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.

Action and Trying

Davidson once memorably said, “We never do more than move our bodies; the rest is up to nature”. This aphorism has the sound of an illuminating truism, but does it stand up to critical examination? Suppose you are suffering from paralysis, total or partial, following an accident. Your physiotherapist asks you to try to move your arm and you find you can’t move it: wouldn’t you think, “I’m trying, but I’m not succeeding; nature won’t let me”? Your body is part of nature and it is not cooperating, so you can’t act; isn’t this just like trying to lift a weight that is too heavy for you? Your act of trying can’t overcome the dictates of nature, whether your own body or the world outside it. Wouldn’t it be more accurate to say, “We never do more than try to do things; the rest is up to nature”? There is our will on the one hand and nature on the other, i.e., the world outside our will. When you lift your arm do you move the bone in your arm but not the clothes on it—is the latter part of nature but not the former? What if your arm is partly prosthetic? What if you always carry a gun in your hand? The distinction between body and nature is artificial, but the distinction between will and nature is not (of course, the will is also part of nature). The correct aphorism is: We never do more than try to do things; the rest is beyond our control. That is, we only have direct control over our will; the rest is a matter of whether the world beyond cooperates. Adopting the terminology of basic actions, we can say that our basic actions are acts of trying (willing); anything else is non-basic, i.e., consequences of acts of trying. The body thus has no privileged position in the philosophy of action. What indeed is the body: does it include the hair on the body, sweat, clothing, tools, machines, other people? Nature and the body merge, with the will attempting to manipulate them. All we really, basically, do is will things; the rest is out of our control—that’s a matter of whether nature chooses to go along with our will. It’s not the body-nature division that matters (not that this is a real distinction); it’s the will-world division. So it would appear.

Is it true that whenever we act we try to move our bodies? And is that really all we try to do? Neither proposition is correct. Surely, we try to do many things beyond moving our bodies—we try to fix things, go places, have careers, find love. The intentionality of trying is wide in scope and not limited to the body’s movements. And do we always try to move our bodies when we try to do these other things? Do we try to contract our muscles, or activate our efferent nerves? We do not—we may not even think about these things. Our mind is not concentrated on our body: you might be trying to score a goal, but you don’t think about your leg at all—you take that for granted. Your brain causes your leg to move thus-and-so, but you don’t give it a second thought—you have your mind set on the goal and goalie. Trying to move your leg in a certain way may hinder what you trying to achieve—there are only so many things you can think about at the same time. Your attention is on the goal not the leg. So, it isn’t that trying to move the body is basic and essential to acting; trying is far more protean and plastic for that to be true. Trying goes with intention and desire, which generally concern ends not means. In principle, you could try without even having a body: you could be a disembodied mind that is suitably causally connected to the external world (isn’t that what God is supposed to be?). What is essential to action is the will (capacity to try) and a causal link to external reality; the body is just one means for getting things done, not a sine qua non. We never do more than will; the rest is up to causality. The basic actions are acts of trying. The body is not at the center of the philosophy of action—it is not even an essential component of the subject. So it would appear.

It might be objected, however, that we are underestimating the role of the brain. Doesn’t the brain (in terrestrial animals) cause and control the body, so it must be occupied about the body, even if the person (or other agent) need not pay the body much attention? This must be conceded: the brain has to have the wherewithal to initiate movements of the body, and this must be detailed and representational. The brain needs a “body image” in order to go about its business. And it must act with that body image “in mind”; it can’t ignore the body as the conscious agent can. Isn’t the brain then part of the philosophy of action of embodied creatures such as ourselves? If so, a philosophy of action that focuses exclusively on what the agent tries to do is incomplete; we need to add what the brain is up to as well. A human action (as well as the actions of other animals) consists of a mental act of trying and a physical act of brain initiation. The combination is the true nature of action as we find it: agent trying plus brain stimulation. In creatures lacking mind (think worms) action is just brain stimulation—efferent nerves and muscles—and such creatures can be said to act equally. Conscious trying is superimposed on this basis, and is presumably a later evolutionary development. Both aspects need to be acknowledged in an adequate philosophy of action—we thus have a double aspect theory of action. Indeed, we have a double ontology theory, since the act of trying is not to be identified with the brain’s action of causing bodily movements. The action of raising your arm consists of two actions—your act of trying to raise your arm and your brain’s action of innervating the relevant muscles. Both need to be described and explained, and integrated. A good philosophy of action has a mental component and a physical component, because two things are involved; an action is both of them together not one separately. The motor part of the brain confines itself to the body, down to the fine details; the conscious agent is more concerned with ends and results and has little time for the physiological machinery. What we call action straddles these two domains and it would be a distortion to limit it to one of them. The agent and the brain are both centrally implicated.[1]

[1] The work of Davidson on action and O’Shaughnessy on the will form the background to this paper. I am adding the brain as an essential component to the story. Both philosophers were too monistic, though in possession of important truths. Human action is Janus-faced. (Let me add that the philosophy of action is a remarkably tricky subject.)

Barbados

I am seriously thinking about moving to Barbados. The political situation in this country has become intolerable and has been for a while (I’m thinking of the political situation in American academic philosophy, but national politics is pretty bad too). Barbados is a British protectorate, not American, so you can get away from that demographic. I was there on vacation twenty years ago, mainly to learn kite surfing, and have good memories of the place and its people. I want to be near the water again. I wonder if any readers have personal experiences of the island which they would care to share.

