In Search of a Theory of Everything
Page 13
Unborn and Imperishable
Parmenides’s philosophical worldview is, so he says, presented to him as a revelation by a goddess and is described in his poem On Nature. The main parts of the poem are the “Way of Truth”16 (which discusses his philosophy) and the “Way of Opinion”17 (which, among other things, discusses the philosophies of other philosophers). His primary goal was not so much to create a specific physical theory that would explain particular phenomena of nature but rather a theory attempting a logical explanation of existence itself: how can something be? It just is, he reasoned, for there is no such thing as nothing. Nature is unborn and imperishable. That which exists can neither be created from nonexistence nor obliterated into nonexistence. If the universe had a beginning, it would mean that it once did not exist—for if it existed it could not begin. But if the universe did not exist, it would have been Not-Being, and so again it could not begin (for Being cannot come from Not-Being). So the only way to explain why the universe is, is to assume that what is, has always been, unborn, without a beginning.
Now, on the one hand, his view of an unborn nature means that nature has not been caused; it does not have a primary cause. On the other hand, the opposite idea is that nature has been caused by a primary cause. This latter view is in a sense antiscientific since the premise of science is comprehensibility. But a primary cause cannot be understood—if it could, we would know what caused the “primary” cause; hence, the “primary” cause would not have been really primary. Conversely, an unborn nature seems, at least at a first glance, to be more in accordance with the scientific premise, because something unborn/uncreated does not require a primary cause (an explanation) of why it exists—for it has always been.
That said, the notion of an unborn (uncreated, uncaused) nature (or, analogously, an imperishable nature having no ultimate purpose) must be examined with more caution. For it does not exclude the possibility of a god coexisting in the whole—as is, in fact, the case of the Parmenidean “Way of Truth,” according to which the apocalyptic goddess Parmenides, and all the rest of nature, all just are. Moreover, an omnipotent and omniscient god could have made nature appear uncreated to us mere mortals. The point is that science cannot prove or disprove the existence of a god, and therefore such a notion, as Parmenides might have put it in his “Way of Opinion,” will always remain a matter of subjective belief. In science we must always begin with an assumed something (a Being)—and if we happen to finally explain such assumed something, we explain it with a new assumed something. Science cannot begin from Not-Being: there is no scientific explanation of a universe coming to be from nothing! Why there is something instead of nothing is scientifically18 unanswerable. Causality in our theories explains only later effects by earlier causes, but it cannot explain the primary cause (the beginning). And as a consequence, there is no way to ever know if there is an ultimate purpose. Even if the truth of the universe is revealed to us, the only way we can know that such truth is absolute is if we ourselves have absolute abilities—so that we can comprehend the absoluteness in the revealed truth. But we do not. And so again, the interpretation of such hypothetical revelation is subjective.
Among our best cosmological models in science, the big bang does not and cannot answer why there is a universe; it only assumes that there is one (that might have begun or might have always been) and then continues to describe it. But it cannot answer why there is what there is. The prediction of the big bang model, that the age of the universe is 13.8 billion years, is only a relative age, namely, that our scientific theories can begin describing the properties of the universe roughly since 13.8 billion years ago. But we emphasize that with regard to what the universe might have been doing before that, we are clueless.
Interestingly, in an effort to avoid the breakdown of the equations at the hypothetical big bang singularity, some cosmological models attempt to model mathematically a self-reliant finite universe with neither space nor time boundaries—that is, an unborn nature having neither beginning nor end.19 A geometrical analogy of such type of universe is the surface of a sphere: its size is finite and no place on it can be considered more of a beginning than an end, more of a center than an edge, or more special in any way. Of course, once more we emphasize that a mathematically/scientifically unborn universe does not say much about the true origin of the universe—that remains a subjective matter—because the truths of mathematics are restricted by its axioms and the truths of science are restricted by its mathematics and its scientific method. Scientific knowledge is provisional.
Lastly, the Parmenidean view is a hopeful philosophy because within its context consciousness is part of Being—I think; therefore, I have consciousness. Hence, consciousness can never become Not-Being, even with the body’s apparent death.
Conclusion
After Parmenides, any new natural philosophy would be considered incomplete unless it could address successfully his various conclusions, which, though unconventional, were logical. And as if that by itself was not a formidable task, Parmenides’s best student, Zeno, assertively supports his teacher’s views by adding to the complexity with his famous paradoxes that question the very nature of plurality, space, time, and the reality of apparent motion.
* * *
1Clement, Miscellanies 6.23, trans. Erwin Schrödinger, Nature and the Greeks and Science and Humanism (Cambridge: Cambridge University Press, 1996), 27.
2Although the bread is three-dimensional (with two space and one time dimensions), the real universe is four-dimensional (with three space and one time).
3Albert Einstein quoted in Brian Greene, The Fabric of the Cosmos: Space, Time, and the Texture of Reality (New York: Vintage, 2005), 139. Also quoted in Carlo Rovelli, Seven Brief Lessons on Physics (New York: RiverHead Books, 2016), 60 (Kindle ed.).
4With the appropriate speed and distance—numbers calculated by the equations of relativity.
