Things then, can’t be purely arithmetical; they are also geometrical. In fact, since in quantum theory the whereabouts of a particle are expressed probabilistically in terms of the wave function, then a particle (a localized entity, a probability number) is also a wave (an expanded entity, a geometrical form). This double description of nature is known as the wave-particle duality (true for both matter and light), but, equivalently, it could have been called the geometrical-arithmetical duality. Plato was the first to challenge the Pythagorean number doctrine with a geometrical theory of matter.
Plato’s Theory of Forms
From Physical Forms . . .
Plato was inspired immensely by the Pythagoreans. But he thought irrational magnitudes destroyed any chance for things to be purely numbers. What then? Look all around you, what do you see? Shapes! Things are shapes for Plato. In his Timaeus,8 he assumes that each form of earthly matter, earth (the solid form), water (liquids), air (gases), and fire, is composed from invisible structures of a unique shape (or form) that control matter’s properties. If we could zoom in on, say, dirt (earth), we would notice that it is composed of microscopic cubes (which easily stack up as solid matter does)—see figure 6.1. Water is composed of icosahedra (relatively the roundest of Plato’s structures so they can roll or slide as water does), air of octahedra (to flow), and fire from tetrahedra (with sharp corners to burn or cut things). Later Aristotle added a fifth essence (element), the quintessential heavenly ether—meaning “blazing” in Greek, found only in heaven9 and thus the matter of shining stars—thought composed of dodecahedra. Our limited senses, Plato thinks, can’t see these structures, only their imperfect bulk.
Figure 6.1 Plato’s theory of earthly matter and Aristotle’s theory of heavenly matter.
The earthly solids have symmetrical faces—squares the cube, equilateral triangles the other three—faces which are constructed from only two types of right-angled triangles, the half-square (having the irrational √2 as a side) and the half-equilateral (having the irrational √3 as a side). Because of their irrational side, Plato considered these triangles to be physically indivisible (atomic) and thus the basic building “blocks” of all earthly matter. From this point on in the history of science, things are a combination of arithmetic (numbers) and geometry (of physical but also abstract forms).
. . . To the Abstract Theory of Forms
With time, the idea that things have a physical unseen basic geometrical form evolved in Plato’s mind into an abstract form, a noetic description of nature known as the theory of “Ideas” or “Forms.”10 According to it, everything we experience—physical objects of sense perception, the ocean, mountains, trees, but qualities, too, friendship, compassion, beauty—is but a mere imperfect copy, a shadow, of a greater truth: an ideal unalterable Idea, a universal Form that represents each particular object or quality. For example, an abstract Form exists that represents the ideal friendship and another Form that represents the ideal right-angled isosceles triangle. Every friendship or triangle of the sort that we have ever experienced, Plato thinks, is an imperfect copy of its corresponding true Form, which is created by God. True knowledge is achieved only with the grasp of these Forms. The quantum mechanical wave functions (which are mathematical forms) may be considered the Platonic Forms of microscopic particles.11
The Value of Mathematics
The theory of Forms may have its origin in Pythagorean mathematics. Plato himself was an accomplished mathematician. The entrance to his Academy had the inscription: “Nobody Untrained in Geometry May Enter My House.”12 Pythagoras showed us that mathematics quantifies nature and that it is the language of science—we can measure nature in order to verify or falsify our scientific hypotheses. And even when the law of a natural phenomenon has not yet been discovered, the law is assumed to exist, as is the mathematical equation that can express it. This, in fact, is the very premise of science. Without such an attitude, science cannot be done and truth cannot be found.
Geocentric Versus Heliocentric: The Relative Truth
Pythagorean Cosmology
Being a great geometer who understood well the relationships of spheres, flat surfaces, and lines, Pythagoras was probably the first to deduce that the earth is spherical. Several observations might have aided him in reaching such a conclusion. During a lunar eclipse, the shadow of the earth on the moon is a circular arc. The masts of receding ships disappear last (and, equivalently, appear first when ships are approaching). Pythagoras himself knew that the evening and morning “stars” are really the planet Venus.
But the most notable achievement of the Pythagoreans in cosmology, often credited to the Pythagorean Philolaus (ca. 470–ca. 385 bce), was when they displaced the earth from the center of the universe and imagined it in motion. So the earth revolves around a center occupied by fire, called Central Fire, and so do the moon, sun, planets (Mercury, Venus, Mars, Jupiter, Saturn—the ones known to antiquity, as only these are visible without a telescope), and the fixed stars—termed so because of their apparently fixed position with respect to one another; on the other hand, planet means literally “wanderer” because planets were changing their position among the fixed stars.13 Central Fire is invisible because the inhabitable hemisphere of the earth faces always away from it, whereas the side of earth that always faces it is uninhabitable because it is too hot. Incidentally the moon’s synchronous motion—according to which its rotational period around its axis is the same with its revolution period around the earth—produces the same effect: the near side of the moon always faces the earth, whereas its far side always faces away from earth, making it always invisible to an earthbound observer. Revolving around Central Fire is another body, the anti-earth, termed so because of its position.14 It is imagined to always be in the same direction as the uninhabitable hemisphere of earth, so, like Central Fire, it, too, is invisible. It is not certain why anti-earth was required (some scholars speculate that it was needed to explain eclipses), or even whether anti-earth was really a planet at all—for due to its position, anti-earth might have simply been the uninhabitable hemisphere of earth.
