Talk:Mathematical beauty

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.)

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)

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)

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)

I don't find many symbolic expressions terribly pretty (except maybe $$e^{\pi i} + 1 = 0$$).

It's hard to explain for me why certain concepts are "pretty". I think its akin to metaphor, or poems. Some poems I find beautiful. The simple, harmonic, clear expression of ideas that relate to many aspects of humanity have many desirable properties. The simple, elegant, clear expression of mathematical concepts is very similar.

There's also a sense of power that some theories seem to lend, and the idea that if I understand a certain theory, then I've empowered myself greatly. In this sense its no different than the appreciation that the character Tim "The Tool Man" Taylor has for his tools, or Martial Artists have for their art (although I'd say theres a kinship between some martial arts and poetry as well). Root4(one) 04:55, 20 January 2007 (UTC)

Look at this. It is not beautiful because there are repeating numbers, or a pyramid shape, but because of the mathematical operation that leads to something so succinct that people find as aesthetically satisfying as art.


 * This is actually a very old idea, traceable back to Ancient Greece: in the Seven Liberal Arts and Sciences, which was the kernel of Classical learning (in "kit form", as it were), we find the idea that Mathematics is equivalent to Geometry, and Geometry to Astronomy, and so again to Music; and, by analogy, Architecture, and all subsequently derived Arts of form and proportion.


 * Nuttyskin (talk) 13:36, 17 December 2018 (UTC)

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)

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 &#9742; 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 &#9742; 20:52, May 19, 2005 (UTC)


 * (End of copied text.)


 * Comments?


 * Paul August &#9742; 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 &#9742; 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 &ne; 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  &#9742; 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 &#9742; 19:56, May 27, 2005 (UTC)

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)

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)


 * The concept of beauty relates to vision and has nothing to do with mathematics. When used as such, it is poetic. Some professionals like to use words incorrectly in order to give the impression of being profound. In so doing, they also become incomprehensible and their words meaningless. But they appear deep. A similar misuse would be that of "algebraic grace."Lestrade (talk) 00:23, 22 November 2012 (UTC)Lestrade


 * Many, many, many mathematicians would disagree, and not just to seem profound. - DavidWBrooks (talk) 00:42, 22 November 2012 (UTC)

In doing so, they would be mistaking a subjective, personal judgment with an objective fact. There is no way that the concept of “mathematical beauty” or “mathematical elegance” can be communicated from one person to another. It will always remain as “I know what I’m talking about, but I can’t make you know what I’m talking about.” In relation to social hierarchies, it becomes, “You don’t understand mathematical beauty and elegance because you’re not as smart as I am.”Lestrade (talk) 14:02, 20 December 2012 (UTC)Lestrade

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)

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.

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)

Deletion of Mandelbrot set
I think the pictures of the Mandelbrot set should be removed. Sure, they are pretty and about mathematics but they don't have much to do with mathematical beauty in the ordinary sense. The pictures seam to belong much better in the article on mathematics and art. At the very least it should be clearly noted that such pictures isn't what is ordinarily meant by mathematical beeauty. {unsigned|217.210.4.149}


 * Okay, I've changed it to a different image. Better ? Gandalf61 21:13, 29 March 2007 (UTC)

Ther's not enough purdy pictures. You have to have the purdy pictures, that's how they sells the telescopes. 83.70.241.213 02:48, 18 April 2007 (UTC)

I think the Mandelbrot set should totally be on this page! Not every person is a math genius, some people (possibly myself) would never understand mathematical beauty in its "ordinary sense". However the Mandelbrot set or other fractals such as the Julia sets speak for them self. They are one of the most beautiful images in existence (I believe) and they are created through mathematics. Th1alb (talk) 22:19, 6 January 2009 (UTC)

Russell's Mistake
In the article, the following words by Bertrand Russell are displayed: "The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics." Here, Lord Russell, speaking in his usual aristocratic manner, has made an error. He is describing the Sublime, not the Beautiful.Lestrade 13:00, 23 October 2007 (UTC)Lestrade