5That is, my baby-self is a simultaneous event relatively to her, not to me. The expression, “you live in the past,” suddenly makes more sense.
6If change is truly an illusion, how is it created giving us a personal view of the universe in which we remember/know only the past (and only part of it) but not the future (an experience that has been called psychological time)? It’s an open question. To the contrary, the moon’s apparent small size, also an illusion, can be explained: it’s due to its distance.
7Karl R. Popper, Unended Quest: An Intellectual Autobiography (London: Routledge Classics, 2002), 148; or see: https://books.google.com/books?id=NyCEnehPMd8C&pg=PA148&lpg=PA148&dq= popper+einstein+parmenides&q=&hl=en#v=onepage&q=popper%20einstein%20parmenides&f=false (accessed January 9, 2017).
8In my opinion, Being’s completeness property may also be interpreted as the “painting” analogy of the block universe (see end of section “Being and the Block Universe”), where each space-time point is “complete” and unchangeable but it is also different than all other space-time points, creating an unchangeable but diversified universe.
9Aristotle, Metaphysics 985b4–20, trans. Daniel W. Graham, The Texts of Early Greek Philosophy: The Complete Fragments and Selected Testimonies of the Major Presocratics (Cambridge: Cambridge University Press, 2010), 525 (text 10).
10Ibid.
11It’s worth emphasizing once more that this answer is expected, for as mentioned, the uncertainty principles, which helped us to arrive to this answer, are in the first place conceived to describe something, not nothing; the notion of nothingness is indescribable.
12Einstein quoted in Joanne Baker, 50 Physics Ideas You Really Need to Know (London: Quercus, 2007), 165.
13Werner Heisenberg, Physics and Philosophy: The Revolution in Modern Science (New York: Harper Torchbooks, 1962).
14Abraham Pais, Subtle Is the Lord: The Science and the Life of Albert Einstein (New York: Oxford University Press, 2005), 5.
15J. J. Sakurai, Modern Quantum Mechanics (Menlo Park, CA: The Benjamin/Cummings Publishing Company, 1985), 226–229.
16Graham, Texts of Early Greek Philosophy, 211–219.
17Ibid., 219–33.
18Within the context of the scientific method.
19Stephen Hawking, A Brief History of Time: From the Big Bang to Black Holes (New York: Bantam Books, 1988), chap. 8.
9
Paradoxes of Nature
Introduction
Through a series of so-called paradoxes, Zeno of Elea (ca. 490–ca. 430 bce) tried to argue for the astonishing conclusion that motion is impossible and plurality is an illusion. Could he be right? We present four of his most daring paradoxes: the dichotomy, the Achilles, the arrow, and the space, which challenge various views on space, time, and motion, and examine them within the context of modern physics. We also refer briefly to the conclusion of his paradoxes on plurality, which deal with whether there are many things or just one.
There is still no commonly accepted resolution for any of Zeno’s paradoxes, a fact that preserves their legacy as the most difficult and long-standing puzzles. Part of the reason for this is the involvement of key notions such as space, time, and matter, of which their true nature is far from known even by the standards of modern physics. The real resolution of the paradoxes might require an even more radical understanding of these notions than the one presently provided by general relativity and quantum theory. Proposed solutions have often aimed to prove that motion is real. Empowered by the uncertainty principle of quantum mechanics, we will argue in favor of Zeno that at best, the phenomenon of motion is experimentally unverifiable!
The Dichotomy Paradox
According to Aristotle’s account, Zeno said, “Nothing moves because what is traveling must first reach the half-way point before it reaches the end.”1 In order to interpret this quote, we must suppose that space is either infinitely divisible (where space is imagined to be divided to ever smaller fractions) or finitely divisible (where space cannot be divided beyond a fundamental length).
Infinitely Divisible Space
The paradox can be interpreted two different ways, both of which are essentially the same. In the first interpretation, the question is this: can a traveler ever start a trip? To begin a trip of a certain distance, a traveler must travel the first half of it, but before he does that he must travel half of the first half, and in fact half of that, ad infinitum. Since there will always exist a smaller first half to be traveled first, Zeno questions whether a traveler can ever even start a trip.
In mathematical language, the traveler will be able to start his trip only if he can first find the smallest fraction (the “last” term) from the following infinite sequence of fractions of the total distance: 1/2, 1/4, 1/8, 1/16, . . . . But such smallest fraction does not exist; it is indeterminable (in fact, this is what is meant by calling such a sequence of fractions infinite). So the paradox is this: while on the one hand, Zeno’s argument, which questions the very ability to even start a trip, is logical; on the other hand, all around us we see things moving. Hence, either Zeno’s reasoning is wrong or what we see is false.
In the second interpretation, the paradox can be reformulated in a sort of reverse manner. In such case the question will be: assuming a traveler can somehow start a trip, can he ever finish it? To finish a trip of a certain distance, a traveler must first travel half of it, then half of the remaining distance, then half of the new remaining distance, ad infinitum. Since there will always exist a smaller last half to be traveled last, Zeno questions whether a traveler can ever finish a trip.
