Talk:Infinity/Archive 5

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"Subsequent thinkers, finding this solution unacceptable, struggled for over two millennia to find other weaknesses in the argument."

@D.Lazard: I think that this passage, with regard to Zeno's paradox, is written poorly. Specifically, it implies that thinkers spent two thousand years unable to comprehend the fact that men could run faster than tortoises. The actual object of contention was the proper way to formulate a mathematical limit; it goes without saying that the physical reality of things being faster than other things was quite well-understood, even at the time. The bow and arrow was used in warfare, runners competed in athletic competitions, et cetera, without issue. While I initially changed the sentence to say "weaknesses in the mathematical argument", I am willing to amend that to "weaknesses in the philosophical argument", per your revert, but I think there should be at least some clarification. jp×g 14:02, 5 May 2021 (UTC)

It seems not useful to qualify "argument", since it is at the corner of philosophy (existence of motion), physics (velocity comparison) and mathematics (resolving the paradox by understanding properties of infinite sums). D.Lazard (talk) 14:37, 5 May 2021 (UTC)

I am the author of the sentence in dispute. Of course most people, before or after Zeno, will agree that — unless seriously disabled — men can run faster than tortoises. Some who lived after Zeno were therefore motivated to find weaknesses in the argument so as to allow for the possibility of motion, and it was a long time before Cauchy succeeded. The "actual object of contention" had nothing to do with limits, a concept that was not formulated, let alone argued over, until long after Zeno. It would have been helpful to qualify the argument as "mathematical" or "philosophical" if it were necessary to distinguish it from other, non-mathematical or non-philosophical, arguments; as there is no such need, however, the qualification gains nothing. Peter Brown (talk) 17:50, 5 May 2021 (UTC)

There was a discussion at some math article not too long ago where it was claimed that the association of Zeno with considerations of the infinite was a modern misconception, and that what Zeno was actually trying to prove was that there was no such thing as motion, maybe even no such thing as change. That strikes me as a bizarre thing to want to prove, but if that is in fact what he was getting at, then we should be careful about conflating Zeno's arguments with later considerations that looked back to Zeno. --Trovatore (talk) 22:39, 5 May 2021 (UTC)

Yes Zeno was a student of Parmenides, and his paradoxes were meant to be a defense of Parmenides belief in the impossibility of change. Paul August ☎ 00:15, 6 May 2021 (UTC)

Paul August and Peter M. Brown, the name "Zeno" still does not appear on the real number page. I have long thought that this is a very serious omission, given that Zeno's paradoxes are specifically invoked when motivating the most central property of the reals, namely completeness. I would very much like to see a good, accurate discussion of the relationship at that article. It sounds like the two of you are more familiar with Zeno's thought than I am, and even though the key consideration is perhaps not so much Zeno himself as how nineteenth-century mathematicians understood Zeno, that would still be a key component. --Trovatore (talk) 16:12, 6 May 2021 (UTC)

Trovatore, how can the Achilles paradox motivate the completeness of the reals? If the racers' speeds and the tortoise's head start are all given as rational numbers, then the time it takes Achilles to overtake the tortoise will also be rational — there seems to be no requirement, here, for irrational numbers. Peter Brown (talk) 16:52, 6 May 2021 (UTC)

Hmm, yeah, but that's sort of an accident, I'd say. The underlying reasoning is topological, not algebraic. The sequence of positions where Achilles is behind the tortoise is an increasing sequence with an upper bound. --Trovatore (talk) 18:00, 6 May 2021 (UTC)

Peter M. Brown For a good background on Zeno's Paradoxes and their so-called "Standard Solution" (which requires a continuum) see the IEP s.v. Zeno's Paradoxes. Paul August ☎ 18:24, 6 May 2021 (UTC)

The IEP article seems confused. The last paragraph of section 3.a.i. starts with the sentence

The Achilles Argument ... presumes that space and time are continuous or infinitely divisible.

Well, which? "Continuous" and "infinitely divisible" are not the same thing! The rational numbers, or the multiples of 2⁻ⁿ for integral n, are infinitely divisible but not a continuum. I do not see how continuity figures in the "standard solution". Mere density seems to be enough.

Peter Brown (talk) 17:44, 7 May 2021 (UTC)

I also was underwhelmed by the IEP article. In particular the line By “real numbers” we do not mean actual numbers but rather decimal numbers did not especially inspire confidence.

Nevertheless I believe it is true that 19th- and 20th-century mathematicians considered the paradox to be resolved via a notion of real numbers that included all limit points of bounded subsequences. Whether you or I think that's convincing (you might say, what's wrong with Achilles sometimes being behind the tortoise and sometimes ahead, but never exactly even with it?) is not really the point; the point is the contribution to the development of the real-number concept. I would like to see this discussed in a well-sourced way at real number, in part just to learn more about it myself. --Trovatore (talk) 18:16, 8 May 2021 (UTC)

Trovatore, if you're correct that

19th- and 20th-century mathematicians considered the paradox to be resolved via a notion of real numbers that included all limit points of bounded subsequences

then yes, Zeno should be mentioned in Real number § History. Why do you think so, however? We can't mention Zeno in this connection without a reliable source on 19th- and 20th-cantury mathematics.

Peter Brown (talk) 21:31, 8 May 2021 (UTC)