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abstract

Mathematics can help us pursue morally significant ends more efficiently—but it can do more. It can help us elaborate and clarify those ends themselves. For example, while it is not a mathematical theorem that we should deal fairly with one another, mathematics can teach us how best to understand fairness in cases of real consequence. This enlargement of ethical understanding is a social function mathematics shares with literature. As I explore this point of common interest, I will consider three standards for assessing ethical standards: feasibility (a particular concern of Aristotle), testability (a particular concern of John Dewey), and mathematical tractability.

keywords

mathematics, literature, ethics, social choice, John Dewey

That mathematics makes for poor literature is a conclusion as uninteresting as it is inevitable—inevitable because were mathematical prose to score high on a scale of literary value, this result would do more to discredit the scale than glorify the prose. It may, however, help us better understand our cultural landscape if, without attempting a literary appraisal of mathematics or a mathematical appraisal of literature, we search for some community of interest between the formal sciences and the literary arts. This search for common ground will also be an opportunity to learn from [End Page 517] some voices seldom heard in conversations about mathematics. Even at the risk of giving our discussion an antiquated flavor, we will welcome those voices, new to our circle, that have something stimulating to say about our project—whether or not they represent the last word on mathematics and literature. Our new friends will take us in a new direction: whereas it would have been perfectly natural for us to emphasize the aesthetic interests shared by mathematicians and novelists, we will instead explore how both tribes, the mathematical and the literary, contribute to the articulation of values that are not narrowly aesthetic.

The Good or the Beautiful?

When Jeeves looks forward to an evening with an "improving book," he embraces a sober Victorian attitude toward literature that even Henry James, for all his admiration of French naturalism, finds "very noble and defensible" (1888, 50). It was clear to Anthony Trollope, for example, that novelists have a duty to "make virtue alluring and vice ugly" (1905, 192). George Eliot hoped to rouse "the nobler emotions, which make mankind desire the social right" (1885, 330). She would find an ally in John Dewey, at least to the extent that he considered art an agency for imparting "imaginative content and emotional appeal" to "bare ideas" (1939, 150). Novelists such as Alphonse Daudet and Émile Zola, on the other hand, found this moralistic attitude magnificently wrongheaded, convinced as they were "that art and morality are two perfectly different things, and that the former has no more to do with the latter than it has with astronomy or embryology" (James 1888, 302). They find an ally in Roman Ingarden, who insists that novelists with an inclination to preach "misunderstand the essence of a work of art and misuse works of art for various extra-artistic ends" (1973, 173).

If our goal were to discover some community between literature and mathematics by fastening on a notion of literary good transferable with little fuss to mathematical productions, then our first impulse might be to side with the French. Since it is a commonplace that beauty inhabits mathematics, we would expect to find plenty of mathematical material that shows well from a specifically aesthetic or artistic angle. If we worried that our project might miscarry, fearing that mathematics and literature share little more than the terminology of aesthetic appraisal (beautiful proof, beautiful [End Page 518] poem), we could be reassured by ready-made arguments to the contrary (as, for example, in Montano 2014). Siding with the French on this issue is hardly unreasonable. Since, at first glance, mathematics seems more likely to provide aesthetic enrichment than moral improvement, a preoccupation with distinctively aesthetic qualities may be our best bet if we want to show that literati and mathematicians pursue something like a common good.

However, before we abandon sturdy English moralism, we might remind ourselves that our forefather Plato did not just share the Victorians' taste for edifying literature: he floated the remarkable idea that good mathematics can play a consequential role in the moral development of civic leaders. Myles Burnyeat, for one, provides a fine discussion of how Plato might have filled in the details. I am going to worry less about the specifics, exploring whether Plato offers a rough hint about how mathematics and literature might conspire to achieve some social good. The general idea, as Burnyeat expresses it, is that the Republic's program of mathematical education provides "much more than instrumental training for the mind." The real target is "an enlargement of ethical understanding" (Burnyeat 2000, 46). Stated thus abstractly, this is an aim we would expect to find in many departments of our culture.

Enlarged Experience

Amplification in general—of insights, sensibilities, resources, experiences—is, certainly, widely recommended. David Prall notes that literature "expands our vision and enlarges our sympathies, and allows us like gods to view all things natural and human sub specie aeternitatis" (1929, 299). If Dewey is to be believed, William James thought that the "point of philosophy" was "the illumination and enlargement of the human mind on the things that are its most vital concerns" (Dewey 1929b, 112). His brother Henry observes that successful works of art offer "a miraculous enlargement of experience"—or, at least, the appearance of such an enlargement (James 1888, 227–28). George Eliot leaves out the self-defeating qualification, insisting on literature's capacity for "widening the mind to a fuller and fuller response to all the elements of our existence." The elements that are morally pertinent, that form "the direct ground of human love and moral action," are the "specifically human" ones—"pain and relief, love and sorrow"—whose "peculiar history" offers "an experience and knowledge [End Page 519] over and above the swing of atoms" (Eliot 1885, 247–48). Literature can open our minds to the grounds of moral action by expanding our field of experience. Though we do not experience the action of a novel as something physically present to us, our reading of literature is in itself an experience that can leave us with a wider perspective on the human drama of feeling, inclination, judgment, and action. So fiction can provide an "enlargement of experience." As Henry James himself says, it can furnish "a personal, a direct impression of life . . . catching the very note and trick, the strange irregular rhythm" (1888, 384, 398).

