The waltz of reason, p.1
The Waltz of Reason, page 1
Copyright © 2023 by Karl Sigmund
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ISBNs: 9781541602694 (hardcover), 9781541602700 (ebook)
E3-20231110-JV-NF-ORI
Contents
Cover
Title Page
Copyright
Introduction
PART I 1. Geometry
2. Number
3. Infinity
4. Logic
5. Computation
PART II 6. Limits
7. Probability
8. Randomness
PART III 9. Voting
10. Decision
11. Cooperation
12. Social Contract
13. Fairness
PART IV 14. Language
15. Philosophy
16. Understanding
Acknowledgments
Discover More
References
Bibliography
Image Credits
About the Author
Also by Karl Sigmund
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Introduction
Mathematics and philosophy have dated for ages. This book describes in a not-too-serious vein some of their most memorable encounters.
The two fields seem to be made for each other. On the one hand, mathematics is a useful tool for both theoretical and practical philosophy. For instance, the theory of knowledge deals with topics such as space and chance that are at the core of geometry and probability theory; ethics deals with notions such as fairness and the social contract, on which game theory sheds a light; and so on. On the other hand, mathematics itself is among the most puzzling and rewarding sources of philosophical problems. It is obviously not an empirical science, so why is it so practical? Is it invented or discovered? And what makes mathematical knowledge seem so secure?
Such questions will lead to entertaining, informative, and occasionally rambling excursions into the history of human thought, involving some remarkably original characters, all of them celebrities from the Dead Thinkers’ Society. Philosophical and mathematical ways of thinking are utterly distinct, but have often progressed on twin tracks. There were times when philosophers and mathematicians could not be told apart. Those times are receding, but the fact remains that the two fields have wonderful ways of stimulating and often surprising each other. It seems that the two siblings from Greece are fated to remain eternally entangled, in a complex and occasionally dizzying waltz—and not without, sometimes, stepping on each other’s toes. Let us follow their evolutions.
There are many illustrations and very few formulas in the book. It is meant as a sightseer’s guide and no more. The text will not say too much on the philosophy of mathematics itself, an arduous field, but it will make up for this by describing applications of mathematics to all kinds of philosophical questions, from morals to logic, and emphasizing the historical quirks of an age-old quest. That quest is guaranteed to remain open-ended, especially as the current explosion of artificial intelligence is likely to sweep some cardhouses of reason from the table, and deal us a new hand.
The first part of the book has to do with space, number, algorithm, axiom, and proof. It traces the evolution of the self-image of mathematics, a short coming-of-age version of the long official history that leads from Euclid to Turing, or more precisely, from Thales to Hales (no pun intended!). Thales was the semi-legendary figure from Miletus, on the shore of the Ionian Sea, who may (or may not) have first conceived, some hundred generations ago, the idea of a mathematical proof conveying insight and certainty. Thomas Hales is the contemporary US mathematician who has become famous for a proof that was so complex that nobody could be completely sure of its validity. Thereupon, Hales removed all doubts by convincing a computer.
The development of mathematics between these two milestones has involved drastic changes in perspective, which greatly exercised the minds of mathematicians and philosophers alike. On the menu were the role of spatial intuition; the fate of the parallel axiom; the divorce of mathematical and physical space; the nature and purpose of number; infinity, surrounded by its halo of taboos and scandals; and the incestuous relations of mathematics with logic. All these issues underwent centuries of development, punctuated by major revolutions, and all led to strange encounters (almost encounters of the third kind) between mathematicians and philosophers.
The second part of this book deals with chance and the continuum. The latter provided thinkers such as Zeno of Elea with riddles to confuse the best minds and filled ancient mathematics with unease. It was only in the aftermath of the Renaissance, during the heyday of the alchemists, that some intellectual adventurers began to develop an infinitesimal calculus. They aimed to reach for the limit by a daredevil “fast forward”—named “passage to the limit”—and to divide the finite into infinitely many infinitely small parts. At almost the same time, probability was tamed. This was the age when mathematicians, with the confidence of sleepwalkers, went beyond common sense. Nobody understood properly how chance fits with causality, nor how an infinitesimal could be smaller than anything and yet not zero. All that seemed to matter was that the stuff worked. In due time, it emerged that the calculus of chances and the computation of volumes could be handled by the same analytical tools. Mathematicians got used to defying reason. They also began to vex philosophers.
