Integral calculus will solve the paradox only on the assumption that space is continuous. If space is discrete, solution lies in probability nature of quantum mechanics. The arrow paradox, meanwhile, is based on incorrect assumption: uniform motion is normal and selfsustainable and doesn't need to be explained (as proved by Newton). Also note that mentioning Leibniz without mentioning Newton might have some significance, as both are claiming to develop integral calculus. -- Hkmaly (talk) 09:23, 28 December 2012 (UTC) Actually Newton discovered (devised the mathematical method of) fluxions which is similar, but not as elegant, as calculus. He got miffed (and the British science establishment on his behalf) that Calculus was a rip-off. Newton did not publicise his fluxions as he firstly thought that they were a 'fix' that having found a solution needed to be shown by Euclidean Geometry. Secondly it have him an analytical edge over his contemporaries that he did not want to give up. Interestingly it can be shown that Pythagoras used an integration technique to calculate his formulas for the circle and sphere families, but worked it out the hard way! --126.96.36.199 17:55, 28 December 2012 (UTC)
There is a simple observation that solves the 'paradox'. Yes, the number of steps become infinite, but the span of each successive step, and thus the time required to traverse it becomes infinitesimally small - REGARDLESS of whether Newton's first law holds (provided negative acceleration - if any, is finite, and provided the assumption of nature being continuous holds). Leibniz and Newton's independent discoveries of calculus (the concept of limits actually), and this branch of mathematics' ability to deal with infinites, resolves the paradox, because an infinite sum of infinitesimals (in some cases... Zeno's geometric progression being one of them) can be shown to converge to a finite number.
While it is true that calculus remains applicable only if space and time...actually, it's post annus mirabilis, so let me reframe that. While it is true that calculus remains applicable only if the domain is continuous, the paradox may be solved to a fair degree even if spacetime is discrete. Why? Because if spacetime was discrete (ref. Planck Length. It's fascinating! :D ), successive division of the remaining length to be traversed would bring you down to the fundamental quantum at some point. Now, unless you violate Newton's law and suddenly bring the moving object to a standstill, it will continue moving. But the only way it can do that is by moving atleast by one fundamental unit (which we'll assume is the Planck Length), which is the same as the distance that it has just translated! [Pedantically, one doesn't even have to wait until the step size reaches such scales. We have this contradiction even if the macroscopic distance is a prime multiple of the Planck Length - but that's a trivial consideration, because one may just as well expand Zeno's paradox to not just halves, but the whole set of fractions of inverted-natural numbers]. But you'll note I said discredited, instead of solved, because rejecting the notion of no-motion-possible would atleast require an acknowledgement of the validity of Newton's first law of motion. But if there's no motion possible, then Newton's laws would be meaningless too.
Maybe that's why the prosecution only called on Leibniz. If the lawyer (or Zeno) had gone on to (chronological aberrations notwithstanding) to discredit Newton, they might've been ostracised as a result of a vengeful Isaac's social engineering. But let's find a more charitable (to the Principia author) explanation. Newton was notoriously taciturn. He was a Member of Parliament for a good while, but his only recorded comments were to have the windows closed because there was a draught. (ROFLMAO) It's unlikely that he'd ever get talked into being a witness, or even a subject matter expert for a criminal trial.
Also, IMHO, the "approach the bench", "-but never reach it!" might be a reference to the fact that the lawyer thinks he will probably never become a judge, and will be ignored, as punishment for his impudence. Leibniz and Newton were the brightest of the luminaries, and their scientific propositions may have been considered as sanctimonious as judgments delivered by the highest court of law. But in contemporary or even historic scientific circles, I guess Zeno was never accorded any more respect, than that set aside for 'a simple αστός'. Add to that the almost truant nature of his proposition, and it's as likely that it'd rile up those in charge of barrister promotions, as a smart question riles up a tired teacher. :) 188.8.131.52 19:29, 28 December 2012 (UTC)
- Planck length. Ref done.
- Note that Newton laws doesn't really work on Planck lengths. See Heisenberg principle. Actually proving that macroscopic object move on the time and distances close to Planck would probably require more computing power that we have. -- Hkmaly (talk) 13:02, 31 December 2012 (UTC)