Read Is God a Mathematician? Online
Authors: Mario Livio
Is God a Mathematician? (5 page)
Another striking demonstration of Plato’s appreciation of mathematics comes in what is perhaps his most accomplished book,
The Republic,
a mind-boggling fusion of aesthetics, ethics, metaphysics, and politics. There, in book VII, Plato (through the central figure of Socrates) outlined an ambitious plan of education designed to create utopian state rulers. This rigorous if idealized curriculum envisaged an early training in childhood imparted through play, travel, and gymnastics. After the selection of those who showed promise, the program continued with no fewer than ten years of mathematics, five years of dialectic, and fifteen years of practical experience, which included holding commands in time of war and other offices “suitable to youth.” Plato gave clear explanations as to why he thought that this was the necessary training for the would-be politicians:
What we require is that those who take office should not be lovers of rule. Otherwise there will be a contest with rival lovers. What others, then, will you compel to undertake the guardianship of the city than those who have most intelligence of the principles that are the means of good government and who possess distinctions of another kind and a life that is preferable to political life?
Refreshing, isn’t it? In fact, while such a demanding program was probably impractical even in Plato’s time, George Washington agreed that an education in mathematics and philosophy was not a bad idea for the politicians-to-be:
The science of figures, to a certain degree, is not only indispensably requisite in every walk of civilized life; but investigation of mathematical truths accustoms the mind to method and correctness in reasoning, and is an employment peculiarly worthy of rational being. In a clouded state of existence, where so many things appear precarious to the bewildered research, it is here that the rational faculties find foundation to rest upon. From the high ground of mathematical and philosophical demonstration, we are insensibly led to far nobler speculations and sublimer meditations.
For the question of the nature of mathematics, even more important than Plato the mathematician or the math stimulator was Plato the philosopher of mathematics. There his trail-blazing ideas put him not only above all the mathematicians and philosophers of his generation, but identified him as an influential figure for the following millennia.
Plato’s vision of what mathematics truly is makes strong reference to his famous Allegory of the Cave. There he emphasizes the doubtful validity of the information provided through the human senses. What we perceive as the real world, Plato says, is no more real than shadows projected onto the walls of a cavern. Here is the remarkable passage from
The Republic:
See human beings as though they were in an underground cave-like dwelling with an entrance, a long one, open to the light across the whole width of the cave. They are in it from childhood with their legs and necks in bonds so that they are fixed, seeing only in front of them, unable because of the bond to turn their heads all the way around. Their light is from a fire burning far above and behind them. Between the fire and the prisoners there is a road above, along which we see a wall, built like the partitions puppet-handlers set in front of the human beings and over which they show the puppets…Then also see along this wall human beings carrying all sorts of artifacts, which project above the wall, and statues of men and other animals wrought from stone, wood, and every kind of material…do you suppose such men would have seen anything of themselves and one another, other than the shadows cast by the fire on the side of the cave facing them?
According to Plato, we, humans in general, are no different from those prisoners in the cave who mistake the shadows for reality. (Figure 9 shows an engraving by Jan Saenredam from 1604 illustrating the allegory.) In particular, Plato stresses, mathematical truths refer not to circles, triangles, and squares that can be drawn on a piece of papyrus, or marked with a stick in the sand, but to abstract objects that dwell in an ideal world that is the home of true forms and perfections. This Platonic world of mathematical forms is distinct from the physical world, and it is in this first world that mathematical propositions, such as the Pythagorean theorem, hold true. The right triangle we might draw on paper is but an imperfect copy—an approximation—of the true, abstract triangle.
Another fundamental issue that Plato examined in some detail concerned the nature of mathematical proof as a process that is based on
postulates
and
axioms
. Axioms are basic assertions whose validity is assumed to be self-evident. For instance, the first axiom in Euclidean geometry is “Between any two points a straight line may be drawn.” In
The Republic,
Plato beautifully combines the concept of postulates with his notion of the world of mathematical forms:
Figure 9
I think you know that those who occupy themselves with geometries and calculations and the like, take for granted the odd and the even [numbers], figures, three kinds of angles, and other things cognate to these in each subject; assuming these things as known, they take them as hypotheses and thenceforward they do not feel called upon to give any explanation with regard to them either to themselves or anyone else, but treat them as manifest to every one; basing themselves on these hypotheses, they proceed at once to go through the rest of the argument till they arrive, with general assent, at the particular conclusion to which their inquiry was directed. Further you know that they make use of visible figures and argue about them, but in doing so they are not thinking about these figures but of the things which they represent; thus it is the absolute square and the absolute diameter which is the object of their argument, not the diameter which they draw…the object of the inquirer being to see their absolute counterparts which
cannot be seen otherwise than by thought
[emphasis added].
Plato’s views formed the basis for what has become known in philosophy in general, and in discussions of the nature of mathematics
in particular, as
Platonism
. Platonism in its broadest sense espouses a belief in some abstract eternal and immutable realities that are entirely independent of the transient world perceived by our senses. According to Platonism, the real existence of mathematical objects is as much an objective fact as is the existence of the universe itself. Not only do the natural numbers, circles, and squares exist, but so do imaginary numbers, functions, fractals, non-Euclidean geometries, and infinite sets, as well as a variety of theorems about these entities. In short, every mathematical concept or “objectively true” statement (to be defined later) ever formulated or imagined, and an infinity of concepts and statements not yet discovered, are absolute entities, or
universals,
that can neither be created nor destroyed. They exist independently of our knowledge of them. Needless to say, these objects are not physical—they live in an autonomous world of timeless essences. Platonism views mathematicians as explorers of foreign lands; they can only discover mathematical truths, not invent them. In the same way that America was already there long before Columbus (or Leif Ericson) discovered it, mathematical theorems existed in the Platonic world before the Babylonians ever initiated mathematical studies. To Plato, the only things that truly and wholly exist are those abstract forms and ideas of mathematics, since only in mathematics, he maintained, could we gain absolutely certain and objective knowledge. Consequently, in Plato’s mind, mathematics becomes closely associated with the divine. In the dialogue
Timaeus,
the creator god uses mathematics to fashion the world, and in
The Republic,
knowledge of mathematics is taken to be a crucial step on the pathway to knowing the divine forms. Plato does not use mathematics for the formulation of some laws of nature that are testable by experiments. Rather, for him, the mathematical character of the world is simply a consequence of the fact that “God always geometrizes.”
Plato extended his ideas on “true forms” to other disciplines as well, in particular to astronomy. He argued that in true astronomy “we must leave the heavens alone” and not attempt to account for the arrangements and the apparent motions of the visible stars. Instead, Plato regarded true astronomy as a science dealing with the laws of motion in some ideal, mathematical world, for which the observable
heaven is a mere illustration (in the same 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 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 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 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 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.”