Zeno's Paradox: a technical note

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In the video Ian and Martin Conversations 02 (on the Philosophy and Critical Thinking course page), Martin makes reference to Zeno’s paradoxes of motion, and claims that to date, there is no agreed philosophical resolution to these paradoxes.

In essence, Zeno paradoxes hinge on the concept of continuity. But of course, Zeno did not have the concept of infinitesimally small, which is the informal concept used by Leibnitz and Newton in the development of calculus in the late 1600’s. Their calculus was a technique for treating continuous motion as being composed of an infinite number of infinitesimal steps. But the calculus of Newton and Leibnitz did not have (according to the standards of the early 20th century) a sufficiently rigorous concept of continuity. Indeed, in the 19th century, infinitesimals were eliminated from standard real analysis, as the use of infinitesimals in mathematical analysis was at the time regarded as “non-rigorous”. In particular, Weierstrass developed the epsilon-delta method in which there was no longer a requirement to conceive of a quantity as actually varying (Potter, 2009). And it was this epsilon-delta method that allowed mathematicians to treat continuity of the real number line WITHOUT reference to the nebulous concept of “tending to a limit”.

In the 1960s, Abraham Robinson and others developed a non-standard analysis approach in showing that there was a conception of continuity of the real numbers, in which there are infinitesimal quantities lying in the real number line.  

The crux of Zeno’s argument in the Achilles and the Tortoise Paradox is the assumption that a completed infinity of places is too many places for Achilles to go to in a finite time in order to catch the tortoise. However – as demonstrated by standard or non-standard 20th century analysis approaches, the measure (such as length) of a continuum is not a matter of adding up the measures of its points nor adding up the number of its points (Dowden, 2017). And hence the time taken by Achilles to catch the tortoise is a temporal interval, a linear continuum of instants. A finite time.

However, McLaughlin (1994) demonstrated how Zeno’s paradoxes could be treated using infinitesimals, and claimed his approach to the paradoxes was the only successful one. This claim has been challenged by other mathematicians. See for example the discussion outlined in Dainton (2010).

References

Dainton, B. (2010). Time and Space, Second Edition, McGill-Queens University Press: Ithaca.

McLaughlin, W. (1994). “Resolving Zeno’s Paradoxes,” Scientific American, vol. 271, no. 5, Nov., pp. 84-90.

Potter, M. (2009). Set Theory and its Philosophy, Oxford University Press

"Zeno’s Paradoxes," by B. Dowden, The Internet Encyclopedia of Philosophy, ISSN 2161-0002, http://www.iep.utm.edu/zeno-par/, 3rd August 2017.  

 

Article written by:
Ian Tame
Senior Tutor