way that geometrical figures drawn on papyrus only illustrate the true figures).
Plato’s suggestions for astronomical research are considered controversial even by some of the most devout Platonists. Defenders of his ideas argue that what Plato really means is not that true astronomy should concern itself with some ideal heaven that has nothing to do with the observable one, but that it should deal with the real motions of celestial bodies as opposed to the apparent motions as seen from Earth. Others point out, however, that too literal an adoption of Plato’s dictum would have seriously impeded the development of observational astronomy as a science. Be the interpretation of Plato’s attitude toward astronomy as it may, Platonism has become one of the leading dogmas when it comes to the foundations of mathematics.
But does the Platonic world of mathematics really exist? And if it does, where exactly is it? And what are these “objectively true” statements that inhabit this world? Or are the mathematicians who adhere to Platonism simply expressing the same type of romantic belief that has been attributed to the great Renaissance artist Michelangelo? According to legend, Michelangelo believed that his magnificent sculptures already existed inside the blocks of marble and that his role was merely to uncover them.
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 declaredthat “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 10 17 (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 2 3 and 9 3 2 . In 1844, the Belgian mathematician
Plato’s suggestions for astronomical research are considered controversial even by some of the most devout Platonists. Defenders of his ideas argue that what Plato really means is not that true astronomy should concern itself with some ideal heaven that has nothing to do with the observable one, but that it should deal with the real motions of celestial bodies as opposed to the apparent motions as seen from Earth. Others point out, however, that too literal an adoption of Plato’s dictum would have seriously impeded the development of observational astronomy as a science. Be the interpretation of Plato’s attitude toward astronomy as it may, Platonism has become one of the leading dogmas when it comes to the foundations of mathematics.
But does the Platonic world of mathematics really exist? And if it does, where exactly is it? And what are these “objectively true” statements that inhabit this world? Or are the mathematicians who adhere to Platonism simply expressing the same type of romantic belief that has been attributed to the great Renaissance artist Michelangelo? According to legend, Michelangelo believed that his magnificent sculptures already existed inside the blocks of marble and that his role was merely to uncover them.
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 declaredthat “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 10 17 (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 2 3 and 9 3 2 . In 1844, the Belgian mathematician
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