In the minds of those who are conventionally thought of as the world's governors or leaders, the recent crisis of financial capitalism is now behind us. Indeed, it is as though the crisis never occurred. The task of putting the global financial system back on track has been successfully carried out. If the economists are to be believed, the causes of the crisis have by and large been elucidated; everything that happened, or almost everything, has been explained. And yet the crisis took the world by surprise. Just as the persons who planned the attacks of 11 September 2001 may not have anticipated that the impact of the hijacked airliners on crashing into the Twin Towers would cause them to collapse, hardly anyone imagined during the summer of 2007, or even in the spring of 2008, that a seemingly quite localized crisis in the American housing market would shake the foundations of the world financial system. It would be unkind to recall the complete blindness of the experts and the utter failure of their forecasts; unkind, too, to recall the irresponsible endorsement of these forecasts by the relevant authorities, among them a former head of the Federal Reserve Board and the current head of the International Monetary Fund, as well as a number of finance ministers and chiefs of state.

We now find the same people proudly congratulating themselves that the worst did not come to pass, scarcely suspecting that the very rails on which capitalism once again rides, owing to their noble efforts, led it to the edge of the abyss. It is for this reason, considering the clear and present danger to which their willful ignorance exposes mankind, that their smug self-regard is intolerable, their complacency ridiculous, and their optimism obscene.

No doubt these people will reject the charge out of hand. I am the one who is guilty of recklessness and irresponsibility, they will retort, for doomsaying sows fear and makes reasoned analysis impossible. To this it is tempting to reply that between the optimist and the pessimist there is, at bottom, no difference-other than that the pessimist is better informed. Instead, let me pose two disturbing questions. First, could it be that optimists owe it to themselves to dread the prospect of catastrophe, precisely because they are optimists? Second, could it be that they owe their optimism to the fact that, perhaps without knowing it, they themselves are doomsayers?

On the Probability of Extreme Events

Not only are the ecological and economic crises of our day connected in ways that reinforce each other; they have in common a number of quite singular features. The one that interests me here has become familiar from attempts to estimate the consequences of climate change. It has not yet acquired the same notoriety in relation to the present financial situation, but that will not be slow in coming. If the rate of global warming begins to markedly increase, as recent evidence seems to suggest may already be happening, extreme events— cyclones, storms, rising sea levels, floods, droughts, and so on-will occur more frequently and with greater intensity. Note that our intuitive conception of the extreme combines two distinct aspects: the magnitude of a phenomenon and its degree of rarity. Since the events that we are considering are random, they can be analyzed using a statistical tool known as a frequency distribution curve (or, what for present purposes we may assume to be the same thing, a probability distribution curve) that, for any given degree of intensity, indicates the probability of a phenomenon's occurrence.

The most familiar of such distributions is the famous "bell-shaped" curve (also known to probability theorists as a "Gaussian" or "normal" distribution). If I toss a coin two thousand times in a row, the number of heads will, on average, be a thousand. Probably it will not be exactly a thousand, but it will very likely be located within a relatively narrow range lying on either side of this number. Extreme events— only a few hundred tails, or else fifteen hundred tails, out of two thousand coin tosses— cannot be excluded, of course, but the bell-shaped curve assigns an exceedingly low probability to them.

Elementary events (in this case successive coin tosses) are observed to converge upon a normal distribution when they are causally independent of one another. The probability of the next toss being tails will always be equal to one-half, even if the preceding tosses have overwhelmingly been tails. All this is well known, at least tacitly, even by those who are bored to tears by statistics. In modern democracies, where no issue can be decided without a sampling of public opinion having been made first, statistics are unavoidable. But now things become both more complicated and more interesting. Much more interesting.

