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Older days scientists were universalists and philosophy, physics and mathematics were a part the same question - understanding the world. Nowadays one may get feeling that the role of applications in development of modern mathematics is negligible - of course it depends on the field. And aim of the question is to get different opinions from different points of view.

Question 1 What is the role of applications in modern mathematics ?

Question 2 Different countries have different mechanisms to stimulate interaction between mathematics and applications - what are these mechanisms and what are their advantages and disadvantages ?

Question 3 (for pure mathematicians) What is your personal stance on applications ? Is it out of your scope of interests or you are have (trying to have) some contact with applications ?

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closed as not constructive by Federico Poloni, Douglas Zare, Felipe Voloch, Emil Jeřábek, Mark Sapir Mar 9 '13 at 1:48

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It might be better to ask each question separately, since the questions are very different. –  Ben McKay Mar 6 '13 at 20:16
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META discussion tea.mathoverflow.net/discussion/1551/… –  Alexander Chervov Mar 9 '13 at 7:37

4 Answers 4

Question 1 What is the role of applications in modern mathematics ?

I don't know if there is such a thing as THE role of applications, but I think it's a great motivation for developing a theory and/or solving a particular problem if it has an application outside of mathematics.

Then again, the same can be said about applications to different branches of mathematics. Besides, applications are not the only source of motivation to do mathematics. Perhaps, one thing real-life applications play an important role in is to make it easier for non-mathematicians to see that mathematics isn't pretentious junk art with no practical or intelectual value. This isn't to say that mathematics with no real-life applications is junk art. But if it has a practical application, it'd be a lot easier to defend your own work from such criticisms from outsiders.

Question 2 Different countries have different mechanisms to stimulate interaction between mathematics and applications - what are these mechanisms and what are their advantages and disadvantages ?

I don't know much about other countries and other branches of mathematics, but it appears that in Japanese universities discrete mathematicians tend to work at non-math departments (typically but not exclusively departments of computer science or computer engineering) slightly more often than in some other countries, regardless of whether their work has immediate applications to the respective fields. And when I say discrete mathematics, I include some branches that some may not consider discrete mathematics, such as number theory and algebra.

From graduate students' perspective, this is a great thing because you can work on "mathematics for the sake of mathematics" if that's your thing while naturally getting a lot of exposure to the applied side of mathematics. For example, I finished my undergraduate study at a traditional mathematics department and went to the graduate school of mathematics of the same university. Then a couple years later, I transfered to the information science department at a different university. I didn't change my research topic during my Ph.D. program, but this transfer definitely influenced my view and attitude toward mathematics and gave me a lot of opportunities to learn stuff outside mathematics, which I don't think would have occurred, at least not to the same degree, if I stayed at the math department.

The negative side effect I hear from faculties is that it makes it harder to get graduate students with strong backgrounds in mathematics because math majors tend to apply for math graduate programs like I did at first.

Question 3 (for pure mathematicians)

Hmm... I don't know how you define that pure mathematicians thing, but looking at my own publication list, I guess I'm not pure or innocent anymore. Oh, well.

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These questions are too broad, and not very clearly stated. First of all what's "an application"? Application to what? Is application to some "physics" which itself has little or no connection to real world (like string theory) counted as an application? We all know examples when this kind of "physics" stimulate a lot of interesting mathematics. Is an application in a completely different area of mathematics counted as an application?

Or only an application which brings profit is counted?

Anyway, no matter how one defines an application, my answers are these.

  1. Outside applications play an important role for mathematics as a whole. Probably as important as it was in the past. There is a huge (and not very well defined) part of mathematics which is called "applied mathematics". This usually does not include "mathematical physics" (at least in the US. But this is a matter of labels).

What goes under the label of pure mathematics, also frequently has applications, sometimes very important.

Second question. The mechanism that I know is financial. Those who distribute money want "aplications". In many cases they do not really understand what it means but they want it to be called this way. Various financing is available for applications, real and imaginary. Sometimes, there is an administrative pressure. But some money is usually involved behind it.

Third question. I qualify myself as a "pure mathematician". I define this as follows: the main criterion for choice of a problem to work on is usually aesthetic; I just like the problem. An alternative consideration is that a problem has potential applications in the real world. This is rarely a motivation for me.

Sometimes my results find applications in various areas, like material science, computer science (always unexpected to me). By "applications" here I mean that scientists from other sciences (who do not call themselves mathematicians) sometimes cite and use my results. I understand that parts of other sciences can be also very remote from the real world. But I am always pleased when people from outside use my results.

Sometimes an applied mathematician or non-mathematician asks a math question. I always try to help if I can. Sometimes I can. Sometimes the problem is even mathematically interesting. This is also very pleasant. In my youth, I was sometimes involved in "applied research" for money and other benefits. I did not like it. I'd rather teach to make my living.

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Answer to Question #3: As a pure mathematician I know that what I do has an application, but I don't know what that is. Would I be curious how it's applied? You're damn right I would, for knowing would help me to live and better judge my own place in history. Considering that the old dichotomy of the contexts of justification and discovery has been shown to be a false way of looking at the sciences, even the purest of mathematicians should be interested in the history of their own ideas!

Paul

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The question is vague, and the answer could be used against "pure" mathematicians, especially by people in administrative positions.

First of all, let me say that as far as "applied math" is concerned, I am on the same page with the late V.I. Arnold who liked to say that there is no applied math, there are only applications of math. I say this as a person who started as a hard core "applied mathematician", doing optimal control.

At some point I found the experience intellectually unsatisfying and I moved on to "pure" math. This is not a value judgement, it's a matter of taste. I still keep an eye open towards "applications", because I sometime see glimpses of interesting math.

Like other poster's I am confused by the term application. Does applying analysis to answer e.g. a famous topology question count as application? (Perelman comes to mind.)

Do Persi Diaconis' card tricks (backed by highly nontrivial math) count as applications?

Do we rank applications according to the number of zero's in a research grant?

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Thank you for the answer. My question is about applications of math outside math. I do not see the sense of specifying the "application" very precisely - hope everybody understands vague meaning and that is enough. I "assume a good will" - if some one thinks it is worth to write an in the answer about what he thinks deserves to be shared with the community - go on... –  Alexander Chervov Mar 7 '13 at 11:33

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