Physics and philosophy

Massimo Pigliucci ("Is Physics Turning into Philosophy," May/June 2006) decries "the anthropic principle," which he defines as "the idea that the universe is fine-tuned to allow for the appearance of life." If, by this definition, he means that some entity did the tuning, he's right that it's closely related to intelligent design. But there are sensible ideas unrelated to religion that also go under the name "the anthropic principle." In particular, if the nature of the universe varies from place to place on very large scales, and if life is possible only in a few of those places, we should not be surprised that we live in such a place. Thus k can appear to us that the universe is fine-tuned for life without the universe as a whole being tuned at all.

Ken Olum

Department of Physics and Astronomy

Tufts University

Medford, Massachusetts

Massimo Pigliucci's article, "Is Physics Turning into Philosophy," has many points of dubious philosophical merit, but I'll stick to a simple but important error in fact. He claims that mathematics is a "branch of logic." While this was widely thought to be true in 1930, it has in fact been mathematically proven to be false.

Alfred North Whitehead and Bertrand Russell, in their monumental treatise, Principia Mathematica, sought to show that all of mathematics could be derived axiomatically using the laws of logic. While they were partially successful, their results were incomplete (self-referential statements were particularly vexing). Given the magnitude of the undertaking, this could hardly be considered a criticism. However, soon after, a young mathematician named Kurt Godel proved that it must be incomplete. Mathematics, even in simple forms such as arithmetic, is logically incomplete, that is, there are true mathematical statements that cannot be derived from the axioms. Further, such a mathematical system derived from logic could never be proven to be consistent, meaning the assumed axioms would never lead to a logical contradiction.

The power of mathematics to model the real world is not in doubt. But this power does not derive from being a branch of logic, with its inherent limitations. Like physics itself, the proof of mathematical efficacy is in the pudding--predictions tested by observation.

Chris Mack

Austin, Texas

Massimo Pigliucci responds:

I most definitely agree with Ken Olum that there are sensible ideas related to the anthropic principle but unrelated to religion. Unfortunately, these are also rather trivial ideas. If all the anthropic principle says is that the reason the universe appears to be fine-tuned to produce us is because universes that follow very different laws of physics wouldn't produce us, well one would want to quote the immortal Homer Simpson: Duh.

As for Chris Mack's point, I differ on the interpretation of what Russell and Whitehead, and also Godel, have actually accomplished. What these people's work shows is that any logical system--including mathematics--is intrinsically incomplete, that is, it cannot be made to stand entirely on its own. We will always need assumptions or axioms that cannot be justified from within the system. This, of course, doesn't mean at all (as it is sometimes claimed) that logic or math are arbitrary constructs, but it does pose interesting limits to the power of deductive logic.

None of this, however, has anything to do with my (not original) claim that math can be thought of as a branch of logic, in the sense that it is entirely based on deductive reasoning and that it follows the same basic structure of formal logic. Indeed, one can read Russell Whitehead, and Godel as having shown that math suffers from the same sort of limitations that affect formal logic, which isn't surprising at all if one thinks of the first as a subset of the second.

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