For several years now, another type of distribution has occupied the attention of specialists. It is found everywhere catastrophic natural events threaten to occur: raging rivers in Europe and North America, hurricanes in the Caribbean basin and the Gulf of Mexico, volcanic eruptions and tsunamis in the Indian Ocean, forest fires in California and along the Mediterranean. It is also found in the world of finance, with the expansion and bursting of speculative bubbles. Although this type of distribution attaches a relatively low probability to extreme events, the chance is nonetheless considerably higher than the one assigned by the bell curve. The statistical weight of a random event, as we have seen, is the product of its magnitude and its probability. If events of very great magnitude are assumed to have a small— though by no means infinitely small— probability, the prospect of a major catastrophe, though it remains comparatively unlikely to occur, cannot help but weigh heavily in our assessment of future risk. Inevitably, the shadow cast by its very possibility darkens our outlook.

In order to see why this distribution, which I have so far refrained from identifying by one of its several names, should be, if not universal, then at least characteristic of the events that are of most pressing concern to us today, let us engage in a thought experiment. Imagine that ten thousand coins rain down from the sky, falling uniformly over an area in which a hundred buckets have been placed to catch them. If coins are assumed to land independently of one another, the distribution of coins per bucket will conform to the bell curve. The number of coins landing in most of the buckets will be close to the average, which is to say a hundred; only a few buckets will contain several dozen coins or, by contrast, several hundred of them. Let us now change the conditions of the experiment, and assume that the larger the number of coins found in any given bucket, the greater the chance that more coins will fall into it. Under this assumption, the distribution of coins over the set of buckets as a whole now takes on an entirely different appearance: the deviations from the mean allowed by the bell curve become amplified by a self-reinforcing mechanism, with the result that the probability of extreme events occurring is considerably increased.

This distribution takes one of its names from the Italian sociologist Vilfredo Pareto, who together with the French-born economist Léon Walras formed what came to be known as the Lausanne School, the cradle of neoclassical economics. Pareto was interested in comparing national patterns of personal income distribution. In every country, and on every scale of wealth, he observed that the ratio of the expected value¹ of individual incomes above a given level to that level is constant. Let us assume the ratio is 1.3. This means that the expected value of incomes higher than the minimum wage, for example, is equal to 1.3 times this wage; and also that the expected value of incomes greater than the salary of a trader at a Wall Street investment bank, for example, is equal to 1.3 times this salary. The shape of this distribution corresponds exactly to the one generated by our thought experiment of the coins and the buckets, which in turn casts light on what gives the Pareto distribution its air of universality. Recall the dynamic at work in that experiment: the more coins a bucket already has, the more new ones it will attract. A Pareto distribution can therefore be seen as the consequence of a rule that will surprise no one, namely, that the wealthier you are, the greater your chance of becoming wealthier still.

Let us now consider the matter from a geometrical point of view. A figure that is self-similar on every scale of observation is said to be "fractal." This concept is due to the French mathematician Benoît Mandelbrot, certainly one of the most powerful and original minds of our time. A Pareto distribution, it turns out, is a kind of fractal distribution.² From this fact flow a number of very remarkable properties.

A distribution is fractal if the ratio of the expected value of magnitudes higher than a given level to this level is constant. Consider the distribution of life expectancies in a country where many children die at birth or in the first months of life. If a child manages to make it through this early period, his survival is likely to be the sign of a particularly robust constitution. It is therefore probable that his life expectancy from this point forward, which is to say with regard to the life that he has yet to live, is large. This relationship is typically fractal. Alas, it will not go on indefinitely. Inevitably there arrives an age when each additional year of life, far from being accompanied by the expectation that the number of years yet to be lived is increasing, signals that life is inexorably drawing nearer to its end.

