Wikipedia:WikiProject Mathematics/Motivation

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This is an archive of past discussions in WikiProject Mathematics. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page.

In my opinion, one serious problem with many of the mathematics articles is that they do not contain any explanation of why their topic is interesting or relevant. In the most extreme cases, they are nothing more than lists of definitions. Sheaf and Adjoint functor are examples. Here are some discussions I would like to see in such cases: Who invented the idea of a sheaf, and why? What were they studying, and how did the idea of the sheaf help them solve their problems? Were sheaves a generalization of some more specialized construction, or an attempt to formalize something people had been doing before? In what situations do sheaves arise? What other concepts are sheaves related to? In the context of what sorts of problems do people think about sheaves? The current article tells me everything about sheaves except why I should care.

Many mathematical objects are studied because they have intrinsic interest. For example, people now study topological spaces simply because topological spaces are known to be interesting. But it wasn't always so. The idea of the topological space was invented in the late 19th and early 20th century as an attempt to generalize the idea of the metric space, and the 'open set' was a generalization of the 'open ball' of a metric space; metric spaces in turn were a generalization of Euclidean spaces, invented in an attempt to get better insight into problems of real and complex analysis. Topological spaces continue this investigation.

I think many of the existing mathematics articles could be improved with the addition of some context of this type.

Dominus 18:49, 2 Nov 2003 (UTC)

Of course. In the case of sheaves, though, the story is quite complex; just because Leray was inventing the spectral sequence at the same time. There is a great deal to be said for the 'genetic' method. On the other hand clear statements are important too. It's only reasonable to say that there is a great deal of work to do.

Charles Matthews 21:07, 2 Nov 2003 (UTC)

Both motivation and clear definitions are necessary. Genetic surveys and motivation with ambiguous, vague, and imprecise definitions is interesting but confusing and ultimately useless; clear, crisp, formal and terse definitions with no idea why anyone would care is logically flawless but does nothing to promote problem-solving or connections with anything else. The sheaf article is a bad article to pick on, because that's one of the most tortuous definitions around. I seem to remember it taking almost an entire class period to just define it. Revolver 07:09, 20 Mar 2004 (UTC)

This might just be because I am fond of history, but I think putting the history of the subject near the top of the article as opposed to near the bottom goes a long way towards solving this problem. -- Miguel

I agree that that is sometimes a useful thing to do. I would like to contrast an older version of the bernoulli numbers article with a revised version.

I think the revised version of the article is substantially better, because it explains right away why the Bernoulli numbers are of interest and why they were studied. (They are coefficients of the closed form of sum(iⁿ) for various n.) The older version of the article starts of by saying they were named after Jakob Bernoulli by Abraham de Moivre (which is not particularly important) and then follows with a definition of the numbers via an exponential generating function.

This is a good example of the problem that I think many of our mathematics articles have: an overreliance on definition. In some sense, once you have the generating function, you know everything there is to know about the Bernoulli numbers. But really you know nothing of value. It is easy to imagine an identical article defining the Glubbernog numbers as being defined by some other, slightly different generating function, say one where the denominator was e^x-2 instead of e^x-1. There would be nothing in this article to suggest that the Glubbernog numbers were more or less important or interesting than the Bernoulli numbers.

It would be similarly peculiar to have an article about the Fibonacci numbers that began by describing them as being defined by the generating function 1/(1-x-x²). An article should begin with the most important fact about its subject. Even if the generating function allows convenient calculation of the numerical values, the numerical values are not what makes the Bernoulli or Fibonacci numbers important. In the case of the Fibonacci numbers, the important thing is the recurrence relation; in the case of the Bernoulli numbers, it is the Bernoulli polynomials and the sums of sequences of consecutive integer powers.

I am discussing this in detail not to criticize anyone's work on this article, but to try to make clear a specific way in which I think many other Wikipedia mathematics articles could be substantially improved.

Dominus 08:03, 6 Nov 2003 (UTC)

Some general comments about wiki work are: (a) telling others how to write may just not work; and (b) it's hypertext. On (b), objecting to definitions by themselves, and ignoring the backlinks, sometimes doesn't work. A page may be there because someone want to refer to a Laguerre polynomial, say, in some other article, where the definition would be intrusive. In fact that's how many pages are created.

Charles Matthews 09:20, 6 Nov 2003 (UTC)

I am not telling anyone how to write; I am not telling anyone how to do anything.

I have no constructive response to your hypothetical Laguerre polynomial example because there is no such article and I don't think it is useful to discuss these matters in the abstract. Perhaps if there were such an article, I would agree with you, or then again perhaps not.

Dominus 11:15, 6 Nov 2003 (UTC)

I think mathematics deserve their own mediawiki site. In that way the search would not be disturb by noise and classification would be proper. Anonymous.

I disagree. The same arguments could be made for a number of disciplines besides mathematics (that they deserve their own mediawiki sites because their terminology introduces "noise" into search results). I'm not sure what you mean by classification; the mathematics articles are fairly well classified in categories and lists. In my opinion, part of what makes Wikipedia great is that all of the encyclopedic information is in one place instead of scattered across separate wikis. - Gauge 02:07, 18 December 2005 (UTC)

[edit] Original research wiki

I have created a discussion page for the implementation of a wiki, Wikipolis, allowing for dynamic collaborations, original research, and some form of peer-review. I invite you all to add your ideas!--Hypergeometric2F1[a,b,c,x] 10:05, 15 December 2005 (UTC)