Of course, elegance , as an aesthetical judgement , is very subjective. Log in or register to write something here or to contact authors. In mathematics , a proof's validity is based upon whether it consists of a sequence of logical steps that can be deduced from one another. This is an objective process. However, mathematicians, surprisingly to some, have a second criterion by which they judge proofs: elegance. Sign in Login Password remember me Lost password Sign up.
Fri Sep 22 at In particular, there have been some really elegant and surprising proofs. There must many such proofs that most of us have missed, so I'd like to see a list, an MO Greatest Hits if you will. Please include a link to the answer, so that the author gets credit and maybe a few more rep points , but also copy the proof, as it would nice to see the proofs without having to move away from the page.
I realize that one person's surprise may be another person's old hat, so that's why I'm asking for proofs that you learned from MO.
You don't have to guarantee that the proof is original. In this fantastic answer , Ashutosh proved that the Axiom of Choice is equivalent to the assertion that every set admits a group structure. Brilliant proof of fundamental theorem of algebra by Gian Maria Dall'Ara Ways to prove the fundamental theorem of algebra.
Some proofs of quadratic reciprocity: What's the "best" proof of quadratic reciprocity? I especially liked that one: What's the "best" proof of quadratic reciprocity?
Nullstellensatz using model theory What are some results in mathematics that have snappy proofs using model theory? An infinite dimensional vector space have smaller dimension than it's dual. Slick proof? Topological proof that Z is a Bezout domain. There are now various methods of estimating where a given person lies within this space e. For example Big Five profiles predict, among other things, life satisfaction [ Boyce et al.
Our conjecture is that classifying mathematicians' perceptions of the qualities of mathematical proofs is an analogous problem to the challenge of characterising human personalities. Thus we followed a research strategy analogous to that used by social psychologists interested in personalities.
First, we produced a list of adjectives which have often been used to describe mathematical proofs. Second, we asked a large number of mathematicians to think of a proof that they had recently read, and to state how accurately each adjective described that proof. Finally, we subjected these ratings to an Exploratory Factor Analysis EFA , to determine on how many broad dimensions mathematicians' perceptions of proofs vary.
We were particularly interested in using these findings to interrogate the various accounts of mathematical beauty described above. Our research strategy does not, of course, allow us to draw any conclusions about objective qualities of proofs or indeed, to say whether or not proofs have objective qualities , and neither does it allow us to understand which proofs have which qualities, or whether there are between-mathematician differences in the assessment of mathematical quality.
It does, however, allow us to investigate the structure of the language with which mathematicians characterise the qualities of mathematical proofs. What is distinctive about Naess's approach is his emphasis on empirically investigating the use of philosophically significant terms as a guide to their meaning, an approach we share. We differ from Naess in the more quantitative character of our methodology and our focus on the practice of a specific community, research mathematicians, rather than the population as a whole.
Of course, the strategy of looking for meaning in use has a much broader philosophical pedigree. It may be traced back even further to the American pragmatists of the nineteenth century, and especially to Peirce.
Our approach could be seen as an attempt systematically to study the language game which takes place when mathematicians evaluate proofs. Our first task was to select a long list of adjectives which have been used to describe mathematical proofs. Using Tao's [ ] list of mathematical qualities as a starting point, we formed a list of eighty adjectives that we conjectured may be used by mathematicians to describe the traits of mathematical proofs.
These are shown in Table 1. Like earlier researchers interested in studying empirically research mathematicians' behaviour [ Heinze, ; Inglis et al. All mathematics departments with graduate programmes ranked by U. If the department agreed, they forwarded an email invitation to participate to all research-active mathematicians in their departments.
As with all research which requires participants to give informed consent prior to participation, our participants were self-selected and so cannot be said to be a truly random sample. The email gave a brief outline of the purpose of the research, and provided a web link to the location of the study.
Participants who decided to take part first saw an introductory page which again explained the purpose and nature of the research. On the second page participants were asked to select their research area applied mathematics, pure mathematics, or statistics , and state their level of experience PhD student, postdoc, or faculty.
On the third page participants were given the following instructions: Please think of a particular proof in a paper or book which you have recently refereed or read.
Keeping this specific proof in mind, please use the rating scale below to describe how accurately each word in the table below describes the proof. Describe the proof as it was written, not how it could be written if improved or adapted.
