Why I Publish Here

The reasons are pretty obvious. First, I have been academically cancelled, so I find it difficult these days to get my work published in the usual places. Second, and equally important, the form of my recent writing is not standard academic form: it is concise, reference-free, and straight to the point. Third, there is just too much of it: no publisher could take on such a large quantity of writing, especially in the form in which it is written. Fourth, I get to write in exactly the way I want to write without having to cater to referees and editors. Fifth, the publishing process is instantaneous; no waiting a year or more till it comes out. Sixth, I simply have too many ideas to write them all up in standard academese; it would take a lifetime. Seventh, this way I can publish all over the world and not just locally. Fortunately, the internet exists, or else I would have thousands of pages of unpublished material just lying around (or I would just not have bothered). What is not true is that this blog is a kind of diary or just whimsical reflections; it is primarily a place for me to publish my philosophical results (as they used to say). And the truth is that I mainly write for posterity these days.

Mathematical Ethics

You might not think that ethics is a very mathematical subject, unlike (say) physics. But we do have the felicific calculus which purports to quantify the rightness of an action according to the amount of happiness (pleasure) produced by it. The quantified variables include intensity, duration, extent, and correlated pain. And this idea is not so far removed from common ethical sense: we constantly compare the quantity of pain and pleasure likely to be caused by an action. We decide how much punishment a criminal action warrants by assessing its hedonic consequences; we might even make ratio judgments (this crime is twice as bad as that crime). It may not be very exact, but it is mathematically formulable. Couple this with the analogy often drawn between ethics and mathematics: both are said to be knowable a priori. Thus, we might say that a given action is known a priori to be twice as bad as another action, say because double the number of people will be harmed. We seem to be able to rank actions according to their degree of moral rightness and express this in quasi-mathematical terms. It isn’t all vague feelings and fuzzy logic. Our mind does seem to be functioning mathematically when thinking ethically.[1]

But there is another mathematical dimension to ethics less remarked upon: the number of ethical principles that comprise morality. We are all aware of the ten commandments, which give the impression that morality can be summed up in ten separate principles, ten being a nice round number (eleven would seem strange). God, it is felt, settled on ten and only ten—not a lot for lackadaisical humans to remember, though requiring some memorization. Philosophers, too, have enunciated lists of principles or edicts to which a definite number can be attached. Thus, W.D. Ross offers seven prima facie duties as comprising the total content of a deontological ethics: fidelity, reparations, gratitude, justice, beneficence, non-maleficence, and self-improvement. The magic number is seven, neither more nor less. G.E. Moore proposed four intrinsic goods: pleasure, knowledge, friendship, and esthetic appreciation. Jeremy Bentham got it down to two: promoting happiness and preventing pain. Kant reduced it still further to one: the good will, respect for the moral law, the categorical imperative. The only basic good is willing what can be universalized. I am not here concerned with the substance of these doctrines, only with their arithmetic: 10, 7, 4, 2, 1. This has a bearing on moral psychology, since moral agents have to be aware of, and be able to use, the precepts in which the right and the good consist—five hundred would be quite infeasible. One is ideal, but difficult to achieve theoretically. The arithmetic matters.

The trouble with any list longer than one item is that the principles may conflict, and often do. Which one do you choose to act on at any given moment—minimizing pain or maximizing pleasure? The utilitarian moral system doesn’t tell you. Moore doesn’t tell us which of his goods is the most important, and by how much. Should you spend your time making friends or appreciating art or producing pleasure? Ross notoriously conceded that in a case of conflict you just have to use your judgment, with no overarching ethical principle to guide you. This can leave you paralyzed ethically. In fact, this problem puts morality in peril, because you are always in danger of acting immorally; you can’t be morally perfect. Indeed, it is hard to avoid being morally shabby much of the time, since conflicts inevitably leave you acting immorally in some respect. You break a promise in order to work on your self-improvement or increase your beneficence or prevent a war. You strive to maximize happiness in some at the cost of allowing pain in others. It becomes impossible to be moral, i.e., to do the right thing all the time. But then, being completely moral isn’t an achievable goal—you are going to act badly no matter what you do. This may lead to skepticism or cynicism about morality, or atheism if you believe that morality derives from God. You want to act morally all the time, but morality itself makes this impossible, according to these pluralistic moral systems. If you can’t do right most (or all) of the time, then what point is there in it? Is it that your actions are always partly right and partly wrong? How do you decide when it is right to break a promise or to lie? Isn’t morality supposed to enable you to decide such questions? Pluralistic moralities make moral dilemmas irresoluble. This is an old story. But monolithic moralities don’t work; they are too simple. Ethics totters. Are we supposed to just muddle through or “go by our gut”? Where is the guidance that morality is supposed to provide?

