Motion, Time and Space through the Ages
" time is not composed of indivisible moments"
(Aristotle)
The first time! The first time
makes even trifles seem sublime.
It doesn't last,
Just when it happens
it's in the past.
(Henrik Wergeland)
The Greeks
Thales from Miletus was the first, and the greatest of the old sages. He was a statesman, a scholar, a philosopher, an astronomer and a mathematician. He discovered theorems in geometry that we still teach our students in high school 2500 years later,
His greatest gift to posterity was to separate reason from mythology. Reason, he said, was the only way to understand the mysteries of Nature. No longer would a
deus ex machina but in and change the course of the play.
His successors followed his script. There were a few relapses; Parmenides took a trip to the Underworld to find Truth but his mentor there did not interfere with the happenings up above. And Aristoteles, although he denied the possibility of divine interference, had to invent a
Prime Mover to explain his understanding of motion; but this mover was not an Olympian God. Rather, it was something undefined, somewhere in the universe. In our age, Newton considered the possibility of divine intervention to keep the Solar System stable. It fell to Laplace to emulate Thales and declare it was an hypothesis he did not need.
The Sages did not march in lockstep. Parmenides saw the World as one whole, and he denied the existence of the Void. Democritus said the world was a collection of tiny particles, separated by a void. Heraclitus said that all was motion, Parmenides said that motion was an illusion and did not exist. (Diogenes did not say anything, he got up and started walking.)
It fell to Aristotle to sort, digest and present to the world the sum total of Greek philosophy. Unlike his predecessors he could write plain English; the sayings of the others are often Greek to me. Like Thales, he was a polymath but unlike Thales he was not a mathematician. His beef with the Pythagoreans, the Italians he called them, was not about irrational numbers but about cosmology.
And unlike the others, whose sayings have reached us by word of mouth or in fractured copies of their texts, a large portion of the writings of Aristotle is intact. In spite of his errors, he is still relevant and revered.
Parmenides
CONCERNING TRUTH
Come now I will tell thee-and do thou hear my word and heed it-what are the only ways of enquiry that lead to knowledge. The one way, assuming that being is and that it is impossible for it not to be, is the trustworthy path, for truth attends it. The other, that not-being is and that it necessarily is, I call a wholly incredible course, since thou canst not recognise not-being (for this is impossible), nor couldst thou speak of it, for thought and being are the same thing.
In pedestrian terms: The World is one, it has always been and will always be. It cannot have been created from nothing and cannot dissolve into nothing. Nothing comes from nothing. The void i
s not; therefore it does not exist.
Conservation laws and the first law of thermodynamics have deep roots.
Since there is no void, what exists must be infinitely divisible. In rebuttal, Leucippus and his student Democritus developed the atomic theory. The world is built from small indivisible particles. Different forms arise from arranging and rearranging these atoms.
Aristotle sided with Parmenides and discarded the atomic theory; it was not revived until our time.
Based on his concept of one, immutable world, Parmenides concluded that motion did not exist; it was an illusion. This argument was expressed with more precision by his student
Zeno of Elea in his paradoxes. Regarding motion, the most succinct is the Arrow Paradox:
“ If everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless. ”
It is a statement that is easy to dismiss, - and miss the point.
Zeno’s statement is wrong, yet profound. It touches the core of logic, of mathematics. Dismissing it proves nothing, teaches you nothing.
Aristotle
Although Aristotle sided with Parmenides on many issues, he did not give him a free pass. In particular, he disagreed with the notion that
motion is an illusion:
Nature has been defined as a 'principle of motion and change', and it is the subject of our inquiry. We must therefore see that we understand the meaning of 'motion'; for if it were unknown, the meaning of 'nature' too would be unknown.
He thus sets himself two tasks: to find ‘the meaning of motion’, and then prove that Zeno’s argument is false:
When we have determined the nature of motion, our next task will be to attack in the same way the terms which are involved in it. Now motion is supposed to belong to the class of things which are continuous; and the infinite presents itself first in the continuous-that is how it comes about that 'infinite' is often used in definitions of the continuous ('what is infinitely divisible is continuous'). Besides these, place, void, and time are thought to be necessary conditions of motion.
While the converse is true, his statement '
what is infinitely divisible is continuous' is false. Rational numbers are infinitely divisible but not continuous. Fortunately, he has a better definition of continuum:
Partitions of a continuum have common boundaries. If you want a specific example: on the real number line, zero is the common boundary of positive and negative numbers.
He fails miserably in the task of defining motion, He denies the possibility of the principle we call inertia, and says:
Everything that is in motion must be moved by something.
