The topic of this year’s Foundational Questions (Quextions?) Institute essay contest is, “How can mindless mathematical laws give rise to aims and intention?”
My answer: They don’t. But not for the reason that most physicists who bother with such speculations probably think.
As a university graduate and veteran of many science-blog comment threads, I’ve seen my share of arguments about the mind-body problem, the Hard Problem of Consciousness and all that. The older I get, the more I feel that the way these arguments set up the problems automatically forbids a solution, and then passes this failure off as profundity. One defines Matter as something entirely unlike Mind, and then one argues about how the latter can or cannot arise out of the former. After enough rounds of this, any elegant formulation of words that appears to provide a solution immediately becomes suspect: If it appears to make the problem more tractable, then it must be treating Matter and Mind insufficiently differently.
When I was younger, I embraced the idea that reality—you know, all the stuff out there and in here that is, like, real and all—is at bottom a mathematical entity of some kind. What can I say? I also listened to Foreigner and thought that racism in America was on the wane. Now that my day job is actually doing physics, as opposed to reading breathless books about it, I find that this attitude, the instinctive identification of the physical and the mathematical, is entirely optional. It says more about your own temperament than pretty much anything else, nothing in particular compels it, and you can cheerfully do research without it. Countervailing influences, some of which I also carried with me since childhood, eventually led me to find it a rather empty business!
My colleague Marcus Appleby wrote an essay about this general topic a while back, an essay called (straightforwardly enough) “Mind and Matter” (arXiv:1305.7381). It prompted me to formulate a slogan, which goes like this:
Cognitive science led me to reject the Cartesian idea of “Mind.” Quantum physics makes it legitimate to reject the Cartesian idea of “Matter.” Confusion reigns when we discard one pole but not the other!
Above and beyond that, I have grown to doubt that the imagery of “mindless mathematical laws” is sustainable, once we confront the daily practice and the philosophy of mathematics. To get a tiny taste of this, consider the following puzzle, posed by John Baez:
You might say the standard natural numbers are those of the form 1 + ··· + 1, where we add 1 to itself some finite number of times. But what does ‘finite number’ mean here? It means a standard natural number! So this is circular.
So, conceivably, the concept of ‘standard’ natural number, and the concept of ‘standard’ model of Peano arithmetic, are more subjective than most mathematicians think. Perhaps some of my ‘standard’ natural numbers are nonstandard for you!
One can remove the ambiguity in the informal definition of “natural number,” but not in a unique way. The resulting multiplicity is irrelevant for practical purposes—ah, but tonight, we’re concerned with impractical purposes! We can shove this befuddlement under the rug and try to move on to defining the real numbers, only to find that a similar difficulty arises there. There is a stupendous diversity of viable candidates for the position of “the real numbers.” Technically, they are known as real closed fields. They are equivalent to the extent that if you can prove a mathematical statement involving the numbers 0 and 1, the operations of addition and multiplication, and the less-than-or-equal comparison in one such arena, you can prove it in any other. But two different real closed fields do not have to be isomorphic: There might not be a one-to-one mapping between them that respects the structure of arithmetic.
Mathematics is that domain of life in which “two minds can know one thing,” as best as that is possible—but you have to work painfully hard to subtract the minds from that image, and the result might leave you with nothing at all.
What does this have to do with physics? Perhaps nothing, except for the mental entertainment and recuperative value obtained by taking a break from physics! The mathematics that finds application in physics is peculiarly robust against changes in the underpinnings—to how we choose to define the real numbers, for example. On the other hand, if we’re being honest, this might be an artifact of what kinds of problems the physicists have chosen to work on! And we have to admit the vexing hassles that even our best physical theories run into because we assume that spacetime is a continuum—i.e., because we habitually think that events happen along a line of real numbers.
Another thing I’ve lost my taste for with the passing years is the blog comment section. So, I don’t have a fantastically strong urge to engage in the FQXI essay contest officially. Instead, I’ll close with a pointer to a manifesto I posted recently, and in particular to the wild speculations in the latter sections.