The Logic of Miracles

Making Sense of Rare, Really Rare, and Impossibly Rare Events

Laszlo Mero

• Publisher: Yale University Press
• Year: 2018
• ISBN:
  9780300238488
• Topic: Mathématiques
• Subtopic: Probabilités et statistiques

"Consistently surprising… The Logic of Miracles breaks new ground in the relationship of probability, fate, and the ability of human beings to behold them."—Douglas Rushkoff, author of Team Human We live in a more turbulent world than we like to think. Yet the science we use to analyze economic, financial, and statistical events mostly disregards the world's essentially chaotic nature. We need to get used to the idea that wildly improbable events are actually part of the natural order. Here, a renowned mathematician and psychologist explains how the wild and mild worlds (which he names Wildovia and Mildovia) coexist, and that different laws apply to each. Even if we live in an ultimately wild universe, he argues, we're better off pretending that it obeys Mildovian laws. Doing so may amount to a self-fulfilling prophecy and create an island of predictability in a very rough sea. Perched on the ragged border between economics and complexity theory, the author proposes to extend the reach of science to subjects previously considered outside its grasp: the unpredictable, unrepeatable, highly improbable events we commonly call "miracles." "It's hard to see how miracles and math fit together. But if you accept László Mérö's invitation, you will enter a world where miracles are normal and the predictable sits side-by-side with the unpredictable. Along the way, he unveils the mathematics of the stock market and explains, in a playful yet mathematically accurate way, the roots of market crashes and earthquakes, and why 'black swans' are not just calamities but opportunities."—Albert-László Barabási, author of Linked

PART ONE

Secular Miracles

The world may be one, but it has endless ways of revealing itself.

1

On the Existence of Miracles

The Big Bang, luckily for us, is not a reproducible event. If it were, we would all be dead.

My childhood friend Alex, who later became a successful businessman, doesn’t believe in miracles. Nevertheless, one of his major ambitions, which he considered a matter of patriotism toward the small Central European country where we grew up, was to register a Hungarian enterprise on the Nasdaq stock exchange. It didn’t matter whether the company dealt in medical technology, energy supply networks, or computer games. It just had to be new, original, and marketable worldwide. If he saw any possibility in a firm, he was willing to invest in it. And that was how he began to grow a little business of mine that had been quietly struggling to survive.

Alex took over the management of my firm, and it was then that I saw how amateurishly I had been doing things. He didn’t, however, interfere in technical matters, although I made severe mistakes in that area as well. His attitude was the same as that of IBM founder Thomas J. Watson, who was once approached by an executive confessing to an error that had cost the company $10 million. “Go ahead,” said the executive. “Fire me. I deserve it.” “Fire you?” Watson responded. “I just spent ten million dollars educating you.”¹

When something went wrong, Alex would quote his business mentor, who was fond of saying, “There is always the next challenge.” The chance that a miracle was waiting just around the corner kept him working with enormous enthusiasm and devotion. Of course, Alex wouldn’t have called what he was waiting for a “miracle.” I don’t think a week has gone by in the last thirty years without his saying at least once, “There are no miracles.” Perhaps that is why the miracles he sought so passionately never materialized. He never did get a Hungarian enterprise listed on the Nasdaq, although there is one now, called LogMeIn. Nevertheless, his efforts launched many enterprises, institutions, and investments that have proven successful and sustainable.

I maintain that miracles exist in our world; if so, it means that they should, in theory, be accessible to scientific study. But how are they to be studied if what makes them miraculous is exactly that they are one-time events and thus by their nature irreproducible, and unreachable by scientific analysis?

What is known as the scientific method relies on the formulation of models that can be verified through systematic observation, measurement, and experimentation under reproducible conditions. If the experiments confirm the model and the model produces miracles—that is, unexpected, extremely unlikely phenomena—then the analysis of such a model should provide a purely scientific way to study what we are calling miracles. For example, the Big Bang, luckily for us, is not a reproducible event—at least not by us. If it were, we would all be dead. Nevertheless, scientists are able to study the Big Bang with the aid of models that can be formulated, tested, and refined by reproducible experiments. We will take a similar course in this book, hoping, perhaps, to convince even Alex that there are in fact such things as miracles, even if we don’t quite believe in them.

Black Swans

Nassim Nicholas Taleb, author of The Black Swan and other successful books, hails from Lebanon, a country that in my youth was known as the “Switzerland of the Middle East.” For many years, Lebanon was an island of peace and prosperity in a region rife with blood and turmoil, and it remained so while the two great Arab-Israeli wars of 1967 and 1973 raged next door. In 1975, a riot broke out, but the Lebanese elite, including Taleb’s family, failed to recognize its significance. They assumed that the riot would run its course and order would be restored in a few days. But order was not restored. Lebanon descended into civil war, and Taleb and his family eventually left the country. To this day, it is a source of wonder how a flourishing, prosperous country could have collapsed overnight.

