To Explain the World: The Discovery of Modern Science (4 page)

There remains a poetic element in modern physics. We do not write in poetry; much of the writing of physicists barely reaches the level of prose. But we seek beauty in our theories, and use aesthetic judgments as a guide in our research. Some of us think that this works because we have been trained by centuries of success and failure in physics research to anticipate certain aspects of the laws of nature, and through this experience we have come to feel that these features of nature’s laws are beautiful.
¹⁷
But we do not take the beauty of a theory as convincing evidence of its truth.

For example, string theory, which describes the different species of elementary particles as various modes of vibration of tiny strings, is very beautiful. It appears to be just barely consistent mathematically, so that its structure is not arbitrary, but largely fixed by the requirement of mathematical consistency. Thus it has the beauty of a rigid art form—a sonnet or a sonata. Unfortunately, string theory has not yet led to any predictions that can be tested experimentally, and as a result theorists (at least most of us) are keeping an open mind as to whether the theory actually applies to the real world. It is this insistence on verification that we most miss in all the poetic students of nature, from Thales to Plato.

2

Music and Mathematics

Even if Thales and his successors had understood that from their theories of matter they needed to derive consequences that could be compared with observation, they would have found the task prohibitively difficult, in part because of the limitations of Greek mathematics. The Babylonians had achieved great competence in arithmetic, using a number system based on 60 rather than 10. They had also developed some simple techniques of algebra, such as rules (though these were not expressed in symbols) for solving various quadratic equations. But for the early Greeks, mathematics was largely geometric. As we have seen, mathematicians by Plato’s time had already discovered theorems about triangles and polyhedrons. Much of the geometry found in Euclid’s
Elements
was already well known before the time of Euclid, around 300 BC. But even by then the Greeks had only a limited understanding of arithmetic, let alone algebra, trigonometry, or calculus.

The phenomenon that was studied earliest using methods of arithmetic may have been music. This was the work of the followers of Pythagoras. A native of the Ionian island of Samos, Pythagoras emigrated to southern Italy around 530 BC. There, in the Greek city of Croton, he founded a cult that lasted until the 300s BC.

The word “cult” seems appropriate. The early Pythagoreans
left no writings of their own, but according to the stories told by other writers
¹
the Pythagoreans believed in the transmigration of souls. They are supposed to have worn white robes and forbidden the eating of beans, because that vegetable resembled the human fetus. They organized a kind of theocracy, and under their rule the people of Croton destroyed the neighboring city of Sybaris in 510 BC.

What is relevant to the history of science is that the Pythagoreans also developed a passion for mathematics. According to Aristotle’s
Metaphysics,
²
“the Pythagoreans, as they are called, devoted themselves to mathematics: they were the first to advance this study, and having been brought up in it, they thought its principles were the principles of all things.”

Their emphasis on mathematics may have stemmed from an observation about music. They noted that in playing a stringed instrument, if two strings of equal thickness, composition, and tension are plucked at the same time, the sound is pleasant if the lengths of the strings are in a ratio of small whole numbers. In the simplest case, one string is just half the length of the other. In modern terms, we say that the sounds of these two strings are an octave apart, and we label the sounds they produce with the same letter of the alphabet. If one string is two-thirds the length of the other, the two notes produced are said to form a “fifth,” a particularly pleasing chord. If one string is three-fourths the length of the other, they produce a pleasant chord called a “fourth.” By contrast, if the lengths of the two strings are not in a ratio of small whole numbers (for instance if the length of one string is, say, 100,000/314,159 times the length of the other), or not in a ratio of whole numbers at all, then the sound is jarring and unpleasant. We now know that there are two reasons for this, having to do with the periodicity of the sound produced by the two strings played together, and the matching of the overtones produced by each string (see
Technical Note 3
). None of this was understood by the Pythagoreans, or indeed by anyone else until the work of the French priest Marin Mersenne in the seventeenth century. Instead, the Pythagoreans according to Aristotle judged
“the whole heaven to be a musical scale.”
³
This idea had a long afterlife. For instance, Cicero, in
On the Republic
, tells a story in which the ghost of the great Roman general Scipio Africanus introduces his grandson to the music of the spheres.

It was in pure mathematics rather than in physics that the Pythagoreans made the greatest progress. Everyone has heard of
the
Pythagorean theorem, that the area of a square whose edge is the hypotenuse of a right triangle equals the sum of the areas of the two squares whose edges are the other two sides of the triangle. No one knows which if any of the Pythagoreans proved this theorem, or how. It is possible to give a simple proof based on a theory of proportions, a theory due to the Pythagorean Archytas of Tarentum, a contemporary of Plato. (See
Technical Note 4
. The proof given as Proposition 46 of Book I of Euclid’s
Elements
is more complicated.) Archytas also solved a famous outstanding problem: given a cube, use purely geometric methods to construct another cube of precisely twice the volume.

The Pythagorean theorem led directly to another great discovery: geometric constructions can involve lengths that cannot be expressed as ratios of whole numbers. If the two sides of a right triangle adjacent to the right angle each have a length (in some units of measurement) equal to 1, then the total area of the two squares with these edges is 1
²
+ 1
²
= 2, so according to the Pythagorean theorem the length of the hypotenuse must be a number whose square is 2. But it is easy to show that a number whose square is 2 cannot be expressed as a ratio of whole numbers. (See
Technical Note 5
.) The proof is given in Book X of Euclid’s
Elements,
and mentioned earlier by Aristotle in his
Prior Analytics
⁴
as an example of a
reductio ad impossibile,
but without giving the original source. There is a legend that this discovery is due to the Pythagorean Hippasus, possibly of Metapontum in southern Italy, and that he was exiled or murdered by the Pythagoreans for revealing it.