Now how might such an enlargement of experience lead to moral improvement? As we have already seen, it might rouse the nobler emotions that make us desire the social right. It might help us "feel truths as well as see them" (Mill 1950, 96). This could mean that our feelings are made to cooperate with an earlier, more cerebral recognition of what is right. Trollope suggests another moral inroad. Yes, literature can strengthen existing convictions by charging ethical ideals with emotional energy; but it can also win new converts to generally accepted standards through agreeable moral instruction. By both poetry and prose, "may true honour, true love, true worship, and true humanity be inculcated; and that will be the greatest teacher who will spread such truth the widest" (Trollope 1905, 188–89). Literature can both teach the right and move us to pursue it. A novelist, such as Trollope, can be all the more effective at such teaching because "he charms his readers instead of wearying them" (1905, 192).

Henry James, the last person to insist that novels be fraught with ethical purpose, hints at yet another morally significant role for literature. If a moral precept is to have a lawlike form, it must refer to types of action, character, feeling, and intention. Literature that expands our vocabulary of such types also boosts our powers of ethical penetration: it strengthens our capacity to formulate principles for the improvement of social life. It is a weakness in the works of Charles Dickens that "his figures are particular without being general; because they are individuals without being types; because we do not feel their continuity with the rest of humanity" (James 1888, 318). This is emphatically not the case with George Eliot: "We feel in her, always, that she proceeds from the abstract to the concrete. . . . The world was first and foremost, for George Eliot, the moral, the intellectual world; the personal spectacle came after; and lovingly humanly as she regarded it we constantly feel that she cares for the things she finds in it only so far as they are types. The philosophic door is always open, [End Page 520] on her stage, and we are aware that the somewhat cooling draught of ethical purpose draws across it" (James 1888, 50). Similarly, Constance Fenimore Woolson "had a fruitful instinct in seeing the novel as a picture of the actual, of the characteristic—a study of human types and passions, of the evolution of personal relations" (James 1888, 187). Such artists, by adding to our stock of ethically significant categories, or by deepening our understanding of categories already available, make us better able to formulate standards with the logical form required of precepts—standards at the right level of generality. This too is an enlargement of experience: the sort of thing that makes you a more experienced person with a more ample ethical understanding.

On at least one occasion George Eliot puts the precept before the type. About a quarter of the way through Middlemarch, our narrator observes, "We are all of us born in moral stupidity, taking the world as an udder to feed our supreme selves" (Eliot 1947, 225). That is an observation, not a prescription, but some imperatives follow readily. Don't be stupid! Be an adult! Leave infantile egoism well behind! Much later we have the uncomfortable experience of learning that an archetype of moral stupidity has been with us for most of the novel. The experience leaves us more experienced, with a better understanding of how, for a practiced mind, the injunction against moral stupidity might serve as a definite guide to action. As Dewey puts it, the precept becomes a "living thought" that "represents a gesture toward the world, an attitude taken to some practical situation in which we are implicated" (1942, 54). We better understand how the deepest moral stupidity is compatible with the most exquisite decorum—as long as the world cooperates. We also see how, when the world turns uncooperative and infantile egoism asserts itself in action, morally stupid people can retain the firmest faith in their own moral rectitude (Eliot 1947, 713). The upshot: we have both a category, moral stupidity, and an enlarged understanding of how instances of the category operate in the world.

Standards for Standards

I have identified three moral purposes that might be served by a work of fiction. The work might (1) invest moral ideals with emotional force; (2) disseminate moral ideals by presenting them in vivid, engaging ways; or (3) contribute to the formation or elaboration of moral ideals themselves. [End Page 521] Since the first two projects seem entirely alien to mathematics, we will concentrate on the third. Is there any way that mathematical insight might be a guide in determining morally significant ends, not just an instrument for attaining them? It will help us answer this question if we take a slight detour—beginning with some observations about Plato's prize pupil.

