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This question made me think a bit about how mathematics can be used in such a way that society benefits from it.

I think there are quite a lot of good answers to the aforementioned question. Still, it seems to me that most of the ways to do mathematics "for good" involves doing some sort of applied mathematics, which means you often have to worry about other things rather than the mathematics itself. For me, however, part of the appeal to mathematics stems from the fact that you can work on a problem without having to resort to making observations of "the real world". I guess it's a form of escapism, it's just you and the problem and while your working on it, there aren't a lot of other things on your mind.

Even though the pursuit of solving a purely mathematical problems can be very stimulating and satisfying if you manage to solve it, I sometimes wish that I could work on something of which I would know it would have a positive impact on society. Sure, you know that there's a chance it will be applied to solve a problem in the empirical sciences, but you don't know that for sure. Hardy didn't know his contributions to number theory , which he did not consider to be useful ("Nothing I have ever done is of the slightest practical use"), turned out to come in very handy when designing cryptographic algorithms (See the RSA algorithm).

I was wondering if there are any purely mathematical problems and conjectures of which it known to have a positive impact on society, once they are solved. By a "mathematical problem" I don't mean "Try to find ways to approximate the solution to some set of partial differential equations more quickly, because it would be useful to predict the weather". That's too vague. I'm talking about genuine, purely mathematical problems and conjectures. Please describe exactly how the resolution of the problem and/or conjecture would be of any "good" to society.

Thanks in advance

Perhaps I should include a partial list of things that I would consider "good":

  • Improving on ways to model the human body through scans. Are there any open mathematical problems in Tomography?
  • Are there any mathematical problems that relate to the the rise of global warming and how we can prevent it?
  • Can we solve parts of the energy crisis through pure mathematics? The research area dynamical billiards comes to mind. Perhaps we can formalize the idea of "trapping" a light ray in a solar cell, enabling us to generate more electricity through these cells.
  • Are there any mathematical problems related to the process of Nuclear Fusion?
  • Which conjectures could enable us to create the Smart grid, once there is a resolution of these conjectures?
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I don't know the details, but I heard that many conjectures in number theory guarantee the efficiency of crypting algorithms that are currently used. so if one were to come up with a counter example, he could (potentially) hack a lot of stuff! (I don't know about "positive" though =) ) –  Glougloubarbaki Dec 7 '11 at 16:26
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In my opinion this is extremely subjective and should be closed. Moreover, it is essentially impossible to answer; we can never know the net effect of anything. For example, dynamite was used to kill plenty of people; but it's also helped us perform tasks that used to have to be done with manual labor. But is that a good thing if people lost their jobs because of it? And so on. Mathematics and its applications provide us with tools (physical, algorithmic, etc.). Any tool can used for "good" or for "bad" ends; that is up to us. There is no (and cannot be) a direct line math $\rightarrow$ good. –  Zev Chonoles Dec 7 '11 at 16:27
    
@ZevChonoles: Why is this too subjective while the other question I gave a link to isn't? Should I perhaps include a list of things that I would consider to be "good"? –  Max Muller Dec 7 '11 at 16:30
    
I think that question essentially "got away" from us, and there were luckily no major arguments. When a question is going strong I'll admit it's more appealing to leave it open. Perhaps we should close it now (23 answers is more than enough anyway). anon's top comment on the question illustrates the problem I am concerned about, that it would not only turn into a discussion about whether something is good or not, but additionally what the "total" effect of it would be. In this case, I would like to get the input of others on the matter before taking action on either question. –  Zev Chonoles Dec 7 '11 at 16:35
    
I think the question would be fine if somewhat rephrased. If it were more about "what practical problems are unsolved because of a lack of mathematical tools?" or something of that nature it would probably be a good discussion. "good of society" is subjective/argumentative, "allows us to do something we couldn't before" is not, or at least much less so. –  Robert Mastragostino Jan 18 '13 at 18:33

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