Modeling matter as discontinuous (atomic) constituted the very first quantum theory, the precursor of the modern. In modern quantum theory, both matter and energy are quantized (discontinuous): matter is composed of disconnected elementary particles, the quarks and leptons, and energy comes in discrete (quantum) bundles (e.g., photons are the particles of the energy of light).
Conclusion
All challenges of rarefaction and condensation could be accounted for only through atomism (to be introduced fully in chapter 12), since such a great idea required first the development of all other great ideas conceived by Democritus’s predecessors, but also in the light of mathematics (Democritus, the principal contributor of the atomic theory, was a brilliant mathematician). The significance of mathematics, not just as an abstract field of knowledge but also as a practical method to describe nature, had been realized early on, especially by the great Pythagoras as a consequence of his passion for numbers.
* * *
1Simplicius, Physics 24.26–25.1, Theophrastus frag. 226A. Or see Daniel W. Graham, The Texts of Early Greek Philosophy: The Complete Fragments and Selected Testimonies of the Major Presocratics (Cambridge: Cambridge University Press, 2010), 75 (text 3).
2Known as the bottom-up interpretation of nature. The top-down philosophy is introduced in chapter 10.
3Aëtius 1.3.4, trans. John Burnet, Early Greek Philosophy (London: A & C. Black, 1920), chap. 1.
4Hippolytus, Refutation 1.7, trans. Burnet, Early Greek Philosophy, chap. 1.
5Aëtius 3.3.2, trans. Burnet, Early Greek Philosophy, chap. 1.
6This view is also expressed in Erwin Schrödinger, Nature and the Greeks and Science and Humanism (Cambridge: Cambridge University Press, 1996); Werner Heisenberg, Physics and Philosophy: The Revolution in Modern Science (New York: Harper Torchbooks, 1962).
7Sextus Empiricus, Against the Professors 7.135, trans. Schrödinger, Nature and the Greeks, 89.
8Aristotle, Metaphysics 985b4–20, trans. Graham, Texts of Early Greek Philosophy, 525 (text 10).
9Ibid.
10Carl Sagan, Cosmos (New York: Random House, 1980), 181.
11Schrödinger, Nature and the Greeks, 62–65, 84–86, 157–162.
12Ibid., 160.
13Ibid., 62.
6
Numbers and Shapes
Introduction
Pythagoras of Samos (ca. 570–ca. 495 bce) initiated the mathematical analysis of nature, a cornerstone practice in modern theoretical physics. “Things are numbers” is the most significant Pythagorean doctrine.1 While its exact meaning is ambiguous, it signifies that the phenomena of nature are describable by equations and numbers—or that they are a self-organization of patterns, not chaotic or random. Therefore, nature is quantifiable and potentially knowable through the scientific method. Based on this, the underlying principle of nature is not material (e.g., water, air) but is rather a mathematical form (an equation). Since the mathematical aspect of nature is not readily realized, the doctrine was emphasizing that sense perception was merely revealing an untrue version of nature (reality), a truer version of which could be glimpsed by the intellect through modeling nature mathematically.
The Pythagoreans quantified pleasing sounds of music, right-angled triangles, even the motion of the heavenly bodies. The “Copernican revolution” (heliocentricity) is traced back to Pythagorean cosmology. But, at last, Einstein’s theory of relativity shines the truth on a popular misconception related to it: that “the earth revolves around the sun (heliocentricity) is correct,” and that “the sun revolves around the earth (geocentricism) is incorrect.”
Plato was inspired by Pythagorean mathematics, but he replaced their “things are numbers” with things are shapes, forms, Forms, an abstract interpretation of nature called the theory of “Forms.” The quantum-mechanical wave functions—mathematical forms that describe microscopic particles—may be considered the Platonic Forms of quarks and leptons.