It was evident to Aristotle that the good for human beings must be attainable by human beings. To measure human creatures by a standard of unreachable perfection would be like saying that a cow is a fine example of its kind only if it can jump over the moon. This stance supplied Aristotle with a standard for evaluating moral standards. The question is feasibility as judged by people with wide experience of human affairs. Of course, feasibility is hardly sufficient: it is, for example, humanly possible to spend much of one's time in a drunken stupor, but that would not count as a display of distinctively human excellence. Feasibility is, however, necessary and is a consideration that would have made the doctrine of the mean particularly attractive to Aristotle. A tightrope walk keeping excess on one side and deficiency on the other may be difficult, but it is a practical possibility. So the old maxim "Nothing too much" passes an essential test: it does not demand either too much or too little from our species. The better we know humanity, the better we can judge what is feasible for us. The best, most systematic knowledge of human capacities and social relations would be an indispensable aid to the most responsible determination of what counts as moral worth.

Dewey had his own standard for assessing moral standards. Yes, a moral or political ideal should be feasible, but we can reasonably ask for more. We can favor ideals that call us to definite actions whose consequences will have a discernible effect on human prosperity. A moral or political end passes the critical test if it inspires a social experiment that helps us understand the conditions under which people lead better lives. It is a point in favor of a "construction of the good" if it lends itself to such an experiment. This sort of testability is desirable because it contributes to the rational development of our ethical understanding. A moral or political precept should stand ready to face the tribunal of experience: "[Such a precept] is not something to swear by and stick to at all hazards; it is a formula of the way to respond when specified conditions present themselves. Its soundness and pertinence are tested by what happens when it is acted upon" (Dewey 1929b, 278).

Of course, testability can be trumped by other considerations. For example, even before the world understood concretely the horrific depravity [End Page 522] of the Nazi system, Dewey (1942) had no trouble explaining why the social experiment embodied in the Third Reich represented, on its face, a hideous wrong. That is, it was deeply wrong to try the experiment in the first place. So it is perfectly clear that a moral or political outlook can be depraved on its face even when armed with a cognitive content and a relation to existing conditions that make it concretely effective and, so, assessable through its effects.

That said, however, it remains entirely reasonable, other things being equal, in the absence of any successful facial challenge (as the lawyers say)—any binding consciousness that an outlook or injunction is wrong on its face—to seek out ideals and principles that can be judged by their effects. It is reasonable to promote moral schemes aiming at goods that might plausibly be "the fruit of intelligently directed activity": the goal being to "develop a system of operative ideas congruous with present knowledge and with present facilities of control over natural events and energies" (Dewey 1929b, 284, 286). So a responsible assessment of claims about moral value "must depend in a most intimate manner upon the conclusions of science" (Dewey 1929b, 274). Instead of "confining intelligence to the technical means of realizing ends . . . intelligence must . . . devote itself as well to construction of the ends to be acted upon" (Dewey 1942, 142). We can, in particular, apply scientifically informed intelligence to the question of whether a precept or ideal is likely to be testable through a course of deliberate implementation and careful, disinterested assessment of results. Whereas Aristotle insists on feasibility, Dewey calls for testability.

Mathematical Tractability

If Dewey had better understood mathematics, he might have taken a further step: he might have observed that mathematical standards and insights can help us sort out our value commitments by pointing us toward morally significant ends that can be intelligently pursued because they are subject to mathematical analysis. Alongside feasibility and testability, he might have suggested a standard of mathematical tractability. Economists might welcome this new standard. Indeed, anyone who practices a mathematically informed science of human behavior will be strongly attracted to measurable properties subject to formal analysis and may well count mathematical tractability as a point in favor of a moral ideal. This notion of a [End Page 523] mathematical standard for standards conjures up an image that, admittedly, is not so realistic: mathematically informed experts filling out score sheets to determine the best channels for social currents. In the messy world we actually inhabit, masses will move without the approval of economists and mathematicians. When such movement already has a tendency of its own, the task for the mathematically minded is not so much to set goals as to interpret them, reconstruct them, wrestle them into shape.

For an example of an idealistic scheme that, at least at first blush, appears unfeasible, untestable, and mathematically intractable, consider a statement by a particularly effusive supporter of a presidential candidate here in the United States (as reported on National Public Radio, May 27, 2015): "I love the man. He's awesome. He's in my heart, and I want him to win so this whole world can be beautiful." There is a moral precept lurking here: act so as to make this whole world beautiful! Aristotle would probably worry that this injunction does not qualify as feasible, because it places too great a burden on humanity. From his point of view, the cosmos will be beautiful to the extent that it is well ordered. Human beings can, indeed, make their own distinctive contribution to this good order—we have our own "deed unto the general nature" (as Emily Dickinson puts it). If the idea is to carry on the work allocated to our species, nothing could be more reasonable. If, as seems likely, rather more is expected of us when we are called to make this whole world beautiful—for example, if we are called to make a plain world fine through some sort of exceptional activity that may or may not find adequate support in the usual human endowments—then the injunction is not to be taken so seriously. Dewey, furthermore, might worry that the injunction does not qualify as testable, because we have so little idea of what it asks us to do and have so little assurance that, having done it, we would be able to determine whether we have bettered our condition. As for mathematical tractability, cosmic or planetary beauty is a notion so vague that we might despair of ever subjecting it to mathematical analysis.

