# Formal definition of 'useful' ?

Has anyone worked out a formal, general-enough definition of what is 'useful', so that it could reflectively be used in mathematics? I am aware of the work in utility theory from economics (but originally from the Bernoullis and improved by von Neumann, so very much 'mathematical'). Such a formalization should be adequate to decide if a particular definition (or theorem) is considered 'useful'.

Note that I fully expect utility to be a relative notion, in other words I don't expect anything to be 'universally useful'. I have some tentative definitions, but before I spend too much time working this out, I would like to know if this has already been done mathematically (as the work of economists on this is [expectedly] too biased towards economic utility).

A concrete example: 20 years ago, elliptic curves would have been considered 'not useful' in the context of cryptography, now it is considered 'useful'. This can be made completely formal. [In other words, my question is about what has been done before, not a discussion of what this is, which if off-topic for MO].

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If we do not know what this is, how can we tell if this has been done before? – Mariano Suárez-Alvarez Apr 13 '10 at 18:49
Sounds like you want Google page rank for references to theorems. It could be pretty interesting to see what rank various theorems get. – Dan Piponi Apr 13 '10 at 18:53
Possible duplicate: mathoverflow.net/questions/17964 – Joel David Hamkins Apr 13 '10 at 19:05
I don't think a formal definition of "utility" would be of any utility. – Qfwfq Apr 13 '10 at 19:16
@sigfpe: I had not thought of it that way, but that is a good analogy. One way to think of it is in term of Kolmogorov Complexity: a theorem is useful if it allows the 'compression' of the formal development of some further pieces of mathematics. It is 'useful' because it expresses an idea which can be re-used. In this sense, Group Theory was shown very useful a long time ago, and now Monoids and Monads are exploding in CS. – Jacques Carette Apr 13 '10 at 19:45