Motion was of two kinds, natural and violent. Natural motion has two forms: linear and circular. Linear motion is always vertical.
It is frustrating reading. You know he is wrong, and yet, if you play the ‘if, if’ game the next paragraph could be Newton’s first law of motion. If he had accepted inertia, and if he had considered the straight line as the natural motion, read the paragraph but replace ‘ be moved’ with ‘move’:
Further, no one could say why a thing once set in motion should stop anywhere; for why should it stop here rather than here? So that a thing will either be at rest or must be moved ad infinitum, unless something more powerful get in its way.
In order to counter Zeno’s argument, he had to stipulate that bodies, space (or place) and time were continuous. It forced him to reject the theories of Democritus:
… it is impossible for anything continuous to be composed of indivisible parts … a line cannot be composed of points, the line being continuous and the point indivisible.
… It is clear, then, from these considerations that there is no separate void.
The question of the nature of time gave him more concern:
… one suspect[s] that [time] either does not exist at all or barely, and in an obscure way. One part of it has been and is not, while the other is going to be and is not yet. Yet time-both infinite time and any time you like to take-is made up of these. One would naturally suppose that what is made up of things which do not exist could have no share in reality.
… But neither does time exist without change
He reached no firm conclusion; “it is a difficult question”, he said. But he continues to treat time as if it were real and continuous; “you cannot stop time”, he said, and “now” is not an interval but the common boundary of “past” and “future”.
For what is 'now' is not a part: a part is a measure of the whole, which must be made up of parts. Time, on the other hand, is not held to be made up of 'nows'. Zeno's reasoning, however, is fallacious, when he says that if everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless. This is false, for time is not composed of indivisible moments any more than any other magnitude is composed of indivisibles.
Hence Zeno's argument makes a false assumption in asserting that it is impossible for a thing to pass over or severally to come in contact with infinite things in a finite time. For there are two senses in which length and time and generally anything continuous are called 'infinite': they are called so either in respect of divisibility or in respect of their extremities. So while a thing in a finite time cannot come in contact with things quantitatively infinite, it can come in contact with things infinite in respect of divisibility: for in this sense the time itself is also infinite: and so we find that the time occupied by the passage over the infinite is not a finite but an infinite time, and the contact with the infinites is made by means of moments not finite but infinite in number.
Thus spake Aristotle, and numerous philosophers through the ages have spent countless hours and barrels of ink discussing the existence of, and he nature of, time and void.
As a pragmatic approach, it is helpful to treat both as if they were real:
It is the mark of an educated mind to be able to entertain a thought without accepting it.
Aristotle
Newton
Newton had his own idea about the nature of motion. To calculate the speed at any given time he built on Aristotle’s concept of infinites. He invented some entities he called fluxions existing in the shadowland between infinites (infinitesimals) and zero.
Fluxions were real in the sense that their ratios were real; but their powers did not exist. (He knew it was fuzzy logic, and never based a proof on this reasoning.) Bishop Berkeley had a field day ridiculing it.
The problem was that Newton’s infinitesimal calculus worked. Later generations of mathematicians managed to put it on a solid logical footing. Fluxions were replaced by differentials, and their ratios by derivatives, i.e. ratios between limits of functions.
For astronomers and engineers, then, Zeno’s problem has been resolved. We can, with a straight face, talk about “instantaneous velocity” and get on with our job.
Philosophers have a harder time. Motion has two components, speed and direction. Calculus gives us a number, “speed” and a direction, a tangent. But the tangent is a straight line, and you cannot construct a non-linear trajectory as a succession of straight lines. “Instantaneous velocity”, then, is a potential, not a reality. It is the velocity a body would have if all constraints were suddenly removed.
Quantum theory offers a surprising solution. It does not tell us what motion is but what we can know about it. Change the Zeno paradox to: “when a body is in a place we cannot know how it moves, if we know how it moves we cannot know where it is”.
Yet, if we are willing to compromise, we can know; sort of. It is a compromise, not forced upon us by the inadequacy of our instruments, but as a law of nature.
Heisenberg’s
uncertainty principle in it original form reads:
The product of the fudge factors for position and momentum is larger than Planck’s constant.
How can we function in such an uncertain world? Because Planck’s constant is so small, that’s how. If it were big enough, photons would hit us with the impact of bullets. On a less lethal level all would be chaos. The hunter would come empty-handed home from the hill. He saw a buck running across the field but did not know where to aim. In the forest, trees act as diffraction grids, and he saw numerous deer but did not know which one was the target.
We should all give thanks to Planck for making his constant so small.
