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At some point this provocative question came to my mind:

Are mathematics mostly driven by applications?

I am taking into account some of the comments made to my original question so I want to make the question more precise. To me it is clear that the answer is NO when we consider only applications outside mathematics. But let us consider also applications inside mathematics.

Are mathematics mostly driven by applications: whether inside or outside mathematics?

I think that even Hardy would agree that it can become an interesting question if we allow for a wide definition of application. I believe this is in part true since many of the "beautiful" results that cannot be used to create new mathematics in other areas or even in the same field are only collected and stored as "mathematical culture". Solving big conjectures on the contrary usually requires mathematicians to develop new powerful tools that are then exploited in other areas.

One historical example of the dissipation of a hot topic from the main mathematical arena is the case of analytic sets. They are still somewhat important in Set Theory, yet, some 80-60 year ago they encouraged incredible mathematicians such as Lusin, Sierpinki or Kuratowski to write long monographs since they believe their existence (which by the way was found by Suslin who found an error in a manuscript by Lebesgue) was of huge importance. It seems that today only set theorist and some analysts are well-acquainted with the existence of this sets. In the same fashion, the importance of studying Baire functions and meager sets seemed to decay outside set theory as more powerful tools for studying continuity emerged. I also believe Measure Theory became so dominant because it was useful to the theory of integration. So,

What are examples of discoveries in your fields of study in mathematics that created a revolution (or at least were and are still very important) and were made by the sole motivation of that we call mathematical beauty?

I apologize in advance if the question is a nonsense. But I am really interested in trying to answer what is mathematical beauty and how it relates to beauty in other arts and I would like to know what are the topics and who are the mathematicians that did their job not believing it was important for something else but because they found it beautiful.

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You don't use the "¿" sign in English. –  Twink Sep 3 '13 at 3:04
    
Indeed, I am so sorry. I've changed it. –  Mauricio G Tec Sep 3 '13 at 3:06
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Folks around here often say that they do mathematics just "because it's fun", and that applications are neither envisioned nor desired. That's fine with me -- I think mathematics is entertaining and beautiful, too. But I wonder how John Q. Public would react if he found out that his taxes pay for these guys to sit around all day having fun. –  bubba Sep 3 '13 at 5:01
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You may be interested in this paper: A Definition of Mathematical Beauty and Its History –  Viktor Blasjo Sep 3 '13 at 5:49
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@Qiaochu -- "it's pretty important to distinguish applications within mathematics and applications outside of mathematics". Indeed. So-called "applications" within mathematics always feel like cheating, to me -- a bit too parochial and incestuous. Theoretical areas of physics and economics aren't much better. An "application" ought to be something that makes a difference in everyday life -- our cars look nicer, our planes fly faster, with less fuel, our bombs explode better, our diseases are diminished, etc, etc. These (for better or worse) are "applications", in my view. –  bubba Sep 3 '13 at 6:38
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