Category mistake
It is a category mistake to attribute beauty or elegance to mathematics. Those two predicates are only meant to describe an experience in which an observer stares in admiration at something that gives pleasure through its fine qualities. But we don't gape at mathematics. We use mathematical symbols to measure and organize quantities. The only way that the characteristics of beauty and elegance could legitimately be used to describe mathematics is through poetry, which allows category mistakes. Mathematicians assert that they experience beauty and elegance in mathematics, but they never explain or try to make sense of their assertion. They want to convince listeners that they are speaking of a mystical experience, unavailable to the lay public. It is another example of the very ancient use of hieratic priestly mysteries that are beyond demotic comprehension.Lestrade 14:34, 23 October 2007 (UTC)Lestrade


 * I hope it's not beyond demotic comprehension to cite a definition from the OED: "2. That quality or combination of qualities which affords keen pleasure to other senses (e.g. that of hearing), or which charms the intellectual or moral faculties, through inherent grace, or fitness to a desired end; cf. BEAUTIFUL a. 3." One of the usage examples is quite appropriate in my opinion - "1860 EMERSON Cond. Life viii. 168 We ascribe beauty to that which is simple; which has no superfluous parts; which exactly answers its end." Speaking of those "hieratic priestly mysteries" you bring up, maybe you would agree with Augustine of Hippo when he said "The good Christian should beware the mathematician and all those who make empty prophecies. The danger already exists that the mathematicians have made a covenant with the devil to darken the spirit and to confine man in the bonds of hell," seeing as we have made such a grave mistake as to categorize mathematics as a thing of beauty. :-) ugen64 03:08, 29 October 2007 (UTC)


 * With regard to the OED's statement that beauty "charms the intellectual or moral faculties," I can't help wondering why it confuses beauty with charm. Beauty is not associated with sexual attraction, whereas that is the essence of charm. I stopped submitting to the OED's authority after I read its definition of "oxymoron" as "pointed foolishness." As far as the Augustine quote is concerned, I believe that you included it in order to provoke ridicule. It is not pertinent to the category mistake that I have explained.Lestrade 12:26, 29 October 2007 (UTC)Lestrade


 * "But we don't gape at mathematics. We use mathematical symbols to measure and organize quantities."

That is nonsense. You don't gape at mathematics. You may perhaps use it only for those utilitarian purposes. Speak for yourself. Michael Hardy 22:23, 31 October 2007 (UTC)
 * "Mathematicians assert that they experience beauty and elegance in mathematics, but they never explain or try to make sense of their assertion. They want to convince listeners that they are speaking of a mystical experience, unavailable to the lay public."
 * "Mathematicians assert that they experience beauty and elegance in mathematics, but they never explain or try to make sense of their assertion. They want to convince listeners that they are speaking of a mystical experience, unavailable to the lay public."

Sigh.... The word "never" here is greatly exaggerated. It is true that mathematicians are generally quite inept at explaining this phenomenon. But in fact most mathematicians would love to make it broadly available and very many of them attempt to do so. Sometimes it's done very well by skillfully exhibiting examples (I'm thinking of Stanley Ogilvy's book Excursions in Geometry and some others like it), but rarely does any mathematician do a good job of explaining it. Some of Gian-Carlo Rota's comments in his book Indiscrete Thoughts (not to be confused with his other book Discrete Thoughts) do far better at explaining it than most mathematicians have done. In particular he makes a careful distinction between beauty and elegance. (He ends up at a bottom-line conclusion that I don't agree with, though.) Michael Hardy 22:32, 31 October 2007 (UTC)
 * If you say that mathematics can have beauty and elegance, then you are talking about inner, subjective experiences that cannot be shared. It is my subjective opinion that there are no such experiences and that it is a misuse of words to attribute those predicates to mathematics. It is incumbent upon the person who makes a positive statement to provide objective proof.Lestrade 01:26, 1 November 2007 (UTC)Lestrade

