Mathematical Ethics

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.