Ever heard of Aleph-nought or the Hilbert’s Hotel and your inquisition got the better of you? Then you probably would know about infinity. How can you enclose infinity? And can the infinity find the ‘finitude’ somewhere? Isn’t it against the glory of infinity? Because doing so jokes with “infinity”. Have we been defining infinity the wrong way? Why did Newton bring fluxions? Of course you know the Archimedes’ Method of exhaustion and Euclid too proves six of his propositions in his Elements utilizing it.
Some mathematical problems may be extremely enigmatic and therefore are without a solution up to now, but one day someone may come up with a brilliant solution (that will bring all to a common page). Fermat’s last theorem went unsolved for three and a half centuries, when instanced. Andrew Wiles was then able to solve it circa 1994. Also we have the ‘continuum hypothesis’, a problem which cannot be solved with available methods that we today have. Equivalently, the age-old ‘trisection’ problem i.e. can we trisect a given angle by using just a ruler and compass (like Euclid, in his Elements)? The Hellenic people were very mystified by how to make such a trisection, and not surprisingly, in the nineteenth century it was proved impossible—yes! Impossible it was with merely a ruler and compass (Galois Theory). Now halt at Zeno’s Paradox!
Zeno of Elea, said to have been son of Teleutagorus, precursor of Socrates and a disciple of Parmenides. Zeno’s work could not survive and according to Proclus, his one book that he wrote before visiting Athens was having forty paradoxes about continuum. According to some narratives, he is considered to be the defender of his mentor, Paramenides, who said some counter-intuitive and tedious things. And in doing so, he concocted (rather devised) some paradoxes that stirred never ending debate and tussle.
The paradoxes that Zeno gave regarding the motion are more startling. Aristotle’s take on four of Zeno’s arguments; The Dichotomy, The Achilles, The Arrow, and The Stadium in Physics is worthy. For the dichotomy, with which I am to deal with, Aristotle put it into words as:
“There is no motion because that which is moved must arrive at the middle of its course before it arrives at the end.”
In order the traverse a line segment it is necessary to reach its midpoint. To do this one must reach the quarter, to do this one must reach the quarter’s half (=1/8) point and so on ad infinitum. Therefore the motion can never begin. The argument here is not answered by the well-known infinite sum:
(1/2) + (1/4) + (1/8) +… = 1
Zeno’s argument based upon two principles i.e. Infinite Divisibility Principle and the Infinite Sum Principle. In reviewing it using our mathematical approach, it can be said that he gave riveting reasoning for the first, but does not even mention the second. Utilizing them, conclusion comes that ‘every magnitude is infinitely large’.
This argument is valid, but ‘shaky’ for the Infinite Sum Principle is false. Infinite Sum Principle can be fixed by restricting it to infinite sets with smallest elements. The amended principle is true, and so the resulting argument’s premises would both be true. But this amended argument is invalid. For the revised principle requires that there be smallest parts, and the Infinite Divisibility Principle does not guarantee that there are such parts – it allows the parts to get smaller and smaller, ad infinitum. We can make Zeno’s argument valid, but then one of its premises is false. Or we can make both of its premises true, but then it is invalid. Either way, Zeno’s argument is wobbly and shaky.
Take an example of a marathon runner. If he were to run an infinite distance in constant time for each sub-interval, in that sense and case there would have been enough infinite time. It further unfurls that if finite distance is infinitely divisible, why should not the time be infinitely divisible (and this is discerned nowhere in paradox).
Was Zeno talking of physical infinity or some metaphysical infinity?
What if you keep dividing your paper sheet into half and half and half so on? The page remains a page and will never convert into some infinite sheet! Why not the infinite series’ that converge are perfectly valid explanation? Is there any inconsistency involved in it? Some say you are considering convergent sums and the finitude, so no paradox here. Some would say Zeno proved the motion was impossible. Some would arrive with allegation, ‘he was unaware of ‘theory of limits.’’ But some still discuss it regarding it a paradox and a balmy topic because Aristotle did not debunk it wholly (rather called them ‘fallacies’).Pragmatically speaking in a physicist’s tone, if this division process goes on until order of a quantum scale or ‘Planck’s distance’, it would not be viable to go on more because of the un-cogency of classical geometry.
Broadening our horizons and continuing proliferation, let us delve into Physics!
Using the Tenseless Theory of Time (TTT), it may seem quite un-instinctual, which says something like future is pre-determined and the past has not really volatilized out of existence. Time flow is felt by us and our forefathers had also been of same fact. The motion is continuous and sequential changes in position as time “passes by”, roots of paradox in review. Our 3-tense-pronged theory of time dazzles. This is what Einstein said. This is what special relativity to me. If populace sees that Betelgeuse died in some supernova explosion “today”, for those extraterrestrials close to Betelgeuse, such catastrophic event is buried 640 years in the past (being 640 light-years away from Earth)—The relativity of simultaneity! Zeno probably denied the motion continuity because of being from such School of Thought where ‘one universe—no change, no motion’ was indoctrinated.
As the QFT (Quantum Field Theory) has its terminologies of ‘unitarity’ and ‘locality’. Both of them negate or go at odds with ‘existent or identity law’ (i.e. an existent is always what it is). The ‘locality’ suffers the same trouble as the Zeno’s Paradox, but existence says that limits (or bounds) are defined and so nothing impossible in journeying from a point to other even the distance has become the Planck’s length or of its order (by dividing it continuously, what Zeno does). In the like manner, ‘unitarity’ comes face to face with ‘existence concept’ (it can be explained in quantum probabilistic lexicon).
Feynman lover like me would opine: “The ‘paradox’ is only a conflict between reality and your feeling of what reality ought to be.” The solutions to paradoxes lie in evincing the fallacy in either the principal assumption or the intermediary conclusions/lemmas. That is why everyone comes with own solutions.
Utterly discombobulated by the Paradox, readers are suggested to give a read to the book, ‘The Universal Book of Mathematics: From Abracadabra to Zeno’s Paradoxes’ by David Darling. He has touched newly born Astro-biology as well as the Astronomy, General Science, spaceflight with special and lucid expositions of Quantum conundrum with its metaphysical and philosophical implications. But the main point is, till now, and after 2500 years, there is no agreement that those Paradoxes are resolved in mathematics. We should now prepare ourselves for these contretemps between the mathematicians and the physicists because they would always be squabbling over it and these paradoxes now could only be used in jokes or the allusions to foregoing philosophers. So no verdicts here please but establishing a rapport with these furors is endorsed.