Wednesday, June 20, 2007

Does infinity paralyze motion?

kw: book reviews, nonfiction, mathematics

", does Achilles ever catch up to the tortoise?" you hear young Zeno ask. Questioning a bystander at the agora you find the visitor is in the midst of discussing his second paradox already. The first was this:
"How did you come here this morning? First you came halfway, perhaps as far as the temple on yon corner. Then you had to come halfway between the temple and this spot. But you were not here yet. You had to come half of the remaining distance, and so forth...then half of that. How many such intervals did you cross? An infinite number? If so, why are you here? How could you cross an infinite number of intervals?"
The tortoise question is more subtle. Achilles can run ten times as fast as the tortoise, so he gives the tortoise a 10-stadia head start. Once he has run ten stadia, the tortoise has gone another stadion. So he has to run another stadion, only to find the tortoise is still ahead, by one-tenth. Will he ever catch the tortoise?

You see the slower overpass the faster every time a race is run. So why is this stranger's reasoning causing such a sensation? But wait! He has two more conundrums yet to pose...

As reported by Joseph Mazur in The Motion Paradox: The 2,500-Year-Old Puzzle Behind the Mysteries of Time and Space, the four known paradoxes of Zeno of Elea (not Zeno the philosopher) are still mysterious, forcing us to think about the continuity—or not—of time and space. The third paradox, the Flying Arrow, attacks time directly:
It is impossible for a thing to be moving during a period of time, because it is impossible for it to be moving in an indivisible instant.
Zeno called the first two paradoxes Dichotomy and Achilles. The fourth, Stadium, is a subtler version of Flying Arrow:
Half a given period of time is equal to the whole of it; because equal motions must occupy equal times, and yet the time occupied in passing the same number of equal objects varies according as the objects are moving or stationary.
Let us pick apart the last thus. It is an unbalanced syllogism. The first statement assumes constant velocity, as we would say today (velocity was not defined for centuries after Zeno). Well and good. Let us consider three objects, in addition to ourselves, the observers. One is stationary, one moves to the right, and the other moves to the left. Each is a long scale, say a meter stick (or yardstick). Just by watching the marks on the sticks, we can see that the end of one moving stick passes the marks on the other moving stick faster than it passes the marks on the stationary stick. Their relative motions differ. Zeno's language implies that the relative motion between any pair of moving objects is the same, which is a fallacy.

This then exposes the fallacy of the Flying Arrow. If time is continuous, there is no "indivisible instant", so the proposition is without meaning. If I am walking at a steady five km/hr (about three mi/hr), then in one hour I cover five km; in a tenth of an hour I cover five-tenths of a km. The ratio between distance covered and time spent is the same. A similar analysis disposes of the Dichotomy.

Why is this so simple for us, but so hard for the 4th Century BCE Athenians? They would not consider dividing space by time, or time by space, because they are different in kind. As we would say, to the Athenian mind, one cannot divide apples by oranges. Pythagorean mathematics was intensely rooted in pure number, and all other calculations were intensely rooted in the quality of the objects counted. Five sheep plus five goats didn't give you "ten livestock" because they didn't think that way. They could only say "five sheep and five goats". And you couldn't consider dividing the flock in half in any way but by separating the sheep from the goats...and that wasn't really "half" anyway. Ratios of sheep to goats were unthinkable.

In The Motion Paradox Dr. Mazur takes us through the history of mathematics with a particular focus on the handling of motion through the 24 centuries since Zeno. When you get right down to it, mathematics is all about motion, either physical motion in space or conceptual motions of various kinds. For without motion, nothing happens; nothing is worth calculating.

In every case, as our mathematical tools have increased in power and subtlety, we have found ways of handling—or sidestepping—Zeno's paradoxes. Yet the author claims we haven't tackled them head-on. Calculus, now three-plus centuries grown, uses the concept of a limit to define such things as velocity or acceleration at a given point in time. If we think about it at all, that "point in time" is not Zeno's "indivisible instant", for we have no interest in dividing it anyway. It is a defined item with numbers attached.

Time is not a thing. It is a convention. Time cannot be defined in isolation, but in terms of changes in some quantity of interest. It is a measure of process. First we decided that certain processes have a constant rate—within some criterion of precision that satisfies us for the nonce—then we compare other processes to these "standard processes". Today's standard process is a certain vibration of a Cesium atom near the top of a parabolic arc in free fall, at a specific ambient temperature and near-zero pressure. Thus time itself is not defined at all, ...or it is defined circularly if you prefer. Suppose, if there is hidden somewhere in the Universe a truly steady, constant "standard process", an Ur-process, and our best "standard process" is wildly unsteady as measured by the Ur-process: We have no way of knowing.

Even the coupling of time with space in Special Relativity does not alleviate the problem. It also couples space with time, so that spatial measurements are as circularly defined as temporal ones.

I find it passing strange that the author didn't mention the one mathematician who gave us a clue to what is going on: Kurt Gödel, Einstein's best friend. His "Incompleteness Theorem" states that every self-contained mathematical system can be used to generate statements that cannot be proven true or false within that system. Zeno's first two paradoxes are the earliest statements known of propositions that could not be tested for truth within the mathematical system in which they were stated. Indeed, some say that we still don't have a sufficiently robust mathematics, that Calculus simply sidesteps the issue.

The author also states that much of modern mathematics lacks rigor when compared to Euclid's system. Rigor requires axioms as a basis, and not all Euclid's axioms are applicable to Calculus, in particular the non-Euclidean geometries and non-mensurable Topologies needed for much of today's work. New math requires new axioms. In my opinion, Calculus does not sidestep Zeno, it defines the Limit and the Tangent that are required to perform its operations, which results in rather simple concepts that make Zeno's paradoxes into fallacies. Based upon the appropriate axioms, we have rigor aplenty.

The biggest mathematical/physics conundrum we presently face is a new Dichotomy, between General Relativity and Quantum Mechanics. One requires continuous space and time, the other requires that space at least be discontinuous, and implies that time also might be discontinuous. For example according to QM, the shape of the P-orbital in a neutral atom in isolation (true isolation is unachievable in practice in a material Universe, but we can come as close as we like and can afford)...I say, the P-orbital has two lobes separated by a plane. The probability of an electron being in that plane is zero. Effectively, it cannot cross the plane. Yet, electrons cross from lobe to lobe millions of times per second, which gives rise to certain hyperfine spectroscopic lines is low-density, hot gases.

The "tunneling" phenomenon on which Scanning Tunneling Microscopy is based requires an electron to jump about a nanometer in exactly zero time. Millions do so every second while the scientist operating the instrument is recording a STM image. This is impossible in General Relativity, but commonplace in Quantum Mechanics.

Once a consistent system is produced that encompasses the phenomena of both GR and QM—I don't say it will encompass both systems, that may be impossible; but it would encompass all the phenomena of each—we may have a language that allows us to speak of motion in a completely unambiguous way.

PS: I don't think "string theory" will get us anywhere. To me it smacks of the Epicycles of Ptolemy, an elegant but effectively useless complication to paper over problems, but not solve them.

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