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If mathematics is about finding solutions to well-defined problems, then philosophy is about finding problems in what previously we thought were well-settled solutions. Mark Steiner's The Applicability of Mathematics As a Philosophical Problem mirrors both sides of this statement, admitting that mathematics is the key to solving problems in the physical sciences, but also asserting that this very applicability of mathematics to physics constitutes a problem.
What sort of problem? According to Steiner, the reigning "ideology" or "background belief" for the natural sciences is naturalism. Typically naturalism is identified with the view that nature constitutes a closed system of causes that is devoid of miracle, teleology, or any mindlike superintendence. An immediate consequence of naturalism is that it leaves humanity with no privileged place in the scheme of things. It's this aspect of naturalism that Steiner stresses. Naturalism gives us no reason to think that investigations into nature should be, as Steiner puts it, "user-friendly" to human idiosyncrasies. And yet they are.
Steiner's point of departure is Eugene Wigner's often reprinted article "The Unreasonable Effectiveness of Mathematics in the Natural Sciences." Wigner concludes that article with a striking aphorism: "The appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve." Throughout the article Wigner refers to the "miracle" and "mystery" of mathematics in solving the problems of physics. Yet although Wigner leaves the reader with a sense of wonder, he does not indicate how this sense of wonder translates into a problem that demands resolution. Enter the philosopher Mark Steiner.
Steiner's project is to take Wigner's pretheoretic wonder at the applicability of mathematics to physics and translate it into a philosophical problem for naturalism. The applicability of mathematics to physics is not a problem for a mind-first Platonic world-view or a math-first Pythagorean world-view or a Logos-first theistic world-view. It is, however, a problem for a nature-first impersonal world-view. According to Steiner, naturalists are in no position to expect that, much less act as though, mathematics should assist in the discovery of physical insights. That naturalists do counts against their naturalism.
It is important to understand that Steiner is not simply appealing to the success of mathematics in resolving the problems of physics. It is not the isolated successes of mathematics as applied to problems in the physical sciences that for Steiner constitutes a philosophical problem (after all, there are many instances where mathematics has failed to be successfully applied to problems in physics). The problem, rather, is the global success of mathematics as a research strategy for facilitating discovery in the physical sciences.
This is a subtle point, and one impossible to convey without actual case studies from mathematics and physics. Indeed, much of Steiner's book consists of such case studies. Consider, for in stance, the physicist Paul Dirac's discovery of the positron and antiparticles more generally. The positron is a particle just like an electron, only with a positive charge. Yet when Dirac proposed the positron, there was no experimental evidence for it. Indeed, there was no reason even to expect its existence. Why, then, did Dirac propose such a particle?
Dirac was at the time trying to understand the Klein-Gordon field equation and the energy levels it assigned to certain quantum systems. He wanted to extend this equation relativistically to the electron, but he found that the only way to do so was by factoring it. Unfortunately, the equation resisted factoring over the real and complex numbers. Dirac therefore "brute-forced" the factorization by introducing higher dimensional "number-like" objects (the property where these objects differed from ordinary numbers was commutativity of multiplication).
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