The Waltz of Reason, page 31
The basic message for moral philosophy is that social norms can develop even in the absence of rationality. Konrad Lorenz, one of the founders of the field of animal behavior (and an adherent of the flawed “good of the species” thinking) described such social norms as “moral-like behavior.”
Interestingly, as soon as we allow for asymmetries between the contestants, very different norms can evolve. Indeed, let us assume that the players can be in different roles: strong or weak, young or old, male or female, resident or intruder. In such cases, conditional strategies can evolve, for instance: “If you are the stronger, escalate; if you are the weaker, run away.” This is clearly a Nash equilibrium strategy: it does not pay for the stronger to deviate from it, and it pays even less for the weaker.
There just remains a small point: namely how to decide who is the stronger and who the weaker. A large part of the behavior of animals in inner-species conflicts can be viewed as a series of tests to find out just this. Rival stags, for instance, ogle each other, do a lot of roaring, and run for a while close to each other on parallel course, almost shoulder to shoulder—all this helps them learn about their rival. Such rituals can last for a long time. They usually end with one of the contestants deciding to concede and give up. On the rare occasions that this is not the case, the last resort is a direct trial of strength, a pushing match with interlocked antlers: again, this does not lead to fatal injuries, in general.
Game theory has been used to describe contests between owners of a territory and intruders. The conditional strategy “if owner, escalate to the hilt; if intruder, concede” leads, once again, to a Nash equilibrium. It does not pay for the owner to deviate from it. It does not pay for the intruder either: any deviation can lead to death or serious injury—and the price isn’t worth it.
This conditional strategy, which John Maynard Smith facetiously named Bourgeois, has been observed in many species, be they birds, mammals, insects, or fishes. Even butterflies display territorial behavior, when “territory” is nothing but a small sunny spot in a forest. Whichever butterfly is the first to discover such a spot will act as if it owns the place, and chase away all would-be intruders. But if the biologist craftily covers the resident butterfly with a dry leaf, for example, then the next butterfly alighting nearby will, within a few minutes, feel at home. If the leaf, then, is removed, both butterflies seem to think that the place is theirs, and try to repel the other. This leads usually to escalated contests: protracted spiral flights that cost both butterflies a lot of energy, somewhat like a war of attrition. Similar experiments on humans can be conducted on the sun terraces of a holiday resort. Guests going for a swim, or for a drink at the bar, tend to leave a towel on their deck chair to indicate that it is theirs. Remove the towel behind their back, and wait for the next guest to occupy the deck chair. When the previous holder returns from the pool, this may lead to dark looks, and occasionally even to an escalated conflict.
The Bourgeois strategy describes a very familiar human trait. Most of us are ready to defend our property and to respect that of others. Where does this social norm come from? Many thinkers have dealt with this question.
Jean-Jacques Rousseau ascribed to the notion of property a truly crucial role in history:
The first man who, having enclosed a piece of ground, bethought himself of saying “This is mine,” and found people simple enough to believe him, was the real founder of civil society. From how many crimes, wars and murders, from how many horrors and misfortunes might not any one have saved mankind, by pulling up the stakes, or filling up the ditch, and crying to his fellows: “Beware of listening to this impostor.”
In the eyes of Karl Marx, private property was the main tool for dominating others. It had to be destroyed. “Expropriate the expropriators!” was the conclusion of his reasoning. His anarchistic predecessor Pierre-Joseph Proudhon blew into the same horn: “Property is theft!” Both philosophers rebelled against the seemingly unbeatable phalanx of all those who held ownership rights as natural and obvious, or at least as useful, if not downright sacred.
A well-known slogan states that “property is nine-tenths of the law.” It is not merely that property and ownership are secured by law and custom, but by a web of social regulations. Economic experiments have documented a deep-seated endowment effect on the individual level. An object becomes more valuable in our eyes when we own it. Normally, I would not be ready to pay $200 for an opera ticket. But if I got it as a gift, I would not be willing to sell it for $200, either. This is a common tendency. Richard Thaler received the Nobel Prize for Economics for discovering it. Before him, the endowment effect was more or less taken for granted. We prize our property, in part because it is valuable, no doubt, but in part also because it is ours. The sparrow in my hand is better than the dove on the roof. The trait must have to do with loss aversion: the widespread propensity to count losses as higher than gains.
Small children grasp at anything. The instinct to hold tight may be a leftover from the tree-dwelling past of our ancestors. The writer Elias Canetti sees in that instinct the germ cell of grab-and-hold capitalism. Be that as it may, kids have no problem grasping the notion of “mine.” Moreover, many animal species seem to have a sense of ownership. This holds particularly for those species that own a territory, a nest, or a cache. In this respect, Rousseau is wrong when he thinks that property is the invention of some scoundrel. It is quite conceivable that a tendency to claim something as “one’s own,” and to defend it, is not a mere cultural trait but has a biological basis.
This leads us back to evolutionary games. We have seen that in an owner-intruder game, the Bourgeois strategy entails a Nash equilibrium. It fits with our social norms. Curiously enough, there is another equilibrium, its mirror image, which is also stable: escalate if you are a challenger, concede if you are the owner. It serves just as well to avoid escalated conflicts. Moreover, it can even be supported by an appeal to fairness. You have had time to enjoy your property, now it is another’s turn.