Taleb reached America and studied at the Wharton School, one of the top institutions for business education, where he was taught the latest theories of modern finance. But there was much that he disagreed with. He believed what had happened to Lebanon could happen at any time to any complex system—including financial markets—and as an investor he had no intention of letting his portfolio end up like Lebanon. When he had made enough money to launch his own investment firm, he adopted an unorthodox strategy. He figured he could win big by betting that sooner or later, there would be a sudden collapse somewhere in the world economy, like the 1975 collapse in Lebanon. (In chapter 6, “The Sources of Equilibrium,” I will discuss in detail the exchange structures on which Taleb’s strategy was built, and I will give a detailed analysis of their advantages and disadvantages in chapter 10, “Life in Wildovia.”)

Taleb used what he had learned at the Wharton School in a way that put him totally at odds with most investors. He wasn’t interested in earning comfortable short-term profits by taking advantage of daily fluctuations in the markets. His goal was to reap enormous rewards should the economy suffer a major crash, while keeping his daily losses to a minimum until such an event occurred. He would not suddenly go bankrupt, he said, although he might slowly bleed to death if nothing cataclysmic happened for a long time.

Taleb’s strategy is far from new. It has been viewed with suspicion for centuries. Various forms of short-selling—betting on the decline in value of a financial instrument—were first prohibited in the 1600s, and whether or to what extent short-selling should be permitted is a disputed question today. A number of countries instituted restrictions on short-selling during the financial crisis of 2008–2009.² The justification for such regulation is that a strategy of short-shelling can act as a self-fulfilling prophecy, itself contributing to a financial crisis. In chapter 11, “Adapting to Wildovia,” we shall see that a total ban on short-selling can hurt the economy more than it helps, yet some regulation is reasonable. Regulation has received increasing emphasis in recent years, but Taleb still has plenty of scope for continuing his investment strategy.

On September 11, 2001, when global stock markets crashed in the wake of the attack on the World Trade Center, Taleb became a very wealthy man overnight. He continued to employ his investment strategy while seeking a better understanding of market crashes, and he continued to develop the software he had designed to implement his strategy. He could now afford to play for higher stakes, and he could afford greater short-term losses. The global financial crisis of 2008 made him a billionaire.

Taleb stated in 2008 that although he has made billions, there is much about which he remains as ignorant as before.³ He still doesn’t understand why Lebanon collapsed in 1975 or why the global economy did so in 2008. He knows only that one must be prepared for the occasional occurrence of unpredictable, even almost inconceivable, events. Taleb is now an academic researcher who studies catastrophic events—let us call them negative miracles—but his ideas apply to many positive miracles as well.

Taleb calls such negative events “Black Swans.” The Black Swan came out in 2007 and topped the New York Times bestseller list for seventeen weeks. (I should note that the book has nothing to do with the like-named 2010 Darren Aronofsky film or the 1954 novella by Thomas Mann.) Despite the efforts of hardworking editors to make it more or less presentable, Taleb’s book is a confusing, poorly written gallimaufry of ideas, thrusts, and parries. The secret of its success is that behind the confusing ramblings on financial philosophy and the flood of arrogance pouring out of the author’s hyperinflated ego (every Nobel laureate in economics is an ignorant quack, George Soros and all the other Wall Street hotshots owe their success to sheer luck, and so on) is a man who, one cannot help but feel, knows something important about the world. His execrable style aside, Taleb is genuinely smart, and he knows his stuff, including the mathematics.⁴

Taleb uses the idea of a black swan to represent things that, though almost inconceivable, nevertheless occur and have a significant impact on the world. The black swan is an unfortunate symbol, for it is something that we can easily imagine, even if we never saw one and had no idea such a thing exists, since we consider it part of a swan’s essence to be white. Nonetheless, we have no difficulty picturing a black swan: it is just like a normal swan, only black. And why not? Indeed, there is a species of black swans living in Australia, but that is beside the point. The book’s German publisher didn’t even put a black swan on the cover. The cover of Der Schwarze Schwan boasts an origami swan—in hot pink.

Nowhere in more than five hundred pages does Taleb define the term “black swan.” Instead, we are given numerous examples of what he considers “black swans,” including the collapse of Lebanon, the dissolution of the Soviet Union, the fall of the Berlin Wall, the birth of Islam, and great stock market crashes. But there are also smaller-scale examples. The inexplicable success of a book can be a black swan, as can an unexpected flop. Other black swans include the many lovers of Catherine the Great (Taleb tells us in a footnote that the precise figure is twelve, not that many by today’s standards); the 2004 tsunami in the Indian Ocean; the invention of the computer, the laser, the Internet, and even the wheel; the discovery of America and that of antibiotics. To Taleb, anything that is unpredictable via common sense and science, but that occurs nonetheless and has a great impact on the world, can be a black swan.