Mandelbrot has tried to convey a sense of the special character of fractal distributions by means of a very fine fable. He asks us to imagine a land that is permanently covered by disorienting mists and fog, and which contains a great many bodies of water. Some are no more than ponds; others resemble lakes, still others seas. An expedition of surveyors and cartographers is sent out from a neighboring land of lakes whose distribution with respect to size is fractal. Coming upon what appears to them to be a lake, they set out to cross it in a boat. Owing to the fog it is impossible to see the far shore. They assume from their experience of their own country that the size of the body of water they are rowing across likewise obeys a fractal distribution.³

The longer the explorers row across the lake without reaching the opposite shore, the larger they are justified in believing the remaining distance to be— as if the horizon were receding faster than they were approaching it. They reason in the following way: the already considerable amount of time that has gone by without the opposite shore coming into view suggests that we are crossing one of those extremely large expanses of water to which the fractal law assigns a sizable weight; therefore it is likely that there is still a much longer way to go than we had thought at first. The idea of an eternally receding horizon is an illusion, of course, since the explorers cannot doubt that the lake has a definite size and that the distance separating them from their destination is a fixed quantity. |They know that sooner or later the opposite shore will come into view. Yet when they pause to compute the expected value of the distance still to be covered, they are bound to conclude that they are further away than the last time they estimated this distance.

Which should they trust: their knowledge that the lake is a finite expanse or their rational expectations? The stark contrast between knowledge and expectation in this case is a vivid illustration of the devilish properties of fractal distribution.⁴ The explorers' confusion reaches its height just as they are on the verge of seeing the opposite shore, for it is then that they believe they have never been further away. What is more, the longer they have been rowing, the more startled they will be when this moment actually arrives. The fog in Mandelbrot's Parable of the Receding Shore is the equivalent of what the German philosopher Günther Anders called "the blindness before the Apocalypse."⁵

Surely this, or something very much like it, must have been Bernard Madoff's state of mind as he sailed the high seas of financial banditry. The broader the base of his pyramid became, with the increasingly successful recruitment of new clients, the greater his confidence that the rewards of his scheme would continue to grow— so long, of course, as he was not caught. And yet he could not be unaware that one day the end would come, and that his scheme would collapse like a house of cards. The longer the scheme worked, the more terrible the surprise was bound to be.

It would be both unfair and misleading to place special emphasis on a single swindler, however. Mandelbrot has shown empirically that speculative phenomena generally are governed by a fractal law. In the euphoric "boom" phase, as the bubble begins to expand, the more optimistic investors are, the greater their reason for looking forward with still more optimism. Indeed, it is just when the bubble is about to burst that the euphoria reaches its highest point.⁶

The theory that I have just sketched is hardly new, and it has been confirmed by events many times over. Even so, it is known only to a few many influential figures in the financial world. This has led some experts (including Mandelbrot himself) to insist that not to be acquainted with it is proof of shameful ignorance.⁷ However this may be, one would like to know whether familiarity with the theory causes a person to change his behavior. This question, like the paradox to which it gives rise, inspires a sense of foreboding. Prudence therefore dictates a maxim: the greater one's reasons for optimism, the more one owes it to oneself to fear catastrophe and to guard against it, for the end is undoubtedly near.⁸ As a theoretical matter, this double bind is resolved by recognizing that, while optimism is rational at one level, doomsaying is rational at another, which transcends the first, for it looks out upon the future from the point of view of the end of the voyage that lies ahead, and not of the voyage as it unfolds. I call this form of prudence "enlightened doomsaying."⁹ It involves an act of imagination by which one looks ahead, to a moment after the extreme event has occurred, and contemplates the path leading to it from a perspective that combines surprise at the event's occurrence with foreknowledge— which is to say, certainty— of this surprise.

For a philosopher, the idea of telling someone he is going to be surprised calls to mind a famous paradox.¹⁰ Here is one of its forms. On a Sunday a man is sentenced to death and told that he will be hanged one morning in the coming week, without the day being named. This warning is accompanied by a prediction, which will turn out to be a diabolical trap: on the day chosen for the execution, when the executioner comes at dawn to bring him to the scaffold, the condemned man will be surprised. He is then taken back to his cell, where he racks his brain in the poisonous hope of discovering when his existence will come to an end. It seems obvious to him that it cannot be the following Sunday. For in that case he would still be alive at noon on Saturday, and therefore in a position to deduce that his hanging will take place the following morning-in which case he would not be surprised. He therefore crosses Sunday off the list of possible occasions. But now it is Saturday's turn to be dismissed. Since Sunday is no longer a candidate, the very same logic will hold good at noon on Friday, assuming the condemned man is still alive then. On applying this reasoning to each of the remaining days of the week, he becomes convinced that none of them can be the day, and therefore that he will not be executed. And so when the executioner comes to get him at dawn on Thursday, he is completely taken by surprise— just as he had been told he would be.