So that you can describe the proof in an honest manner, you will not be asked to identify it or its author, and your responses will be kept in absolute confidence. Please read each word carefully, and then select the option that corresponds to how well you think it describes the proof. Emphasis in the original. Participants were then shown the list of eighty adjectives, presented in a random order, and asked to select how well each described their chosen proof using a five-point Likert scale very inaccurate, inaccurate, neither inaccurate nor accurate, accurate, very accurate.
Finally participants were thanked for their time, and invited to contact the research team if they wanted further information. A total of mathematicians participated in the study, consisting of PhD students, 23 postdocs, and 86 faculty.
Sixteen participants did not respond to one or more adjectives, resulting in a total of 20 missing values 0. Prior to conducting the main analysis, the suitability of participants' ratings for factor analysis was investigated. Thus both tests supported the use of an EFA. Participants' ratings for the 80 adjectives were entered into an EFA, using the maximum likelihood method.
Both the Scree Test and Parallel Analysis are approaches which attempt to find a balance between extracting sufficient factors to explain a high proportion of the original variance, and extracting so many that closely related latent constructs are represented. Loadings for the five extracted factors are given in Tables 2 and 3.
These numbers describe how well each adjective describes each factor: so if an adjective has a high positive loading then it is very representative of that factor, if it has a zero loading then it is independent of that factor, and if it has a high negative loading then it is very unrepresentative of that factor. Adjectives which loaded strongly onto Factors 1 aesthetics and 2 non-proofs. Adjectives which loaded strongly onto Factors 3 intricacy , 4 utility , and 5 precision.
Before describing the factors in detail we first note that Factor 2 appeared to be somewhat different from the other factors. We therefore concluded that Factor 2 was not a true dimension upon which proofs vary or at least, proofs which mathematicians have recently read or refereed and chose to think about do not vary substantially on this dimension. Consequently we do not discuss Factor 2 in the remainder of the paper. Repeating the EFA on the 60 adjectives which had mean ratings significantly greater than 2 on the 1 to 5 Likert scale resulted in four factors which were essentially identical to Factors 1, 3, 4 and 5 discussed below.
The table of factor loadings for this analysis is given in Appendix Table A1. The relationship between each adjective's mean rating, and its loading onto Factor 2.
We refer to this factor as the aesthetics dimension. We refer to this factor as the intricacy dimension. We refer to this factor as the utility dimension. We refer to this factor as the precision dimension.
We structure our discussion of these findings in five sections. First we summarise our main results, then we discuss the implications of these findings for the three accounts of mathematical beauty discussed in the introduction. Finally, we discuss the role of empirical data in discussions of mathematical practice, and argue that reflective reports from mathematicians must be treated with appropriate caution. Our analysis indicated that mathematicians' appreciation of the qualities of mathematical proofs can be reasonably well understood using four-dimensional space.
Looking directly at the correlates of beauty revealed that there appears to be no relationship between a proof's perceived beauty and its perceived simplicity, contrary to the classical view espoused by many mathematicians and philosophers. As noted in the introduction, the classical idea that mathematical proofs tend to be regarded as beautiful if they are simple has been supported by many notable mathematicians and philosophers [ Engler, ; Wells, ; McAllister, ; Tappenden, ; Chandrasekhar, ; Cherniwchan et al.
We found no evidence for this view. How plausible is this latter suggestion? We do not believe that this is the case. This says that if you have a planar graph a network of vertices and edges in the plane that stays connected if you remove one or two vertices, then there is a convex polyhedron that has exactly the same connectivity pattern.
And each of these three has variations. So, having several proofs leads you to several ways to understand the situation beyond the original basic theorem. And it was an amazing surprise — why should this topological tool prove a graph theoretic thing? This turned into a whole industry of using topological tools to prove discrete mathematics theorems.
And now it seems inevitable that you use these, and very natural and straightforward. On the other hand, we are always happy if we manage to prove something with one surprising idea, and proofs with two surprising ideas are even more magical but still harder to find. So a proof that is a hundred pages long and has a hundred surprising ideas — how should a human ever find it? This is a hundred pages, or many hundred pages, depending on how much number theory you assume when you start.
And my understanding is that there are lots of beautiful observations and ideas in there. By definition, a proof that eats more than 10 pages cannot be a proof for our book.
0コメント