The third part of the book turns to practical philosophy: morality and economics, politics and law. Plato had once proposed that ideal rulers should start out by doing mathematics for ten years. Mercifully, this suggestion fell flat. But 2000 years later, some mathematicians did indeed turn to reflect on what was good and desirable. This started harmlessly enough, with the investigation of voting schemes, at a time when democracy was the pipe dream of radicals and the only republic was the “republic of scholars.” A little later, the Benthamite notion of a “felicity calculus” attracted ridicule, but that of “utility” took hold of economics. Eventually, utility turned on itself and cast a shadow on our optimistic self-image as “rational beings.”
Toward the middle of last century, under the innocuous heading of “theory of games,” mathematics began to investigate conflicts of interest. Such conflicts are the raison d’être for all morals and laws. Today, it seems hard to understand how philosophers could ever have cogitated on selfishness, cooperation, or the social contract without resorting to the Prisoner’s Dilemma or the Stag Hunt game. Similarly, the notion of fairness or the evolution of ownership norms are by now fully established as mathematical issues (which does not necessarily mean that we understand them any better).
The fourth and last part of this book tries to approach mathematics from the outside, as if landing on an unknown shore and fraternizing with the native tribe. The first chapter takes a look at the language of mathematics, or more precisely, at its writing. This quasi-graphological approach reveals that the tribe is currently undergoing a change, one that seems to accelerate at a dizzying pace. This upheaval is caused by digitalization, of course (a word that refers to the fingers used for counting). The computer, that mathematical brainchild, is radically transforming mathematics in more ways than can be reckoned yet. The next chapter is a tip of the hat to the philosophy of mathematics, a venerable discipline that seems, today, to deal almost as much with itself as with mathematics. And the last chapter turns to what may appear, to many eyes, to be the greatest riddle: namely, why does mathematics provide so much delight for some (but only some) of us?
As for myself, I have loved mathematics since as far as I can think back. I well remember the cozy evening at home when little me carefully measured the angles of a triangle, added them up, and discovered that my dad had been right! While growing up in Vienna, I soon crossed traces of Ludwig Wittgenstein, Kurt Gödel, and the Vienna Circle, and I could not help wondering about their disparate views. There were other formative experiences. A large part of my professional life (maybe the best) was
This being said, I must confess that I have occasionally experienced, while writing this book, a sort of The Old Man and the Sea feeling: namely, that I had hooked a fish who is far, far too big for me and drags me and my skiff, hell knows where.…
I can only take it philosophically.
PART I
1
Geometry
Memories of a Nameless
The Art of Unforgetting
The opening scene is set in Athens, in the villa of a shady politico named Anytus. Young Meno, an up-and-coming military leader, is on a visit. Socrates also happens to be around. Meno is quick to seize the occasion and asks him whether virtue can be taught—a bait that never failed to hook Socrates. The gambit secured Meno’s passport to eternity. He died soon after, in the Persian Wars, under dubious circumstances, but a Platonic dialogue was named after him.
Figure 1.1. Socrates (469–399 BCE).
Figure 1.2. A page of Meno.
Meno is one of the earliest works of Plato. This is where, for the first time, the philosopher presents one of his favorite ideas: There is a knowledge that we learn by remembering. Our immortal soul has known it all along. We only need to dig it up.
The Greeks have a word for this recovery of buried knowledge: anamnesis. The notion may strike us as hopelessly outdated, a leftover from an age of superstition. However, it led to one of the most venerable, and indeed thrilling, encounters between philosophy and mathematics. This came about when Socrates proposed to defend his curious idea by performing, for Meno’s benefit, an experiment on total recall.
One of the slaves hanging around is asked to approach, a boy whose utter lack of education is beyond any doubt. By skillful questioning, Socrates leads him to the discovery of a geometric theorem that the boy had certainly never been told before. Hence, concludes Socrates, he must have known it all along. The boy had merely been unaware of it. With gentle probing, his submerged knowledge has come to light. In modern parlance, part of the subconscious had become conscious, like in a session on the couch of Dr. Freud. Socrates himself likened his role to that of a midwife—he merely had helped the boy unforget what he had forgotten.
