Talk:Infinity/Archive 5

@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)

An editor has identified a potential problem with the redirect Unendlichkeit and has thus listed it for discussion. This discussion will occur at Wikipedia:Redirects for discussion/Log/2022 February 12#Unendlichkeit until a consensus is reached, and readers of this page are welcome to contribute to the discussion. ~~~~
User:1234qwer1234qwer4 (talk) 02:05, 12 February 2022 (UTC)

In the pages History section the Brihadaranyaka Upanishad is cited as showing how ancient Indians understood the concept of mathematical infinity. I found no such support for this claim given in the English translation for this and instead it only refers to infinite from a spiritual sense, not specifically within the context of an abstract mathematical or philosophical concept. I don’t claim to be familiar with this source but if we do keep this source it would follow we must also discuss all other ancient cultures who made reference to the idea of the eternal or everlasting as equally being aware of the idea of infinity.

Perhaps we need to better define whether we are covering the history of mathematical infinity or infinity in a broader perhaps more spiritual sense. This way we can narrow down what should and should not be included in this section. 121.98.205.163 (talk) 06:34, 14 September 2022 (UTC)

I thought that looked familiar. That section is very similar to what was discussed here, which ultimately (IMO as a participant in that discussion) ended in a consensus to not include the text, and with the OP receiving an indefinite block which is still in place. Looks like it was re-added in February, but with no better sourcing than before, so I've re-removed it. Writ Keeper ⚇♔ 13:59, 14 September 2022 (UTC)

The redirect Taylor Archibald has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at Wikipedia:Redirects for discussion/Log/2023 July 13 § Taylor Archibald until a consensus is reached. —Lights and freedom (talk ~ contribs) 07:52, 13 July 2023 (UTC)

In this edit, Panamitsu changed "...that which is..." to "...something which is..." in the first sentence.

I think I understand the concern — that which is could be seen as excessively flowery or old-fashioned. But something which is does not strike me as an entirely adequate replacement, for a couple of reasons:

• First, I know this is a bit of an American preoccupation, but it ought to be something that is if anything
• More seriously, the reader is tempted to view this "something" as referring to some specific thing, and that is not what the article is about

Lacking a better suggestion, I would prefer to restore that which is, which is not that exotic and which fairly elegantly solves the specificity problem. But maybe we can come up with a third option which is ha, I did it myself better still? --Trovatore (talk) 19:20, 8 August 2023 (UTC)

@Trovatore Oh, sure, I'm just a bit confused about your concern about the article not being about a specific thing. Are you able to elaborate? In my mind, endless or infinite can only be used to describe something (whatever that may be), and that "something" in the article says "an arbitrary thing." Just like how the Oxford dictionary defines it as "a thing that is unspecified or unknown." Panamitsu (talk) 21:13, 8 August 2023 (UTC)

My intuition is that this "something" can be read in two ways — it could be a "generic" something, or it could be some particular thing the speaker already has in mind (that would still be "arbitrary" in the sense that the word places no restriction on what the speaker might have in mind). For that reason I still like that which is better. --Trovatore (talk) 21:21, 8 August 2023 (UTC)

@Trovatore I disagree about your point about the reader already having something in mind, considering that it is at the start of the article. Pulling from the Oxford dictionary, how about we use something similar to "the quality of being endless"? Panamitsu (talk) 08:30, 9 August 2023 (UTC)

No, not the reader; the writer. It's like if I say "I have something to tell you". What you hear is that I have something specific already in mind, not that I just want to speak. So it could be interpreted as "there's this thing called infinity, and now I'm going to tell you some particular things about it, namely that it's boundless etc".

"Quality" seems to point too much away from interpretations that are objects (not that qualities can't be objects, but it's not what you think of).

What's really wrong with that which is? I thought it was kind of a nice solution. --Trovatore (talk) 16:04, 9 August 2023 (UTC)

@Trovatore I've personally never heard of "that which is" before, so it doesn't seem to make sense to me. Perhaps it's a technical term that I just haven't been exposed to. Panamitsu (talk) 22:34, 9 August 2023 (UTC)

"That which is" sounds perfectly fine to me, and is a common expression. Also, I don't think "something" is likely to be misunderstood in this context at the beginning of the article, though it does sound a bit awkward. Both solutions sound acceptable to me, though I prefer "that which is". seberle (talk) 10:55, 10 August 2023 (UTC)

@D.Lazard: We are in agreement that Achilles takes ${\displaystyle {\frac {10}{0.99}}}$ seconds to overtake the tortoise. As a repeating decimal, that's 10.101010... seconds. You claim that this is ${\displaystyle 10{\frac {100}{99}}}$ seconds, but that works out to 11.010101... seconds. My replacement, ${\displaystyle 10{\frac {10}{99}}}$ yields 10.101010..., which is what we want.

Peter Brown (talk) 02:23, 23 July 2021 (UTC)

${\displaystyle 10{\frac {10}{99}}={\frac {100}{99}}=1.010101...}$ not ${\displaystyle 10.101010...}$ Oh, you do not mean a multiplication, but the vulgar fraction ${\displaystyle 10{\tfrac {10}{99}}}$. IMO, even correctly formatted, this must be avoided, as vulgar fractions are not commonly used in many countries. I'll add a multiplication sign. D.Lazard (talk) 08:20, 23 July 2021 (UTC)

Yes, I was interpreting ${\displaystyle 10{\tfrac {10}{99}}}$ as a mixed number. Now, it isn't obvious to me, or probably to the general reader, why ${\displaystyle {\frac {10}{1-0.01}}}$ should be equal to ${\displaystyle 10\times {\frac {100}{99}}}$ or why anyone should care. I am accordingly omitting this step. Peter Brown (talk) 18:15, 23 July 2021 (UTC)

I am not persuaded that this mathematician solved Zeno's paradox in 1821. At minimum I recommend editing this to say that he claimed to solve the paradox. First, the only citation is to an original document in French. If this mathematician really did solve Zeno's famous paradox, an English citation or description of the work to show that it has stood up to peer review would be more persuasive. Second, and perhaps more compellingly, how does a repeating value solve a paradox that is primarily concerned with the problem of infinite regression? On the face of it, this does not seem to offer a solution to the paradox, but only supports the challenge/dilemma of the paradox further. Empiric78 (talk) 13:39, 9 October 2024 (UTC)