Abstract Objects Beyond Logic

Argument

A broader ontological issue concerns abstract objects generally: numbers, propositions, sets, and modal truths. These entities appear to be non-physical, non-spatiotemporal, and (in many accounts) necessary rather than contingent.

Naturalism, especially materialism, faces a dilemma: either deny such entities (at high cost to mathematics and rational discourse), reduce them to physical states (which struggles to preserve necessity and universality), or accept them as brute “platonic” realities (which undercuts strict physicalism).

Theism supplies a comparatively unified account: abstracta can be understood as grounded in a necessary intellect (e.g., as divine ideas), preserving their necessity without positing an autonomous realm of abstract entities that simply “exists.”

Even if one does not press the theistic identification immediately, the argument pressures atheistic materialism to clarify whether it can accommodate necessary abstract truth without quietly abandoning its own ontology.

Formal Argument

P1. Rational discourse presupposes necessary truths (e.g., logical laws, mathematical relations, propositional entailments) that are not dependent on any particular physical token or contingent physical arrangement.

P2. A strictly physicalist ontology contains only contingent spatiotemporal particulars and their causal relations.

P3. Contingent spatiotemporal particulars cannot, by themselves, ground the necessity and universality of abstract truths without either (i) reducing necessity to contingency (loss of necessity), or (ii) smuggling in an irreducible abstract realm (abandoning strict physicalism).

C1. Therefore, strict physicalism is metaphysically incomplete: it cannot account for necessary abstract truth without revision.

P4. If necessary abstract truths exist, then either (i) Platonism, (ii) divine conceptualism (necessary mind), or (iii) a non-theistic hybrid that is not strict physicalism must be true.

P5. Platonism posits an autonomous abstract realm that lacks a unifying explanation of its existence and normativity (its “authority” over rational agents).

P6. Any hybrid view that treats abstract truths as brute necessities likewise lacks a unifying explanation of why such necessities obtain and why they are normatively authoritative.

C2. Therefore, non-theistic accounts (Platonism or brute-necessity hybrid views) leave necessary truths and their normativity as unexplained primitives.

P7. A necessary mind grounds necessary truths as contents of intellect and explains their normative authority in a unified source.

C3. Therefore, divine conceptualism (a necessary mind) is the most unified explanation of necessary abstract truths and their normativity.

Analogy

A library can contain many copies of a statute, printed on paper in different fonts and editions. The statute’s authority and content are not identical to any one copy: you can burn a particular book without destroying the statute, and you can reprint the text without altering the law’s meaning.

If you tried to identify the statute with ink patterns on pages, you would be unable to explain how the “same law” persists across different physical tokens. Likewise, mathematical and logical truths persist across different inscriptions, minds, and media. Treating them as merely physical patterns confuses the vehicle with the content and struggles to preserve the invariance we actually rely upon.