Degrees of Mind and Body

We are accustomed to a sharp and rigid distinction between mind and body, between “the mental” and “the physical”. We also tend to think that these are absolute concepts: something is either one or the other with no degrees. One thing can’t be more mental than another, or more physical. These categories don’t admit of gradation; there is no sliding scale of mentality or physicality. But is this right? It seems to me that it is just another example of the human tendency to prefer discontinuity over continuity—to be uncomfortable with variations on a theme and happier with simple dichotomies. Actually, the mental and physical (mind and body) do come in degrees and things do vary in their status as mental or physical. This is in fact easy to prove and conceptually quite liberating.

As usual, the dictionary will put us on the right track (there is more philosophical wisdom in a good dictionary than in Kant’s Critique of Pure Reason). For “mental” the OED gives us “relating to, done by, or occurring in the mind”: clearly, that admits of degree. Something could be more or less closely related to the mind, more or less distant from the mind, more or less done by the mind, and partially occurring in the mind. But we need to know what “the mind” is before these linguistic intuitions can be accorded conceptual (ontological) weight. For “mind” the OED has “the faculty of consciousness and thought” and “a person’s ability to think and reason; the intellect”. This is commendably definite and quite restrictive: the mind is equated with conscious reason, intellect, thought—not perception, emotion, sensation, or character. These things are thus not deemed “mental”, since they are not done by, or occur in, the intellect. Memory is not included, but we can suppose it closer to the mind, as so defined, than sense perception and bodily sensation. Character traits are quite far from the mind (intellect) and are therefore only weakly mental; perception and sensation more strongly so, emotion more strongly still, memory almost there. We have grades of mentality—a sliding scale. In humans, thoughts about thoughts, the intellect directed onto itself, might qualify as the most strongly mental; sensations in the bowel as the most weakly mental. It all depends on proximity to the intellect—similarity to what the intellect is or does. Thus, mentality comes in degrees—some things are more mental than others.

The story with “physical” is very similar. The OED gives us “relating to the body as opposed to the mind”, citing the phrase “a physical relationship”. Again, that can intuitively admit of degrees—things can be more less related to the body, more or less connected to it, or like it. But we need to know more about what exactly the body is before we can have a definite idea of what “physical” means. The dictionary obliges us with “the physical structure, including the bones, flesh, and organs, of a person or animal”, adding as “technical” “a material object”.  Clearly, things can be more or less similar to a body as so defined: trees, rocks, liquid water, clouds, volumes of air, regions of space. Some things are more body-like than others in the animal-body sense. Thus, some things are more “physical” than others—trees more than clouds. What about the technical sense? Here we need to move a bit towards physics itself. A material object is understood to be a discrete bounded solid thing located in three-dimensional space—as opposed to such things as heat, light, radiation, gravity, magnetism, and fields of force. It used, indeed, to be debated whether such items were really physical at all, given the paradigm supplied by the material object. Such debates presuppose a dichotomous attitude to nature—isn’t it better to say that physicality comes in degrees? Some things are more like material objects than others—with neutrinos and force fields at some distance from the paradigm (electrons and protons closer). Thus, physicality comes in degrees. Black holes (misnamed—they aren’t holes in anything) are frightfully physical, being of enormous mass and density, stars somewhat less so, oceans still less, air even less, empty space hardly physical at all. It is pointless to try to force some sort of dichotomy onto this sliding scale; better just to speak of degrees of physicality (body-hood). This spares us pointless verbal quibbles and pseudo-questions. The dichotomous use of “physical” belongs to an earlier stage of physics, i.e., Cartesian mechanism (the dichotomous use of “mental” is similarly rooted in the past, mainly religious, i.e., incorporeal soul versus material substance).

It would be nice if the two scales overlapped—if one graded smoothly into the other. Could the least physical thing also be the least mental? Could there be a borderline case? I think this has been supposed by some very free thinkers: thus, we hear talk of “energy fields” that are vaguely spiritual, light is supposed to be mindlike, the impalpably physical has been likened to the soul. Ghosts, if such there be, are thought to hover on the boundary between the mental and the physical. This would be intriguing if true, because it might give us a handle on the mind-body problem; but alas, it is just loose talk, metaphor, misty poetry. There is really nothing that qualifies as genuinely both mental and physical—though some things combine the two. Nothing is both like the animal body and also like the intellect—both flesh and bone and also thought and reason. True, actions can be described physically and mentally—as material movements of the body and as intentional. But this is because they are double-aspect things: the aspects are not midway between the mental and the physical. The movement isn’t like a thought and the intention isn’t like a limb. Still, this supposed middle ground is worth pondering as a conceptual possibility, because it offers the prospect of psychophysical linkage—intelligible emergence. There could perhaps be something, now unknown, that straddles the divide, something close enough to both paradigms. Anything like the intellect and also like the animal body would be a candidate for being the key to unlocking the mind-body problem.[1]

[1] One result of adopting a degree conception of the mental and physical is that we get different strengths of materialism. Suppose the conscious intellect were reducible to the gross anatomy of the brain: that would be an extremely strong form of materialism, because it would take the paradigm of the mental and reduce it to the paradigm of the physical. The strongly mental would reduce to the strongly material (I don’t think anyone has ever championed such a view). By contrast, it might be claimed that memory or unconscious belief is reducible to electrical properties of neurons—these both being relatively weak instances of the mental and physical. Or again, it might be held that character traits consist of energy fields (unknown to current physics) surrounding the pineal gland—where these are understood as quite remote from the usual paradigms. In other words, we can envisage different degrees of materialism, depending on what mental and physical phenomena we are considering.