The Man
Pythagoras founded a school in Croton in southern Italy, open to both men and women, where he and his students pursued various studies, including religion, philosophy, science, mathematics, and music. They practiced a common way of life: asceticism (through body exercise, a vow of silence, a special diet that avoided meat and fish) and secrecy (probably to keep their discoveries exclusive to their students in order to attract more members to their school). Therefore, it has always been difficult to distinguish exactly what philosophical views belong to him or to some other Pythagorean. Aristotle avoids such difficulty by often referring generally to “the Pythagoreans.” Plato in The Republic writes specifically about Pythagoras and says that he was uniquely respected and loved by his students, not only for his knowledge but also for teaching them “the Pythagorean” way of life, best known for its high ethical standards. Wisdom, justice, and courage were among the sought-after virtues. Friendship was also highly valued. Pythagoras’s dogmatic influence on his students was evident by their reference to his opinions as prophesies with the characteristic phrase “He himself said so.”2 Pythagoreanism had remained influential in Greek philosophy continuously for more than 800 years (from its birth, the end of sixth century bce, down to third century ce).
From Earthly Harmonies to Cosmic Symphonies
Mellifluous Sounds of Music
Pythagoras’s first application illustrating the role of numbers in nature was the mathematical description of mellifluous sounds of music. First, he discovered that in stringed instruments the sound of a plucked string depends on its length and tension. For example, the sound of a plucked guitar string is of a higher pitch as it is pressed down and made shorter by your finger. Then he observed that the blended sound produced by two plucked strings of the same tension is more pleasing when their lengths are in ratios of small integers—for example, 2:1 is the octave, 3:2 the fifth, 4:3 the fourth, and 5:4 the third—thus numbers forming a discrete, quantum set, which for the example given is the set of 1, 2, 3, 4, and 5 (obtained by arranging in sequence the numbers of the aforementioned ratios).
The Pythagorean theory of music was a significant milestone in the evolution of science from two points of view. First, since the phenomenon of sound can be quantified, that is, it is represented by mathematical formulas, why not all phenomena? Second, if all things truly can be represented by numbers, then mathematics is the underlying and unifying principle of every natural phenomenon, even the seemingly dissimilar. With this in mind, everything may somehow be related at least mathematically—in other words, a kind of master mathematical equation may be invented that can describe everything, the goal of a theory of everything. So phenomena that have no apparent relationship with one another, on a deeper level, mathematically, may prove to obey the same mathematical principle and thus have something subtle in common. We have already spoken about unification efforts undertaken nowadays in search of a theory of everything. But the first step of cosmic unification in search of a universal law was taken with the intellectually bold Pythagoreans when they connected mathematically two seemingly unrelated phenomena: their earthly harmonies with the heavenly motions. How did they do this?
The Music of the Spheres
First, they supposed that, similar to the way an object on earth moving through air can produce a sound (slow movement making a low pitch, fast movement a high one), the stars (including the sun), moon, and planets (including earth), moving through ether (the purer air believed then to fill the universe) can produce their heavenly sounds. But these sounds must blend into a song harmoniously. They reasoned that the ratios of the length of strings that produce the harmonious sounds in string instruments must be the same as the various ratios formed by the speeds of the revolving heavenly bodies. This requirement restricts basically the speeds and orbits of the heavenly objects to certain discrete, quantum numbers, an idea that resonates with the essence of modern quantum theory!
Relative speeds of heavenly bodies could be easily deduced by comparing each body’s rising time.
