Abstract (and Other) Objects
Abstract (and Other) Objects
I propose to discuss an incredibly difficult question and to suggest an incredible answer to it—an answer that only ineluctable logic could recommend to our modern sensibility. It concerns the threefold distinction between the physical, the mental, and the abstract. Let’s first try to get the abstract in our sights (this is hard enough), starting with the OED: the adjective “abstract” is defined as “theoretical rather than physical or concrete”, and “derived, extracted”. Thus, the abstract is defined by contrast with the physical or concrete, and implies an operation of derivation or extraction. We may gloss this as: “derived or extracted from something physical or concrete”. This corresponds with the idea of abstraction as a mental act—the abstract is what is derived byintellectual abstraction. It is thus something secondary, dependent—both ontologically and epistemically. The abstract object depends for its existence and nature on the concrete object from which it is an abstraction, and we acquire the concept of the abstract object by a mental act of abstraction. For example, the number 2 depends on the (concrete) objects so numbered, and we have the concept 2 by a mental act of abstraction (derivation, extraction). This is clearly not the notion of the abstract entertained by Pythagoras and Plato (and Frege and Godel); they viewed abstract objects as primary existences—as more real than concrete perceptible objects, or at least equally real. If anything, Pythagoras and Plato viewed the concrete as dependent on the abstract, ontologically and epistemically. And there have been plenty of mathematicians who would endorse this picture—for them, mathematical objects are directly apprehended and depend for their existence on nothing else. So, the dictionary definition is quite wrong as far they are concerned; or rather, they would demur from describing numbers etc. as “abstract” in the dictionary sense. They would choose to describe them in other terms—as Plato described them as Forms. In truth, it is hard to find satisfactory language for the doctrine in question—quite possibly because the conception is hard to pin down. The concept tends to be defined negatively, by contrast with “physical” or “mental”. If we imagine alien thinkers who are natural-born Pythagorean Platonists, but shaky on the mental and the physical, we might suppose them to have a robust ontology of the “abstract” (while not using that term), and only negative definitions of the mental and physical (what is not what mathematical objects are). In any case, our definition of the abstract would not be natural to them, and is conceptually lacking in our own case. But let’s suppose such entities to exist in whatever sense Pythagoras and Plato supposed, as autonomous, independent constituents of reality, even if we find it difficult to get a conceptual-linguistic handle on them. What, then, about the other two categories—are they properly defined?
Here the situation is not much better than with the concept of the abstract: neither concept is clearly defined. The concept of the “physical” is hopelessly undefined; this is a familiar story that I won’t repeat. The OED gives us “relating to the body as opposed to the mind”. That is about the best we are going to get, but everything hangs on the word “mind”: do pain and other bodily sensations relate to the mind or the body? Surely, the body, but they are usually classified as mental by theorists. What about the senses, particularly touch? The body, of course. Thought relates to the mind, but most of what we habitually call “mental” today does not. I would say that all of the mind relates to the body, particularly the brain, as opposed to some supposed immaterial entity (the “soul”); so, the mind is all physical, according to the dictionary. So be it: let the mind be physical in that sense—the distinction then vanishes. Nor is the concept of the mental well-defined: are character traits mental, or the unconscious, or skills, or language? There is no “criterion of the mental”. Both concepts are ill-defined, virtually meaningless. I think they should be abandoned in theoretical taxonomic contexts; they are obsolete holdovers from religious doctrines of the soul and immortality. The body and brain have various kinds of attributes—that’s all we need to say.[1] From the point of view of nature, no deep ontological division exists; there is no natural kind consisting of all and only (so-called) mental things. Nor is there a natural kind of (so-called) physical things. We could instead speak simply of organic and inorganic and give up on the (metaphysical) mental-physical distinction. This is a Spinoza-like position: the body has the power to produce all the attributes we and other animals possess, including consciousness—there is just one substance. It is also Darwinian: everything evolves by natural selection acting on the body. There is no separate soul, no supernatural basis for the mind. Accordingly, the traditional threefold distinction is defective beyond repair: each concept is deeply flawed and should not be used to shape our considered ontology. We need a new taxonomic scheme.
So, what would the new and improved ontology look like? Granted we accept the reality of mathematical objects, it will be a dualist ontology: Forms and Bodies, basically. We will have geometry and arithmetic on the one hand and organic and inorganic bodies on the other. Simplifying, we have material things and immaterial things—Forms not being material. Spinoza plus Plato. There is no Cartesian dualism, but there is a Platonic dualism. There is no Mind-Body dualism, but there is a Body-Form dualism. This means that the problems of dualism will carry over to the new dualism: how exactly are the two spheres of reality related? We can envisage a familiar array of options: attempts to reduce one sphere to the other, eliminative positions, and interaction problems (Plato was already aware of these). The two realms have quite different natures (they are different metaphysical natural kinds) yet they come together, because mathematics applies to the natural world. None of this means that the so-called mind-body problem will disappear; but it does mean that we cannot frame it in the post-Cartesian (pre-Darwinian) style. Trying to conceptualize this new dualism will not be easy, because we don’t have concepts naturally suited to the task, particularly concepts of the “abstract”. There is much here that is “wide open and extremely confusing”.[2] Bodies are spatiotemporal but Forms are not; bodies change and evolve but not Forms; bodies can be perceived via the senses but Forms cannot. Could we get by with an ontology of bodies and sets of bodies? Can our current conceptual scheme take the strain? Could there be a deeper unity that encompasses both? Could bodies have tiny Forms in them that combine to create larger Forms (“pan-Formism”)? Are some things intrinsically more mathematical than others? Are bodies appearances of Forms? Is it possible to be an idealist about Forms? Is the philosophy of mathematics really the most basic area of philosophy (the “mathematical turn”)? Is the concept of body inescapably anthropocentric? Is the Body-Form problem soluble? Is there anything it’s like to be a number?[3]Does the brain have a hidden mathematical essence? And so on.[4]
[1] We can certainly raise questions about the relation between these attributes (say, consciousness and neural states), but we won’t be treating these attributes as “mental” in some inclusive sense (Wittgenstein would approve).
[2] I am quoting Kripke.
[3] This may seem like a wild question, but is it any wilder than asking the same question about atoms? If numbers are real constituents of reality that enter into bodily facts (including facts about conscious experience), then we can envisage a metaphysics that seriously ponders the question.
[4] Much of metaphysics has centered around the relation between the “mental” and the “physical”, poorly defined as those words are (religion surely comes into this). According to the new dispensation, Plato’s metaphysics replaces this preoccupation: we really should be concerned with the relation between the Formal (mathematical, logical, moral, aesthetic, universal) and the Bodily (spatial, perceptible, concrete, evolving, particular). It won’t be easy to make this switch. This is why I said that the answer I am sketching is incredible (to us, now). But these are early days. Who knows what philosophical convulsions might lie in wait for us?
