Zeno famously said an arrow cannot be in motion as it occupies some precise position at some precise time. And is thus at rest.

Modern physics resolves this by stating the arrow has momentum. It is its momentum that carries from one place to another. At any precise time it has a specific velocity. So it is not at rest.

To simplify our presentation let us reduce the arrow to a point, and suppose it to move in a straight line with no forces acting on it.

One then can identify the continua of time with the continua of the line.

So motion is time. And time is motion.

But it seems to me that the paradox of rest remains. Which in this pictures is this. That a continuum of only points is actually not a continuum. There is no cohesion.

Modern studies of the real line remove this paradox by stipulating that a topology is present and this binds together points. One of course then notices that the points are actually not neccessary, only the continuum itself, that is the topology.

That is pointless topology (pun presumably intended by the author of the coinage).

This then removes Zenos first prescription to be able to specify an exact moment in time. In this picture this cannot be done.

But if one wishes to retain points, one could try a different logic, say by saying the point is both a point and not a point - this resolves to false in classical logic, but if one drops the law of the excluded middle, then this is not so. Following this line of thought for intuitionistic logic, gives the notion of the rigid, infinitesimal line. Which is not a point - it is a line, but it is also a point - it is infinitesimal.

Does this work as a solution towards resolving Zenos paradox?

  • Seriously, we're going to re-litigate Zeno today? What's your definition of an infinitesimal? There are no infinitesimals in the real number system. Non-standard analysis is highly technical. And nobody really thinks there are infinitesimals in the physical world. Have you got a definition of infinitesimal other than that it's "a point and not a point?"
    – user4894
    Mar 17, 2014 at 0:23
  • @user4894: for Robinsons non-standard analysis, you can just take an axiomatic description of the non-standard reals plus the transfer principle. You don't need the nitty-gritty of model theory except to justify it, and don't we do the same for the reals? You don't start looking into set-theoretical foundations when you look at the calculus? Mar 17, 2014 at 1:17
  • I don't, but other people have "An infinitesimal space is supposed to be a space whose extension is infinitely small, yet not necessarily perfectly small". The point of putting it in the terms that I did, was to connect it with Zenos argument and Aristotles. Mar 17, 2014 at 1:21
  • yes, it's point's all the way down. Look @ Nassim Haramein - Crossing the Event Horizon for a more detailed reasoning behind this Mar 19, 2014 at 16:37
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    @user132181: Thanks - seriously :). Apostrophes are confusing, so I've followed the last dictum - 'when in doubt don't use an apostrophe'. I'll try to be more careful from now on. Mar 23, 2014 at 10:08

2 Answers 2


In Achilles and the Tortoise, when Achilles pursue the tortoise, he is not "counting" the point in space.

In the linear space continuum, points are not "isolated" entities : they are cuts. We have a "place before" (i.e. all numbers before SQRT(2)) the cut, and a "place after" the cut (all numbers after SQRT(2)) [this is Dedekind' analysis of the continuum : see The Continuum and the Infinitesimal in the 19th Century].

Achilles will overtake the tortoise simply running (for example) twice faster than the tortoise: we do not need to assume that he has to run faster and faster ("count the natural numbers by counting faster and faster and faster").

We need to use a device to measure the progress of the run: if we use the heartbeat, and assume that the tortoise starts an heartbeat before Achilles, after the first interval in time dt, she [he, it ?] will have traversed a certain amount of space ds. With the second heartbeat Achilles will start, and we assume that he runs twice as faster as the tortoise. After the second heartbeat (i.e.after 2 x dt), both Achilles and the tortoise will have traversed a space equal to 2 x ds. After the third heartbeat, Achilles will have definitively outrun the tortoise (he have traversed a 4 x ds, while the tortoise has only traversed 3 x ds): this has required a finite amount of time (3 x dt) and without the need of an "unlimited increasing" speed.

If we model the race with the mathematical continuum we must not make the mistake of describing the progress of the runner as made of successive move from on point to "the next": in the real number line, a point has no "next".

We may say that Achilles will win because he is not counting the point in space; he is "traversing" intervals in time.

My personal "feeling" with this paradox is the same as the proposed solution in Wiki Zeno's paradoxes :

Pat Corvini offers a solution to the paradox of Achilles and the tortoise by first distinguishing the physical world from the abstract mathematics used to describe it. She claims the paradox arises from a subtle but fatal switch between the physical and abstract. Zeno's syllogism is as follows:

P1: Achilles must first traverse an infinite number of divisions in order to reach the tortoise;

P2: it is impossible for Achilles to traverse an infinite number of divisions;

C: therefore, Achilles can never surpass the tortoise.

Corvini shows that P1 is a mathematical abstraction which cannot be applied directly to P2 which is a statement regarding the physical world. The physical world requires a resolution amount used to distinguish distance while mathematics can use any resolution.

See also Zeno's Paradox for further refences and Wesley Salmon, Zeno's Paradoxes, (2nd Ed - 2001).

