Argument from Contingent Abstract Objects

When we do mathematics, we seem to be discovering facts about objects like the number 7 or the set of prime numbers, rather than inventing arbitrary conventions. These objects, if they exist, are strange: they are not located anywhere in space, they do not participate in causal chains, and yet mathematicians and scientists treat truths about them as objective and often surprising. This apparent realism about mathematical objects, often just called Platonism, has been remarkably durable in the philosophy of mathematics despite its metaphysical oddity.

A similar puzzle arises for propositions, the things we say are true or false, believed or doubted, asserted or denied. The proposition that Paris is the capital of France seems to be a single, shareable object that many different sentences in many different languages can express. Propositions, like numbers, seem to be abstract, necessarily existing entities, yet they also carry semantic content: they are about something, and they have truth conditions. This combination of abstractness and intentionality is what makes propositions especially interesting for natural theology.

If numbers and propositions exist, they do not seem to exist contingently in the way that tables and planets do. It is not merely actually true, but necessarily true, that 2 plus 2 equals 4 and that nothing can be red and green all over in the same respect. This necessity is part of what makes mathematics different from empirical science: mathematical truths are not confirmed by observation but proven by demonstration, and once proven they hold in every possible circumstance.

Necessity of this kind raises a distinctive explanatory question. Contingent objects invite causal explanations: why does this object exist rather than not? But necessary objects, by definition, could not have failed to exist, so a causal explanation in the ordinary sense seems inapplicable. The question shifts from 'why does it exist' to something more like 'what is its ultimate ground or nature,' and this is precisely the kind of question theistic conceptual realism tries to answer.

Contemporary philosophers such as Greg Welty and Paul Gould have developed and defended a view called theistic conceptual realism, according to which abstract objects like numbers, properties, and propositions are best understood as divine ideas, the necessary contents of an eternal and necessarily existing mind. On this view, realism about abstracta is fully preserved (numbers and propositions really exist, objectively and mind-independently of any human mind), but their ultimate ontological ground is a necessarily existing divine intellect rather than a free-floating realm of Platonic forms.

This view has ancient roots. Augustine argued that the eternal truths and exemplars that Plato located in a separate realm of Forms are better understood as ideas in the mind of God, since a mind is a more intelligible bearer for such content than an impersonal abstract realm. Contemporary conceptual realists update this Augustinian move with more precise treatments of modal logic, the semantics of propositions, and the metaphysics of properties.

The central appeal of the view is that it explains why abstracta bear content at all. A bare Platonic object, on the traditional view, is just an inert abstract particular; it is puzzling why such an object would be 'about' anything, in the way propositions are about states of affairs. Minds, by contrast, are the paradigm bearers of intentionality. If propositions must be about something, and intentionality requires a subject, then propositions plausibly exist as the thoughts of a mind, and since they exist necessarily, that mind must exist necessarily too.

Not every realist about abstracta accepts a theological grounding. Mark Balaguer has defended what he calls full-blooded or plenitudinous Platonism, according to which every mathematically consistent structure exists as an abstract object, with no further explanation needed or possible; mathematical existence is simply brute. On this view, asking 'what grounds mathematical objects' is a category mistake, since necessary objects require no further explanatory ground beyond their own internal consistency.

Theistic conceptual realists respond that even if bare mathematical objects like numbers could plausibly be treated as brute, this strategy is much less plausible for propositions specifically, because propositions are not merely abstract but representational. A number does not seem to be 'about' anything, but the proposition that snow is white clearly is about something, namely snow being white. This representational or intentional feature is what conceptual realists argue calls out for a mind as its ground, in a way that mere numerical structure may not.

Not all theists accept the conceptual realist framework. William Lane Craig, in his extensive work 'God Over All: Divine Aseity and the Challenge of Platonism,' argues that theists should instead be anti-realists about abstract objects altogether, denying that mathematical and semantic discourse commits us to their real existence. On Craig's neutralist or fictionalist approach, sentences like 'there is a prime number between 4 and 6' can be true without any real, existing number being referred to, much as fictional discourse about Sherlock Holmes can be meaningful without Holmes existing.

Craig's motivation is partly theological: he worries that if abstract objects exist necessarily and independently of God, this compromises divine aseity, the classical doctrine that God alone exists in an underived, self-sufficient way. Conceptual realists respond that their view actually protects aseity better than Platonism, since abstracta on their view depend entirely on God's mind rather than existing as brute uncreated objects alongside God; the dependency, they insist, runs from abstracta to God, not the reverse.

The debate between theistic conceptual realism, Platonism, and theistic anti-realism ultimately turns on prior commitments in philosophy of mathematics that are contested independently of theism. Mark Colyvan and other naturalist philosophers defend indispensability-based realism on purely secular grounds, arguing that our best scientific theories commit us to mathematical objects regardless of any theological framework, which shows that realism about abstracta does not by itself favor theism over naturalism.

What the argument from contingent abstract objects shows, at its most modest and defensible, is a conditional claim: if one accepts realism about necessarily existing, content-bearing abstract objects, theism offers a more unified and arguably more intelligible metaphysical home for them than either brute Platonism or a naturalistic multiverse of abstracta. Whether that conditional claim moves an audience depends heavily on how compelling they already find realism about propositions and their intentional content.