In this book, I will use the term “miracle” in a sense similar to Taleb’s use of “black swans.” Like Taleb, I will not give a precise definition, so suffice it to say that a miracle’s most important characteristic is that it is a one-time, unpredictable and irreproducible event. Our miracles need not be as grandiose as Taleb’s black swans: they do not necessarily change the world. There are major miracles and minor ones, but all of them are miracles nonetheless.

Miracles and Black Swans

Just because it is a one-time and irreproducible occurrence doesn’t mean that a miracle cannot occur again. It may reoccur, as mysterious and irreproducible as it was the first time.

I am inclined to think that every person is actually a minor miracle, being a one-time and unrepeatable occurrence. I am also inclined to agree with the scholar Martinus Biberach, known for having copied out a well-known quatrain in 1498, but also with Martin Luther, who dubbed the poem the “rhyme of the atheists” and revised it to be more theologically correct. Here is the original poem:

I live and know not how long,

I die and know not when,

I travel and know not whither,

I marvel that I am happy.

And Luther’s riposte:

I live as long as God wills,

I die when and how God wills,

I travel and know precisely whither,

I marvel that I am sad.⁵

Biberach’s happiness and Luther’s sadness can both be considered minor marvels—or miracles—though my own worldview puts me closer to Biberach.

A universal characteristic of black swans is that after one has occurred and become known, it may in retrospect seem inevitable. We generally hear many explanations after the fact. Although black swans and miracles are similar in many ways, they differ in this respect: a miracle frequently defies all logical explanation, even in hindsight.

For the faithful, miracles in the strong, theological sense of the word serve as evidence of the power of God, so the believer has no reason to doubt their existence. If something previously considered a miracle (such as solar eclipses) is found by science to be a consequence of the laws of nature, the believer’s faith is unshaken. Man is fallible and may mistake one of God’s laws for one of God’s miracles. Nevertheless, there are many miracles that defy explanation, such as the virgin birth, and one must, in the words of Tennyson, “by faith and faith alone embrace, believing where we cannot prove.”⁶

For the nonbeliever, what we call miracles only demonstrate the limits of our knowledge. There have always been phenomena that our current knowledge cannot explain. But that doesn’t make them theological miracles. Sooner or later, science will find an explanation. But science has also proved, while providing explanations for more and more natural phenomena that previously defied understanding, that there will always be phenomena that science, however much it advances, cannot explain. I will discuss this in a later chapter, where I will also define more precisely what we mean by miracles.

Essential Reading

Douglas R. Hofstadter’s Gödel, Escher, Bach ranks among the books that have had the greatest effect on my thinking, even though I thoroughly reject the author’s premise. The book argues, over seven hundred–plus pages, that the system of logic discovered by the Austrian logician Kurt Gödel (which I will describe in detail in chapter 3, “The Source of Miracles: Gödel’s Idea”) is a sufficient basis for creating a full-fledged artificial intelligence. Moreover, Hofstadter had the chutzpah to write in his introduction, “In a way, this book is a statement of my religion.”⁷ Let’s just say that I attend a different house of worship. Nevertheless, most of what I know about Gödel’s theorem and logic in general comes from Hofstadter, for it was he, unlike the hordes of experts whose articles he cites, who told the story of Gödel’s theorem in a way that captivated me. Hofstadter revealed for me the source of miracles, and I could not have written this book without having read his.

Everything you need to know about Gödel’s theorem can be obtained from Hofstadter’s book. You don’t have to reinvent the wheel when it comes to understanding scientific facts that have been established beyond doubt. You simply have to learn them, and from books like Hofstadter’s if possible. The Bible is also a primary source for the way I think about the world, even though I am certain that the birds and the beasts were not created as described in Genesis. We simply cannot think about flora and fauna as we did before Darwin, just as we cannot think about the Sun and Moon as we did before Copernicus. In fact, every one of the books that has shaped my worldview—including fictional works by such writers as Géza Ottlik and J. K. Rowling—is a work that I disagree with in some fundamental way. But I still take great pleasure in letting these stories shape my thinking.

The same goes for The Black Swan. I fundamentally disagree with Taleb’s point of view. Possibly it is just my prejudice, because my home country, Hungary, has never collapsed overnight, at least not in the past four or five centuries. Then again, it has also never been the Switzerland of Central Europe. I don’t believe that...