John Maynard Smith named this the Proudhon strategy. In real human societies, the Proudhon strategy seems never to have emerged, except fleetingly. There are many attempts to explain its glaring lack of success—none generally accepted. Obviously, evolutionary game theory still has some way to go.
To apply mathematical reasoning to ethics is an old dream. Baruch Spinoza, for instance, wrote a treatise on ethics entitled Ethica, Ordine Geometrico Demonstrata (or Ethics, Demonstrated in Geometrical Order). Up to a few decades ago, it was taken for granted that such a mathematical underpinning of ethics must be based on rationality—for isn’t reason the domain of mathematics? Evolutionary game theory, however, is mostly based on a naturalistic view of ethics as an anthropological or even biological phenomenon. This perspective threatens to loosen the grip of philosophy on ethics, and to hand it over to the sciences. Such a science-based perspective—ethics as a branch of psychology—is hardly new, but it is surprising that so much of its recent progress makes use of mathematical tools that do not even mention rationality.
In the early years of game theory, it was frequently discussed whether it was a normative or a descriptive theory—whether it applies to what agents should do, or to what they actually do. Game theory, however, is a branch of mathematics. As such it is as little normative or descriptive as, for instance, algebra. It merely helps to explore the consequences of this or that assumption. That players are rational is one such assumption.
The rationality doctrine leads to many consequences that are nowhere to be found in real-life interactions. One telling example is backward induction. If two players know that they will play a donation game for six rounds, the outcome of the last round seems a foregone conclusion: both will defect, since this is their dominant strategy—it is better no matter what the co-player does. Nothing in round five will affect this outcome. To all intents and purposes, it is as if round five were the last round. But then, its outcome is also a foregone conclusion: both players defect. And on it goes, step by step back to round one. Backward induction dictates that rational players will never cooperate, and remain stuck with payoff zero. In experiments, however, this outcome is rare. Players will mostly cooperate, with the possible exception of the last or next-to-last rounds. In such a game, it is simply too silly to be rational.
The prominent role of rationality is due to a force of habit alone. Some 200 years ago, mathematicians lost their creed in a unique set of geometrical axioms. Why should anyone expect a unique set of axioms for game theory?
PART IV
14
Language
Speaking in Ciphers
A Double Bill
In the spring term of 1939, the University of Cambridge offered two distinct courses on “Foundations of Mathematics”—an intellectual extravagance of sorts. However, there was little danger of waste, even less of redundancy: the two lecturers, Ludwig Wittgenstein and Alan Turing, were each known to go their own ways.
Figures 14.1 and 14.2. Ludwig Wittgenstein (1889–1951) and Alan Turing (1912–1954) on parallel courses.
Ludwig Wittgenstein was turning fifty, an expat professor of philosophy in Cambridge. Alan Turing had not yet reached his thirties. He was a fellow at King’s College who now held his first lecture course, for the modest fee of 20 British pounds. His topic was the “foundations” in the classical sense, as understood by modern mathematicians: meaning axioms and logic. In the wake of Hilbert and Gödel, Turing had electrified the field with his seminal paper on computation and the decision problem. Here was a worthy successor to the Cambridge trio of Russell, Whitehead, and Ramsey, who had done so much of the groundwork.
Wittgenstein went after a different game. He had no truck with the usual spiel that mathematicians dished out to each other about the foundations of their science, perhaps even fooling themselves to believe it. Wittgenstein was from Vienna, and as little inclined to take words at their face value as Sigmund Freud or Karl Kraus. Wittgenstein wanted to know what mathematicians really do.
Everything in Wittgenstein’s biography is spectacular. His father was a steel baron and patron of the arts, Habsburg’s answer to Andrew Carnegie. Ludwig grew up in a palace, as the youngest of eight. He started out in aeronautics at a time when the conquest of the air took wing. In 1912, however, the uncommonly intense young engineer switched tracks and enrolled in the University of Cambridge for philosophy. His teachers were Bertrand Russell and George W. Moore, the heralds of analytical philosophy. Within a few months, they took down his dictations on logic. Shortly after, he retired to an isolated hut in Norway to pursue his thoughts undisturbed.
When World War I started, Wittgenstein volunteered for the Austrian Army. In between spells at the front, he finished his Logical-Philosophical Treatise, coolly stating in the preface that he considered its truth to be “unassailable and definitive.” After the catastrophic defeat of the Central Powers and a one-year spell in an Italian POW camp near Monte Cassino, he returned to forlorn, destitute Vienna, and donated his vast inheritance to his surviving siblings (three of his brothers had taken their lives). He earned his living as a teacher, at elementary schools located in the backwoods of Lower Austria. His booklet, renamed Tractatus Logico-Philosophicus, appeared after agonizing delays.
Wittgenstein was done with philosophy. Hadn’t he solved the problems, in their essentials? Refusing to engage with busybodies, he snubbed the persistent attempts of the Vienna Circle, that avant-garde group of philosophers and mathematicians, to get close to him and soak up his words.