I don’t think this problem has a straight solution, i.e., a solution that identifies some new moral precept, similar to the ones already enunciated, that provides a moral algorithm. Pluralism is unavoidable, but pluralism leads to moral conflicts, and moral conflicts produce moral uncertainty, indeed indeterminacy, which can lead to moral nihilism. Morality seems to consist of a bunch of moral values that form no natural unity and can easily clash with each other. Truth-telling can conflict with compassion, promise-keeping with beneficence, knowledge with pleasure, esthetic appreciation with justice. We seem headed for moral skepticism and moral inertia. But the lack of a straight solution is not the lack of any solution; perhaps we can provide something workable but not quite up to what we were hoping for. Let me suggest an analogy: the nature of existence. We wonder what existence is (what “exists” means) and look for a property like those already recognized—shape, color, size, solidity, etc. But we come up short: existence isn’t anything like being perceived or thought about or spatial or material or mental. It doesn’t seem like a regular property at all. In desperation we might then settle for a primitivist theory: existence is an indefinable simple irreducible property. Then it occurs to us that it is not like the familiar properties of things—a first-order property, as we like to say. It is a second-order property—the property of a property having instances. For red things to exist is for the property of redness to have the property of having instances. This is a kind of skeptical solution to our problem; we change our expectations about the form of the property (sic) of existence. We settle for less. In effect, we treat “exists” as a quantifier. It thus has a kind of mathematical character: “There is at least one object x such that…”.

By analogy, we can treat “right” in a similarly second-order quantificational way: it doesn’t denote a specific first-order moral duty or value, but generalizes over the domain of such duties or values. To say that a particular action is (or was) right is to say that it satisfies most of the precepts we accept as morally correct—or at least many, or anyway the important ones. It is a property of a collection of duties not a duty in that collection. For example, I go to an art gallery with a friend of mine in order to show gratitude for a favor he did me, thus producing knowledge and pleasure. The action is right because it instantiates the full range of Moore’s first-order types of good: esthetic appreciation, friendship, knowledge, and pleasure (as well as Ross’s gratitude duty). In fact, it had no moral downside—assuming I didn’t break a promise in order to go with my friend to the gallery, or lied about where I would be spending the afternoon, or neglected my duty to help people in need. That’s why I said most, because it can easily be argued that my action was morally wrong, given the amount of suffering in the world that I should be out there alleviating. All we can realistically expect is that our actions are instances of many of our prima facie duties. This may not add up to what we were hoping for—a recipe for moral perfection—but it is better than an abject admission that all we can achieve is a sorry mixture of right and wrong. It is a skeptical solution (like Hume’s treatment of causal reasoning)—a surrogate for the straight solution we were seeking. It enables us to say that our actions are largely right, if not entirelyright. And, like the existence case, the form of the concept of right is second-order; it is a kind of meta concept. Existence requires us to go up a level and speak of first-order properties; moral rightness requires us to go up a level and speak of first-order moral duties. To do our moral duty tout court is to maximize the number of prima facie duties that we fall under: all of them ideally, but failing that most, or at any rate many of the most important ones. We maximize the quantity of moral goodness in the action we perform. This won’t resolve all moral dilemmas and conflicts, but it is better than nothing; it mitigates the problem of moral plurality. If we combine it with a quantitative conception of individual values, we get a modestly mathematical model of moral reasoning: we are trying to maximize the number of values instantiated by our actions and their individual quantity of value. For instance, we are trying to foster friendship, show gratitude, produce pleasure, and appreciate art—all to a reasonably high degree. There was a fair amount of pleasure, much esthetic appreciation, and a decent quantity of friendship enhancement. Thus, we are thinking mathematically—in terms of number and quantity. We aren’t just blindly obeying some specific moral rule without any kind of computation—as it might be, “Produce pleasure!”. The missing ingredient in the traditional lists of goods and duties is “Think mathematically!” The ten commandments need an eleventh commandment, viz. “When obeying these commandments maximize the number and quantity of goods produced”. This is a meta and mathematical commandment. And let’s not forget that ethical reasoning can be explicitly and complicatedly mathematical, as when deciding about government policy concerning large groups of people, probabilities, and degrees of goodness (e.g., taxation policy). How much money, if any, should I give to charity? Should I provide an expensive treatment for my cat at the vet’s? Should I go and visit a distant relative? Ethics is often hard because the math is hard (in this respect it is like prudence).

