Monday, July 2, 2012

How Peircean Continuity Defeats Zeno’s Paradox

Zeno’s paradoxes argue that motion is impossible. For example, if an arrow is to move through space, and space is a (discrete) continuum, then for the arrow to move through any interval it must move through an infinite number of discrete points. However, if the arrow travels no distance during any instantaneous moment, and there are an infinite number of such moments, then how is the arrow to pass through any of them? That is, to move, it must somehow escape the instantaneous moment, but if space is infinitely divisible per continuity, then any such escape will be met with another moment ad infinitum. The key to this argument is that if an arrow does not move in any instant, but moves only through a succession of instants, then it cannot move at all since we can always find another instant that it must pass through in order to move to the next.

Peirce’s simple solution, which I follow, is that there is no such thing as an discrete instant of time. If there is an instant, it must maintain a real relation to its past and future. I have argued elsewhere about what these relations must include if we are speaking of a thing existing in time: e.g., about only the past being fully actual and determinate, etc. Against what I propose, if one insists that instants are discrete or fully “present,” then defeating Zeno’s paradox in a satisfactory manner will become difficult if not impossible.

Prima facie, it seems that perdurance theories can handily defeat Zeno’s paradox, though I would question the stipulations required, but I have difficulty imagining how endurance theories do so. They conceive instants as fully actual; e.g., the present is all that exists.

A last note. The definition of “continuity” that I see people use is not the mathematical one that Peirce, Whitehead, or myself would invoke. When people use the word in my experience, they almost always mean “discrete continuum of absolute points.” The mathematical conception invokes a limit-concept that cannot be absolutized or made fully actual. (For mathematicians: the part of the definition that invokes an epsilon-neighborhood is what I am talking about as being unable to be absolutized or made fully actual.)


  1. Of math I know nothing it's not in my makeup but I fully agree with what I understand of your your POV despite that. And with immanent transcendence which I believe Peirce makes possible.

  2. Stephen,

    It is good of you to visit; I see you on the Peirce-L a lot.

    Peirce has a modern mathematical conception of continuity, and he gives a simple definition in his manuscripts that I link elsewhere on the site. However, since then, the contemporary notion is far more precise and invokes "episolon neighborhoods," which is another way of saying that a point is not an isle unto itself, but always a neighborhood of points.

    As for immanent transcendence, I tend to think it like this: the immanent is the now and the actual that time transcends. Where does it lead? Another immanence.

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