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I'm a physics student, and I know that, in physics, value is considered for compliance with experiment. No theory, no matter how beautiful, is taken serious without experimental proof. People publish papers in physics, when they find new applications of physics or when they give new meanings to some physical entities through their new theories. What result is valuable in Mathematics? Do you call it new, for example, when someone presents a fast way to compute number of primes, or when someone presents a complicated formula but a precise one, or just the beauty of the approach is valued?

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    $\begingroup$ I like the spirit of this question, but I'll warn that it it risks being closed as too broad. $\endgroup$ Sep 12, 2016 at 21:07
  • $\begingroup$ "[In physics] value is considered for compliance with experiment". True, but it's not the only form of value. If you asked a fair sample of physicist “What results are valuable in physics?” then I'd bet a good percentage would state mathematical results that at present have no relation to experimental evidence (e.g. string theory). Also that mantra you have been told that "No theory, no matter how beautiful, is taken serious without experimental proof" is not as true as it once was. $\endgroup$
    – Winther
    Sep 12, 2016 at 21:38
  • $\begingroup$ The most valuable theorems are the most general ones. $\endgroup$ Sep 12, 2016 at 21:39
  • $\begingroup$ Is there experimental proof of string theory? $\endgroup$
    – David K
    Sep 12, 2016 at 22:09
  • $\begingroup$ you are right, maybe I should have said "If one obtains experimental results opposing to those expected from the theory, the theory has no value". $\endgroup$
    – N.S.
    Sep 13, 2016 at 7:38

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One way to be valuable is to be aesthetically pleasing: $$e^{i\pi}=-1\quad \text {(Euler)}$$ A second way is to solve a problem which had baffled great mathematicians: $$x^n+y^n=z^n \;\text {has no non-trivial solutions in integers for } n\gt2 \quad \text {(Wiles) } $$ A third way is to prove a difficult and useful result in some branch of mathematics: $$ \text {Every algebraic variety in characteristic zero has a resolution of singularities } \text {(Hironaka) } $$ A fourth way is to introduce a powerful new concept: $$\text {Logarithms } \:\text {(Napier) } \quad\text {Flatness } \:\text {(Serre) } \quad \text {Distributions } \; \text {(Sobolev, Schwartz) } $$ ...and there are many, many other ways.

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I'd say it depends on the context and no real objective answer can be drawn but it's quite an interesting question.

To my mind, though it can be debated, the function of mathematics is to answer questions in the language of mathematics, which may have come from the physical world or more abstract settings. This would suggest that a result is valuable if it provides an meaningful answer to a question. The result's value may be enhanced by its clarity, simplicity, beauty and rigor but none of those characteristics are inherently necessary.

As to whether a result of fast computations of primes is valuable, per se. The speed of the algorithm would be valuable to the computer scientist but only indirectly valuable to the mathematician as a tool for their work. The real value to the mathematician comes in the answer to the question of "how fast can we make prime finding algorithms" and "what is the next biggest prime?".

The question of whether we call a mathematical object "new", is rooted in the timeless, philosophical debate of whether mathematics is invented or discovered. Which is interesting but ultimately probably useless.

As to whether complicated but precise formulae are valuable, yes, in the sense that they have precision in answering questions, but they would be more valuable if they were simpler. Ideally, mathematics shouldn't fill blackboards with long-winded, formulae that go on forever, it should get to the point in an articulate and straight-forward way.

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