A Physicist's Philosophy of Mathematics
Converging Realities: Toward a Common Philosophy of Physics and
Mathematics. Roland Omnès. xviii + 264 pp. Princeton
University Press, 2005 (first published in 2002 in France under the
title Alors l'un devint deux). $29.95.
It is refreshing to find a physicist joining today's ongoing public
conversation about the nature of mathematics. Roland Omnès is
a prominent expert in quantum mechanics who has written several
books explaining that difficult and important subject. He also knows
a lot of mathematics and is aware that there is much disagreement
about what we know in mathematics, how we know it and even whether
we really do know it. As a physicist who loves mathematics, in
Converging Realities he comes forward with a new idea,
which he proposes to call "physism."
He quotes philosophers who have said that mathematics is "a
miracle"—that it's miraculous that mere humans can come
to have knowledge about things they have never seen or touched, and
that this knowledge is more clear, more certain, than any knowledge
of the visible and the tangible.
It has also been said, by physicists if not by philosophers, that
the existence of the laws of nature is a miracle.
"Physism does not try to explain why there are such miracles,
but reduces them to a unique one," says Omnès, because
mathematics is "the laws governing the universe and
There is a difficulty, he acknowledges. Not so much with what's
meant by mathematics, or what's meant by the laws.
The big difficulty is with what's meant by is.
Omnès argues convincingly that a lot of what theoretical
physicists do is very much like a lot of what mathematicians do.
Both go from examples and observations to guesses or hypotheses
about patterns or formulas. Both then proceed to reason out the
consequences of their hypotheses, guided by the law of contradiction.
Omnès says that "When a physical theory . . . requires
mathematics in the formalized corpus . . ., one can make the axioms
necessary for the theory explicit, at least in principle, and follow
the unfolding of ideas from these axioms into the mathematical
corpus." He goes on to claim that "The physical laws, as
we know them presently, reach through this process into an extremely
large part of the mathematical corpus, and maybe all of it."
Indeed, he says,
If one reduced mathematics to the unique and modest role of
a language of the laws of physics, the present state of theoretical
physics and simple consistency requirements would not appreciably
modify the present corpus of mathematics.
. . . The consistency of mathematics is therefore tantamount to the
existence of mathematically expressible laws of nature.
Minor glitches can be shrugged off. (A few oddball branches of math
like higher set theory and nonstandard logics may not be physical,
but who cares?)
Omnès's amplified statement of physism contains these assertions:
There are basic axioms for logic and mathematics. These
axioms are laws of physics. . . .
Conversely, they generate every possible field of mathematics.
New laws, new axioms, new fields are possible. . . .
In a chapter on the philosophy of mathematics, Omnès observes
that "The essential qualities of mathematics are consistency
and fecundity. They are also the two main features necessitating an
explanation." And, he believes, physism explains the fecundity
and consistency of mathematics. Nature is necessarily consistent,
and she is also, for whatever reason, fecund.
But he weakens his case by adding this:
What exactly is the extent of the present mathematical
corpus that is in relation with the mathematics of physics? I cannot
say I have analyzed this question carefully, but I considered it
from time to time when reading papers in theoretical physics or
Physism avoids two tough questions—What is the nature of
mathematical objects? and What is the nature of physical reality?
But at least it's better than formalism, logicism, intuitionism,
constructivism or Platonism.
Omnès's philosophy reminds me of a provocative remark by the
famous Russian mathematician Vladimir Arnold. (Arnold works
half-time in Paris, so the two may even be acquainted.) Arnold has
written, "Mathematics is part of physics. Physics is an
experimental science, a part of natural science. Mathematics is the
part of physics where experiments are cheap."
I hope Omnès's daring proposal is taken seriously by those
interested in these matters and receives the analysis and critique
Apart from the philosophical adequacy of physism, I have qualms
arising from exposure to the "modeling" going on nowadays.
People employ computing machines to generate vast numbers of
"networks," according to some suitable rules. Their
"results" consist of asymptotic and statistical analysis
of these computer-generated networks. This activity is funded by
some actual agency or institution, so it could influence some actual
concrete policy decision about some part of the "real world."
The zeroth principle of mathematical modeling is "Don't confuse
the model, which is an artifact, with the real world, which was here
prior to and is independent of the model." Only by maintaining
that distinction can one intelligently improve the model, or compare
it to alternative models.
Omnès knows all that. I guess he is sure that his idea that
"Mathematics ‘is' physics" will not contribute to
confusion between mathematical models and the physical world.