You're saying people half no inner subjective experiences?
 * I'm saying that the words beauty and elegance cannot be objectively used to describe mathematics. Those words are used for sensual experiences through sight, sound, smell, taste, and touch. Mathematics is a logical, not a sensual, experience.Lestrade 13:21, 1 November 2007 (UTC)Lestrade

Many people have tried to share mathematical beauty with those who have not experienced it. Take a look at Ogilvy's book that I mentioned. Michael Hardy 02:36, 1 November 2007 (UTC)
 * What does Ogilvy say? Can his claim be summarized in a few words? Was he talking about geometric forms, such as triangles and circles, that might be said to produce aesthetic pleasure, because of their simplicity or perfection, when they are viewed?Lestrade 13:25, 1 November 2007 (UTC)Lestrade

I don't recall that he makes explicit claims on the subject. He only gives examples. Michael Hardy (talk) 06:40, 18 December 2007 (UTC)

Simple is simple. Beauty is beauty.
Why is "beauty" used as a synonym for "simple"? They are not equivalent. The number "one" is simple. Is it therefore beautiful? In addition, the word "elegant" does not correctly describe any mathematical entity. "Elegant" means graceful and refined. "Beauty" and "elegance" are used by mathematicians incorrectly and are only an attempt to seem arcane and mysterious. This is as old as humanity. It is called hieratic language and is meant to separate and distinguish a priesthood from lay people who use demotic language. Many mathematicians might consider restricting themselves to numbers, in which they are competent, and leaving words to those who can employ them correctly for the purpose of the communication of concepts.Lestrade (talk) 12:53, 20 February 2008 (UTC)Lestrade

I saw this section while passing through, and had to comment. This is by means of personal explanation, not as something to be added to the article, so it is certainly POV. I hope that's alright here, and it seems to be based on earlier comments. Beauty in mathematics is never used to mean 'only' simple, exactly for the reason you pointed out. It is true that many mathematicians tend to find things which aren't simple to also be ugly, but it is silly to think that beauty and simplicity are the same concept, and in fact the article you are commenting on makes it fairly clear that while beautiful things are often (but not always) simple, simple things are generally not beautiful.

I feel it is also necessary to make a short edit on the category mistake section. When mathematicians talk about beauty, it is normally in the same sense that one might talk about a philosophical novel being beautiful. This may not be kosher under your definition of beauty, though (as the OED reference was mentioning), it is certainly kosher under common use. To further explain the above comment, we might call something like "Nausea" beautiful if it was well-written, and also if it contained interesting ideas. To appreciate the beauty, it is unfortunately necessary to understand French fairly well, and something similar is true in mathematics. You must be able to read algebra to understand why it is prettier to prove things about normal forms of matrices using the theory over PIDs rather than directly, even though the latter proofs can be longer and certainly contain a lot of extra information that makes them *less* simple. Unfortunately this means that most people will never understand this feeling of beauty. But the fact that we require interpretation doesn't mean nothing is happening. As a final attempt to convince you, we know that little kids can't understand or appreciate modern jazz, and they almost certainly feel nothing - jazz music requires interpretation. But it is a little silly to think that we should judge beauty based on this small amount of interpretive ability.

Finally, I understand what you mean when you claim that we are using beauty in a way that is a little bit incorrect, in the sense that most people don't have this experience. In that sense I completely agree with you. But this is a feeling almost universal among people who do study mathematics (I've been a mathematician for quite a while and never run into another mathematician who would deny this), and beauty seems like the easiest word to use. I don't think anybody would mind if you added a section to the article explaining this. It might be true that a made-up word like snazzle should really be used to describe beauty-requiring-heavy-interpretation as opposed to gloomph for beauty-as-an-immediate-sensation. It also seems like a good idea to add references to these sorts of things if there are philosophy sections on wikipedia. As it is, though, this article seems like it is describing what most people (or at least most mathematicians) mean when they say 'mathematical beauty', and that should probably be the main focus of an encyclopedia entry. —Preceding unsigned comment added by 64.230.45.234 (talk) 18:27, 7 August 2008 (UTC)

