Addendum

I had expected Professor Salmon's analysis of Austin's paradox to produce many letters of disagreement, but evidently Salmon argued his case skillfully because I received not a single one. Of course the debate is largely verbal, a question of what sort of language to use in making the problem and its solution precise.

Many other problems are analogous to Austin's dog in the sense that there is a precise answer in forward time, but hopeless ambiguity when the event is time reversed. Consider for instance a point starting at the earth's equator and moving due north with uniform speed along a loxodrome. It will circle the north pole a countable infinity of times, reaching the pole at a precise instant. But time-reverse the event and the point can cross the equator at any spot. Because there is no "last" revo lution around the pole, there is no precise beginning of the time-reversed event that will determine a unique spiral path.

Mathematics Magazine, which originally published Austin's paradox as problem Q503 (January 1971), returned to the paradox in its September issue by publishing comments by four mathematicians, all of whom considered the problem self-contradictory. The magazine did not publish Salmon's reply to one comment. I reproduce it below:

In the September-October number, Lyle E. Pursell comments on Quickie Q503 (Austin's boy-girl-dog problem) as follows:

The author's solution to the problem looks like a proposal to sum an infinite series by starting at the "last" term! Since, if the latter three reverse their motions as the author suggests in his solution, then the dog must reverse his direction infinitely many times before the boy and the girl get back to the starting point.

While no original texts have survived to the press date, it seems plausible to suppose that Zeno of Elea (circa 500 b.c.) might have made a similar comment about Achilles:

Since Achilles must run half of the racecourse before he can run the whole, and he must run a quarter before he can complete the half, etc., it is evident that Achilles must run infinitely many distances before he can have reached any point, however near, beyond his starting point. To say that Achilles has run any finite (i.e., nonzero) distance looks like a proposal to sum an infinite series starting at the "last" term!

Although Austin's dog must reverse his direction between segments whereas Zeno's Achilles keeps going in the same direction, does this difference really have any bearing upon the absurdity involved in the "proposal to sum an infinite series by starting at the 'last' term!"? It appears that Austin's dog exhumes Zeno's old regressive dichotomy paradox. If Achilles can run a racecourse, why cannot Austin's dog do what is required of him?

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