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It seems to me, perhaps naïvely, that Aristotle resolved Zenos' famous paradoxes well, when he said that,

Time is not composed of indivisible nows any more than any other magnitude is composed of indivisibles,

and that Aquinas clarified the matter for the (relatively) modern reader when he wrote

Instants are not parts of time, for time is not made up of instants any more than a magnitude is made of points, as we have already proved. Hence it does not follow that a thing is not in motion in a given time, just because it is not in motion in any instant of that time.

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For me, any further attempts at explanation and resolution, such as Russell's, (bar rather tenuous appeals to connections with quantum physics) simply reiterate, and rather labour Aristotle's initial erudite explanation.

What serious arguments exist that counter Aristotle's objection, if any, and why is Aristotle's objection not considered a resolution to this paradox?

  • 1
    I saw this question after I'd asked this one; see the suggestion where I say that Aristotle is correct. – Mozibur Ullah Aug 14 '15 at 19:07
  • @MoziburUllah some nice points made here, whichg appear to agree with Hurkyl's observation below in the comments. Do you not think your question would serve better as an answer in this thread? – martin Aug 14 '15 at 21:41
  • well yes, but I'm more interested in that question in what Socrates means when Zeno is denying plurality, the contrapositive of Parmenidian Unity. – Mozibur Ullah Aug 15 '15 at 15:23
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    I've added it below. – Mozibur Ullah Aug 15 '15 at 15:33
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    Perhaps worthy of note on this question, is that Newton's calculus doesn't require that the points exist. They simply give us the limit of invisibility, an ideal which continuous motion approaches if we are to project reality down to states at given moments. Planck length and Planck time tell us such ideal is not true to reality, but sufficiently close at least at the human scale. Furthermore, p-adic arithmetic has its own calculus less dependent on the distancelessness between transcendental numbers and repeatedly proves itself a superior measuring system though less intuitive initially. – samerivertwice Jul 29 '17 at 1:36
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Aristotle's solution was largely accepted until the end of 19th century when Cantor and Dedekind formalized the notion of continuum in terms of set theory. Under their interpretation time is in fact composed of indivisible nows, just like a line is composed of points, and any other magnitude is composed of indivisible elements as well. It does not mean that Aristotle is "wrong", but it does mean that his interpretation of time/continuum is at odds with modern mathematics, and its usefulness is thus severely diminished. Since mathematics is used to describe time and motion in physical theories one would prefer a solution that accomodates its premises.

Of course, purely mathematical problems of motion are resolved by calculus, but from a philosophical point of view the nature of instantaneous velocity, a shadow of motion where there can be no motion, is puzzling. And the fact that even classical physical description requires not the sensible physical space, but the hidden configuration space with twice as many dimensions (to account for velocities and make Zeno's arrow move) is even more puzzling. When it comes to time, position, and velocity things get even more puzzling in quantum theory.

Thinking back, one realizes that Aristotle's solution was always incomplete: he says what time is not, which is enough to dismiss the paradox, but not what it is, which is needed to explain the puzzles of motion that give rise to it. Under different assumptions, physical theories give much more pointed and detailed answers. It is possible that some future theory would vindicate Aristotle somehow, but its description of time would have to be far more elaborate than his.

  • a very interesting Historical viewpoint. Thank you :) – martin Aug 11 '15 at 23:33
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    I think the description of modern mathematics is a bit misleading. The line isn't composed just of points, because how those points are related (e.g. a topology on the line) is an essential ingredient. If you just take the points without the 'glue', it doesn't resemble a continuum at all. – Hurkyl Aug 12 '15 at 7:02
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    An I'd add that the more modern approach is to see 'infinitesimals' as a consistent class of numbers, with strange arithmetic rules governing them. Infinitesimals are not zero, and they are not indivisible. en.wikipedia.org/wiki/Non-standard_analysis – mousomer Aug 12 '15 at 17:52
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    @Hurkyl I agree. The difficulty seems to be in relating the glue to the instants, non-standard analysis sort of does it with infinitesimal halos. Interestingly, Aristotle only denies Zeno's second premise, dissociation into instants, but not the first, equivalence of arrows at rest and in flight within an instant. But both versions of analysis do the opposite. I think elementary topoi capture Aristotle's "holistic" view better than sets with glue, points there are external morphisms, not constituents, and they are known to model intuitionistic "fluid" continuum. – Conifold Aug 13 '15 at 23:35
  • Theres also the notion of diffeology which in a sense is just a modification of the notion of topology; I'd have to agree with Hurkyl that Aristotle was on the ball; were the continuum modelled exactly by a set of points there would be no need for a topology - but in fact we find the notion helpful. – Mozibur Ullah Aug 14 '15 at 19:13
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How does it actually address the problem? There is no contention in the question itself that time is composed of points in any way. Each of these points in time at which we sample the distances does, in fact, exist, and we can take each of these ratios of distances. So this argument misses the point entirely.