Perhaps this can shed some light on the simplicity v beauty issue: certain beautiful things may be simple but boring. In his work on low-complexity art, Jürgen Schmidhuber described a simplicity-based algorithmic theory of beauty which takes the subjectivity of the observer into account: among several observations classified as comparable by a given subjective observer, the most beautiful one is the one with the shortest description, given the observer’s previous knowledge and his particular method for encoding  the data. This is closely related to the principles of algorithmic information theory and minimum description length. One of his examples: mathematicians enjoy simple proofs with a short description in their formal language. Another example describes a pretty human face whose proportions can be described by very few bits of information, drawing inspiration from less detailed 15th century proportion studies by Leonardo da Vinci and Albrecht Dürer. But Schmidhuber's theory explicitly distinguishes between what's beautiful and what's interesting, stating that interestingness corresponds to the first derivative of subjectively perceived beauty, assuming that the observer continually tries to improve the predictability and compressibility of his observations by discovering regularities such as repetitions and  symmetries and fractal self-similarity. Whenever the observer's learning process (such as a predictive neural network) leads to improved data compression such that the observation sequence can be described by fewer bits than before, the temporary interestingness of the data corresponds to the number of saved bits. This compression progress is proportional to the observer's internal reward, also called curiosity reward. A reinforcement learning algorithm can be used to maximize future expected reward by learning to execute action sequences that cause additional interesting input data with yet unknown but learnable predictability or regularity. The principles can be implemented on artificial agents which then exhibit a form of artificial curiosity.

Fleabox (talk) 19:53, 13 August 2008 (UTC)

A mathematician's non-apology
I don't know who wrote this page, but as a reader and a mathematician let me say (with no aplology) that it reads really hopeless to me. It seems like a set of cut and paste summaries from totally haphazard references. If this is in any way about mathematical beauty I should resign from the AMS. This page needs a total rewrite. History2007 (talk) 13:06, 1 March 2009 (UTC)

Go ahead and rewrite it then. 114.30.103.74 (talk) 14:28, 7 March 2009 (UTC) Steven, Cable Guy


 * I will put it on my list. It will be sometime this decade for sure. History2007 (talk) 16:03, 7 March 2009 (UTC)

Section on mathematical information theory does not belong
The section on mathematical information theory is about an application of mathematics to our understanding of beauty, not about the experience of mathematics as beautiful. It is a different topic and should be removed to a different article, such as one on aesthetics or one on information theory. —Preceding unsigned comment added by Calc rulz (talk • contribs) 12:12, 14 July 2010 (UTC)

Pi is ugly.
I see that Euler's identity: e^(pi*i)= -1, is being used as an example. Sorry, but starting with an ugly constant like pi make this equation ugly. See the http://tauday.com/tau-manifesto : e^(tau*i)= 1. — Preceding unsigned comment added by Reddwarf2956 (talk • contribs) 00:17, 30 August 2012 (UTC)


 * David Hume wrote: "Beauty in things exists in the mind which contemplates them." ("Of Tragedy," I. b.) The same is true of ugly. It’s all personal, subjective opinion. "Everyone carries his own inch–rule of taste&hellip;." (Henry Adams, The Education of Henry Adams, ch. 12) What is your personal, subjective judgment regarding Sarah Jessica Parker or Whoopi Goldberg?Lestrade (talk) 16:51, 21 December 2012 (UTC)Lestrade

mass-energy equivalence
E=mc^2 has to be in here. It is a simple formula with big implications that, with a small amount of explanation, everyone can understand and be taken aback by.

