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A Physicist's Philosophy of Mathematics

Reuben Hersh

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.

Two miracles!

"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 its particles."

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 mathematics.

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 it deserves.

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.

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