Economists, however, are not so easily put off. In the face of a real prospect that effusions about cosmic beauty will become policy, economists would set to work trying to associate those effusions with something measurable. The unwelcome alternative is that we are deprived of a powerful instrument of intelligent control: control over natural and social forces and control over our own conceptual apparatus. Mathematical analysis can help us achieve our goals and can also help us understand those goals. It can [End Page 524] enhance both the efficiency of our action and the precision of our thought. This means that mathematical tractability—or a stubborn determination to beat injunctions into mathematically tractable shape—can open up a particularly promising road to testability. Mathematical analysis can give us a very definite idea of what we are called to do, can show us how to do it, and can help us measure the extent to which our actions have nurtured an identifiable good. Amartya Sen argues that, in particular, the mathematically informed work of economists can enrich our ethical understanding: "There is something in the methods standardly used in economics, related inter alia with its 'engineering' aspects, that can be of use to modern ethics. . . . The framework of consequential reasoning and pursuit of interdependences extensively developed in economics in many different contexts . . . provide many insights into pursuing the inescapable problems of interdependence involved in valuing rights in a society" (1987, 9, 73). It will be old news to students of welfare economics and social choice theory that formal economic analyses can illuminate questions of fairness and social justice.

We conclude, then, that mathematical tractability, a property detectable by mathematical insight, can be a point in favor of a moral precept. Of course, all our reservations about the testability standard apply with full force here. Add mathematical sophistication to the fanatical application of a depraved standard, and you multiply horrors. So I must repeat my earlier qualifications: other things being equal, in the absence of a successful facial challenge, it is reasonable to seek out and promote constructions of the good that submit to mathematical analysis. This consideration gives—indeed, has already given—the mathematically adept a seat at the table as we struggle to understand our moral responsibilities.

As for the exact meaning of these qualifications, what the "other things" might be, in what sense they may or may not be equal, and what makes a facial challenge compelling—none of this is obvious (at least to me). If the injunction "Be kind to one another" inspires no mathematical acrobatics, should it give way in the face of more mathematically tractable precepts? That cannot be right. As the narrator of Middlemarch observes, "There is no general doctrine which is not capable of eating out our morality if unchecked by the deep-seated habit of direct fellow-feeling with individual fellow-men" (Eliot 1947, 664). The point is not that tender feelings will make virtuous an action whose atrocious consequences were foreseeable or that the absence of such feelings will make vicious a beneficial action done for a right reason. The issue is that the disinterested choice and application [End Page 525] of moral standards is an expression of good character—including kindly dispositions. We need good habits supported by appropriate feelings to overcome, in J. S. Mill's words, "the common tendency of man to make a duty and a virtue of following his self-interest," to beat down "the artifices by which we persuade ourselves that we are not yielding to our selfish inclinations when we are" (1950, 89).

Spontaneous sympathy, revulsion in the face of cruelty—these too act as standards for judging standards and their applications; indeed, they act as an indispensable check on the "general doctrines" of this very essay. There were two articles to Bertrand Russell's (1998, 394) creed: kindness and clear thinking. Our standards of feasibility, testability, and mathematical tractability foster clear thinking and, for that very reason, can serve the cause of kindness, but only if we let fellow feeling operate as a force of its own. As Dewey remarks, "We are sure that the attitude of personal kindliness, of sincerity and fairness, will make our judgment of the effects of a proposed action on the good of others infinitely more likely to be correct than will those of hate, hypocrisy, and self-seeking" (Dewey and Tufts 1932, 265). Samuel Scheffler adds, "Without a strong general aversion to harming others . . . one might be tempted to inflict harms not only when that would produce optimal results, but also when it would merely secure some personal advantage" (1988, 7).

I had better end the sermon here before we are drawn even further into waters I am poorly equipped to navigate. We turn now to an example of a mathematically tractable problem: a case where mathematical analysis helps us both understand and achieve a goal that, at the start, is far from precise.

Optimality

We make the following assumptions about the city of Galesville's four hospitals:

Each hospital has a spot for one intern.

The same four candidates, and no others, apply for each of the four spots.

The hospitals rank the candidates first through fourth without ties.

The candidates rank the hospitals first through fourth without ties. [End Page 526]

The hospitals confer and, after much discussion, agree to let us assign them interns. They stress the fragility of their consensus and urge us to act in a way that will encourage everyone to abide by our decision. They particularly insist that we be guided by the preferences expressed in the rankings—both because they consider these preferences worthy of respect and because our attention to them should help to maintain the consensus. To no one's surprise, they ask us to give their own preferences greater weight than those of the candidates. So we need to find an assignment that preserves the consensus and respects everyone's preferences but gives greater weight to the preferences of the hospitals.

