Eugène Charles Catalan (1814–94) conjectured that among all the possible powers of whole numbers, the only pair of consecutive numbers (excluding 0 and 1) is 8 and 9. In other words, you can spend your life writing down all the pure powers that exist. Other than 8 and 9, you will find no other two numbers that differ by only 1. In 1342, the Jewish-French philosopher and mathematician Levi Ben Gerson (1288–1344) actually proved a small part of the conjecture—that 8 and 9 are the only powers of 2 and 3 differing by 1. A major step forward was taken by the mathematician Robert Tijdeman in 1976. Still, the proof of the general form of Catalan’s conjecture stymied the best mathematical minds for more than 150 years. Finally, on April 18, 2002, the Romanian mathematician Preda Mihailescupresented a complete proof of the conjecture. His proof was published in 2004 and is now fully accepted. Again you may ask: When did Catalan’s conjecture become true? In 1342? In 1844? In 1976? In 2002? In 2004? Isn’t it obvious that the statement was always true, only that we didn’t know it to be true? These are the types of truths Platonists would refer to as “objective truths.”
Some mathematicians, philosophers, cognitive scientists, and other “consumers” of mathematics (e.g., computer scientists) regard the Platonic world as a figment of the imagination of too-dreamy minds (I shall describe this perspective and other dogmas in detail later in the book). In fact, in 1940, the famous historian of mathematics Eric Temple Bell (1883–1960) made the following prediction:
According to the prophets, the last adherent of the Platonic ideal in mathematics will have joined the dinosaurs by the year 2000. Divested of its mythical raiment of eternalism, mathematics will then be recognized for what it has always been, a humanly constructed language devised by human beings for definite ends prescribed by themselves. The last temple of an absolute truth will have vanished with the nothing it enshrined.
Bell’s prophecy proved to be wrong. While dogmas that are diametrically opposed (but in different directions) to Platonism have emerged, those have not fully won the minds (and hearts!) of all mathematicians and philosophers, who remain today as divided as ever.
Suppose, however, that Platonism had won the day, and we had all become wholehearted Platonists. Does Platonism actually explain the “unreasonable effectiveness” of mathematics in describing our world? Not really. Why should physical reality behave according to laws that reside in the abstract Platonic world? This was, after all, one of Penrose’s mysteries, and Penrose is a devout Platonist himself. So for the moment we have to accept the fact that even if we were to embrace Platonism, the puzzle of the powers of mathematics would remain unsolved. In Wigner’s words: “It is difficult to avoid the impression that a miracle confronts us here, comparable in its striking nature tothe miracle that the human mind can string a thousand arguments together without getting itself into contradictions.”
To fully appreciate the magnitude of this miracle, we have to delve into the lives and legacies of some of the miracle workers themselves—the minds behind the discoveries of a few of those incredibly precise mathematical laws of nature.
CHAPTER 3
MAGICIANS: THE MASTER AND THE HERETIC
Unlike the Ten Commandments, science was not handed to humankind on imposing tablets of stone. The history of science is the story of the rise and fall of numerous speculations, hypotheses, and models. Many seemingly clever ideas turned out to be false starts or led down blind alleys. Some theories that were taken to be ironclad at the time later dissolved when put to the fiery test of subsequent experiments and observations, only to become entirely obsolete. Even the extraordinary brainpower of the originators of some conceptions did not make those conceptions immune to being superseded. The
Some mathematicians, philosophers, cognitive scientists, and other “consumers” of mathematics (e.g., computer scientists) regard the Platonic world as a figment of the imagination of too-dreamy minds (I shall describe this perspective and other dogmas in detail later in the book). In fact, in 1940, the famous historian of mathematics Eric Temple Bell (1883–1960) made the following prediction:
According to the prophets, the last adherent of the Platonic ideal in mathematics will have joined the dinosaurs by the year 2000. Divested of its mythical raiment of eternalism, mathematics will then be recognized for what it has always been, a humanly constructed language devised by human beings for definite ends prescribed by themselves. The last temple of an absolute truth will have vanished with the nothing it enshrined.
Bell’s prophecy proved to be wrong. While dogmas that are diametrically opposed (but in different directions) to Platonism have emerged, those have not fully won the minds (and hearts!) of all mathematicians and philosophers, who remain today as divided as ever.
Suppose, however, that Platonism had won the day, and we had all become wholehearted Platonists. Does Platonism actually explain the “unreasonable effectiveness” of mathematics in describing our world? Not really. Why should physical reality behave according to laws that reside in the abstract Platonic world? This was, after all, one of Penrose’s mysteries, and Penrose is a devout Platonist himself. So for the moment we have to accept the fact that even if we were to embrace Platonism, the puzzle of the powers of mathematics would remain unsolved. In Wigner’s words: “It is difficult to avoid the impression that a miracle confronts us here, comparable in its striking nature tothe miracle that the human mind can string a thousand arguments together without getting itself into contradictions.”
To fully appreciate the magnitude of this miracle, we have to delve into the lives and legacies of some of the miracle workers themselves—the minds behind the discoveries of a few of those incredibly precise mathematical laws of nature.
CHAPTER 3
MAGICIANS: THE MASTER AND THE HERETIC
Unlike the Ten Commandments, science was not handed to humankind on imposing tablets of stone. The history of science is the story of the rise and fall of numerous speculations, hypotheses, and models. Many seemingly clever ideas turned out to be false starts or led down blind alleys. Some theories that were taken to be ironclad at the time later dissolved when put to the fiery test of subsequent experiments and observations, only to become entirely obsolete. Even the extraordinary brainpower of the originators of some conceptions did not make those conceptions immune to being superseded. The
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