Talk:Mathematical beauty

From Wikipedia, the free encyclopedia

Contents

[edit] This article is rather poor

This article is rather poor, and needs a complete overhaul! (unsigned comment by anon 194.51.2.35 on 10:48, Aug 22, 2003 (UTC).)

Even though I'm responsible for most of it, I have to agree.
First of all, we need to see if we can clarify the distinction between beauty and elegance. Then, the fact that correct mathematical results can be judged independently on their importance, utility and aesthetic impact needs to be stressed (and each of these criteria for judging mathematics deserves its own NPOV development). Philosohical stances such as Platonism, the Pythagoreans, etc probably should be mentioned, but really belong in an article on the philosophy of mathematics or on the influence of mathematics on philosophy. The beauty of mathematics also plays a role in the teaching of mathematics.
It might be that creating this article in the first place was an outstandingly bad idea, but part of the point was to have a page which discusses whether, when and why mathematicians see beauty in mathematics, and how or whether aesthetic considerations play a role in the development of mathematics. Like I said before, importance deserves an entirely separate discussion along the same lines, as well as utility. (I am interpreting importance to be internal to mathematics and utility to be external, but I don't know if that distinction is useful).
Miguel (on 14:19, Aug 22, 2003.)
Suggestion: write mathematical elegance if you think it's a distinct or internal consistency concept - its own branch of aesthetics maybe? Google seems to back you that it's distinct from beauty, it doesn't turn up the same type of references quite.
Suggestion: write foundations of measurement and deal with the issues now covered in mathematical fetishism which are the "dark side of mathematical beauty". That will make clearer what is actually a quasi-empirical issue and what is within axiomatic mathematics itself. (See quasi-empiricism in mathematics if you are unclear what that is). With the philosophy of mathematics issues out of the way, or at least clearly stated, you will find it easier to state what the elegance, beauty, utility questions really are. The treatment of these issues here now is pathetic - you are to be commended for actually making a start on it.
(Unsigned comment by anon 142.177.94.99 on 14:30, Aug 22, 2003.)

[edit] Two suggestions

Wow, I can't believe this article was ever even considered for deletion! Not that it's a tremendous article as it stands (it needs a lot of work), but the topic itself is certainly justifiable as an encyclopedic entry. Whoever suggested otherwise and thought it should be obviously deleted is just ignorant. There is a ton of stuff out there on this topic, not just current, but lots of writings by mathematicians throughout history, who have always stressed this aspect of their work. Just the question of what results in math are considering important or elegant and how this is decided is a question becoming more addressed in math philosophy. I just have 2 suggestions, that I think would greatly help:

  1. Move all the stuff about math and art to another page. This really is not the same topic at all. There is a difference between the aesthetics of math (why mathematicians think their work is pleasing, what counts as elegant, etc.) and the application of math to understanding art, music, architecture, etc. I don't think it's wise to confuse the two. There should be two separate articles. The latter could be called "mathematics and art" or "application of mathematics to the arts" or something similar.
  2. I think "Aesthetics of mathematics" would be a much more general, encompassing, and NPOV title for the article. JMO.

Revolver 19:15, 31 Jan 2004 (UTC)

[edit] Proposed rewrite

I am proposing to undertake a re-write of this page, in three stages :-

  1. Move the stuff about mathematics and art to its own page (already done that).
  2. Re-word some of the remaining material, but retaining existing contents and order as much as possible (done that too - Jun 26).
  3. Re-order the material under the headings Beauty in experience, Beauty in method, Beauty in results and add new material (done that too - Jul 1).

If you are watching this page, please add comments or alternative proposals below. Gandalf61 13:13, Jun 25, 2004 (UTC)

[edit] Numbers aren't beautiful

Numbers aren't beautiful; symbols are pretty and expressions are beautiful. And Galileo was as wrong as Kepler. lysdexia 01:36, 17 Oct 2004 (UTC)

The beauty that mathematicians find in mathematics is obviously altogether different from any esthetic charm anyone finds in symbols or expressions. Michael Hardy 19:30, 17 Oct 2004 (UTC)
it's more to do with the concept, what it actually means rather than the direct answer.Wolfmankurd 19:52, 26 May 2006 (UTC)

[edit] About Users Critical and CStar

For the record, the user Critical ( talk, contributions), who slapped the "disputed NPoV" sticker on this page, has made his or her first edits tonight (or today) and within less than two hours has attacked eight articles for PoV, including (ironically given the CStar example given on the Logical fallacy talk page), Physical law. These were the only "edits" (plus weak justifications on talk pages in the same vein as this one). I don't think the PoV claim has merit. We may ask if this series of attacks is to be taken seriously.