For example, the moon rises about 50 minutes after the stars (some reference group of them that on some day rises together with the moon), but the sun rises only about 4 minutes after the stars, well-known facts in antiquity. With this in mind, the apparent revolution speed of the stars is the fastest, of the sun the second fastest, and of the moon the slowest. Since their speeds were different, the sounds they would produce as they move through ether would also be different, to say the least. But they also had to be harmonious, the Pythagoreans conjectured, since nature was a “cosmos,”3 a term credited to Pythagoras himself, a beautiful and well-ordered universe for which a cacophonous music of heavens was unaesthetic. The music of the heavenly bodies is inaudible, the Pythagoreans explained (as Aristotle tells us), because it is continuously playing and “the sound is in our ears since our birth, thus it is indistinguishable from its opposite silence; sound and silence are distinguishable only via their mutual contrast.”4 In a parallel example, a cook does not smell his own food after a few hours of cooking it. The earthly string harmonies, which could be heard, inspired the Pythagoreans to deduce by analogy the heavenly harmonies, which could not be heard. This type of approach, to come up with a general law by analogy of something specific, is common practice in science.
Such unusual interconnection was celebrated first in 1619 with the harmonic law of Johannes Kepler (1571–1630) when the astronomer discovered that, as planets revolve around the sun in their elliptical orbits, the ratios formed by each planet’s fastest speed at perihelion (a planet’s closest distance from the sun) over its slowest at aphelion (its greatest distance from the sun) are very close to the Pythagorean ratios of pleasing harmonies in stringed instruments. In his book The Harmonies of the World, Kepler wrote, “The heavenly motions are nothing but a continuous song for several voices, to be perceived by the intellect, not by the ear.”5
Moreover, in the beginning of the twentieth century, the seminal era of quantum theory, physicists Niels Bohr (1885–1962) and Arnold Sommerfeld (1868–1951) conceptualized the atom as a miniature solar system, with the electrons orbiting the nucleus of an atom like the planets are orbiting the sun. In their theory, however, the orbits of the electrons are quantized; they are restricted to certain discrete speeds and sizes (as were the heavenly bodies in the Pythagorean theory) that are expressible in terms of specific integers called quantum numbers that “display a greater harmonic consonance than even the stars in the Pythagorean music of the spheres [heavenly bodies].”6 Remarkably, unlike the Pythagorean theory of planetary motion, which was quantized, the Newtonian theory was not: planets, according to Newton’s theory of gravity, do not have a restriction in their speeds or orbital sizes. But they should, according to quantum theory, although their quantum behavior is negligibly small because of their large mass.
Even more so, according to the latest developments in string theory and in the words of string theorist physicist Brian Greene (1963–), “everything in the universe, from the tiniest particle to the most distant star is made from one kind of ingredient—unimaginably small vibrating strands of energy called strings. Just as the strings of a cello can give rise to a rich variety of musical notes, the tiny strings in string theory vibrate in a multitude of different ways making up all the constituents of nature. In other words, the universe is like a grand cosmic symphony resonating with all the various notes these tiny vibrating strands of energy can play.”7 A subtle cosmic interconnection between all things in nature, describable mathematically, was an idea envisioned by the great Pythagoras and has been consistently reaffirmed by modern physics. Mathematics nonetheless is not always rational.
The Irrationality of a Number
The proof of the Pythagorean theorem—that in a right-angled triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides—was the epitome of the newly born notion of mathematical deductive reasoning, in which general theorems are proven starting from the least number of axioms. It was especially encouraging to the most important Pythagorean doctrine, “things are numbers.” But it ended up also being a bad omen. For soon after the theorem’s proof, its application on a special kind of right triangle—the isosceles, with its equal sides having a length of one unit—led to the discovery of a new type of number, the irrational number, which perplexed the Pythagoreans and shook the very foundation of their number doctrine. It was found that the length of the hypotenuse of this right triangle is equal to the square root of two, that is,
√2 = 1.4142135623 7309504880 1688724209 6980785696 7187537694 8073176679 7379907324 7846210703 8850387534 3276415727 . . . ,
which does not have a precise numerical value; it can only be approximated—shown here it is truncated to 100 decimal places, for there is literally not enough paper in the entire universe to write such a number completely! That is, one cannot write down a precise number for the length of such a hypotenuse, only an approximate one, but we must emphasize that an approximate number is only approximate; it represents not the true length of the hypotenuse but only an approximate length. So how can things be numbers when some things cannot be assigned a precise number? To answer, we need to understand irrational numbers a bit more.
