is worth preserving. There is a sense in which mathematical truths are not merely conventional, not merely useful, not merely about marks on paper or sounds in the air. The persistence of mathematical truth across cultures and millennia, the convergence of independent investigations on the same theorems, the resistance of mathematical structure to revision in ways that empirical theories are revised, each speaks to something real about the subject matter of mathematics. The account of [§2.6](<2.6 Triconditionality of Nature.md>) is consistent with that sense. Mathematical structures are real structures; their reality is the reality of what the structural conditions afford, the count and pattern of every possible collection, described maximally and changelessly. The Platonist who reads the changelessness as exemption from [§2.6](<2.6 Triconditionality of Nature.md>) has misread the changelessness; what does not change is not a timeless object floating free of the conditions but the description of what every possible collection, every possible structure, would be, and that description does not change because the possibilities it describes do not. The stability is the stability of the possible, not the serenity of an abstract object.

Frege's *Grundlagen* (1884) was printed on paper that has decayed in the intervening century and a half. The arithmetic he was investigating did not decay. What did not decay, on the present reading, is not a timeless object exempt from [§2.6](<2.6 Triconditionality of Nature.md>) but a truth about every possible collection: that seven of anything is the successor of six of it. The number 7 is not somewhere because it is not the kind of thing that could be anywhere: it is the count any seven-membered plurality has, and pluralities are where their members are. It does not move or decay because descriptions of what is possible do not move or decay; the possibilities they describe are fixed by the structural conditions that govern the possible. The Platonist's number is not a counterexample to the present account, and it is not a hidden instance of it either. It is a description of what reality, under the conditions of [§2.6](<2.6 Triconditionality of Nature.md>), makes countable.


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← [4.2.3 Emergent Time](<4.2.3 Emergent Time.md>) | [4.2.5 The Unreality of Time](<4.2.5 The Unreality of Time.md>) →