For the following reasons I am thinking that these pages has been the victim of a tiresome semi-sophisticated troll and the PoV sticker should be removed sooner rather than later, if not immediately. We may note that CStar ( talk, contributions) after making edits, paused during the period user Critical made edits, and then CStar took up responding to these edits after the series of user Critical edits ends, as if there is only one user involved, and the user logged out, changed cookies and logged back in. Further, user CStar left a note on Charles Matthew's talk page, Chalst's talk page, and Angela's talk page pointing to a supposed PoV accusation placed on the Logical argument page, when in fact no such sticker has been placed. Perhaps the irony regarding the Physical law page is not so ironic. Hu 05:18, 2004 Dec 1 (UTC)

I have responded to this on the logical fallacy talk page, as well as on the pages of the above mentioned users. It does appear that these pages were as Hu suggests the victim of a tiresome semi-sophisticated troll. But I wasn't the perpetrator. This suggestion appears to have been an honest mistake, I consider the matter closed, and it appears that Hu does as well. CSTAR 01:43, 2 Dec 2004 (UTC)
Has Critical explained anywhere why he thinks this article is POV ? If not, it is difficult to see how to respond or amend the article, other than by just removing the notice. Gandalf61 09:30, Dec 1, 2004 (UTC)

[edit] Move of "Mathematical beauty" to "Aesthetics in mathematics", comments?

The following discussion, concernong the move of mathematical beauty to Aesthetics in mathematics (which has now been moved back) was first begun on User talk:R.Koot, then was continued here. I am now moving the discussion to this page.