Whatever may be the logical virtues or defects of this argument, it plainly depends on the existence of a known end or term: the life of the condemned man will not last longer than the next Sunday. But it is precisely this condition that does not obtain in the capitalist world. Torn between hope and despair, Madoff expected nevertheless that the stream of new clients would continue to grow. Speculators, for their part, counted on the subprime bubble to go on expanding forever, and the desparate Americans who mortgaged their entire future in order to buy a house looked to the unlimited increase in its value in order to be able to pay for it. What makes capitalism possible is the belief that it is immortal. What may be called the original sin of capitalism lies concealed in the fact that the future must endlessly stretch out before it if, at any given moment, it is to be able to deliver on its promises. This is the source of the cult of growth. For the capitalist system to function satisfactorily at any given moment— which is to say, for full employment to be achieved— agents must anticipate that its expansion will go on indefinitely. The lesson of Mandelbrot's parable is that the longer the final reckoning is postponed, the more surprising its inevitable occurrence will be.

The world's leaders have, as I say, succeeded in putting the capitalist locomotive back on track. For the moment its progress is halting; but as it picks up speed, the more hopeful they will become and the more firmly they will believe in a radiant future. It is at precisely this moment that they ought most to distrust their reasons for optimism. For catastrophe may be lying in wait for them just around the bend.

The Economy of Apocalypse

My second question follows naturally from the first, while at the same time pointing in the opposite direction. It proceeds from the uneasy suspicion that the optimism of the prime movers of the crisis, among them the world's governors, is fed by a catastrophism that dares not speak its own name.

The answer I wish to propose here, very tentatively, was first formulated by one of the most insightful commentators on the crisis, the financier Peter Thiel.¹¹ As a young man he co-founded PayPal, the world's largest e-payments company, before going on to become a principal investor in Facebook. Thiel's perspective is that of an enlightened doomsayer, but, unlike armchair philosophers like myself, he has put to the test of experience investment decisions that are based on explicit assumptions and rigorous analysis (and not, it should be added, on mathematical models so complex and so opaque that they have acquired something like an autonomous power of decision).

Thiel is struck first of all by the wholly novel character of speculative bubbles over the last twenty years or so, with regard both to the circumstances under which they have formed and to the violence with which they have burst, one after another. The euphoric phase, no less than the crash itself, displays the characteristic features of extreme events— so forcefully, in fact, that even the laws of fractal distribution seem incapable of accounting for them. Just before the Japanese real estate bubble burst in the late 1980s, the capitalization of the Tokyo Stock Exchange represented half the world's market capitalization. The Land of the Rising Sun seemed destined to rule the earth, or so many believed. The brief reign of Japan, Inc. was followed by the still vaster Internet bubble of the late 1990s— the most enormous boom the world had ever seen. No one could have imagined that it would be succeeded in its turn, five years later, by a global real estate bubble of even greater magnitude.

Some analysts see such events as the result of the "irrational exuberance" of markets; many more blame the greed of traders and their love of profit— as if this were something new. On all sides, intellectual laziness and incuriosity take refuge in moral indignation. We will be better served by trying instead to uncover the deeper causes of our current predicament.