I am proposing a theory of the logical form of an ethical proposition: the logical form of the proposition that x is right (where x is a particular action) is given by the proposition that the number of duties or values is maximized by x. Like Russell’s theory of descriptions, the analysans contains a quantifier, here ranging over duties or values, as well as the concept of maximization. It doesn’t take “right” as a logical primitive, and it doesn’t interpret “right” as we naturally would in “gratitude is right”. That use of “right” is not a quantified meta use but a first-order use. The difficulty comes (as Ross realized) with all-out moral judgments (not prima facie judgments); here we run into problems of interpretation and conflict. The skeptical solution is intended to rescue morality from the problems posed by its inherent plurality. Really, what other kind of solution could there be? If morality consisted in a single all-encompassing precept, wouldn’t we have discovered it long ago? Once we determined that the rule “Do what God commands!” is not sustainable, we knew we were in for a rough ride. Morality simply doesn’t have a nice unified shape, a common essence, a single magic formula. This is why we need a bit of mathematics to manage it. It’s like predicting the weather—complex, multifaceted, fallible.[2]

[1] Is it true that all a priori thought is overtly or tacitly mathematical, while a posteriori thought is not? Logic is clearly mathematical (it can be used to formulate mathematics) and analytic propositions take the form of equations (the meaning of “vixen” is identical to the meaning of “female fox”). We count meanings by reference to such equations. And many philosophers have taken mathematical truths to be analytic. The exception would seem to be ethics, but if I am right ethics is quite mathematical in its own way. Certainly, mathematical certitude has been an ideal of moral reasoning. The senses, by contrast, are neither a priori nor mathematically ratiocinative.

[2] I have never been able to take moral anti-realism very seriously, but moral anti-coherentism troubles me: that is, the idea that it is not possible to make morality consistent. Duties so easily conflict; no simple rule is free of counterexample. We can’t even say it is wrong to lie without needing to append qualifications. The need for the “prima facie” operator is immediately disconcerting. The clash between consequentialism and deontology seems irremediable. It is just not a well thought out coherent body of doctrine, but a kind of ragbag of rules of thumb. One longs for some moral rigor. But I would not draw anti-realist conclusions from this.

Metaphysics of Shape and Color

How many shapes are there in the universe, and how many colors? A standard answer is infinitely many in both cases. This answer is not incorrect, but it doesn’t go deep enough. Metaphysically, we want to know how many irreducible shapes and colors there are. As a matter of basic ontology, how many shapes and colors do we need to recognize? And is reductionism the right approach? How unified are the various shapes and colors—what kind of “ism” applies to them (monism, dualism, pluralism, etc.)? Is there a magic number or is it arbitrary? Is there an essence shared by all shapes, and ditto for colors? What is the metaphysics of shape and color?

