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)

"Unendlichkeit" listed at Redirects for discussion

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)

History section - Dubious Tag Discussion

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)

"Taylor Archibald" listed at Redirects for discussion

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)

that which is

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)

Achilles v. Tortoise race duration

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

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

$10{\frac {10}{99}}={\frac {100}{99}}=1.010101...$ not $10.101010...$ Oh, you do not mean a multiplication, but the vulgar fraction $10{\tfrac {10}{99}}$. 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 $10{\tfrac {10}{99}}$ as a mixed number. Now, it isn't obvious to me, or probably to the general reader, why ${\frac {10}{1-0.01}}$ should be equal to $10\times {\frac {100}{99}}$ 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)

exacting and ultra-exacting cardinals

"The discovery of exacting and ultra-exacting cardinals represents a significant advancement in set theory and large cardinal theory."

https://www.popularmechanics.com/science/math/a63121596/exacting-cardinal-infinities/

https://www.newscientist.com/article/2459158-mathematicians-have-discovered-a-mind-blowing-new-kind-of-infinity/

https://arxiv.org/pdf/2411.11568

Juan Aguilera, a co-author of the paper from Vienna University of Technology

69.181.17.113 (talk) 02:09, 13 December 2024 (UTC)

I would suggest starting with the large cardinal page rather than here. This article should probably mention large cardinals at a high level (which currently it doesn't seem to), but not get into too much detail. --Trovatore (talk) 02:16, 13 December 2024 (UTC)

Agreed on both. - CRGreathouse (t | c) 15:47, 13 December 2024 (UTC)

Triangle image

I'm still trying to figure out if I think the image of the Sierpiński triangle is a good idea. I have to admit I can't think of anything better off the top of my head. It's certainly better than just an image of the symbol, and probably better than at least the current image of the reflecting mirrors, which I think is visually kind of confusing and noisy.

But in any case I copyedited the caption. I don't know what it means for something to "contain an infinite amount of itself". I tried to replace it with something meaningful and accurate; not sure if it's too wordy. --Trovatore (talk) 19:30, 16 January 2025 (UTC)

I think the Sierpiński triangle is a good image here, and I think your text is fine. Personally I'd scrub the "because of its recursive pattern" to keep it snappy but I don't mind the current version. - CRGreathouse (t | c) 02:42, 17 January 2025 (UTC)

Yeah, I considered that. With your encouragement I'll go ahead and kill it. --Trovatore (talk) 02:45, 17 January 2025 (UTC)

We could use its animated GIF or another fractal zoom GIF which I think makes the recursion a lot more self-evident if that is the way we want to go, but in my opinion having a vanishing point image like File:Ambigram tessellation Milan - concentric circles.png portrays it very well without being overly distracting. It would work great for the printable version of the article too. I'd be against using any real-world example like the last image because there are none that truly exist. --– Mullafacation {◌͜◌ talk} 19:20, 18 January 2025 (UTC)

It is currently unknown whether the entire universe (as opposed to the observable universe) is infinite.

I'm not a huge fan of animations in articles, but there's something to be said for the Sierpiński zoom. Could we make it sub in the static version for printable?

For the Milan thing, to be honest, I'm not sure what I'm looking at there.

As to whether there are any "that truly exist", that's sort of an open question. I've kind of toyed around in my head with the image at right (read the caption). I'm sure we could find a prettier one if we wanted to go that way. But the main thrust of this article seems to be mathematical rather than physical, so I kind of think probably not, but it's worth throwing out there. --Trovatore (talk) 19:45, 18 January 2025 (UTC)

If we go with the animated GIF, the printable version will be the first frame which looks the same as the image we have. And there's nothing special about the Milan image, it just looks like a tunnel with no ending. The pi image does the same thing but it is also serves as a practical mathematical example so I would prefer that. I did see in the article that it's an open question about the universe but I meant we can't show a picture that truly shows the infinity of anything in the real world. If it's an open question, it's not the strongest example we can choose. I don't know if there's any examples of images that suggest that the universe might be infinite but the image there doesn't. With pi there's no obvious reason why it's infinite while with the Sierpiński triangle you can see by looking at it how it is able to fit inside itself and go on forever. – Mullafacation {◌͜◌ talk} 21:54, 18 January 2025 (UTC)

OK, I'm on board with something like the Sierpiński animation, but I would prefer higher resolution and maybe a little pause on the first frame. Right now it kind of looks "off-balance" as it immediately descends into a non-centered part of the image. I wonder if someone has come up with an animated version of SVG? --Trovatore (talk) 22:10, 18 January 2025 (UTC)

I do not think that the Sierpiński triangle ia a good first image, because the reader must have a rather good knowledge of infinity to understand where is the infinity of this image. IMO, this image is much more pedagogical:

D.Lazard (talk) 20:17, 18 January 2025 (UTC)

I think that's too symbolic, too text-based for my liking. The Sierpiński image at least moves away from symbolic representation.

We could also consider the zero option. When it's too hard to find an appropriate image, it's worth thinking about whether an image is really needed. Shoving in images purely pro-forma doesn't appeal to me.

That said, I do think the Sierpiński triangle is better than nothing, and I also prefer it to the decimal representation of π. --Trovatore (talk) 20:41, 18 January 2025 (UTC)

I agree. I don't like putting the decimal approximation of pi. There are already too many people that have the misunderstanding that "pi is infinite." seberle (talk) 23:21, 20 January 2025 (UTC)

Infinity, documented by Vedic text Yajurveda (c. 8th century BC)

I noticed that infinity in this article is associated with Greek history, primarily, which documented it around 610–546 BC. Their concept was Apeiron (the infinite) as cosmic origin. However, Yajurveda documents it around 8th century BC or earlier. Their definition was in Infinity you minus or add parts, it still remains infinity.

