An Audacious Solution to the Mind-Math Problem

An Audacious Solution to the Mind-Math Problem

The mind-math problem is the problem of explaining how the mind and mathematical reality manage to come together: how do numbers and geometric figures get to be apprehended by the mind? Suppose we adopt a Platonic view of mathematical reality—it consists of abstract objects, existing outside space and time, independent of the physical and mental worlds. Suppose too that we regard the mind as existing in space and time, concretely, embodied in the physical brain. How, then, can the mind make contact with the mathematical realm? There can be no interaction, no contact, no common ground. If the mathematical were mental (or physical), then the mind would have a chance of becoming acquainted with it; and if the mental were mathematical, some sort of communion would be possible. But the two belong to different worlds, almost different universes. We would not be surprised, on metaphysical grounds, to learn that the twain never meet, so ontologically remote are they; and yet they manifestly do meet, intimately so. For we have mathematical knowledge, mathematical perception (apprehension), and mathematical beliefs (and even desires). It seems easy for them to meet. But how is that possible, given that there is no causal connection? Physical objects cause our knowledge of them (we are told), but abstract objects can’t do that—they have no causal powers (we are told). This problem then leads to attempts to close the gap: we can reduce the mathematical objects to something closer to the mind (ideas, notation), or we can equip the mind with special non-causal faculties that permit quasi-mystical communion with the Platonic world (intangible telescopes etc.) Neither approach meets with universal acceptance, and indeed are generally acknowledged to be far-fetched and revisionary. We clearly have such knowledge, but we can’t fit it into our preconceived assumptions—Platonism about mathematical truth, naturalistic causalism about knowledge in general (propositional and objectual).[1]

What notion of causality is in play in these reflections? The kind derived from modern science (allegedly) and the kind derived from common sense (allegedly). We call this “mechanism”—causation by proximal contact, impact, bodies in motion touching each other. That kind of causation is clearly inapplicable to the math-mind relation. But mechanism has long been out of favor and now looks like common sense gone awry, ever since gravitational action at a distance became accepted as real. However, gravity is not the right model for mathematics either, because numbers and geometric forms don’t have mass and don’t exist in space either, according to Platonism. Still, might there not be a broader notion of causality that applies to the relation between math and the mind? In earlier papers[2], I have suggested as much: logical relations, particularly entailment, can be viewed as a species of causal relation. I won’t repeat the arguments here, but their relevance to the present issue is immediate: mathematical reality, construed Platonically, causes mathematical belief, in the extended sense of “cause”. Moreover, it causes the brain to be configured in a certain way—that is, it is (part of) the causal explanation of the brain’s structure.[3] It is because numbers and figures are a certain way that people have the mental and brain states that they have. This is a far cry from mechanistic causation by proximate interaction; it is a sui generis type of causation or causal explanation. We can say that mathematical truth gives rise to mathematical knowledge, has it as a consequence. It is, indeed, hard to see how this could not be so: for it is scarcely conceivable that mathematical truth plays no role in the etiology of mathematical knowledge, as if it had nothing to do with what people believe mathematically. It is because 2 + 2 = 4 that people believe that proposition, to put it crudely. How could they come to know it by some other means, such as sensory perception of material objects? There must be some sort of causal generative connection here. The numbers must be exerting some sort of “force” that produces beliefs about them, though not any physical force with which we are familiar. We might call it the “mathematical force” just to have a name (or “mathemity” to mimic “gravity”). It is defined as whatever it is about numbers that makes them able to command belief—their propensity to invite belief. Once we apprehend them, they induce us to form certain beliefs about them and not others.

It may be said that this is all very mysterious and should not be entertained for that reason. But this is a bad argument: even mechanical causation is mysterious, as we have known since Hume. All causation is mysterious, but it doesn’t follow that it doesn’t exist. Thus, the way is open to accepting mysterious causal Platonism (we already accept mysterious Cartesian causal mechanism). This theory enables us to respond simply to the initial problem: there is no incompatibility between mathematical Platonism and a broadly causal conception of knowledge. We just need to jettison old-fashioned ideas about what causation can be. True, the result is pretty mysterious, but no more so than causation in general; and isn’t it really quite commonsensical, given that we have no trouble with the proposition that we believe what we do mathematically because of how things are mathematically?  It certainly isn’t because of anything else (sensations of color, aches and pains, the sound of number words). I will even venture to suggest that the ability of this view of causation to solve the mind-math problem, which has hitherto proved intractable, puts the underlying metaphysics of causation in a stronger light.[4]

[1] Paul Benacerraf’s well-known paper “Mathematical Truth” is the locus classicus here, but the problem is as old as Plato.

[2] See my “A New Metaphysics”, “Causal and Logical Relations”, and “Because”.

[3] This causal explanation may trace back to genetic selection: the genes make the brain they do because of certain mathematical truths, thus installing innate configurations. That is, we have basic mathematical knowledge innately in virtue of mathematical facts; similarly for basic physical knowledge.

[4] We could take a similar view of ethical knowledge: ethical facts cause ethical belief, though not in the mechanistic sense but the “giving rise to” sense. We have the ethical beliefs we do because of the ethical facts; these are the origin of the causal chains that lead up to ethical belief (at least some of the time). It is the badness of pain that makes me think that pain is bad, not (say) the emotion that pain produces in me or what people tell me. The explanation of ethical belief involves ethical truth (though other factors can come into it)—sometimes, if not always.