In the kingdom of money, as Peter Thiel is well qualified to observe, the apocalyptic perspective has few champions; indeed, there is still less sympathy for it there than in society as a whole. What appeal could the idea that capitalism is mortal possibly hold for an investor? If capitalism were to die, nothing would any longer have value. If its end were to be predicted, and the date of its demise announced, the prophecy would be immediately falsified, for in that case, the possibility of indefinitely extended existence having been denied, catastrophe should strike at the time of the prediction and not at the predicted time. The alternative, a far more attractive one, is to act as though capitalism is immortal. And yet as Thiel shows, while at the same time introducing a new paradox, this does not mean that the apocalyptic perspective has not exerted an immense influence on the calculations and behavior of investors. Quite the contrary.

The survival of capitalism is today indissociably linked to the success of globalization. But what would the failure of globalization signify? That the forces of anti-globalization had triumphed after all? Thiel dismisses this suggestion, for in his view anti-globalization is part and parcel of globalization, and cannot exist apart from it.. Paraphrasing Tocqueville, one might say that globalization draws its strength from everything that opposes it. Globalization bears the marks of a providential dispensation: it is universal, it is enduring, it escapes human power at every turn; events, no less than human beings, all serve to promote its development. If globalization fails, this can only be the result of a major catastrophe, whose collateral damage will include the end of capitalism. This catastrophe would resemble more or less closely the one whose broad outlines have been sketched by doomsayers such as myself.¹² The human destruction of nature and the ceaseless escalation of violence among human beings conspire to cast the very survival of humanity in doubt, with the most terrible threat remaining the threat of widespread nuclear conflict.

Thiel does not believe that economic and financial agents permit themselves to contemplate the prospect of catastrophe directly. They eliminate it from their calculations, on the ground that it is too horrible to bear serious examination. But it is precisely in removing it that they give it a place; in fact, a quite considerable place. In trying to make sense of this paradox, it will be instructive to make a simple calculation. Imagine an investor who is keenly aware of the threat, but who nonetheless does not wish to factor it into his assessment of future outcomes. Intuitively he understands that the path capitalism must travel in order to survive is like the crest line in an alpine landscape, beyond which lies the abyss. Let us suppose that the probability this investor tacitly assigns to the optimistic scenario— the survival of commerce under successful globalization— is 10%. If he anticipates that a certain business will one day be worth $100 per share, as long as the optimistic scenario comes to pass, what price should he be willing to pay for it today? Presumably 10% of $100: multiplying probability by magnitude, he arrives at a valuation of $10. Note that this calculation completely neglects the other branch of the alternative, which implies disaster for all investors. If a 90% probability of global catastrophe were taken into account, the expected value of the share would not only be negative; the anticipated loss would be infinitely negative! This deliberate neglect lies at the heart of Thiel's paradox. During the most recent great bubbles, investors have not valued such shares at $10, but at much higher amounts, no doubt in many cases close to $100. Indeed, disregarding the catastrophic scenario altogether has the consequence that in any possible world in which investors survive, the share price will turn out to be $100. In that case its anticipated value should logically be $100 as well.

This sort of reasoning calls to mind a humorous advertisement for the French national lottery some years ago that cited a statistical study showing that 100% of past winners had bought a ticket. Thiel insists, however, that one must not lose sight of the psychological context surrounding the formation of recent bubbles. If investors risked so much of their money on the success of Internet firms in the late 1990s, it is because the alternatives seemed to them frighteningly bleak. If America's new class of paupers rushed to take advantage of subprime mortgages, it is because they saw this as the only way to avoid certain destitution in old age. Could it be that these people, in conducting their own optimistic thought experiment, showed more foresight than their neighbors? That in projecting themselves into the only conceivable future that was not catastrophic, and thereby granting it the probability of a sure thing, they did what had to be done in order for it to have a chance of coming about?

We are therefore now in a position to formulate Thiel's paradox: only by adopting the apocalyptic perspective can optimism be given its fullest possible expression. What we have witnessed during the past decade is the enactment of this paradox, and with it the emergence of the double bind that has left so many people stranded between hope and despair. I very much fear that these two developments confirm the wisdom of the method that I recommend under the name of enlightened doomsaying. In combination, they enable us to see that the embrace of unrestrained optimism grows out of a diffuse, unreflective catastrophism, and that this in turn justifies a rational form of catastrophism— one that calls upon us to act with the aim of nullifying the apparent inevitability of catastrophe.

