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
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).
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 insight 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.
In place of naturalism, Steiner therefore opts for an anthropocentrism which
affirms that humans do have a privileged place in the scheme of things.
Steiner's anthropocentrism falls short of a full-blown metaphysical position
like Judeo-Christian theism, Platonism, or Pythagoreanism. But it stands
sharply against the widely held evolutionary view that humans are mere
accidents of natural history.
The Applicability of Mathematics As a Philosophical Problem is a
technical contribution to analytic philosophy that presupposes not just a
background in philosophy but also extensive exposure to mathematics and
physics. Readers without the relevant technical background should be prepared
to find no more than 20 percent of the book intelligible. Even so, Steiner's
challenge to naturalism is accessible, powerful, and well worth pondering.
William A. Dembski is a fellow of the Discovery Institute. He is the author
of The Design Inference (Cambridge Univ. Press) and the editor of Mere
Creation (InterVarsity). His book Intelligent Design: The Bridge
Between Science and Theology is forthcoming from InterVarsity.
Copyright © 1999 by the author or Christianity Today, Inc./Books &Culture