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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

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It seems to me that you have two questions here.

First, you inquire about a formal account of "usefulness". I believe that this is already provided by the formal mathematical accounts of utility in utility theory. The concept of utility in that theory is extremely flexible, not limited to economics or any other specific endeavor. Thus, it seems able to provide for any account of "usefulness" you may have in mind. Let's just say that the utility provided by a given thing is equal to the "usefulness" you had in mind for it.

Your second question is more directly aimed at analyzing the usefulness of various specific mathematical ideas. For this question, I'm not sure that what is lacking is a formal definition of usefulness. After all, even if one knows a lot of formal utility theory, it doesn't help you to find out which flavor of ice cream your child likes best. Rather, what one would seem to want is ways of measuring various specific measurable aspects of that utility function. Thus, it is a problem of measurement, rather than formal theory. In the case of measuring the importance of utility or usefulness of various mathematical theorems or definitions, several people have suggested a page-rank type calculation, based on citation statistics, which I find interesting.

Another approach to this second question is the one I described in my answer to the question here, which is to analyze the mathematical relationships between the various theorems of mathematics, over a very weak base theory. This subject is known as Reverse Mathematics, and one of the most surprising conclusions (not at all obvious) of this research effort is that the great majority of classical mathematical theorems (and contemporary ones as well) fall into one of five equivalence classes. That is, most theorems turn out to be logically equivalent to one of the big five. This kind of analysis may lead you to abandon what might otherwise have been a tempting principle: that logically equivalent theorems should be equally useful.

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[Thanks for the reminder to look more closely at Reverse Mathematics]. I agree that 'logical equivalence' is too crude a metric: the length of the necessary contortions (the computational content of the proof?) to use one of the 'equivalent' theorems should be counted as part of the utility measurement. Utility is related to convenience, and it is inconvenient to have to translate. Your point that this is a problem of measurement is well-made, I will have to think about that some more. –  Jacques Carette Apr 13 '10 at 20:18
But why should utility just measure convenience? The point of utility theory is that the utility function is measuring some kind of value, whatever notion of value is desired. If you want a formal theory of "usefulness", then let utility measure it and you have a formal account of usefulness. –  Joel David Hamkins Apr 13 '10 at 20:49
Not just 'convenience' is important. And utility theory is too general, one needs to narrow down the notion of 'value' more for this to be meaningful. So part of the question is to also understand what should be the notion of 'value'. PageRank and Kolmogorov Complexity are two guides here. –  Jacques Carette Apr 13 '10 at 22:35

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