I plan to make edits to this effect, and give equal or more weight to this concept being a Indian (both Hinduusm, and Jainism) concept.

From the vedic period the philosophical nature of infinity has been the subject of many discussions among philosophers.

And obviously add a full section on Indian vedic concept. Buddhimatta (talk) 06:28, 3 September 2025 (UTC)

So first of all, given reliable sources that have made the connection, I do think it may be reasonable to treat ancient Indian concepts to some extent. We do already have a small subsection of the History section, called "Early Indian", which could probably be profitably expanded.

That said, it seems to be the consensus that this article is to be largely focused on the notion of infinity in mathematics. We have a separate infinity (philosophy) article that might be a better home for some of this history. I don't know that I completely agree with that division but it is the current one.

You should also be aware that there is a bit of a history of editors attempting to expand the treatment of the Jain notions for what seemed likely to be nationalistic (or possibly just anti-Western) reasons. There was an editor called Jagged 85 who got community-banned in the most overwhelming way I've ever seen, more than a decade ago, for widespread misrepresentation of sources, especially on video games, but at some point s/he also turned his attention to the transfinite number article, and was pushing the Jain stuff hard. This left a fairly bad taste that has not entirely gone away. That doesn't mean the material shouldn't be covered, but it would be a good idea to find good sources (if possible, ones without an axe to grind, though this is not an absolute requirement) in advance of editing. --Trovatore (talk) 21:12, 3 September 2025 (UTC)

Thanks a ton for that thoughtful response and the historical context, @Trovatore

While I do belong to India, I am not religious. I have to acknowledge that I learned about concept of infinity early on due to my parents being religious (Jains). But as a editor, I am here to make wikipedia more trustable and factual. I wanted to focus on expanding the documented history, which from my understanding starts with the Hindu Vedas.

I have put together draft timeline that I was hoping to get a consensus on and then build the article modifications or new sections accordingly.

I also agree with you that the "mathematical concept" and the "philosophical concept" are so intertwined that it should be one article: Philosophy gives meaning to infinity, mathematics gives structure. Both are two ways of talking about the SAME concept. Kinda like "∞" is same as "Infinity".

Infinity Original Documented Timeline:

1. Yajurveda (Vedic India) Approximate Date: 8th century BC or earlier (~2800–2900 years ago from 2025) Trusted Sources: Chapter 2: Vedic and Puranic Cosmology https://www.researchgate.net/publication/2180120_Concepts_of_Space_Time_and_Consciousness_in_Ancient_India Atharva Veda (Vedic India) Approximate Date: 1000–800 BC (2800–3000 years ago from 2025) Trusted Sources: https://www.britannica.com/topic/Vedic-religion
2. Anaximander (Greek) Approximate Date: 610–546 BC (~2550 years ago from 2025) Trusted Sources: https://www.researchgate.net/publication/312595450_FROM_THE_INFINITY_APEIRON_OF_ANAXIMANDER_IN_ANCIENT_GREECE_TO_THE_THEORY_OF_INFINITE_UNIVERSES_IN_MODERN_COSMOLOGY Anaxagoras (Greek) Approximate Date: 500–428 BC (2450 years ago from 2025) Trusted Sources: https://books.google.co.in/books?hl=en&lr=&id=240MF0fzc8wC&oi=fnd&pg=PR1&ots=Hxow0A6spu&sig=k7fwH1SNAJTOb0xBAWT2XTMFPss&redir_esc=y#v=onepage&q=infinity&f=false
3. Jainism (Mahavira) Approximate Date: 6th century BC (~2600 years ago from 2025) Trusted Sources: https://pluralism.org/mahavira + https://books.google.co.in/books?redir_esc=y&id=jdjNkZoGFCgC&q=infinity#v=onepage&q=infinite&f=false + https://www.scribd.com/document/776077970/The-BIG-BOOK-of-Jain-Concepts#content=query:infinity,pageNum:48,indexOnPage:0,bestMatch:false
4. Aristotle (Greek) Approximate Date: 384–322 BC (~2350 years ago) Trusted Sources: https://plato.stanford.edu/entries/aristotle-mathematics/

Buddhimatta (talk) 15:37, 4 September 2025 (UTC)

@Buddhimatta: Please, do not use AI to reply to others. These comments may be collapsed per WP:AITALK. Leonidlednev (T, C, L) 15:50, 4 September 2025 (UTC)

Hi @Leonidlednev, Your analysis is False Positive. How does my hand written 2.5 hours of research get flagged as AI generated?? Are you using some AI tool which is actually not working. Will await your answer. Many thanks in advance. Buddhimatta (talk) 07:54, 5 September 2025 (UTC)

@Buddhimatta: Sorry for the late reply (and the FP). I originally thought that the paragraph was generated by ChatGPT due to the presence of utm_source=chatgpt.com in the source URLs. Due to that, it triggered the filter, and made me check out the comment. In the future, make sure to check the sources and remove the tracking parameters from the URLs (right-click > copy clean link) so they are not tagged as being generated by AI. Leonidlednev (T, C, L) 18:31, 7 September 2025 (UTC)

thank you Buddhimatta (talk) 06:37, 9 September 2025 (UTC)