Mathematical Existence

The Platonist asks us to accept that mathematical entities exist independently of human thought in much the same way that material objects do. The central claim of both types of realism is that the entities in question pre-exist the existence of minds and would continue to exist even if minds did not. They do not derive their being from mental acts. Material objects are said to exist in space and time, while mathematical objects are said to exist outside of space and time; still both exist separately from minds, occupying their own region of reality. In both cases we are thought to have a conception of what this independent existence consists in—we understand what each type of realism is saying. In the case of material-object realism we can also envisage minds without external objects, as with brain-in-a-vat scenarios or dreams or hallucinations or simple illusions: the mind might be in the same state as it is now and yet no material objects answer to its representations. This possibility provides a clear sense in which material objects exist independently of minds: their existence is not entailed by the existence of corresponding mental states. So minds don’t entail a material world and a material world doesn’t entail minds. Material objects exist over and above minds. According to most views, matter and mind causally interact, which demonstrates their ontological independence—though it is possible that the two merely exist in parallel consistently with realism.

But here a disanalogy appears: for there is no analogue for mathematical objects of the brain-in-a-vat scenario. We can’t envisage a situation in which mathematical thoughts occur and yet correspond to no objects: the brain in a vat can’t be thinking mathematical thoughts and yet these thoughts are not about what they purport to be about. For example, you can’t be thinking that 2 + 2 = 4 and yet there are no such numbers as 2 and 4  (and no such things as addition and equality). We can’t subtract the objects from reality and leave the mathematical mind intact. You can’t hallucinate the number 3 as you can hallucinate a table, having merely an impression of that number. In the case of mathematics mind entails existence: there can be no mathematical mind without corresponding mathematical objects. Of course, this doesn’t yet tell us what the nature of those objects might be; what it does tell us is that the analogy with material objects breaks down at a crucial point. Mathematical existence is bound up with mathematical thought, while material-object existence is not bound up with material-object thought. Thus you can be a skeptic about the material world but not about the mathematical world: material objects might not exist, consistently with the existence of minds apparently about them, but mathematical objects must exist if there are minds that think about them.[1] To put it differently, mathematical intuition implies that there are suitable objects for it to be about (whatever their nature may be). You can’t think about numbers and yet there are no numbers for you to think about—but you can think about material objects and there not be any.[2]

This presents a challenge to Platonism. According to Platonism, numbers exist independently of minds, so it should be possible to separate them in thought from the minds that apprehend them; but that appears not to be possible on reflection. At the very least, the model of material-object existence breaks down. We can’t conceive of the mind as occupied about matters mathematical—performing a calculation, say—and yet no numbers are being apprehended and none exist; but we can conceive of a mind seeming to perceive material objects and yet failing to because there are none to be perceived. This is part of what realism about material objects asserts, plausibly so, but realism about mathematical objects cannot make a parallel assertion. It looks as if mathematical cognition entails mathematical existence, which is not what Platonism suggests: that is far more consonant with forms of mathematical mentalism. To be sure, it doesn’t entail that metaphysical position, but it does present a puzzle to the Platonist: why the failure of symmetry with the material object case if mathematical reality is as fully independent of the mind as Platonism suggests? The mathematical anti-realist will argue that mathematical objects without mathematical minds is a fantasy, and that the entailment from mathematical thought to mathematical existence shows that mathematics must be ultimately mental. How is the Platonist to explain the fact that there can be no mathematical thought without mathematical objects? Precluding skepticism in this way is a symptom of anti-realism, since it prevents us severing mathematical reality from mathematical cognition. We can’t say, “It might turn out that there are no numbers, despite my having mathematical thoughts concerning them”. And we can’t say, “Mathematical reality might be a complete hallucination” as we can say, “Material reality might be a complete hallucination”. But then we can’t articulate Platonism in mathematics in the way realism about material objects is typically articulated.[3]

[1] I put aside here fictional views of numbers; my point is just that we can’t get an analogue of a skeptical scenario for mathematics, i.e. the idea that our mathematical thoughts, which purport to be to be about real things, are actually about nothing. We couldn’t just be dreaming that numbers exist as we could be dreaming that a material world exists.

[2] Morality belongs with mathematics in this respect: no moral thoughts and feelings without moral values. We can’t envisage a world in which there are no moral values and yet people think of them. Again, this presents a challenge to moral realism, since one would expect the analogue of material-object realism to obtain: for it encourages the idea that moral values are mental constructions or projections not external realities not entailed by internal facts. The case of realism about other minds, on the other hand, mirrors realism about material objects, since we can envisage a situation in which we have the same thoughts about other minds and yet there are none (we are surrounded by zombies).

[3] Could we limit Platonism to the claim that mathematical objects can exist without being known? But it is difficult to see how that is consistent with the admission that mathematical thoughts entail the existence of mathematical objects—how could something whose nature is entirely non-mental be guaranteed to exist by mental acts?  Certainly we don’t think that is possible in the case of material objects.