  • Great answer! I deleted mine b/c yours is better.
    – user5172
    Mar 17, 2014 at 16:59
  • Dedekind cuts is just one way to construct the real line by way of the rationals. You could use Cauchy completion. Completion is about points - no points in a precisely defined way are missing. Mar 20, 2014 at 1:38
  • If the topology on the line is discrete, then there is no sensible idea of moving from one point to another - recall, that a discrete space is extremally disconnected; what I'm trying to point out in the question, or at least one part of it is this, is that topology or cohesion matters - which isn't seen in the standard solution. Mar 20, 2014 at 1:41
  • I'm also, not talking about points as in the classical definition; of having no extension whatsoever; but as an infinitesimal extension: a point that is not a classical point, but in a sense is just pure extension. Mar 20, 2014 at 1:46
  • Although isn't the particular problem of Zeno that I was discussing; it can be solved quite straight-forwardly using the synthetic plane geometry of Euclid - two straight lines that are not parallel must meet at a point, using this as the model for the problem gives us the time t when Achilles run past the tortoise; and redoing it in the analytic geometry of Descarte gives us the same answer - so long as we use the classical topology of the real line; using the discrete topology means there is no proof that the lines intersect, as the topology isn't connected. Mar 20, 2014 at 2:05

The suggested solution is very similar to Whitehead's argument presented in Part I of Process and Reality and presupposed in the Theory of Extension in Part IV. If you recall from Aims of Education he holds that a line is a "moving point" while a "plane is a moving line." Along with von Neumann's work on mathematical rings, Whitehead relies on a continuous, point-free projective geometry of non-finite dimension. Whitehead's "extensa" or extensive structures are logically and metaphysically prior to time and space as topological relational systems (mathematically speaking, they may be seen as spatial, but this pertains to "spatial reasoning"). This allows for the openness in which "universal relativity" (not to be confused with Einstein's as only a single application) and "atomicity" are compatible. He writes:

The creatures are atomic [i.e., undivided]. In the present cosmic epoch there is a creation of continuity. Perhaps such creation is an ultimate metaphysical truth holding of all cosmic epochs, but this does not seem to be a necessary conclusion. The more likely opinion is that extensive continuity is a special condition arising from the society of creatures which constitute our immediate epoch (PR, 35-6).

This compatibility allows for the "becoming of continuity" and not a "continuity of becoming" as is assumed in Zeno's paradox. Under the conditions of our epoch

The extensive continuity of the physical universe has usually been construed to mean that there is a continuity of becoming. But if we admit that ‘something becomes,’ [in the physical sense,] it is easy, by employing Zeno’s method, to prove that there can be no continuity of becoming. [Where we are speaking exclusively of the way that actual entities, considered physically rather than logically, divide the extensive continuum there is a becoming of continuity, but no continuity of becoming. [Under those assumptions, the actual occasions are the creatures which become, and [to that same extent, in the physical sense only] they constitute a continuously extensive world. In other words, [physically speaking,] extensiveness becomes, but ‘becoming’ is not itself extensive (PR, 35).

Therefore, Zeno's paradox is a "mathematical fallacy" as Whitehead explains:

In his ‘Achilles and the Tortoise’ Zeno produces an invalid argument depending on ignorance of the theory of infinite convergent numerical series. Eliminating the irrelevant details of the race and of motion—details which have endeared the paradox to the literature of all ages—consider the first half-second as one act of becoming, the next quarter-second as another such act, the next eighth-second as yet another, and so on in-definitely. Zeno then illegitimately assumes this infinite series of acts of becoming can never be exhausted. But there is no need to assume that an infinite series of acts of becoming, with a first act, and each act with an immediate successor,† is inexhaustible in the process of becoming. Simple arithmetic assures us that the series just indicated will be exhausted in the period of one second. The way is then open for the intervention of a new act of becoming which lies beyond the whole series. Thus this paradox of Zeno is based upon a mathematical fallacy. The modification of the ‘Arrow’ paradox, stated above, brings out the principle that every act of becoming must have an immediate successor, if we admit that something becomes. For otherwise we cannot point out what creature becomes as we enter upon the second in question. But we cannot, in the absence of some additional premise, infer that every act of becoming must have had an immediate predecessor.The conclusion is that in every act of becoming there is the becoming of something with temporal extension; but that the act itself is not extensive, in the sense that it is divisible into earlier and later acts of becoming which correspond to the extensive divisibility of what has become (PR, 69, emphasis added).

  • Whitehead speaks as though continuity of becoming is natural & obvios, and Zeno shows that it is not. It would be worth spelling this out in more detail. Mar 17, 2014 at 1:36
  • I revised my answer and added the full quote from Whitehead on Zeno to make it explicit. I think it is a rather cool argument that I hadn't heard until about three years ago. Something to consider at least. Thanks for another great question and I hope the clarification to what Whitehead thought about this as a mathematician and then metaphysician. Mar 17, 2014 at 1:42
  • @Mozibur, he specifically says that there is NO continuity of becoming. Therefore, it's not "natural & obvious" in the way there is a becoming of continuity. He follows Locke's notion of "perpetual perishing" to describe the processes of transition and concrescence that makes reality "incurably atomic." This is why he is famous for the explanation that we experience "drops of experience." Mar 17, 2014 at 1:45
  • I was focusing on this part of his exposition 'The extensive continuity of the physical universe has usually been construed to mean that there is a continuity of becoming.' Mar 17, 2014 at 1:47
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    @Jackson: ok, that fits in with Russells dismissal of the Philosophy of Antiquity, and Wittgenstein was Russells student...possibly further impoverishment there. The Tractatus, his early philosophy makes no mention of any antecedents by name, presumably, seeing itself as sui generis; possibly picking up Russells bad habits in relation with the Tradition; Whitehead understands that in mathematics that the aximatic is a contingent, whereas Russell & Wittgenstein take it as an End. Mar 17, 2014 at 2:38

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