In fact, in Nonstandard Analysis we do exactly that, simplifying Calculus by constructing infinitessimal elements and modeling a continuum that is in fact made up of points that do in fact add up to the magnitude, so we know that this way of looking at geometry is not really wrong in an absolute sense.

The problem is dividing zero by zero and getting a definite non-zero and non-infinite number. This is somehow disquieting enough that the sham solution that does not engage the argument at all was more acceptable to ancient and medieval thought.

But, since people like Archimedes of Perga, Albertus Magnus and Newton, we are used to the idea that the continuum just naturally heals over division by zero and allows limits of ratios to be well-defined.

It is still a place where one can easily get trapped in nonsense, if one does not handle it very carefully, so, although we accept a solution for it, it deserves to be marked out as problematic territory, and calling it a paradox is not out of order.

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    I am inclined to disagree, but that is a purely personal view. I welcome you answer, though I don't think that division by zero is the core issue here. On the contrary, I believe Zeno's argument is calculus in reverse (ie, attempting to approach a "limit" from zero, as opposed to from a convergence, but +1 anyway for enriching the argument :) – martin Aug 11 '15 at 23:37
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    Zeno does need time to be composed of moments for his argument to work:"if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless", without moments there is no part of time where Zeno can assert that the arrow is at rest to begin with. So if we accept that time is not composed of moments and line of points Zeno's reasoning fails. This was a common view until the late 19th century, Euclid's word for "point" even translates as "mark", something placed on the line externally, set theoretic view is very recent. – Conifold Aug 12 '15 at 0:04
  • I think NSA is better described as trying to get all the advantages of the infinite divisibility point of view even when you limit yourself to only dividing the line into (hyper)finitely many line segments. – Hurkyl Aug 12 '15 at 6:49
  • @Conifold Newton would not accept that time is broken down into moments. Yet the point of view of Newtonian physics freezes time as needed, folding the continuous aspects of motion into a separate register, as derivatives. So to claim you need some artifice to see the arrow as motionless, basically says classical physics is cheating. Points in time exist whether or not time is by nature a composite of points. Inertia and derivatives, which break down the whole Aristotelian notion of the distinction between actual and potential attributes, are what first really address this issue seriously. – jobermark Aug 12 '15 at 15:29
  • @Hurkyl The point is that it demonstrates that the two models do fit together into something reasonable, and that fully rejecting either of them as nonsense (as Aristotle does, and Analysis insisted upon doing, later) is out of order. – jobermark Aug 12 '15 at 15:37
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Aristotle does seem to reason about time with a mathematical model (geometrical magnitude) different from the one usually used today (real number). This is not at odds with modern mathematics as much as with the way time is modeled in modern physics.

The real numbers (and measure theory) lead to paradoxes of their own such as Banach-Tarski (not directly relevant to linear magnitudes, but famous). The constraints in how classical geometry was practiced did not support divison of a magnitude into indivisibles, but only into smaller magnitudes. This put up a kind of barrier to more modern resolutions of the paradox, a barrier that was broken down by calculus. But while these constraints enabled paradoxes like Zeno's, they also suppressed the paradoxes inherent in measure theory.

Aristotle & Aquinas' resolution applies to the geometrical magnitude model of time (in which a magnitude cannot be divided into indivisibles) but not to the modern real-number/calculus model of time (in which a magnitude can be divided into indivisibles, and recovered via integration or measure). However, a newer mathematical model may resolve some old paradoxes just as it introduces new ones.

  • A modern reflection of Aristotle's view is the fact that the Lebesgue measure is only countably additive, so while continuum can be assembled from points as a set it can not be as a measurable set. But this does not help with resolving the puzzles of motion. I think because both Aristotle's and calculus solutions are too formal, they do not get to the root of Zeno's reasoning. Psychologists still puzzle over how our mind establishes temporal succession of events, physicists speculate that time may be emergent, the puzzle will remain until we understand time, subjective and objective. – Conifold Aug 12 '15 at 3:10
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    @Conifold: Measure theory is too weak to express the idea of a continuum; e.g. there's no intrinsic way to distinguish between the interval [0,1] and the union of the intervals [0,0.5) and [1.5,2] if both are equipped with the usual measure. In fact, every separable nonatomic normalized measure algebra is isomorphic to the unit interval equipped with the Lebesgue measure. (although, as you point out, it is enough to tell the difference between 'isolated' points and a continuum) – Hurkyl Aug 12 '15 at 8:06
4

If I wanted to disagree with Aristotle, I would do it on the following grounds.