First shape. We might begin with the idea that geometrical figures fall into a few basic kinds (natural geometrical kinds): triangles, quadrangles, polygons, circles, ellipses; then three-dimensional figures are derived from these. You can combine these to produce infinitely many possible shapes, but they are the basic building blocks; everything shaped is reducible to them. This would be quite an insight and it has shaped (!) human thought from Euclid on. But it doesn’t go far enough: can’t we reduce all possible shapes to variations on straight-sided figures and curved-sided figures? Triangles and squares (etc.) on one side, circles and ovals (etc.) on the other side. The rectilinear and the curvilinear. Once you have these core shapes you can derive the rest: for example, you can simply add a side to a triangle to get a rectangle, and you can squash a circle to get an ellipse; three-dimensional figures are got by analogous means. So, we might endorse a dualistic geometry—roughly, squares and circles. Then the claim would be that all the shapes of nature are reducible to these; and that claim is plausible enough. But have we gone far enough—couldn’t there be a more ambitious form of shape reductionism? Couldn’t geometrical monism be true? For consider: can’t we derive circles from squares by multiplying the sides of the square to infinity, using the concept of a limit? Isn’t there an intelligible procedure that will take you from squares to circles? And what about the idea of taking sections of the circumference of a circle, straightening them out, and putting them end to end? The gulf between squares and circles does not appear unbridgeable (it’s not like matter and mind or fact and value). The circle is a kind of modified square—we can imagine a square growing into a circle (or vice versa). We can always bend a straight line into a curved one and a curved line into a straight one—imagine performing this transformation with a piece of string. Circles and squares are topologically equivalent. Underlying the geometrical dualism, we have a geometrical monism. You can do a lot of work with a straight line (or a curved one). Metaphysically, we have variations on a theme. Ontologically, all shape is based on a single shape. A pre-Socratic might announce “All shape is One”: the many reduce to the one. A theologian might teach that God first created the square and then let nature take care of the rest. A philosopher may proudly call himself a “rectangle monist”. A modal philosopher might go a step further and say that necessarily all shapes reduce to the rectangle—in all possible worlds the only real shape is the rectangle. This might be called “shape minimalism” and Occam’s razor cited piously. And the doctrine is not wildly implausible; in fact, I might include myself under that label. The geometric jungle is analyzable into a single basic figure (I stop at reducing everything to the straight line). I might even be attracted by the idea of reducing every shape to the egg shape, on account of its poetic resonance: all geometry develops from an egg. This is certainly a debatable position in the metaphysics of shape.

But what about color? Here the situation looks very different: the colors are not all derivable from a single color, or even two colors. The basic colors are standardly said to be red, blue, yellow, green, black, and white. There could be other colors too, and some animals may see them, but they clearly form a plurality. You can’t deform blue to get yellow, or black to get white. These are color primitives, irreducibly different. They can combine to generate potentially infinitely many colors, but the procedure starts with a (finite) inventory of primitive colors. Color pluralism is the indicated metaphysical doctrine. Someone who claimed that all colors are one would be on shaky ground (“Everything is green” sounds like a non-starter). So, we know that the metaphysics of color differs in this respect from the metaphysics of shape. Shape is inherently unitary, but color isn’t. It is in the nature of color that there should be a variety of colors, actual or possible. There is no privileged color that reduces all the rest. So, when you look at an object and see the many shapes that compose it, you see variations on a single shape; but the many colors you see are not variations on a single color. There is geometrical unity but not chromatic unity. The two sorts of quality are intertwined, but their ontology is quite different. Colors form a family; shapes form a continuum. Colors are discrete; shapes are continuous. Colors are pluralistic; shapes are monistic. Visual perception is a mixture of both. And it knows it, tacitly anyway: it knows that shapes and colors are different in this way. We need different color receptors, but we don’t need different shape receptors (rods and cones and all that). The processing of shape is essentially simpler than the processing of color; shape perception no doubt preceded color perception in the evolutionary history of vision. The metaphysics of shape and color influences the cognitive science of human vision (and non-human types of vision). Shape is subjectively monistic, but color is subjectively pluralistic. This is a truth of phenomenology. The perceived world (the visual field) has a mixed metaphysics: partly monist, partly pluralist. We see things in both ways (seeing-as).

How does all this bear on the contents of platonic heaven? Interesting question. With color we would naturally say that it contains the usual six colors and no others—no shades or mixtures. These come from the mixing of colors that occurs in the empirical world. There is no need to stack up endless derivative colors in Plato’s heavenly storehouse. In the case of shape, one gets the impression that Plato favored a well-stocked geometrical heaven—triangles, rectangles, ellipses, circles, etc. But in the light of our reductive efforts, we could cut this down to the basics; and we appear to have a choice—the circle or the square. Circles seem the most appealing choice, in view of their reputed perfection; so, let’s go with that—the Form of Circularity only. Then we obtain the rest by sublunary operations and iterations. There is the Good, the Circular, and the Colors (neatly side by side). Geometry is commendably minimal (Occam-shaved) in its quota of basic Forms. It has a small but powerful cardinality. There is no need to wax extravagant in constructing Plato’s heaven, at least when it comes to shape. One shape will do the job.

