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The factoring worked and gave Dirac the relativistic solution he wanted for the electron. But because the "number-like" objects he introduced also had a higher dimension than the ordinary numbers, Dirac's solution to the Klein-Gordon equation also yielded extra solutions—solutions corresponding to the extra dimensions of his "number-like" objects. One of the solutions suggested a positively charged particle that in every other way was identical to the electron. What started as a mathematical trick designed to factor an equation and yield insight into the electron therefore yielded an entirely new particle and, indeed, an entirely new type of matter—antimatter, the discovery of which fundamentally altered our understanding of the physical universe.
Dirac's mathematical manipulations and physical speculations would have remained just that except for two facts: (1) In 1932 Carl Anderson experimentally confirmed the existence of the positron. (2) In the nineteenth century mathematicians had already constructed the "number-like" objects that Dirac needed to factor the Klein-Gordon equation. They are known today collectively as the Clifford algebra, and Dirac had to reinvent it to get a relativistic equation for the electron.
Where is the philosophical problem for naturalism in examples like this (and Steiner makes clear that such examples are wide spread throughout mathematics in its application to physics)? The problem is that mathematics is a thoroughly human enterprise. Nature may condition us to see patterns that are readily perceived—that, as it were, ride on the surface structure of nature. At the same time, nature should be indifferent to human idiosyncrasies. Thus, the problem for naturalism posed by Dirac's reinvention of the Clifford algebra and subsequent discovery of antimatter is that it occurred entirely through the manipulation of humanly constructed notations, and with attention not to physical reality but to human convenience.
Equations that are factorable are much easier for us to deal with than those that are not. Factorability, however, has no physical significance. A world indifferent to us has no stake in rendering itself intelligible to us by making the equations that describe it factorable through some mathematical device (like the Clifford algebra). And yet precisely such idiosyncratic manipulations of humanly constructed notations result in genuine and previously unsuspected physical insights.
There really is a problem here for naturalism. As Steiner notes, in every other area where human constructions are manipulated according to human convenience, naturalism expects and indeed confirms no profound in sight into the structure of the world. The rules of chess, for instance, do not yield insight into the structure of the atom. The study of palindromes (sentences that read the same backward as forward; e.g., "Madam, I'm Adam") tells us nothing about the first three minutes after the Big Bang.
Indeed, the claim that human constructions manipulated according to human convenience supply insights into reality belongs to what traditionally has been called magic—the view that what humans do in the purely human world (i.e., the microcosm) mirrors the deep structure of the world at large (i.e., the macrocosm). Naturalism has no place for magic. And yet the applicability of mathematics to physics is magic. Ac cording to Steiner, mathematics is the last redoubt of magic, but one that stands se cure and is in no danger of naturalistic debunking. This is a user-friendly world where we humans are the users, and where the tool of discovery that renders the natural world friendly is mathematics.
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