I don’t think math is a science. Math is a valid field in its own right. We don’t have to call it a science to give it validity (though many seem to think calling it something else is somehow a slam against math). But the nature of testing ideas for acceptance is fundamentally different in math than in science.
Science, as I understand it, requires acquiescence to an external universe. As Feynman said “It doesn’t matter how smart you are. It doesn’t matter how beautiful your theory is. If it doesn’t agree with experiment, you are wrong.”[I’m going by memory so that might be perfectly verbatim].
This is qualitatively different from what mathematicians do. In math, the test of validity is completely internal, completely intellectual. In math, you can’t be perfectly logical and end up wrong; in science you can - it happens all the time. Deferring to the judgement of the universe, THAT is the hallmark of science. I mean, astrologers abstract, model and predict, even if they don’t do it very well or very sensibly. I don’t think astrology is just a bad science, though. It isn’t a science at all! That’s because astrology doesn’t defer to the results of the experimental tests.
This, in fact, is one of the problems with modern theoretical physics, in my opinion. When people criticized Heisenberg for his uncertainty principle on the grounds that it did not define position and momentum very well (in a certain sense we need not go into here), he responded that he didn’t HAVE to define them that well, because the UNIVERSE hadn’t: If there is a measurement that cannot be made, a theory NEED NOT POSTULATE such a quantity. (You may postulate such a quantity, but you don’t HAVE to.) A lot of modern physics explores questions whose results cannot be subject to experiment, at least not at this time. In the spirit of Heisenberg, cannot one ask of the most abstract theoretical physics “Why is it necessary that I incorporate these results into my world view, especially when it is (sometimes) so damn complicated to do so?” They reduce to no more and no less than mathematical exercises. One may treat them as important aspects of the truth, but since they have no present experimental ramifications, one need not consider them important. Interesting, perhaps, even important, sometimes, for what they imply must be true (IF we assume everything worked out previously is the complete story and correct) but IMO, from a scientific point of view, a lot of this is just idle curiosities. Well, Okay, I’m exagerating with that statement. Theoreticians tell us what the implications are of our physical laws. If we find a contradiction, THAT’s very important. But if we find out that black holes evaporate due to quanta tunneling their way free (to put it simplistically), well…That makes sense, and it is compatible with everything else we know (or think we know), but is it the Truth ? Is the prediction alone, elegant as it is, sufficient proof? There are arguments on both sides, but the traditional answer in science would be that it must remain no more than a hypothesis until subjected to experimental evaluation. Physicists hate having this pointed out: Their intellectual capacity to model has exceeded their physical ability to keep up through experiments (not in every area, but in a LOT of the hottest fields of theoretical physics). The implications for their field and their careers should this view - that such untestable work is scientifically questionable - become widespread are, shall we say, problematic. Self interest has led theoreticians to do a full-scale rethinking of what science should be; mathematical consistency is sufficient, they often claim now.
I remain very thoroughly unconvinced by this assertion.
So I submit that a refocus on integrative principles rather than further reduction, which would result in predictions on an experimentally accessible scale, would be a boon to physics as well as the other sciences. Or “pseudosciences” to Lib.