What are we to do? The best outcome would be for all parties to get their top choice—but that may be impossible. (One candidate might be the top choice of two hospitals.) The worst outcome would be for all parties to get their last choice—again, something that may be impossible. (One candidate might be the last choice of two hospitals.) That much is clear, but on the big middle ground between these extremes, how do we tell whether one assignment is better than another? We understand that an assignment is bad if it threatens the consensus. Less clear is this business of respecting preferences. What could it mean for one assignment to fit the rankings better than another? The ideally good assignment (top choices for everyone) should count as the best fit, while the ideally bad assignment (last choices for everyone) should count as the worst. Beyond that, things are not so clear.

Suppose we identify some situations that would make an assignment a particularly bad fit and an especially dire threat to the hospitals' consensus (their agreement among themselves to abide by our decision). We might then do a preliminary sort, tossing out assignments guilty of those dangerous misalignments. For starters, here is an unfortunate situation that belongs on our don't list: Lloyd is assigned to Gale Memorial but would have preferred Shapley Clinic; meanwhile, Shapley Clinic gets David but would have preferred Lloyd. Lloyd and the Shapley Clinic might reasonably complain about an assignment that so clearly works to their mutual disadvantage. Such an assignment is said to be unstable because it gives Lloyd and the Shapley Clinic an incentive to thumb their noses at us and cut a deal on their own. An assignment is unstable in this sense if and only if it leaves a candidate and a hospital preferring each other to what they actually got. These assignments are undesirable, first, because they threaten the fragile consensus by encouraging defections: by tempting a candidate and [End Page 527] a hospital to cut a separate deal. They are undesirable, second, because they jar so markedly with preferences we are supposed to respect: avoiding unstable assignments is one way of showing respect for the preferences of both hospitals and candidates.

Our plan was to do a preliminary sort based on a list of dangerous misalignments. If that plan sounds reasonable, our next step may come as a surprise: we declare our don't list complete, throw out all the unstable assignments, and define a notion of optimality that applies to the stable ones. We do this even though there are other misalignments that qualify as bad fits likely to produce discontent: a hospital getting stuck with its last choice, for example. Why pick on the unstable assignments while ignoring other defects? The answer, provided by David Gale and Lloyd Shapley (1962), is that stability and instability are mathematically fruitful notions that deliver a powerful result, supplying us with a reasonable notion of "best possible assignment," guaranteeing that such an assignment exists, and telling us how to find it. It is not, I dare say, intuitively obvious that our preliminary sort should throw out exactly the unstable assignments. It is a substantial mathematical insight that this procedure puts us within reach of a unique assignment that is optimal according to a reasonable standard. Indeed, the insight is substantial enough to have helped Shapley win a Nobel Prize.

Actually, in the universe of stable assignments, there are two notions of optimality, two notions of best assignment, each with its own existence and uniqueness theorems. A stable assignment is optimal for candidates if and only if no candidate would be better off under another stable assignment. A stable assignment is optimal for hospitals if and only if no hospital would be better off under another stable assignment. Since we have been asked to give the preferences of the hospitals the greater weight, we decide that the best stable assignment is the one optimal for hospitals. This does not mean that the preferences of candidates count for nothing: we were already guided by those preferences when we eliminated the unstable assignments. We can now identify the stable assignment optimal for hospitals using an algorithm supplied by Gale and Shapley. If we announce our choice of that assignment and Gale Memorial complains because it got stuck with its last choice (a possibility even in an assignment optimal for hospitals), we will point out that all the assignments in which Gale Memorial fares better are unstable ones and, hence, bad fits that are a particular threat to the consensus the hospitals all wished to preserve. [End Page 528]

Mathematics to the Rescue

Let us recall how we got to this point. The hospitals issued two directives. Directive 1: respect everyone's preferences but give ours the greater weight. Call that the goal of respect. Directive 2: preserve our consensus. Call that the goal of unity. This latter goal—particularly dear to Plato (Burnyeat 2000, 74–75)—was clearly understood from the start. Unity is preserved if and only if every hospital accepts our assignment. The best outcome is no defections. The introduction of the notion "unstable assignment" did not clarify or modify the goal of unity; it just gave us a better idea of how we might achieve it (by giving us a better idea of what we should avoid). So the situation was of this type: we clearly identified one of our goals and used mathematics to make our pursuit of it more efficient.

Once we decided that, overall, with regard to both our goals, the best assignment was the stable one optimal for hospitals, we still had to figure out which assignment that was. To our relief, Gale and Shapley supplied an algorithm for doing just that. So the situation was of this type: we believed that the best outcome was the one satisfying certain conditions and used mathematics to determine which outcome met those criteria. This employment of mathematics did not clarify or modify either the goal of respect or the goal of unity. Now, however, we come to a more interesting application of mathematical insight: an application that helped us make one of our goals more precise.

