# What Result Is Valuable in Mathematics? [closed]

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?

• I like the spirit of this question, but I'll warn that it it risks being closed as too broad. Sep 12, 2016 at 21:07
• "[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. Sep 12, 2016 at 21:38
• The most valuable theorems are the most general ones. Sep 12, 2016 at 21:39
• Is there experimental proof of string theory? Sep 12, 2016 at 22:09
• 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".
– N.S.
Sep 13, 2016 at 7:38

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.