(Beginning of moved text.)
R.Koot moved the Mathematical beauty to Aesthetics in mathematics. But I don't agree with the move. As there was no discussion and there is currently no consensus for the move, I decided to move the page back. But I can't. (I don't know why, I'm sure I've done this before) Apparently an admin needs to do it. So I though I'd bring the matter here, to see if we can reach a consensus about what should be done. I've copied the discussion below from R.Koot's talk page.
(Beginning of copied text.)
Hi R.Koot. Welcome to Wikipedia. I noticed that you moved Mathematical beauty to Aesthetics in mathematics. Was there any discussion which preceded this? I can't find any. I don't think I like the name change, especially as the article uses the term "beauty" throughout. Paul August ☎ 19:41, May 17, 2005 (UTC)
I have not discussed it, but I personally prefer the word aesthetics. It also allows categorization under philosophy of mathematics (aesthetics being a branch of philosphy). I do agree that it mismatches with the article, but I'd think it would be better to change the article than the title.
--R.Koot 20:14, 19 May 2005 (UTC)
Hi Rudy thanks for your reply. I'm afraid I don't agree with your move. Aesthetics and beauty are not synonyms. Aesthetics might roughly be defined as the theory of beauty. Changing the article to match the title is backwards. The title should match the article, not the other way around. The article is about mathematical beauty, not aesthetics in mathematics. Titling the article "Mathematical beauty" does not prevent it from being classified under "philosophy of mathematics". I'm going to move it back. If you want to try to gain a consensus for moving it to "aesthetics in mathematics", I suggest you make a case on either the article's talk page or on Wikipedia talk:WikiProject Mathematics. Paul August ☎ 20:52, May 19, 2005 (UTC)
(End of copied text.)
Comments?
Paul August ☎ 21:13, May 19, 2005 (UTC)
Agree with Paul. That article is about beauty and not aesthetics.
One of course cannot move that page back, since there is a redirect in its place. An admin would need to delete the redirect first. Oleg Alexandrov 21:48, 19 May 2005 (UTC)
Of course generally you can't move a page to an already existing page, but in the special case where the existing page is a redirect to the page you are trying to move, I thought you could. I'm sure that I've done that before. Of course things could have changed and/or I could be losing my mind (not an altogether unlikely possibility). Anyway any admin who's listening care to help out? (Oleg: you should have accepted Charles' nomination, I think you'd make a good admin ;-) Shall I renominate you?) Paul August ☎ 22:03, May 19, 2005 (UTC)
The problem is that "Mathematical beauty" was first moved to "Aestetics (mathematics)" and subsequently to "Aesthetics in mathematics". If the move had been done in one go, it would be possible to undo it. By the way, I agree with Paul, both about moving the page back (aesthetics ≠ beauty) and that Oleg should be administrator. -- Jitse Niesen 22:17, 19 May 2005 (UTC)
Now we are going off topic, but if you would really like a new admin, who is a good mathematician, thoughtful person, and been here for a while, then how about nominating Jitse? Jitse's been here since 2003, does a lot of VfD work (certainly more than other mathematicians I've seen), and is not addicted to Wikipedia, which is a good thing. Paul, what would you think? Jitse, would you turn down such a nomination? As far as me being admin, that time will come (I mentioned to Charles that he could ask me about this again after three months, which would be in July I think)... Oleg Alexandrov 22:59, 19 May 2005 (UTC)
I've moved it back since that seems like what people want. Aesthetics in mathematics now redirects to Mathematical beauty, and Aestetics (mathematics) (sic) is gone. Let me know if anything else needs doing. BTW, I hang out on RFA a lot and I imagine a RFA for Jitse would likely succeed. The low edit count would be offset by the length of time, breadth of contributions, being active in Wikipedia: space, and vandal fighting. I'd vote for him, anyway. CryptoDerk 23:44, May 19, 2005 (UTC).
Thanks Crypto. Ok we're off topic, but my comment was parenthetical. Anyway, now that I think of it, as a categorical topologist, I'm not sure I can, in good conscience, support either of you applied mathematicians for adminship ;-) Yes of course, I would be happy to support Jitse. Paul August ☎ 00:18, May 20, 2005 (UTC)
Oh dear, this is not going as planned. I'm conveniently going abroad for two weeks this Saturday. Prod me again when I return and I'll think about it. -- Jitse Niesen 00:42, 20 May 2005 (UTC)
Sounds good. By the way, about the edit count, 1215 edits looks to me as a reasonable number. Oleg Alexandrov 00:47, 20 May 2005 (UTC)
(End of moved text)

Paul August ☎ 19:56, May 27, 2005 (UTC)

[edit] How many proofs ?

"The theorem for which the greatest number of different proofs have been discovered is possibly Pythagoras' theorem." ... which is (the number, not the theorem of course)?

Probably hundreds - a more precise answer will depend on the definition of different. Elisha Scott Loomis published over 360 proofs in his book Pythagorean Proposition (ISBN 0873530365). This page gives details of 54 proofs. Gandalf61 14:33, 26 October 2005 (UTC)
added citation Wolfmankurd 19:56, 26 May 2006 (UTC)

[edit] What about visual beauty

Beauty is something that man feels and our senses are more than 50% visual.

I must find figures, maybe as simple as a pentagon, to illustrate this. Those figures can be described with some maths but are they purely mathematical ? At least some have been discovered with the use of maths.

Advice would be appreciated : thanks. PS - see also : these geometric patterns--Harvestman 22:36, 6 January 2006 (UTC)

Referencing and posting an image of a, heh, beautiful fractal and some basic shapes, like a torus. Overall I agree, this article is completely missing the beauty of visualization.
Thank you for your support. Harvestman

I don't think visual mathematical beauty can be shown through pictures to someone very well. It would be like trying to explain what six dimensions looks like inside your head. It doesn't come that obviously at first, more that one's appreciation of the beauty deepens as one's personal knowlegde and experience of working with mathematics does. - Anna Gardiner 82.69.33.65 10:06, 27 January 2006 (UTC)