We might today describe this as the discovery that numbers like the square root of 2 are irrational—they cannot be expressed as ratios of whole numbers. According to Plato,
⁵
it was shown
by Theodorus of Cyrene that the square roots of 3, 5, 6, . . . , 15, 17, etc. (that is, though Plato does not say so, the square roots of all the whole numbers other than the numbers 1, 4, 9, 16, etc., that are the squares of whole numbers) are irrational in the same sense. But the early Greeks would not have expressed it this way. Rather, as the translation of Plato has it, the sides of squares whose areas are 2, 3, 5, etc., square feet are “incommensurate” with a single foot. The early Greeks had no conception of any but rational numbers, so for them quantities like the square root of 2 could be given only a geometric significance, and this constraint further impeded the development of arithmetic.

The tradition of concern with pure mathematics was continued in Plato’s Academy. Supposedly there was a sign over its entrance, saying that no one should enter who was ignorant of geometry. Plato himself was no mathematician, but he was enthusiastic about mathematics, perhaps in part because, during the journey to Sicily to tutor Dionysius the Younger of Syracuse, he had met the Pythagorean Archytas.

One of the mathematicians at the Academy who had a great influence on Plato was Theaetetus of Athens, who was the title character of one of Plato’s dialogues and the subject of another. Theaetetus is credited with the discovery of the five regular solids that, as we have seen, provided a basis for Plato’s theory of the elements. The proof
*
offered in Euclid’s
Elements
that these are the only possible convex regular solids may be due to Theaetetus, and Theaetetus also contributed to the theory of what are today called irrational numbers.

The greatest Hellenic mathematician of the fourth century BC was probably Eudoxus of Cnidus, a pupil of Archytas and a con
temporary of Plato. Though resident much of his life in the city of Cnidus on the coast of Asia Minor, Eudoxus was a student at Plato’s Academy, and returned later to teach there. No writings of Eudoxus survive, but he is credited with solving a great number of difficult mathematical problems, such as showing that the volume of a cone is one-third the volume of the cylinder with the same base and height. (I have no idea how Eudoxus could have done this without calculus.) But his greatest contribution to mathematics was the introduction of a rigorous style, in which theorems are deduced from clearly stated axioms. It is this style that we find later in the writings of Euclid. Indeed, many of the details in Euclid’s
Elements
have been attributed to Eudoxus.

Though a great intellectual achievement in itself, the development of mathematics by Eudoxus and the Pythagoreans was a mixed blessing for natural science. For one thing, the deductive style of mathematical writing, enshrined in Euclid’s
Elements
, was endlessly imitated by workers in natural science, where it is not so appropriate. As we will see, Aristotle’s writing on natural science involves little mathematics, but at times it sounds like a parody of mathematical reasoning, as in his discussion of motion in
Physics
: “A, then, will move through B in a time C, and through D, which is thinner, in time E (if the length of B is equal to D), in proportion to the density of the hindering body. For let B be water and D be air.”
⁶
Perhaps the greatest work of Greek physics is
On Floating Bodies
by Archimedes, to be discussed in
Chapter 4
. This book is written like a mathematics text, with unquestioned postulates followed by deduced propositions. Archimedes was smart enough to choose the right postulates, but scientific research is more honestly reported as a tangle of deduction, induction, and guesswork.

More important than the question of style, though related to it, is a false goal inspired by mathematics: to reach certain truth by the unaided intellect. In his discussion of the education of philosopher kings in the
Republic
, Plato has Socrates argue that astronomy should be done in the same way as geometry. According to Socrates, looking at the sky may be helpful as a spur to the
intellect, in the same way that looking at a geometric diagram may be helpful in mathematics, but in both cases real knowledge comes solely through thought. Socrates explains in the
Republic
that “we should use the heavenly bodies merely as illustrations to help us study the other realm, as we would if we were faced with exceptional geometric figures.”
⁷

Mathematics is the means by which we deduce the consequences of physical principles. More than that, it is the indispensable language in which the principles of physical science are expressed. It often inspires new ideas about the natural sciences, and in turn the needs of science often drive developments in mathematics. The work of a theoretical physicist, Edward Witten, has provided so much insight into mathematics that in 1990 he was awarded one of the highest awards in mathematics, the Fields Medal. But mathematics is not a natural science. Mathematics in itself, without observation, cannot tell us anything about the world. And mathematical theorems can be neither verified nor refuted by observation of the world.

This was not clear in the ancient world, nor indeed even in early modern times. We have seen that Plato and the Pythagoreans considered mathematical objects such as numbers or triangles to be the fundamental constituents of nature, and we shall see that some philosophers regarded mathematical astronomy as a branch of mathematics, not of natural science.

The distinction between mathematics and science is pretty well settled. It remains mysterious to us why mathematics that is invented for reasons having nothing to do with nature often turns out to be useful in physical theories. In a famous article,
⁸
the physicist Eugene Wigner has written of “the unreasonable effectiveness of mathematics.” But we generally have no trouble in distinguishing the ideas of mathematics from principles of science, principles that are ultimately justified by observation of the world.

Where conflicts now sometimes arise between mathematicians and scientists, it is generally over the issue of mathematical rigor. Since the early nineteenth century, researchers in pure
mathematics have regarded rigor as essential; definitions and assumptions must be precise, and deductions must follow with absolute certainty. Physicists are more opportunistic, demanding only enough precision and certainty to give them a good chance of avoiding serious mistakes. In the preface of my own treatise on the quantum theory of fields, I admit that “there are parts of this book that will bring tears to the eyes of the mathematically inclined reader.”