In addition to its revolution around Central Fire, earth also rotates around its own axis daily, accounting for the apparent revolution of the sky. This understanding was in audacious opposition to the popular view of an immobile earth at the center of the universe as well as to the evidence of the senses that do not feel earth’s motion. In an analogy, to understand the apparent revolution of the sky, pretend to be the earth and stand at the center of a room. Then begin to rotate around the axis of your body, say, counterclockwise. The walls, which you can think of as the sky (with the sun and stars), appear to revolve around you in the reverse direction, clockwise.
Only Central Fire is self-luminous; all other bodies are shining with reflected light from it. In fact, this might be the justification of its postulated existence: since the moon is shining with reflected light—an ancient knowledge—and by the cycle of day and night, so, obviously, is the earth. But why not the sun, which in many ways is like the moon—in motion, shape, size, eclipses, color—and all heavenly bodies? If yes, a source of light had to be speculated, hence the Central Fire.
That the Pythagorean system is neither geocentric nor heliocentric is actually a quite justifiable cosmological theory. Since every visible celestial body appears to be moving, perhaps so should the earth, the Pythagoreans might have thought. Now the difficulty with imagining the earth in motion has its origin in our deceptive senses, namely our eyes. Our inability to detect the actual distance of what we see, in particular the stars, is tricking us into thinking that all bodies are the same distance from us. Therefore, being also in different directions, they appear fixed on a hemispherical dome, which is part of what we call the sky. As the sky appears to revolve daily around us, the new stars brought into view appear, for the same reason, to also have the same distance from us as all the rest. Thus, we imagine every star fixed on a spherical sky—even though at any one time we see
only a hemispherical sky—with us on earth at its center. In addition to that, the apparent daily revolution of the dome-like sky around us is easily tricking us into thinking that the earth is absolutely motionless at the sky’s center and therefore is at the absolute center of the universe, as if the earth occupies a special position in the universe. This is, in fact, the geocentric view, which, due to our imperfect senses and our initially uninformed intellect, naturally emerged as the first cosmological model. But the Pythagoreans were well aware of the unreliability of the senses, and they were also accomplished mathematicians with sharp, critical minds. Moreover, being people of virtues, such as humbleness, the Pythagoreans had no difficulty displacing the earth and themselves from the center and purpose of the universe.
No Special Center
Influenced by the Pythagorean cosmology, philosopher Heraclides Ponticus (ca. 390–ca. 310 bce) proceeded to devise his own. He explained correctly the varying brightness of Mercury and Venus as the result of their varying distances from earth. The additional observations that these planets seem to always follow the sun—when visible, each of them (but independently of the other) either rises just before the sun or sets just after the sun does, especially so Mercury because it is closer to the sun—prompted him to imagine these two planets revolving around the sun, thus justifying their varying distance from earth and consequently their varying brightness, and the sun revolving around an immovable earth at the center of the universe. This partial heliocentric view—with only two planets revolving directly around the sun—became a full-blown heliocentric theory when Aristarchus proposed that all planets including earth revolve around the sun.15 While this theory was rejected in favor of the geocentric view, it was revived much later by Copernicus.
Traced back to Pythagorean cosmology are the first steps away from the prejudices of the geocentric and anthropocentric worldview and the inspiration for the discovery of the heliocentric worldview. However, perhaps due to scientific misrepresentation of the topic, the popular perception is that the heliocentric model is correct and the geocentric model incorrect. But the profundity of the heliocentric model is really this: (1) it is another point of view as good as the geocentric—though initially, like the geocentric, it, too, was incorrectly perceived as absolute, as if the sun were the absolute center of the universe (as Copernicus held in his On the Revolutions of the Heavenly Spheres, who inspired Galileo Galilei [1564–1642] to hold the same hypothesis in his book the Dialogue Concerning the Two Chief World Systems)—and (2) since another center is as good as the previous one, the notion of an absolute center of the universe is abolished. In fact, in modern physics the any-center view is correct. A particular center is chosen merely for its conceptual and mathematical convenience for the understanding of a physical phenomenon and is not to be misinterpreted as absolute or uniquely correct. This view is supported by special relativity (see next subsection). It is also supported by astronomical observations, including the discovery of numerous new galaxies, each with billions of stars revolving relative to the galaxy’s center, and generally observations indicating that the universe is isotropic (thus, no one location is more special than another). Finally, such a view may be accepted based merely on pure humility, that neither the earth nor the sun should occupy a special center, and in general that no point in the universe should be more centered or privileged than another. The universe has neither an edge nor a center, and the laws of physics apply equally the same everywhere. “The merit of the Copernican hypothesis [that (1) annually earth revolves around the sun, not the sun around the earth and (2) diurnally earth rotates on its axis, not the sky around earth] is not truth, but simplicity; in view of the relativity of motion [from Einstein’s theory of relativity], no question of truth is involved.”16 Equally correct (as will be emphasized a bit more later in this section), we could imagine that (1) annually either the earth revolves around the sun or the sun around earth and (2) diurnally either earth rotates on its axis, or the sky revolves around earth. Space and time were absolute in Newtonian physics but became relative in Einstein’s theory of special relativity. This means that for relativity an absolute frame of reference—a special location of observation that can be used to refer to absolute motion—does not exist. There exist only relative frames of reference that can be used to refer to relative motion. Hence, we can choose any center relative to which something can be at rest or in motion. But a special center for absolute rest or absolute motion is utterly meaningless. Space, time, and motion are all relative. Let us elaborate.