Hello Anna! I have to answer. Baudelaire says "get drunk, be it wine, religion or poetry, but get it". My point is : that mathematical beauty that we experience in our mind is alike a mystical experience, and, being mystical, it is hard to share. One has to go for it.
This is why I asked about pictures, that can be shared and seen and appreciated more easily. This won't take away any of the spiritual beauty, it can lead to it. Now I'm looking for your agreement. --DLL 11:23, 4 February 2006 (UTC)

[edit] The language with which God wrote the universe

Galileo Galilei is reported to have said "Mathematics is the language with which God wrote the universe", a statement which (apart from the implicit deism) is consistent with the mathematical basis of almost all physics since his time.

I'm afraid I don't understand this line of the article; could someome explain it to me? I can see how one might draw a connection between Galileo's spiritual views and the fact that all physical laws are fundamentally grounded in mathematics, but I don't see how "Mathematics is the language with which God wrote the universe" could be "consistent" with physics, which (along with math) neither posits the existence of God nor claims that mathematics was "written" (even in the figurative sense of "created"). I think one reason this line is confusing is that the line makes the disclaimer "apart from the implicit deism"; as deism ("God created the universe perfectly, then left it alone to run its natural course") is a type of theism ("God exists"), and the quotation, at least, seems blatantly theistic, but not really all that deistic, mentioning none of the aspects of deism which distinguish it from any other form of theism (like disbelief in miracles and divine intervention). The implication of presenting a highly (and explicitly) theistic, and only subtly (and implicitly) deistic, quotation, and saying "apart from the implicit deism", is that the deism isn't "consistent with the mathematical basis of almost all physics since his time", but the theism is! This seems profoundly backwards to me, as, if anything, the existence of a deity would be infinitely more consistent with modern physics and mathematics than the existence of non-deistic beliefs in theism like miracles, which by their very nature violate the laws of physics. If the quotation said "apart from the implicit theism", then deism, being a subset of theism, would obviously be disregarded as an aspect of the quotation's consistency with the relationship between mathematics and physics; but by going out of our way to not say "theism", by sticking to only the very specific philosophy of "deism", we implicitly claim that Galileo's belief in God is completely consistent with mathematics and physics (which, even if it were true, is obviously a complete digression, original research, and irrelevant to the topic of this article). -Silence 14:55, 3 February 2006 (UTC)

Your comments are both deep and interesting. I think the problem is that there are at least two huge concepts in Galileo's short statement, a theological one and a physical one, but the quote itself may not be broken into two parts. As I see it, Galileo was stating that mathematics is the language in which the laws governing the physical universe are written, and that God was the writer of the laws. My addition of a second clause to the sentence was intended to comment only on the first of these two ideas, pointing out the striking fact that vast areas of fundamental science of which Galileo was almost entirely unaware have proved to be best expressed in a predominantly mathematical way. The bracketed part of the sentence is merely a clarification that science has, of course, only provided confirmation of the first of these two ideas. I would agree with the same statement using the word "theism" instead of "deism", but the "clockmaker" idea (perhaps "software engineer", in modern terms?), to which the Wikipedia article on deism draws attention, seems closely aligned with Galileo's statement. Elroch 02:33, 4 February 2006 (UTC)

[edit] Empty sentence

From the article: "Of course, this distinction is merely platitudinous; all proofs are deep." This sentence seems either wrong, POV, or empty. If it's backed up with something, it should be left alone. But I'd want some backup.

[edit] Best of

In Reference desk/Mathematics we find that question - Most elegant parts of mathematics : "As a devout follower of the world of math, I always love to see mathematical elegance. What is your favourite part of mathematics and can you suggest some beautiful problems and solutions to me?"

May the answers be integrated in this article, e.g. in a "see also" section ? Also, if math beauty was "ineffable", there would be nothing to say. Aesthetics use criteria such as harmony, symmetry ... Mathematicians love concision, universality, &c. Each example should at least tell why it is a good one and what criteria apply. Thanks. --DLL 15:35, 25 May 2006 (UTC)

Retrieved from "http://en.wikipedia.org../../../m/a/t/Talk%7EMathematical_beauty_b34c.html"