Thursday, December 2, 2010

Is Mathematics a Divine Language?

Is mathematics a divine language? That's the provocative question posed by Mark Vernon in an article in the Big Questions Online blog.

Readers may remember that editing BQO is the gig of Rod Dreher, former editorial board member of The Dallas Morning News. I assume he's still at BQO, but it's hard to say for sure because someone at the Templeton Foundation dropped the cone of silence over Dreher last summer after he posted several articles about the construction of an Islamic center in lower Manhattan. (Dreher took the anti-Muslim position, naturally.) The ability of readers to comment on BQO articles was shut off at the same time. Because BQO doesn't allow me (or anyone else, for that matter) to comment on Mark Vernon's article on the BQO website, I'll do so here. :-)

After the jump, is mathematics a divine language?

Mark Vernon starts out with a simple prediction of how humans might first communicate with extraterrestrial intelligence. It's reasonable to suppose that the first radio messages we send might be something like a binary version of pi (3.14159...), figuring that any intelligent beings would recognize the ratio of the circumference to the diameter of a circle in Euclidean space and conclude that the radio signal was sent by another intelligence.

Vernon then goes out a little farther on the limb, claiming that mathematics seems to be discovered, not created. "The reason to think this is that discoveries made about the physical world are often, first, discoveries made about mathematics." Here's how that works. Scientists come up with mathematical models that seem to fit observations of the natural world. Then they explore those models to see what else might be true if the model accurately captures nature. If experiments support predictions based on the models, it can be said the mathematical discovery led to the scientific discovery.

Where Vernon gets so far out on the limb that it breaks off is when he concludes that this neat ability of scientists to find mathematical models for the natural world "is exactly what you'd expect if the universe were created by a powerful deity, worthy of worship." I dunno, personally I'd expect a lot more randomness or at least a lot more variability over time than our actual universe exhibits. After all, an eternity of nature everywhere following the same preordained laws might be as boring to a powerful deity as a toy train forever circling the same oval track is boring to children.

Vernon does give the opposing view some space. Physicist Robert Penrose's attitude is that the ideal world of mathematics is always only an approximation of the natural world. The closer we look at the natural world (at the quantum level, for example), all of our models begin to break down. Some physicists have hopes of finding new mathematical models that are accurate at the scale of the very, very small, but they recognize that the new models would still only be models -- approximations of reality -- not necessarily reality itself, or the language of God.

My own introduction to Penrose's insight came years ago, in a non-Euclidean geometry class. Euclidean geometry assumes that two lines, each perpendicular to a third line, remain at a constant distance from each other forever (what we call parallel). This isn't proven, it's just assumed as fact. Everything else about the world of Euclidean geometry is deduced from a small set of such postulates. But start with a different set of postulates (for example, that two lines, each perpendicular to a third line, "curve away" from each other), and you can deduce an entirely different world, a non-Euclidean world. Which geometry conforms to the "real" world? For centuries, Euclidean geometry reigned supreme. Everyone thought it was self-evident that the "real" world was Euclidean. Then Einstein figured out that space is not Euclidean and time is not fixed, that spacetime itself is curved. General relativity and quantum mechanics taught us that none of our existing mathematical models is a good fit for all situations at all scales. Some models are good for one thing. Other models for another. The search for models that are good for more and more things simultaneously is the search for the Grand Unified Theory or a Theory of Everything. So far, the search has proved frustrating.

My takeaway from the non-Euclidean geometry class was the insight that mathematical models are more like art or poetry than a fixed attribute of nature waiting to be discovered. There is freedom in the choice of postulates you start with. There is novelty in the mathematical world that results. A mathematical model is the statue a sculptor imagines in a block of marble, not the fossil skeleton a paleontologist discovers in a block of limestone. The mathematical models, like the statue, may be useful, they may even be beautiful, but mathematical models are, at heart, human creations, not reality itself, nor the language of God.