# 1153: Proof

 Proof Title text: The prosecution calls Gottfried Leibniz.

## Explanation

Zeno of Elea was an ancient Greek philosopher who devised several apparent paradoxes of motion called Zeno's paradoxes. Here are the two relevant to the comic:

Arrow paradox: At any instant in time, an arrow suspended in mid-air is no different from an arrow in motion. How, then, can motion occur? (Answer: calculus [all objects have a velocity].) The lawyer presumably intends to use this argument to prove that his client could not have used the arrow to commit murder. Another possibility was that it is impossible to hit a person in motion.
Dichotomy paradox: Suppose I need to go from point A to point B. First I must walk halfway there: half of the distance between A and B. Then I must walk half the remaining distance, which would bring me to three-quarters of the original distance; then I must again walk half the now-remaining distance to reach a point seven-eighths of the way from point A, and so on. Because I would have to take an infinite number of non-zero steps, I will never reach point B. By the same argument, the lawyer in the comic can get closer and closer to the judge's table, but never reach it.

There are two possible law vs math/logic puns in the comic, on the words "approach" and "proof." "Approach the bench" is a legal term meaning to have a private conversation with the judge; approach in calculus means an infinite process where a function value gets closer and closer to a limit value that it never actually reaches, reminiscent of Zeno's paradoxes. "Proof" is also ambiguous, with a different meaning in formal mathematics than in jurisprudence. See Proof (truth) and Mathematical Proof, for example.

Gottfried Leibniz is the co-inventor of calculus (along with Isaac Newton; see 626: Newton and Leibniz). If Leibniz were to testify in this imaginary trial, he might argue that calculus invalidates Zeno's paradoxes, because the moving arrow has a different velocity than a stationary one (the function describing the motion has a nonzero derivative at the point), and the infinite series in the dichotomy paradox has a finite sum. Both Zeno and calculus assume a continuous, infinitely divisible, ideal spacetime (as does quantum mechanics); a different solution would be available if spacetime turns out to be discrete. However, Zeno is arguably not concerned with actually calculating the correct answer. In the real world, Zeno can be trivially disproven simply by moving and reaching a desired target (it is said that Diogenes the Cynic reacted to the paradox by wordlessly walking to a destination, to demonstrate his contempt for it). It remains a question of debate whether a mathematical approach addresses the central points in Zeno's arguments.

## Transcript

Zeno: My client couldn't have killed anyone with this arrow, and I can prove it!
Judge: I'd like to examine your proof, Zeno. You may approach the bench.
Zeno: —But never reach it!

# Discussion

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)

Quantum mechanics is based on differential equations, and also requires that space not be discrete. 199.27.130.192 (talk) (please sign your comments with ~~~~)

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! --90.197.0.66 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. :) 115.242.227.17 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)

The Quantum Zeno effect seems to be a modern-physics reincarnation of the arrow paradox. (Doesn't it feel like every time you think that you have something fiqured out, some quantum phenomenon will suddenly pop up and make matters more complicated than they ever were?) – St.nerol (talk) 00:52, 15 January 2013 (UTC)

The Quantum Zeno Effect has nothing to do with Zeno at all. The authors just thought it sounded more profession than "A Watched Particle Never Boils". The Quantum Zeno Effect is not a paradox at all, but rather just the mathematical implications of quantum entanglement. -- 206.181.86.98 (talk) (please sign your comments with ~~~~)

The explanation previously said that quantum mechanics (in contrast to Zeno and to calculus) says that spacetime is really discrete. It does not. In fact, it inherently assumes that spacetime is continuous. The original formulation of quantum theory can be redone with discrete spacetime, but that breaks special relativity, and cannot be extended to QM. There may be ways around that within QFT (doubly special relativity, continuously variable fields over discrete space, ...), but it's unlikely any of them can be made consistent with our universe. Loop quantum gravity might work, but it still has continuous spacetime, it just means (oversimplifying) that at a small enough scale there's more noise than signal in any measure of distance. 199.27.130.180 14:17, 18 September 2015 (UTC)