SAuhsoj (talk) 22:59, 21 November 2012 (UTC)
 * Here's an equation for you: simple ≠ beautiful.Lestrade (talk) 14:08, 20 December 2012 (UTC)Lestrade

Image of beauty in method
The image in the article exhibits shapes of pastel colors. Are these delicate colors supposed to suggest visual, artistic beauty? If so, then this is not the kind of mathematical beauty that is suggested in the article. The image’s caption reads as follows: "An example of 'beauty in method'—a simple and elegant proof of the Pythagorean theorem." Here, again, we have a statement that suggests that simplicity is the same as elegance, and that both of these are the same as beauty. This is beside the fact that the so–called proof that is "shown" is by no means simple. Words like simple, elegant, and beauty are subjective and perspectival. Anything mathematical can be considered beautiful only by fiat or decree based on authority.Lestrade (talk) 21:26, 18 January 2013 (UTC)Lestrade

Opinionated caption
The caption under the image of a compound of five cubes claims that "There is a certain 'cold and austere' beauty in this compound of five cubes." This is a completely subjective opinion. It might as well be expressed in a private language. What sense of the word "cold" is meant? The word "austere" has many meanings, such as "rigid," "formal," "difficult," "hard," "uncompromising," "unrelenting," "relentless," "ascetic," "stiff," "straight–laced," "harsh," "grim," "stern," "forbidding," "grave," "severe," "sober," "serious," and "simple." The caption might be equally justified in asserting the opposite by claiming: "There is a certain warm and fancy beauty in this compound of five cubes."Lestrade (talk) 00:45, 19 January 2013 (UTC)Lestrade

Math fetishists?
I thought I would find information about math fetishists here. Especially since "math fetishism" redirects to here. There's certainly a category of people who find mathematicians insanely hot. — Preceding unsigned comment added by 198.144.209.8 (talk) 18:53, 13 November 2013 (UTC)

Feynman's jewel?
Per Talk:Euler's_identity and consensus on that page, I have removed the following sentence, which seems to be misattributed:

The physicist Richard Feynman called the equation "our jewel" and "the most remarkable formula in mathematics.'' ref: Feynman, Richard P. (1977). The Feynman Lectures on Physics, vol. I. Addison-Wesley. p. 22-10. ISBN 0-201-02010-6 109.157.83.88 (talk) 12:53, 13 February 2014 (UTC)


 * The Feynman quote refers to Euler's formula. Euler's identity is just a special case of that formula. I have restored the Feynman quote and added a clarification. Gandalf61 (talk) 13:05, 13 February 2014 (UTC)


 * Thanks. Your revised wording fully addresses my concern. 109.157.83.88 (talk) 13:27, 13 February 2014 (UTC)

fMRI study
An interesting primary source picked up by the BBC. I guess WP:SCIRS would be the appropriate sourcing guideline for this. 109.157.83.88 (talk) 13:47, 13 February 2014 (UTC)

Mathematical beauty and the arts?
Although this topic is within the scope of this page imo, I feel the current Mathematics and the arts section, which mainly lists examples of mathematics being used in the arts, doesn't really address the question in hand. 86.173.146.3 (talk) 17:40, 18 February 2014 (UTC)

Adjectives
Regarding that first list, these seem appropriate: Deft Curt Shocking Novel Able

208.54.85.182 (talk) 10:52, 3 May 2014 (UTC)

External links modified
Hello fellow Wikipedians,

I have just added archive links to 1 one external link on Mathematical beauty. Please take a moment to review my edit. If necessary, add after the link to keep me from modifying it. Alternatively, you can add to keep me off the page altogether. I made the following changes:
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Beauty also applies to other sciences
This discussion of beauty should not be limited to mathematics. Einstein's theory of general relativity, Maxwell's equations of electromagnetism and Darwin's theory of evolution are all generally regarded as beautiful. 2.110.93.203 (talk) 10:22, 21 August 2016 (UTC)
 * Certainly, but not here: this article is about mathematical beauty, not scientific beauty. - DavidWBrooks (talk) 12:00, 21 August 2016 (UTC)

Extremely limited article
First, I will say that the subject of mathematical beauty is a very subtle one that is not easy to describe.

This article goes about as far as the beauty that can be found in high school mathematics and no further.

Unfortunately, the most striking examples of mathematical beauty are ones that occur in more advanced mathematics, which the article does not even hint at.

I hope that someone knowledgeable about these things can add some words about beauty in the realm of higher mathematics. 2601:200:C082:2EA0:D438:D1B0:E828:A41D (talk) 16:51, 7 June 2023 (UTC)