The phrase, “the unreasonable effectiveness of mathematics” goes back to the title of an essay by the physicist Eugene Wigner in 1960. He points out that mathematics is developed largely on aesthetic grounds, and yet large parts of it eventually get co-opted by physics and the other natural sciences to formalize parts of their theories. There seems to be no reason to believe that mathematics (especially the limited fragment of mathematics that humans actually get around to developing) should have anything to do with the physical world. He then goes on to point out how surprising it should be that it’s even possible to formulate laws of physics in the first place, let alone that they should be mathematical. And he spends the last little bit of the essay discussing the conceivability both of finding a unified theory to which all our scientific theories are approximations, or of the impossibility of such a theory, which would leave us with multiple contradictory theories, each good for its own domain. The fact that we’ve managed to come so far seems to cry out for explanation.
Wigner seems to miss some aspects of the development of mathematics though. He suggests that mathematicians find something beautiful and develop it, but doesn’t point out that these theories are actually very often developed just to explain things in already-established areas of mathematics. For instance, complex numbers were developed to fill in the steps in the solution of certain cubic equations over the real numbers. At least some of the theory of groups was first developed specifically by Galois and Abel to show why there was no corresponding method for solving quintic equations. If all of mathematics was developed for motivations resembling these (as I think plausible), then once we realize that the very basic parts of mathematics are applicable, it may be no surprise that the rest of it is as well. If the natural numbers apply to some phenomenon, and some other theory was developed to explain the natural numbers, then it seems plausible that this theory would be applicable to the explanation of the phenomenon the natural numbers apply to.
Of course, this still leaves open the question of why so much mathematics seems to apply in contexts other than these. If group theory was developed to explain properties of real numbers and other fields, then why should it apply to the fundamental particles of physics in a context independent of any such field?
Greg Frost-Arnold has a fascinating post suggesting that in fact in pre-Galilean astronomy, the effectiveness of mathematics might not have seemed so unreasonable. After all, they believed then that the “heavenly bodies” had similar properties of permanence and perfection to the objects of mathematics. And if they were all created by the same God, then it would make sense that mathematics and astronomy had a lot of overlap. The effectiveness only started seeming really unreasonable when Newton showed that there were mathematical theories unifying earthly and astronomical motion.
At any rate, this contemporary effectiveness of mathematics, which seems so unreasonable, for some reason hasn’t been a very central question in the philosophy of mathematics. Instead, people have focused on more foundational questions about mathematics, like what the nature of mathematical truth is, and how it is that we have access to it. But I think Hartry Field’s program in Science Without Numbers gives the closest thing to an explanation for the effectiveness of mathematics. His main goal is to prove a certain claim about the ontology of mathematics (namely, that there is none), but I think it’s more successful as an extension of the methods of Krantz, Luce, Suppes, and Tversky in their Foundations of Measurement to explain how mathematics can be applied in a rigorous manner. He formulates the axioms of Newtonian mechanics in a way that the mathematics that is applied to it can be straightforwardly seen to be a conservative extension. Thus, he justifies this application.
Michael Dummett, in “What is Mathematics About?” criticizes this program, saying that “Field envisages the justification of his conservative extension thesis as being accomplished only piecemeal.” Dummett suggests that this would be unsatisfying, because it would never make mathematics completely justified, but only justify particular applications of particular theories. Whether or not he’s right that this is all that Field would accomplish (Field seems to claim to have shown that all of mathematics is conservative over any non-mathematical theory), I think that this is actually almost exactly the goal we should want to achieve. It wouldn’t do to suggest that any mathematical theory can be applied to any aspect of the world – there’s only certain applications that make sense, and only those should be justified. We would still face some puzzles as to why it is that so much mathematics ends up applying to so much of the physical world, but at least each particular application would no longer seem so unreasonable.