In the history of mathematics, integers (. . . , –4, –3, –2, –1, 0, 1, 2, 3, 4, . . .) were supposed to be the only numbers needed, since with their various ratios (fractions) every number that exists (including nonintegers) could, so it was thought, be written down. For example, the positive noninteger 1/3 is expressed as a ratio of two integers, obviously 1 and 3; negative noninteger –5/4 is the ratio of the integers –5 and 4; even 0 may be thought of as the ratios 0/2 or 0/7, and so on; in fact, even integers themselves may be expressed alternatively as a ratio of two integers, for example, 8 = 16/2. Numbers that can be expressed as ratios of integers are called rational. So, for the Pythagoreans (and in general, up to that point in history), only rational numbers were thought to exist.
Since for the Pythagoreans every number was expressible as a ratio of two integers, so, too, should the length of every geometrical line. But they were shocked to discover that the length of the hypotenuse of the aforesaid type of isosceles right triangle could not be expressed as a ratio of two integers! That length was not a rational number. It was equal to the square root of two (√2), which turned out to be an irrational number. Irrational literally means that there is no ratio, none at all, that can provide an exact numerical value for √2. Hence, the √2 can only be approximated. For example, truncated to one decimal place, the √2 is equal to the number 1.4 (which, in this approximate value, can be thought of as the ratios 14/10 or 7/5); to two decimal places, the √2 is equal to 1.41 (which, in this approximation, can be thought of as the ratio 141/100). There are infinitely many irrational numbers, all numerically inexpressible by ratios. The famous number π (pi) is irrational.
The irrationality of the √2 was so shocking to the number doctrine that, according to legend, Pythagoras’s student Hippasus of Metapontum (from fifth century bce), said to have discovered it, was drowned in the deep sea in an act of divine retribution. Irrational numbers have been playing a critical role in the advancement of mathematics and physics since the time of Pythagoras. But they still present an epistemological challenge because they provide only a numerically approximate knowledge of nature. This “approximate knowledge” is a significant point that will be picked up again in chapter 9 in order to try to understand the fascinatingly well-reasoned but paradoxical view of Zeno, that apparent motion is not real—that, an apparently flying arrow, for example, is not really moving!
Is the Universe Arithmetical or Geometrical?
How, then, can all things be numbers if some things cannot be given an exact numerical value? They cannot if exact numbers is all that we have in mind. But they can in some broader sense. First, in general, the phenomena of nature are assigned numbers (e.g., today’s temperature) determined by the various equations of modern physics. These numbers (the phenomena) are in turn compared with experime
ntal data (again, numbers) in order to verify or falsify the hypothesis that predicts them, as required by the scientific method.
Second, we’ll see in the next chapter that, within the context of the most advanced theory of matter—the quantum theory—microscopic particles lost their permanency and distinguishability and, as a result, their properties (e.g., position, velocity, energy, even their very existence) are expressed only as probabilities, only in terms of average numbers calculated from the so-called wave functions (probability functions), solutions to quantum equations. Since every macroscopic object in nature is composed of microscopic particles, then indeed all “things are numbers.” Yet from another perspective, the wave functions themselves are abstract three-dimensional geometrical forms. Yet again, their exact geometrical shape depends on various quantum numbers. For the atoms of the periodic table of chemistry (e.g., carbon, oxygen), these quantum numbers also restrict the number of electrons that could be present in a shell (somewhere around the nucleus) in order to avoid overcrowding—for electrons (kind of like people) need their own space, an idea known as Pauli exclusion principle.
But once more, back to a geometrical representation of the universe, we’ll see also in the next chapter that gravity, in Einstein’s theory of general relativity, is a manifestation of the geometry of space; it is not a force as Newton had declared it.
In Search of a Theory of Everything Page 6