The goal of respect was decidedly unclear at the start. Yes, we should respect everyone's preferences, but how were we supposed to do that? The ideally best outcome (first choices all around) was clear enough, but that outcome might have been impossible. If it was impossible, we faced a daunting problem. Given two less-than-ideal assignments, how were we to decide which was the more respectful? It helped enormously that our job did not actually require us to invent an independent criterion of relative respectfulness. Since our job was to pursue two goals at once, we needed a standard not of respectfulness per se but of respectfulness-consistent-with-unity. The requirement of respectfulness-consistent-with-unity strongly motivated our rejection of unstable assignments and, as it turned out, left us with assignments among which we could make a principled selection of the best.

Note, now, that the Gale-Shapley notion of the best assignment, the notion "stable assignment optimal for hospitals," is a thoroughly mathematical [End Page 529] construction. People without mathematical credentials might have fabricated this notion of optimality, but that would have been strong evidence that those people were blessed with mathematical gifts and a mathematical outlook: they must, in fact, have been able to penetrate the noise about hospitals and interns to get at the relevant structures. For what it is worth, the editor of the American Mathematical Monthly would seem to have agreed that Gale and Shapley's work counts as mathematics—a point on which Gale and Shapley themselves insist. Referring to their proof that stable assignments always exist, they declare that "any mathematician will immediately recognize the argument as mathematical" (1962, 15). One might, I suppose, maintain that "stable assignment" is not a mathematical notion even though it is the subject of a mathematical theorem—but that seems wrongheaded. Take the example of map colorings. We would concede that a kindergartner will have some idea of what it means to color a map, an idea that may not be particularly mathematical. Nonetheless, the "colorings" of the Four Color Theorem will, necessarily, be conceived mathematically since, otherwise, they could not be the subject of a mathematical result.

So our situation was of this type: in our struggle to decide what it would mean for an outcome to count as best, we were guided by mathematical ideas and standards—that is, our very construal of the term best was mathematically informed. We knew all along what it would mean to maintain unity, though it was much less clear how we were to do it. On the other hand, we started out with a very shaky grasp of what it would mean to preserve unity in the way most respectful to the given preferences. With the help of mathematical insight, we won through to a new, clear, principled notion of the best outcome.

According to the standard of mathematical tractability, it is reasonable, subject to some essential qualifications, to seek out constructions of the good that submit to mathematical analysis. As you may already have noticed, the Galesville situation was crafted with this standard in mind: it was designed to be mathematically tractable. The narrative, with its decidedly axiomatic air, posed a problem requiring us to consider relations between structures (the rankings)—a distinctively mathematical exercise if ever there was one. The setup cried out for mathematical analysis, and as was intended all along, that analysis did not just show us how to pursue a goal already well understood; it helped us understand what it would mean for us to realize that goal. Or perhaps it would be better to say that it supplied us with a new, definite notion of [End Page 530] what was best—a notion that counted as a welcome clarification of the original goal.

As contrived as the Galesville fable may seem, we do not have to look far for real-life analogues. As Alvin Roth showed, the algorithm of the National Intern Matching Program (NIMP) is equivalent to the Gale-Shapley procedure. Implementation of the NIMP algorithm created a market notable for its "orderly operation and longevity" (Roth 1984, 991). Mathematical tractability allowed for a mathematical analysis leading to a definite course of action subject to responsible assessment. It does not detract from this point that Gale and Shapley's analysis came a decade after the birth of the NIMP algorithm. That algorithm was, in itself, a mathematically insightful response to a mathematically tractable problem—a response entailing definite actions assessable according to definite standards (such as longevity and participation rate). So mathematical tractability served the cause of testability. To put the point less austerely, it served the cause of clear thinking and effective action in pursuit of a social good.

Just to be clear: mathematical tractability will not make it one bit more likely that an injunction is morally correct, but it may help us tell whether it is morally correct. Mathematical analysis can help us figure out what the injunction means—or ought to mean—and, so, help us understand what it enjoins us to do. If the mathematical analysis points us toward observable, quantifiable phenomena, it may help us determine whether obedience to the injunction betters our condition: whether it makes us better people in a better community with better opportunities.