Lastly, how do we explain these numbers—what do they signify? Or are they entirely arbitrary? Our mystical tendences favor special mystical numbers, but it is hard to see any meaning in the numbers we have arrived at. One is a nice round number for shapes, but six doesn’t sound very meaningful for colors—and anyway there may be other colors perceived by other perceivers. There is nothing Godlike about the number six (three maybe). Could there be worlds that have fewer colors, or completely different colors? That sounds strange, but it may be the result of our limited perception-driven imaginations. Surely, black and white will be universal, and it would be an impoverished world without red and blue. I will leave this question for further research into the mathematical metaphysics of color. The point I have wanted to make is that shape and color have different ontological profiles.[1]

[1] Isn’t it odd that this question has not been pursued in work on shape and color? To my knowledge, the questions I raise (and answer) in this paper have no traditional literature devoted to them, despite the interest in the two topics. The usual focus is on the distinction between primary and secondary qualities, not the question of intra-category reduction and inter-category divergences. Colors have no real essence in common, but shapes do. It is customary to construct geometry from points and lines in an atomistic style, but no such atomism will work for color science.

A Proof of Platonism

I am going to prove that the universal whiteness is not identical to the class of white things. The proof is similar to the well-known proof that pain is not identical to C-fiber firing.[1] Thus: I can think of whiteness apart from the class of actual white things; I can conceive of a possible world in which a different class of things is white. That class is not identical with the class of actual white things, but the same universal is being instantiated. Therefore, whiteness is not identical to the class of actual white things. QED. Compare: I can conceive of a possible world in which pain is not correlated with C-fiber firing, but instead is correlated with D-fiber firing. It is the same sensation but a different physical correlate. Therefore, pain is not identical to C-fiber firing. Pain is modally detachable from C-fiber firing, so it can’t be identical to that. Whiteness is modally detachable from actual white things, so it can’t be identical to them. It can’t be reducible to those things, as pain can’t be reducible to that brain state. What about the claim that pain can exist without being attached to any physical state? That too seems conceivable, at least at first sight (pending a clearer idea of the “physical”). Similarly, it seems conceivable that whiteness could exist in a possible world without any white things; if so, it is detachable from all possible white things. This intuition is reinforced by the following reflection: if I reduce the number of white things in a possible world, I don’t reduce the universal whiteness—it doesn’t get any smaller. Indeed, what would it mean to say that a universal had been reduced in size? Classes can be reduced in size, but not universals. The identity and existence conditions of universals are not the same as those of classes of particulars.

There is a more general point to be made: what explains the ability to think of universals in the absence of particulars instantiating them? The most obvious explanation is that the two are separate and distinct: when I think of whiteness I am not thinking of a class of particulars. The explanation is simply that they are not the same thing. Similarly, for pain and C-fiber firing: the concepts are different because their reference is different. Intuitively, that is very plausible, because a class is a plurality of things but a universal isn’t—it is one thing. Also, particulars have many other properties, but whiteness doesn’t have this kind of heterogeneity; it is whiteness pure and simple. Universals aren’t collections of anything; they are unities. Nor can we say that whiteness is just a mode of presentation of the class of white things, since the same could be said of the class, i.e., that it is just a mode of presentation of whiteness. In fact, it is more plausible to suppose that whiteness is ontologically basic, since it is what unifies its instances into a class. The class doesn’t generate the universal; the universal generates the class. The class would not exist without the universal; it cannot be detached from the universal that forms it. The particular presupposes the general, not vice versa.[2]

[1] See Kripke’s Naming and Necessity.

[2] Actually, I don’t much like the traditional labels “general” and “particular”:  the particular is also general in that it has many properties (it is multiple), and the general is also particular in that it is a specific thing not some sort of generalized nothing. The particular is many and the general is one. We could call the universal whiteness a “particular” because of its specificity, and the particular white thing general because it is host to a plurality of properties. Calling a universal “general” suggests some sort of distributed unspecific nature, but in reality, universals are as particular and specific as one might wish. One is not thinking “generally” if one thinks of the universal whiteness. Still less is one thinking vaguely or indiscriminately or plurally or imprecisely. One is thinking of one specific universal in particular. We must rid ourselves of the idea (prejudice) that universals are somehow cognitively and ontologically improper or badly behaved. Plato elevated them over particulars for a reason.