Newtonian Absoluteness Versus Einsteinian Relativism
Newtonian Physics
In Newtonian physics, space and time are absolute and thus independent of an observer’s relative motion. This means that space distances and time intervals are unchanged by motion. For example, the length and mass of an object are the same for all observers independently of their location or motion relative to the object or relative to one another. The same for them is also the way time passes. Twins, for instance, have always the same age with respect to one another, whether they move or not relative to each other. In general, two events that are simultaneous for one observer are simultaneous for every observer—absolute simultaneity. Space is a kind of preexisting passive (unaffected, in a sense “disconnected” from everything else) immutable playground where objects exist and events occur while time flows steadily in the background as if a cosmic clock existed that showed the same exact time for everyone and every location in the universe. “Now” and “here” are absolute concepts for Newton—that is, everyone agrees when and where something happens. But Einstein’s theory of special relativity proved all these to be false, in spite of the fact that all these are how we experience the world daily.
Special Relativity
In special relativity the speed of light in a vacuum, designated c, is always 671 million miles per hour—put differently, in 1 second light travels as far away as is the distance of eight times around the earth. It is the same in all reference frames (for all observers, moving or not relative to the light source). It is also a kind of cosmic speed limit—absolutely constant—for although it could be approached, absolutely no material object can travel as fast as or faster than light. This fact has nothing to do with engineering. It is not because we don’t have high-powered engines to accelerate an object to the speed of light; rather, this fact is how nature behaves. It is a law of nature that has withstood the scrutiny of experiments since 1905, the year it was postulated by Einstein in his theory of special relativity. If c were not a cosmic constant, causality would be violated and the universe would be paradoxical: a message, travelling faster than c, could be sent to my past to prevent my parents (my cause) from meeting. But how could I (the effect) then exist with my cause eliminated? That I exist is evidence c must be a cosmic constant, but the consequences of that are radical.
Two of the most dramatic consequences concern space and time: they are no longer absolute, they are relative—dependent on an observer’s relative motion. They are combined mathematically (by the so-called Lorentz transformation) into a continuum called space-time. Space distances and time intervals do change with respect to an observer’s relative motion. Relative space means that a moving object contracts in the direction of motion, as seen by (relative to) a stationary observer—a phenomenon known as length contraction. Relative time means no cosmic clock, that the passage of time in a moving clock (say, aboard a moving spaceship) is dilated; it is slower relative to the passage of time in a stationary clock on earth—a phenomenon known as time dilation. To function properly, the Global Positioning System (GPS), a common cellphone app, takes time dilation into consideration. If it didn’t, the GPS receiver in your car or phone would miss your destination. Parenthetically, the GPS must also take into account another time effect, predicted by general relativity. That clocks in orbit, where gravity is weaker compared to the ground, run faster relative to clocks on earth.
Interestingly, time dilation mak
es time travel possible because using it we can travel into the future. Suppose Earthly and Heavenly are twins. Heavenly likes to journey in space, while Earthly prefers to stay on earth. If Heavenly travels at a speed close to c, upon her return to earth she will realize that she has aged less than Earthly (and all the other people or things on earth). How much less depends on the duration of her trip as well as how close her speed was to c. For example, if her speed was 99.5 percent of the speed of light, then for every 1 year that Heavenly ages during her trip, Earthly ages 10 years. But if her speed was 99.99 percent of the speed of light, then for every 1 year that Heavenly ages, Earthly ages 71 years. So a twin who takes a trip into space will age less with respect to the twin on earth. Here we emphasize that Heavenly feels no different, as regards the passage of time, while traveling. The difference in age is noticed when the twins compare notes, for example, meet again.
In general, if you travel at a speed close to the speed of light, the time elapsed for you will be less compared (relatively) to the time elapsed for those not traveling with you. Hence, because of time dilation, you can then travel into the future of those not taking the trip with you; like the astronauts in the 1968 film Planet of the Apes, who aged only 18 months during their near-light-speed journey and returned to find a postapocalyptic earth where the elapsed time since they left was 2,006 years. So by taking a trip at high speeds you may return to earth at some future century of your choice. You may enjoy great developments of a more advanced civilization in that future century. The downside is that, if it is too far into the future, none of your familiar people may be alive to welcome you. Would such a trip be worth taking?
In Search of a Theory of Everything Page 7