The practice of using mathematics to enhance ethical understanding has an ancient pedigree. It was, as earlier noted, endorsed in general terms by Plato. For a particular application, we need look no further than Nicomachean Ethics, book 5, where Aristotle offers mathematical analyses of distributive and corrective justice. Such an approach is reasonable, he says, because justice in these two senses is a matter of proportion and, so, falls within the province of arithmetic—arithmetic being understood not just as the study of abstract number but as the science of quantity in general. Now, before we get too carried away at the thought of a mathematical reconstruction of ethics, we might recall Aristotle's admission, at the very start of the Nicomachean Ethics (1094b14–27), that his treatment of what is "fine and right" will not rise to the level of nomothetic science. Nonetheless, we should do our best, as did Aristotle, with the best tools of the best sciences—mathematics included. [End Page 531]

Aristotle was certainly not the last philosopher to deploy mathematics in the service of ethical insight. R. B. Braithwaite (1955) provides a prominent example. Here is Patrick Suppes's assessment: "Braithwaite is surely on the right track in his direct and unashamed use of mathematics in analyzing a moral problem. . . . It would be a pity if formal developments in ethics during the next few decades were restricted to formal logics when so many relevant mathematical concepts are now at hand from game theory, welfare economics, and statistical decision theory. For . . . the fundamental problems of moral philosophy are conceptual problems of evaluation and decision, individual and social, not problems of language" (1958, 215). John Rawls (1971, 134–35) argues that Braithwaite's analysis is misguided, but not because it leans so heavily on mathematics. Indeed, Rawls remarks: "A correct account of moral capacities will certainly involve principles and theoretical constructions which go much beyond the norms and standards cited in everyday life; it may eventually require fairly sophisticated mathematics as well. This is to be expected, since on the contract view the theory of justice is part of the theory of rational choice" (1971, 47). Amartya Sen (1970, 121–23) reaches similar conclusions both about Braithwaite's analysis and about the prospects for an enrichment of ethical reflection by mathematics, as exemplified in Collective Choice and Social Welfare, chapters 7 and 9. (For an overview of some more recent literature, see Verbeek and Morris 2010.)

Cultural Evolution

Beowulf was a man of surpassing virtue: peerless in battle, freehanded, proud, giving no more quarter in words than in arms. Rejoicing over the bodies of his enemies, he harvested heads and limbs as trophies. Quick to exact vengeance and answer any challenge, exulting in the fruits of his exploits but never miserly, he sought fame and shunned dishonor: "Nor, in his cups, did he ever kill his drinking-companions." All in all, he was a "man of supreme merit" and a favored retainer of the Almighty (Anonymous 1957, 78, 87).

While acknowledging that this picture of manly virtue may reflect a way of life adaptive in its time and place, we can still be glad that humanity has developed modes of living congruent with very different ideals—ideals in which martial prowess and the liberal distribution of spoils are less [End Page 532] prominent, ideals that promote a fuller development of human capacities. We can be glad, for example, that the better angels of our culture preach the dignity of productive labor, the danger of concentrated power, the terrible stupidity of wasted talents, and the surprising wisdom of a well-informed public. We can also be thankful for material and social conditions that give such preaching a chance to influence conduct on a large scale.

We apes have had a long struggle and have come a long way. It was once a great step forward to give a sharp edge to a piece of flint. Now we land our tools on little dots up in the heavens. Our technical advances are especially clear: we have won greater and greater control over the conditions that allow us to flourish. That, however, is not the whole story. We have also groped our way forward to a richer understanding of what it means for us to flourish. This has been a hard nut to crack, this question of how we should live, of the best life for a human being—and our work is not done. Thomas Nagel puts the point with particular force: "In ethics . . . we should be open to the possibility of progress . . . with a consequent effect of reduced confidence in the finality of our current understanding. . . . The idea that the basic principles of morality are known, and that the problems all come in their interpretation and application, is one of the most fantastic conceits to which our conceited species has been drawn" (1986, 186). On a more positive note, Dewey emphasizes the progress made by our forebears and our opportunity, indeed responsibility, to continue their work: "We who now live are parts of a humanity that extends into the remote past. . . . The things in civilization we most prize are not of ourselves. They exist by grace of the doings and sufferings of the continuous human community in which we are a link. Ours is the responsibility of conserving, transmitting, rectifying and expanding the heritage of values we have received that those who come after us may receive it more solid and secure, more widely accessible and more generously shared than we have received it" (1934, 87). Moving this work forward will require imagination and ingenuity in all their forms. Help is welcome from every department of culture. The point of this essay has been that help is forthcoming from two sources not usually allied: literature and mathematics.

Literary imagination can vivify moral ideals and aid in their diffusion. It can enrich our stock of moral categories and, by breathing life into abstract types and setting them loose in a landscape rich in human feeling and purpose, can bless us with an enlarged experience of the dangers [End Page 533] and possibilities of our life together. Literature can strengthen our moral understanding even while investing moral ideas with emotional energy.

The blessings of mathematics lie mainly on the side of understanding. Mathematics helps us see moral truths; it is not clear that it does much to help us feel them. "Poetry," says Dewey, "may deliver truth with a personal and a passionate force which is beyond the reach of theory painting in gray on gray" (1929a, 5). To be fair, mathematics, even while painting in gray, offers exultations of its own, but they are the raptures of finding the answer, solving the problem, cracking the nut. We can feel some of that delight when we use mathematical insight to gain clarity in the moral sphere. That feeling, however, will attach to the act of clarification itself rather than the ideal or precept clarified. It would, to that extent, be akin to the satisfaction Trollope might have felt when he recognized that he had produced not just an improving book but a work of art.

