I am looking for a poll in which mathematicians were begged asked requested to explain what they think that mathematics essentially is and is about. Answers should be rather short, but not necessarily as short as e.g. "the study of abstract structures and their properties and relationships". The more participants of the poll, the better (of course), and the higher their reputation, the better, too.

Does anybody know of such a poll?

(I would not be unhappy if one or the other would give her own opinion here.)

  • $\begingroup$ How about this math.SE thread? Also, "begged" seems like a strange word to use here. $\endgroup$ – Zev Chonoles Jun 19 '13 at 22:09
  • $\begingroup$ I believed "begged" to be more polite. But I changed it to "asked", anyway. $\endgroup$ – Hans-Peter Stricker Jun 19 '13 at 22:13
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    $\begingroup$ Using begged makes it sound like mathematicians are conceited or the asker is desperate. :-P Use requested if you want to sound polite. $\endgroup$ – wjm Jun 19 '13 at 22:17

Some years ago, Giandomenico Sica asked me and various other mathematicians to write definitions of "mathematics", eventually to appear in a book (provisionally) called "The Dictionary". A number of the definitions were available on line shortly afterward, but I can't find them now. What I wrote is undoubtedly longer than what you're looking for, but here it is anyway:

Mathematics is the study of abstract patterns.

Whenever phenomena in nature, in society, even in mathematics itself exhibit similarities, the opportunity arises to abstract from the particular phenomena and investigate the similarities, and the result can be called mathematics. Even such a primitive fact as 2+3=5 is an abstraction, indicating what happens when we count stones, or people, or stars; the mathematical fact persists even when the context changes dramatically.

In order to ensure this persistence, mathematicians generally insist on precise statements of the hypotheses under which any result is applicable, and they insist on airtight deductive reasoning to establish facts and preclude the possibility of overlooking some necessary hypothesis. The deductive approach, in particular, distinguishes mathematics from the natural sciences, where experimental verification is the accepted criterion for validity. Thus, for example, the Riemann Hypothesis, which asserts that a certain infinite sequence of points lies on a single line, has been verified for the first several billion points of the sequence, but, despite this overwhelming experimental evidence, it remains a hypothesis, not an established mathematical fact.

Much of mathematics, having emerged by abstraction from natural phenomena, is in turn applicable to natural phenomena. Of course, in such applications, one must be careful to ensure that the phenomena to which one applies mathematical results are of the appropriate sort. Even the simple equation 2+3=5 requires, for its application to counting objects, the assumptions that these objects do not vanish, merge, reproduce, etc.

Remarkably, the mathematics obtained by abstracting the patterns found in one context often turns out to be applicable to other contexts that, at first sight, look completely different. Even when the original context was within mathematics itself, so that the abstraction was considered pure rather than applied mathematics, applications to natural science often followed.

In addition to its utility, mathematics has an aesthetic aspect. The patterns studied are often beautiful, sometimes in a sense that can be appreciated by almost anyone (e.g., pictures of certain fractals) and sometimes in a sense that can be appreciated only by those with mathematical training. Many mathematicians chose to become mathematicians not because of the prospect of applications but because of the beauty and (perhaps closely related to beauty) the intellectual challenge of the subject.

  • $\begingroup$ Thank you very much, especially for your definition! Where do you think can I find out more about Sica's project? $\endgroup$ – Hans-Peter Stricker Jun 20 '13 at 9:49
  • $\begingroup$ @HansStricker As I said, the project seems to have vanished from the web. Shortly after I wrote this defnition, it and other people's definitions were at polimetrica.eu but my browser now claims this site can't be found. $\endgroup$ – Andreas Blass Jun 22 '13 at 1:23
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    $\begingroup$ @HansStricker Incidentally, while looking for something entirely different, I came across the following definition, attributed to L.E.J. Brouwer: "Mathematics is nothing more, nothing less, than the exact part of our thinking." $\endgroup$ – Andreas Blass Jun 22 '13 at 1:24

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