Paris, November 2009

* Ecole Polytechnique, Paris and Stanford University. jpdupuy@stanford.edu

¹ The expected value of a random variable is the probability-weighted sum of its possible values.

² A Pareto (or fractal) distribution is also called a "power law," especially in mathematical physics, with reference to the scaling exponent of the formula that characterizes it. The so-called "fat tail" of this distribution represents the relatively high weight it assigns to extreme events. Additionally, the term "Lévy flight" is used in the technical literature of probability to refer to a type of random walk in which the length of the steps is distributed according to a power law. The still rather unsettled character of the related terminology is evidence that this form was discovered more or less simultaneously in a wide variety of fields, each of which claimed the right to give it its own name.

³Mandelbrot published this fable for the first time in "Hasards et tourbillons: Quatre contes à clefs," Les Annales des Mines (November 1975): 61-66; available on-line at http://math.yale.edu/mandelbrot/web_pdfs/078hasardsettourbillons.pdf. Modified versions were later published in English; see, for instance, Benoît B. Mandelbrot and Richard L. Hudson, The (Mis)Behavior of Markets: A Fractal View of Risk, Ruin, and Reward (New York: Basic Books, 2004). This premonitory book, published before the crisis, seems to have gone entirely unnoticed by its principal actors. As Mandelbrot remarked in a recent interview ("Il était inévitable que des choses très graves se produisent," Le Monde [18-19 October 2009]), it was inevitable that a very serious crisis would occur. See also Mandelbrot, Fractals and Scaling in Finance: Discontinuity, Concentration, Risk (New York: Springer, 1997).

⁴ In mathematical terms, the "variance" of the distribution (i.e., its spread, or average distance to the mean) is infinite. In philosophical terms, because the distribution is self-similar, it has no scale of its own: the scale is determined by the observer's position, whatever that may be.

⁵ See Günther Anders, Die Antiquiertheit des Menschen: Über die Seele im Zeitalter der zweiten industriellen Revolution (Munich: Beck, 1956).

⁶ See Jean-Pierre Dupuy, La panique (Paris: Les Empêcheurs de Penser en Rond, 2002).

⁷ Christian Walter and Michel de Pracontal, in Le virus B: Crise financière et mathématiques (Paris: Seuil, 2009), show that the world of finance remains incurably attached to the bell curve, or law of normal distribution (the "B" of the book's title refers to Brownian motion, a random walk whose steps are governed by this law). The authors attribute a large part of the recent financial crisis to the chronic and flagrant underestimation of the likelihood of extreme events, which they see as the result less of perverse institutional incentives than of a culpable unwillingness to see the world as it really is.

⁸ We know today that Madoff found himself in precisely this double bind: the surer he was that luck was on his side and that his pyramid would continue to grow, the more terrifyingly certain it became that the end was near and that he would soon be caught. On the living "nightmare" of the eight years before his arrest, see "Lapses Helped Scheme, Madoff Told Investigators," New York Times (31 October 2009).

⁹ See Jean-Pierre Dupuy, Pour un catastrophisme éclairé: Quand l'impossible est certain (Paris: Seuil, 2002).

¹⁰ Known variously as the Paradox of the Unexpected Hanging, the Hangman Paradox, and the Prediction Paradox. W.V.O. Quine, one of the founders of American analytic philosophy, published a particularly subtle commentary on it; see "On a So-called Paradox," Mind 62 (1953): 65-66.

¹¹ See Peter A. Thiel, "The Optimistic Thought Experiment," Policy Review (February-March 2008); available on-line at http://www.hoover.org/publications/policyreview/14801241.html.

¹² See Jean-Pierre Dupuy, La marque du sacré (Paris: Carnets Nord, 2009).