The whole episode took merely a quarter of an hour. With that, the boy returned to his lowly sphere of ignorance. He neither learned what all the questioning had been about, nor that it had brought him his fifteen minutes of fame—or rather (since he had never been asked for his name), fifteen minutes of immortality. Socrates and Meno returned to their discussion of what virtue is about.
Tellingly, Socrates had chosen geometry for his experiment, not any other science such as physics or geography. With his well-honed dialectical skills, he could probably have led the slave boy to also remember that Crete is an island or that everything is made from water, fire, air, and earth. Socrates preferred to focus on a geometric theorem, because nothing can pass more plausibly for an eternal truth.
The boy had been asked to construct, for a given square, a square of twice its area. We know, from school, that the length of its side must be √2 times the length of the side of the original square. It must therefore be as long as its diagonal. But talking of square roots was out of bounds. Actually, Socrates did not even mention a square. He spoke of a quadrilateral whose sides have equal length (Figure 1.3). This is not enough to define a square; but Socrates added—no doubt using a figure such as Figure 1.4—that its diagonals are of equal length. This guarantees that the quadrilateral with equal sides is indeed a square.
Figure 1.3. A rhombus is a quadrilateral whose sides have equal length.
Figure 1.4. A square is a rhombus whose diagonals have equal length.
Socrates could have demanded instead that all angles are equal. This condition would also guarantee that the quadrilateral with equal sides is a square. But he preferred to introduce diagonals right at the start of the Q and A session, in a casual way—a neat trick, as these diagonals will eventually turn out to yield the solution (Figure 1.5). Toward this end, Socrates allowed the boy to follow his own way, gently correcting his mistakes. Double the side length? No, this would not do; it leads to a square having four times the area of the original square. Multiply the side length by one and a half? No, still too large. And so the dialogue winds on, until, in the end, Socrates has coached the correct answer from the boy.
Figure 1.5. The gray square on the right has twice the area of the gray square on the left.
We shall see in the next chapter that the corresponding problem in three dimensions is unsolvable. It is the Delian problem, which owes its name to the tiny island of Delos in the Aegean Sea. Some thirty years before Socrates had his chat with Meno, a plague had raged through Greece. Pericles had died from it. As usual in a pandemic, experts knew exactly what to do. They said, “Go ask an oracle.” Lo and indeed, the oracle offered advice: double the size of the cube-shaped altar that stands in the temple of Apollo, and you will thereby appease the gods. Today, the plague is gone, the altar nowhere to be found. Apollo has quit. But the Delian problem remains unsolved. In fact, we know—we have proof—that it will remain unsolved for all time. How wise of Socrates not to have asked the boy to recall the solution.
Three Angles for Euclid
Geometry was the first branch of mathematics to really flourish. It may have done so because of its obvious use to architects, sailors, and field surveyors. More likely, it flourished because it is beautiful. Even the simplest geometric figures, such as the triangle, are fascinating. In music, the triangle is a marginal instrument hidden somewhere in the back of the orchestra. In mathematics, triangles shine in the front row—the very first objects to fascinate early Greek thinkers, such as Thales of Miletus or Pythagoras of Samos. Triangles are also the first mathematical figures likely to raise a child’s interest.
One of the oldest geometric theorems is that of Pythagoras (Figure 1.6). If a, b, and c are the lengths of the sides of a right triangle (c being that of the hypotenuse, the side opposite the right angle), then a2 + b2 = c2. This fact had been known to Egyptians, Indians, and Babylonians, but Pythagoras was (possibly? probably?) the first to offer a proof.
What is a proof? For the old Greeks, it was an argument to make everyone see why the statement is true—just as the slave boy suddenly came to see why the diagonal of a square is the side of a square of twice the size.
There exist many proofs of the Pythagorean theorem. The most common proof is based on Figure 1.7 (which in the case a = b is more or less straight from Meno). The large square, with side length a + b, is divided into five pieces: the four right triangles and the square formed by their hypotenuses. Each triangle has area , which altogether yields 2ab. We obtain c2 = (a + b)2 − 2ab by simply removing the right triangles from the large square.
Figure 1.6. The theorem of Pythagoras.
Figure 1.7. One step in its proof.
Figure 1.8. Another step in its proof.
Another decomposition of the same “large” square with sides of length a + b (see Figure 1.8) yields the equation (a + b)2 = a2 + 2ab + b2. Substituting this into the previous equation, we obtain c2 = a2 + b2, as had to be proved.
