Is God a Mathematician_ - Mario Livio [19]
Modern-day Platonists (yes, they definitely exist, and their views will be described in more detail in later chapters) insist that the Platonic world of mathematical forms is real, and they offer what they regard as concrete examples of objectively true mathematical statements that reside in this world.
Take the following easy-to-understand proposition: Every even integer greater than 2 can be written as the sum of two primes (numbers divisible only by one and themselves). This simple-sounding statement is known as the Goldbach conjecture, since an equivalent conjecture appeared in a letter written by the Prussian amateur mathematician Christian Goldbach (1690–1764) on June 7, 1742. You can easily verify the validity of the conjecture for the first few even numbers: 4 = 2 + 2; 6 = 3 + 3; 8 = 3 + 5; 10 = 3 + 7 (or 5 + 5); 12 = 5 + 7; 14 = 3 + 11 (or 7 + 7); 16 = 5 + 11 (or 3 + 13); and so on. The statement is so simple that the British mathematician G. H. Hardy declared that “any fool could have guessed it.” In fact, the great French mathematician and philosopher René Descartes had anticipated this conjecture before Goldbach. Proving the conjecture, however, turned out to be quite a different matter. In 1966 the Chinese mathematician Chen Jingrun made a significant step toward a proof. He managed to show that every sufficiently large even integer is the sum of two numbers, one of which is a prime and the other has at most two prime factors. By the end of 2005, the Portuguese researcher Tomás Oliveira e Silva had shown the conjecture to be true for numbers up to 3 1017 (three hundred thousand trillion). Yet, in spite of enormous efforts by many talented mathematicians, a general proof remains elusive at the time of this writing. Even the additional temptation of a $1 million prize offered between March 20, 2000, and March 20, 2002 (to help publicize a novel entitled Uncle Petros and Goldbach’s Conjecture), did not produce the desired result. Here, however, comes the crux of the meaning of “objective truth” in mathematics. Suppose that a rigorous proof will actually be formulated in 2016. Would we then be able to say that the statement was already true when Descartes first thought about it? Most people would agree that this question is silly. Clearly, if the proposition is proven to be true, then it has always been true, even before we knew it to be true. Or, let’s look at another innocent-looking example known as Catalan’s conjecture. The numbers 8 and 9 are consecutive whole numbers, and each of them is equal to a pure power, that is 8 23 and 9 32. In 1844, the Belgian mathematician 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 Mihailescu presented 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