# A Brief History of the Paradox

Writing in your own words, address ALL of the following instructions. [Part A.] On p. 49 of our Sorensen text, A Brief History of the Paradox, author Sorensen presents three “paradoxes of motion” posed by Zeno in Ancient Greek philosophy. These concern respectively: a person walking across a room; Achilles catching the Tortoise; and an arrow at rest while moving toward an archery target. Choose one of these three paradoxes: exactly as Sorensen phrases that proposed paradox. Explain how that paradox develops, expounding on the Sorensen formulation. How does this Zeno line of argument lead into a strict paradox (of form “P and not-P”)? (You’ll be filling in Sorensen’s intuitive presentation so as to lay out Zeno’s line of argument in your own best formulation as you understand Sorensen’s phrasing.) [Part B.] What makes this line of argument a paradox? Briefly explain what constitutes a “paradox”, and show how in your exposition Zeno’s line of argument leads to a paradox. Note that we observe the relevant type of motion in everyday life. In your exposition, where does a notion of “infinity” arise? How does that feature arise as alleged of a finite movement in infinitely many parts? On pp. 323-324 of the Sorensen text, author Sorensen outlines a theory of numbers developed by mathematician Georg Cantor that distinguishes between “natural numbers” and “real numbers”. Cantor proved that the size (or “cardinality”) of the set of real numbers is greater than the size of the set of natural numbers. Today’s mathematicians hold that Zeno’s paradoxes can be resolved if we assume Cantor’s theory of “transfinite” numbers. Explain, briefly, how we can use Cantor’s transfinite number theory to resolve, arguably, the Zeno paradox of motion you have explained. (The aim is not to reproduce Cantor’s mathematics, but to show how the given Zeno paradox may be resolved following Cantor: Where did Zeno go wrong?) [Part C.] On pp. 184-185 of the Sorensen text, author Sorensen discusses what has come to be called “McTaggart’s paradox of time”. Explain McTaggart’s distinction between the A series of time and the B series of time. Why are these two notions of time, allegedly, inconsistent with each other, yielding a paradox of time?  Returning to your chosen Zeno paradox of motion, which of these notions of time do you think is assumed in formulating the paradox? Would the paradox equally arise if you assumed the other notion of time? (The aim here is simply to think in terms respectively of A-series time and B-series time, and see if which you choose makes a difference to your Zeno paradox.) SOURCES: Use ONLY the Sorensen text, the Professor’s Notes on time, and the guest lecture by Stella Moon on the Cantor mathematics as applied to the three Zeno paradoxes. (The aim is to think in these terms as you address the Zeno style paradoxes. There is a huge amount of literature on all these themes, but you may get confused or even misled by reading around, and the aim here is NOT a “report” but an exercise in your own thinking carefully about the issues raised in our Lectures and Discussion sections in relation to just these sources.)

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