how to weigh the conditions given in a proposition

Question

As we can see,there are some conditions given in a proposition. If there are 2 propositions having approximate conclusions.Usually，we can name one propositions gives stronger conditions than the other. My question is that whether there is a good system to weigh the conditions given in a proposition.e.g. a condition "uniform convergence"is stronger than a condition"convergence". Can we give each condition a value to show their sharpness?

Answer 1

I think that one may have there three kinds of valuations for "strength" of conditions in theorem.

• practical ones: for example some conditions may be easier to check, or even define whilst other may be difficult to check or even define in practical causes. Then You may use fro example "computational complexity" of conditions as such kind of valuations for theorems. Conditions which needs more computations or resources or time in order to qualify are less useful. It may be somehow tricky because some conditions may be obvious although complex to check in practice: for example continuity of function is difficult to check in computational way but sometimes is obvious in physics for example. So it should be distinguished if You want build "theory of such valuations" or only measure difficulty in practical cases. In this kinds of valuation You measure practical obstacles connected with some condition. In fact probably "strong" conditions from point of view of pure mathematics, which gives You theorem which may be hard to use in theoretical math because of its requirements, maybe have easier conditions to check computationally, as they will be very specific, and have broad applications in practical computations and physics for example.

• aesthetic one: this is very interesting cause. It is extremely difficult to formalize someone aesthetics. You may see that sometimes aesthetic means to be symmetric but it is somehow non clear what kind of symmetry You mean when we spoke about conditions of theorems. Also it is well known that sometimes very simple axioms for example, may be most restrictive - strong, whilst very complicated one and not so pretty, may be week enough to give very interesting structure for theory for example. So here You will probably do not get any progress...

• last one criterion by size of structure: restrictive in intuitive meaning is as follows: the more restrictive criteria has less positive cases. So it may be measure by size of positive cases space. This is probably the most obvious case, because beautiful and meaningful theorem should have week conditions and strong thesis, so it may concern about big class of objects and say something very certain about them. But if You measure size of structure, You may probably get the same size structures not equally interesting. For example class of continuous functions is very big, but modular functions may be concerned as much more interesting as "single object". So You have to relate "strength of requirements" to "size of set obtained in thesis". Regarding criteria, should You add also criteria defining modular function to the list in this case? You want to compare theorems from different theories or within certain one only?

I suppose that aesthetic conditions are interesting but hard to formalization, whilst practical one are something You may think about in constructive and useful way but probably it is not You are asking for...