I think I am coming to a good, new understanding of the relationship of pure mathematics to the natural sciences. A major concern of mine is just how reliable is rigorous (characteristically "pure") mathematics for applications in the natural sciences.
One very notable epiphany I've encountered recently is in relation to Oystein Ore's commentary about how mathematics has had a tremendous history in the form of "brain teasers" purely for entertainment purposes. [Number Theory and its History by Oystein Ore (Sterling Professor of Mathematics, Yale U. (McGraw-Hill, 1948), pp. 25-26]
This leads me to believe, and to much more readily accept, the idea that, culturally, the primary motivation of modern pure mathematics is simply entertainment as a kind of game and not at all for applications in the natural sciences.
In fact, it seems mathematics faculty in the great universities disdain the fact that they have to teach mathematics courses for physicists and engineers. And likewise, it seems physics textbooks often care very little about mathematical rigor. Generally, it seems physicists care more about arriving at physically testable hypotheses or standard theories than about how they arrived at those hypotheses. But physicists will engage in mathematical rigor if such rigor is within easy reach.
I have seen some literature attempting some sort of an axiomatic development of one or another physical theory. But it seems there has not been much interest in this idea. However, Euclidean geometry stands as an example of the idea of using rigorous argument to develop empirically testable, "real world" theorems.
A key motivation is the hope and hypothesis that a more mathematically rigorous theory will create far more reliable conclusions of a physically testable nature so that there is a far lesser burden on the need for (often rather expensive) experimental testing of conclusions. But is this hope and hypothesis warranted? I am thinking in particular here of what seems an excellent advanced textbook in classical mechanics that nevertheless seems to have some flaws or omissions in rigor: H. C. Corben and Philip Stehle: Classical Mechanics, 2nd Ed. (Dover, 1977). How important is such mathematical rigor from the standpoint of scientific methodology?
An expanded, advanced, well-developed theory of the relationship of mathematical rigor to the complexity and costs of experimental testing seems likely, but it seems the basic questions here need to be addressed before running over hill and dale in the development of expansive, potentially tenuous theories.
So when it comes to the applicability of rigorously derived mathematics to, say, physics, the idea seems to be simply to "play with it" and see if you might come up with some sort of interesting, testable hypotheses. It seems there is very little motivation culturally for thinking that rigor in pure mathematical theory has any bearing on developing good, testable hypotheses in physics.
That seems to be a current fact of life culturally about the relationship of mathematical rigor to credibility or plausibility of physical hypotheses which may result from such rigorous treatment.
Can anyone provide more insight into this matter or help to correct any misconceptions I may have about this?