Trollope could take further satisfaction in helping his readers feel as they should toward virtue and vice. Can mathematicians really join him in this? Plato would say yes because in better understanding virtue and vice, we more clearly see the beauty of the first and the ugliness of the second and, so, are drawn more strongly to one and repulsed more strongly by the other. A psychologist would be in a better position than a philosopher to determine whether this is so.

Individuality

Whatever the verdict of the psychologists, the point stands that both literature and mathematics can be morally edifying. Having hammered away at that point, it would be well, at the last, to emphasize that number theorists and novelists have interests and passions of their own that should not be sacrificed on the altar of moral edification. Henry James speaks up forcefully for the novelists:

The only obligation to which in advance we may hold a novel, without incurring the accusation of being arbitrary, is that it be interesting. . . . The ways in which it is at liberty to accomplish this result . . . strike me as innumerable, and such as can only suffer from being marked out or fenced in by prescription. . . . A novel is in its broadest definition a personal, a direct impression of life: that, to begin with, [End Page 534] constitutes its value, which is greater or less according to the intensity of the impression. But there will be no intensity at all, and therefore no value, unless there is freedom to feel and say. The tracing of a line to be followed, of a tone to be taken, of a form to be filled out, is a limitation of that freedom and a suppression of the very thing that we are most curious about.

(1888, 384)

Dewey makes a broader case, speaking for artists in general, not just novelists: "The artist in realizing his own individuality reveals potentialities hitherto unrealized. This revelation is the inspiration of other individuals to make the potentialities real, for it is not sheer revolt against things as they are which stirs human endeavor to its depths, but vision of what might be and is not. Subordination of the artists to any special cause no matter how worthy does violence not only to the artist but to the living source of a new and better future" (1960, 243).

In their arguments for artistic freedom, James and Dewey make some points that fit mathematics none too well. Mathematicians do not offer anything like "a direct impression of life." As for unrealized potentialities, Dewey is thinking of real conditions favorable to personal growth and positive social change, conditions that make such advances practicable. An artist's prophetic vision can set us on a new path to a better life with some promise of success; it can help us see that we have the personal and cultural resources to progress, to do better individually and collectively. If some inspired mathematics deserves to be called visionary, that will probably be because it shows us how to do better at mathematics. A mathematical mother lode is a vein of problems whose solution enriches mathematics. We should not look there for a stirring vision of improved social arrangements. The rewards of unfettered art are not the same as the rewards of unfettered mathematics.

Nonetheless, as James and Dewey emphasize, there is a common evil in the subordination of novelists and number theorists to systems or causes not of their making. For Dewey, the problem is not that we might deflect a great man from his appointment with destiny. Our best hope, he thinks, lies not in any superindividual but in the electric confusion of profuse and peculiar gifts meeting opportunities for development. A man of destiny will not move a mountain, but the mountain might give ground to a host of individuals if there is "a social medium which will develop their personal capacities and reward their efforts" (Dewey and Tufts 1932, 276). If there [End Page 535] is destiny in human affairs, it consists in social systems that so effectively withhold opportunities for distinctive and unexpected growth that people lose their individuality: "become imprisoned in routine and fall to the level of mechanisms" (Dewey 1960, 241). To demand that creative individuals offer us moral edification is to take a step toward a stale routine offering little prospect of moral progress. Better to follow a precept Henry James attributes to Guy de Maupassant: "He [Maupassant] is plainly of the opinion that the first duty of the artist, and the thing that makes him most useful to his fellow-men, is to master his instrument, whatever it may happen to be" (James 1888, 249–50). Mathematicians too have their instruments. Those with exceptional instruments should feel it their duty to master them subject only to their own standards or the standards of the mathematical tribe.

Now Dewey himself recommended "ardor in behalf of light shining into the murky places of social existence" and "zeal for its refreshing and purifying effect" (1934, 79). Yes, maybe we should all feel more keenly a moral imperative to strengthen our moral understanding: "One of the few experiments in the attachment of emotion to ends that mankind has not tried is that of devotion, so intense as to be religious, to intelligence as a force in social action" (Dewey 1934, 79). Both number theorists and novelists could march under that banner, but we must not insist that they do so. What will best serve the cause of socially active intelligence is good mathematics and good art. Let good mathematics and good art be the ends-in-view, and other goods will follow.

Stephen Pollard
Truman State University

note

. Thanks to my informant on economic matters: Florence Emily Pollard.

works cited

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Dewey, John. 1929a. Characters and Events, vol. 1. New York: Henry Holt.
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