In modern mathematics, we know how to attach a lot more data to individual points; e.g.

  • In physics, at any instant, particles have momentum. Physical laws relate momentum to how the position changes
  • Similarly, in differential geometry, each point has a cotangent space: data from the cotangent space tells us the rate at which a function is varying
  • More generally, each point has a stalk: the germ of a function at a point is enough information to tell us everything about the function in an entire open neighborhood of the point. (although not how big the neighborhood is)

and we've well-studied how to assemble data at individual points into a continuous whole.

We can insist that at each instant - each "indivisible now" - we're not limited to knowing precisely where something is, but instead its entire germ of motion. And consequently, we are able to study motion as a continuum of "indivisible nows".


That said, I think this objection is more of a technicality than disagreeing with the spirit of Archimedes position. My intuition is that the ideas above (and more) are all fleshing out a "fuzzy"* notion of point that somehow has an extension despite** just being a point. And in some sense these fuzzy points still manage to be "divisible"; e.g. by passing from the stalk to the fiber. (i.e. forgetting the "germ" of motion and only remembering the position at that instant).

A more charitable interpretation is that this fuzzy notion of point still satisfies the spirit of Aristotle's position.

*: No relation to fuzzy logic

**: Some even manage it more transparently, such as how nonstandard analysis creates about a standard point an entire halo of "infinitesimally nearby" nonstandard points.


Also, to expand upon something I mentioned in another comment, I would object to the objection above anyways, on the grounds that it still needs more than "points and extra data" -- i.e. there is still additional information (such as a topology) that encodes how the points themselves relate to each other. This additional information is at least as important (and arguably more important) than the points themselves.

2

The following is actually the content of a question asked here; but which has a bearing on the question above:

Zeno is well-known as the storyteller of Achilles and the Tortoise and how the tortoise never catches Achilles; which is against our experience; the question of how to square these two notions generally falls to the theory of infinite series; and this is in fact only a formalisation of the following physical observation:

That the sequences of displacements that Achilles moves is an infinite series; that we know that the total sum of these displacements must add to a finite sum (since his and the tortoises path do eventually cross); the formalisation of this mathematically is technically called the monotone convergence theorem

However when we turn to Zenos appearance in Platos Parmenides we find Socrates saying that:

I see, Parmenides, that Zeno would like to be not only one with you in friendship but your second self in his writings too; he puts what you say in another way, and would fain make believe that he is telling us something which is new.

to which he elaborates

For you, in your poems, say The All is one, and of this you adduce excellent proofs; and he on the other hand says There is no many; and on behalf of this he offers overwhelming evidence. You affirm unity, he denies plurality. And so you deceive the world into believing that you are saying different things when really you are saying much the same. This is a strain of art beyond the reach of most of us.

Which hardly appears to be the content of the above argument; for where is plurality denied there - and which appears to be the heart of Zenos concerns according to Socrates.

note:

A possible suggestion is that both motion in terms of displacement and time are measured using the real line; and this concieved as a plurality of points doesn't allow for motion at all. For how can one move from one point to another? For between a point and another is a void.

This sounds un-natural and unintuitive; but consider the real line with whats called the discrete topology where:

the points form a discontinuous sequence, meaning they are isolated from each other in a certain sense

Visually, its as though we took a magnifying glass to the line and saw between the points a void (its of course a magnifying glass with an unaturally high magnification power and not found in any high street store); suggesitively its also called the totally disconnected line. Topology thus ties all the plurality of points together into a unity; it dispells the void; and allows for motion.

1

There is a difference between a point and an indivisible. I guess the important thing is to consider the distinction. The easy way out here is to consider a point as an object without magnitude and indivisible as an object with magnitude. If we assert that a line is made out of points, that will mean that a string of infinitely many zeros will add up to a unity, which is clearly incorrect. But with indivisibles as objects with magnitude as they were also used by Archimedes, then they will add up to a unity. This is a problem of Newton's time but it goes back to Archimedes to he never had to clear definition of limit and maybe he never even attempted to define it.

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