First “Answer”
First, note that getting up and walking, as Antisthenes the Cynic2 did after listening to Zeno’s presentation and thinking that a practical demonstration is stronger than any verbal argument, is not at all a refutation of Zeno’s paradoxes of motion, because Zeno does not deny apparent motion; he questions its truth. The natural philosophers were well aware of the deceptiveness in apparent reality; what we see happening is not necessarily happening the way we see it.
An “Answer” Based on Simple Mathematics
With the first interpretation in mind, to start the trip, the traveler must first figure out the smallest fraction of the total distance, that is, the “last” term of the infinite sequence of numbers 1/2, 1/4, 1/8, 1/16, . . . . Only then he will know where to step first and begin the trip. But such a term is indeterminable. After infinite subdivisions of the total distance, the “last” term of the sequence is indeed infinitesimally small and approaches zero, though it is not exactly zero: there will always exist a smaller first half to be traveled first. Now since such a term approaches zero, we might want to approximate it to exactly zero. But with such approximation the traveler will step first where he is already, the beginning. This might be interpreted to mean that the trip cannot start; thus, motion is impossible. Nonetheless, this is not necessarily the best conclusion since it is reached only after our convenient approximation of the “last” term with the number zero. Since the actual value of the “last” term is indeterminable, a better conclusion would be that, indeterminable must also be the status of the trip (whether the trip can ever begin). Thus, the notion of motion is, to say the least, ambiguous. The same conclusion is obtained through similar arguments applied to the second interpretation of the paradox.
An “Answer” Based on Modern Mathematics
Often an answer to the paradox is sought through calculus. Suppose the trip distance is 1 meter. Then, as per interpretation two, a traveler first travels half of the trip distance, that is, 1/2 of a meter, then half of the remaining distance, that is, an additional 1/4 of a meter, then half of the new remaining distance, that is, an additional 1/8 of a meter, ad infinitum. To find out if the traveler covers the trip distance of the 1 meter, we must add all the segments traveled by him, that is, 1/2 + 1/4 + 1/8 + 1/16 +. . . . Because the sum of this infinite geometric series converges on 1, some argue that the distance traveled by the traveler after infinite steps is 1 meter; thus, he has moved and the paradox is resolved.
But this argument has a flaw hidden in the details of calculus. To be able to do calculus (i.e., calculate series sums like the one in hand), irrational numbers must be approximated with rational. And there exist infinitely many irrational numbers along any space distance. For example, between the point zero (the beginning of the trip distance) and the point of 1 meter (the end of the trip distance), there are infinitely many irrational numbers—such as √2 – 1 = 0.414213562 . . . , or half of it, or one third of it, and so on—that must be approximated with rational numbers before any sum is calculated. For example, approximated to four decimal places, the rounded-off value of √2 – 1 = 0.4142. Zeno, however, seems to tacitly question these very axioms and approximations that are required in mathematics to make the series convergent to a practical and calculable answer. This is because nature, he would claim, does not have to behave according to the result of such convenient and ambitious human approximations. “Mathematics can never tell us what is, but only what would be if [this or that axiom is assumed].”3 Furthermore, some argue that the convergence method does not address the paradox because it does not explain how an infinite number of tasks (going from the first half of the distance to half of the remaining, etc., ad infinitum) can be carried out in finite time.
Now, in the first interpretation of the paradox, motion can’t start, and in the second interpretation, motion can’t end. These contradictory results hint that the very premise of the paradox, the infinite divisibility of space, might be flawed. Epicurus (we’ll see in chapter 13) assumed that space is finitely divisible, that it has a quantum nature! That resolved the paradox. But to be successful, he had to reimagine space and time to be even more radical than the way Einstein did.
The Achilles and the Tortoise Paradox
“In a race the faster runner can never overtake the slower. Since the faster runner must first reach the point from which the slower runner departed, the slower runner must always hold a lead” (Aristotle’s account of Zeno).4
The paradox say
s that in a race between, say, fast Achilles and a slow tortoise, initially separated by some distance, Achilles can never overtake the tortoise because before he achieves that he must first reach the starting location of the tortoise. But by the time he arrives there, the tortoise will have had the chance to move to a new location forward; and by the time he arrives at the tortoise’s new location, the tortoise will move farther forward to another new location, ad infinitum. Therefore, despite that faster Achilles will be constantly approaching the slower tortoise, still there will always exist some small and ever-decreasing distance separating them (though not necessarily in fractions of half, as in the dichotomy paradox). This is a paradox because, despite Zeno’s argument that a faster runner cannot overtake a slower one is logical, fast runners apparently do overtake slower ones. Is Zeno’s reasoning flawed, or are our senses false?
This paradox is basically the same as that of dichotomy, so everything mentioned earlier applies here. In the dichotomy paradox, the first interpretation (for which a traveler cannot start his trip) seems to deny motion more directly than the second interpretation (for which a traveler is assumed to move, although he cannot ever finish his trip). In the Achilles paradox, Achilles and the tortoise are assumed to be moving, but motion seems to not work in the conventional way, for the faster Achilles cannot overtake the slower tortoise. In the arrow paradox, Zeno is even more audacious, for he directly denies motion by any interpretation. Reconstructing it reads as follows.5