As a teacher, Wittgenstein was highly motivated, but prone to fits of classroom rage: slapping his pupils, or pulling their hair, their ears, whatever came to hand. His career was brought to an abrupt end when he knocked out an eleven-year-old. A chastened Wittgenstein returned to Vienna after six years of school service. Next, he named himself an architect and directed the construction of a modernistic town house for his sister. And at last, having finished with the workers and craftsmen, he condescended to meetings with select members of the Vienna Circle. Some turned out to be worth talking to. It gradually transpired that there was still something left to do in philosophy.
In 1929, now aged forty, Wittgenstein took the train back to Cambridge and submitted his Tractatus for a PhD. Some time before, he had claimed that nobody can do philosophy for more than ten years. Nearer to the truth is that nobody can do without philosophy for more than ten years. All through the 1930s, Wittgenstein wrote and discussed tirelessly, in Cambridge, in Vienna, or back in his old Norwegian hut, but he published nothing. This did not stop the University of Cambridge from appointing him to a chair in philosophy. They knew a legend when they met one.
Only a handful of select disciples were allowed to see Professor Wittgenstein. The rest were kept at bay. “In certain circles,” wrote Ernest Nagel, a young philosopher from the United States, “the existence of Wittgenstein is debated with as much ingenuity as the historicity of Christ has been disputed in others.” Those who wished to attend his lectures had to undergo an interview with Wittgenstein. Nagel was turned down: Wittgenstein said that he wanted no tourists. Turing, however, was accepted: Wittgenstein could make use of a mathematician who was unafraid to come out of his corner and take it. This is how Turing came to stand answer to Wittgenstein for all that had gone wrong with the foundations of mathematics, be it set theory, formal systems, or metamathematics.
Here is a sample of their exchanges:
Wittgenstein asked Turing: “How many numerals have you learned to write down?”
Turing, sensing what was to come, replied cagily: “Well, if I was not here, I would say countably infinite!”
Wittgenstein: “How wonderful—to learn infinitely many numerals, and in so short a time!” And with Turing still so young!
Turing conceded: “I see your point!”
Wittgenstein: “I have no point!”
And so it went on.
Like all the other students, Turing had had to promise beforehand never to skip any of Wittgenstein’s classes (of which there were two per week). On March 19, 1939, however, Turing excused himself. Wittgenstein was nettled, acidly remarking:
Unfortunately Turing will be away from the next lecture, and therefore that lecture will have to be somewhat parenthetical. For it is no good my getting the rest to agree to something that Turing would not agree to.
Turing took it without flinching. He knew how to keep mum. For some time already, he had been earmarked by the British Secret Service. Sometimes he had to leave Cambridge to follow ultra-secret courses on cryptanalysis. Everyone knew that war loomed around the corner, and MI6 was worried about Germany’s military cipher. Turing’s former PhD advisor Max Newman had brought up his name. It proved a brilliant hunch. The two of them would eventually devise some of the first proto-computers, rattling dinosaurs of machinery, to decode German top-secret messages. (See Chapter 5.)
By 1939, however, Alan Turing had only conceived a purely hypothetical computer to investigate the limits of formal systems. The abstruse automaton would play an important role in his lectures on foundations. In the wake of Gödel’s demonstration of undecidable mathematical propositions, these last ten years had seen breathtaking progress.
Forever the contrarian, Wittgenstein saw things in a completely different light. He explicitly stated: “My task is not to speak on Gödel’s proofs etc. but to speak past them.” These last ten years, for him, had been taken up with establishing the philosophy of language.
Wittgenstein’s guiding rule was: “The meaning of a word is its use in the language” (though only “for a large class of cases”). To examine that use more closely, he had devised the method of language games, “to bring into prominence the fact that the speaking of language is part of an activity, of a form of life.” His task, as a philosopher of mathematics, was to describe these games, not to explain them. Games have rules, and players need not always be aware of them. Wittgenstein wanted to uncover them, patiently, one by one. He did not share the view that there lurked a seamless entity behind what is called “mathematics.” Instead, he spoke of the “colorful medley” of mathematics. Just as astronomy deals with a wide variety of phenomena (planets, radio waves, galaxies, dark matter) that have little in common except being out there in the sky, so mathematics cannot be reduced to a single object or a single method. It is a motley.
To follow in Wittgenstein’s wake, we may imagine landing, like ethnographers, on some unknown shore of Archipelago Mathematics, and observe how the natives—the mathematicians—communicate with each other. This may help us grasp the rules by which they live, their “form of life” (to use one of Wittgenstein’s favorite terms).
Speaking Mathese
Each science has its language. So has mathematics, evidently. In addition, mathematics is a language. This is a widely accepted view. Here are quotes from two of the foremost mathematicians of our time, Yuri Manin and Alain Connes. The former says: “The basis of all human civilization is language, and mathematics is a special form of linguistic activity.” The latter goes even further: “Mathematics is unquestionably the unique universal language.”
Ever since Galilei, physicists have taken this view for granted: “The universe is written in the language of mathematics: the letters of this language are triangles, circles and other mathematical figures.”
