The Peculiar Faith of the Mathematician or Scientist

I’m delighted to see that Freddie deBoer and Led-the-commenter have delivered excellent pushback on a weak point of my previous post . Their point in brief is that the mere fact that our contingent minds have had remarkable success in explaining the universe so far is no guarantee that they will continue to experience such success. Moreover, it is possible that there are deep fundamental questions that we will never even be able to posit, much less solve.

On these two points, I concede entirely. Both of them are correct! They are not, however, as disastrous for my argument that epistemic despair is unwarranted as they may at first seem. As I said before:

Contingent minds merely undermine the  necessity of our being able to comprehend the world . . . they leave open, however, the  possibility of contingent minds that “just happen” to be of the sort that can make sense of the universe in which they happen to be located. Nevertheless, Freddie is right about one thing: once we eliminate necessity, we need reasons to think that our minds are of the right sort;

My post was an attempt to provide such a reason. Not a proof, mind you, but suggestive evidence. It may help to consider my approach to be Bayesian. The mere fact that the sun has risen every day that I’ve been alive is not proof that it will continue to do so, but with a fairly modest set of prior epistemological commitments, it’s reason enough to think that my belief that the sun will continue to rise is justified. Similarly, the prior success of science in describing the universe is not a proof that fundamental reality is accessible to us, but it is, to me at least, strong evidence that our minds are not wholly disconnected from reality as some of the formalists and constructivists would have us believe.

Freddie’s and Led’s challenge still warrants investigation, however, and today is a particularly fruitful day on which to consider it. Today is Great and Holy Saturday, when our thoughts are drawn to the small band of disciples who along with Mary gathered outside the tomb of Christ, waiting and hoping for the resurrection of the Lord, their presence motivated by nothing more than a promise. What Freddie and Led have nicely pointed out is that mathematics and science are based on a similar kind of promise.

I recall another Saturday, several years ago, when I was in college and trying to decide whether to take the plunge and become a math major. Late that night, I ran into an inebriated grad student who, as it happens, was writing his dissertation on non-foundational set theory. The two of us chatted, and I explained my dilemma. His first question was blunt, in the manner of mathematicians: “Are you smart enough?”

“I think so,” I replied, “it seems like most of the hopefuls get weeded out by the first class that requires them to do abstract proofs, and I have no trouble with that, so I should be fine, right?”

He smiled drunkenly and shook his head. ”No, proving things is the easy part.”

He was right of course, the difficulty of proof pales in comparison to the difficulty of stating what you wish to prove. Mathematicians since well before Hardy have been publishing paeans to proof as a creative and intuitive process, but trying to determine which questions are mathematically interesting is a far more daring act.  The aesthetic and analytic faculties must operate in full concert, fueled by the belief that what seems like it should be true actually is true . . . and provable.

Mathematicians have struggled with these doubts ever since Gödel showed that all that is true is not provable and, more importantly, since Matiyesevich and Chaitin showed that many interesting true statements are unprovable, rather than just Gödel’s artificial corner-cases. Setting problems and working as a mathematician, however, requires a further faith — a faith in the overall coherence of mathematics and in our ability to apprehend it.

I think the proper scientific analogue is nicely raised in Max Tegmark’s excellent paper on neo-Platonism . In order to work, the physicist must believe that we do not reside within the “physics doomed” quadrant of the diagram on page 12 of that paper. The point is that physics and mathematics are both epistemologically daring activities. I’ll  hasten to add that this in no way implies the truth or validity of the particularly bold prior commitments that the physicist and the mathematician hold, consciously or unconsciously. Freddie and Led have done us a service by reminding us of just how non-foundational these enterprises are. They rest on strong basic beliefs about the nature of the universe and the nature of our minds.

The inevitable response, and one that I expect to see in the comments, is that philosophers of physics and philosophers of mathematics have come up with systems within which these activities make sense even if they are divorced from Truth. Some of these systems even give explanations for the observed coherence, consistency, and success of these fields without making any appeal to correspondence with reality.

This is entirely true, and I won’t contest it, what I will say is that however successful these systems are philosophically, they are laughably out of line with the psychology of actual, practicing mathematicians and scientists. Anecdotally, I have never met a mathematician who, when asked what he does for a living, says: “I shuffle formal symbols in arbitrary patterns that are internally consistent and make sense to me.” Nor have I met a physicist who would reply: “I make tautological statements about internal questions related to the socially constructed version of reality that I’ve received.”

Is it psychologically likely that somebody who holds such beliefs would go through the trouble of doing mathematical or scientific research? I doubt it, in my experience they tend to become philosophers of science or mathematics instead. For a less anecdotal take on this, I’d recommend Feyerabend and Norris .