Uncertainty

The title of this post is not a reference to quantum uncertainty, but rather to the notion of Platonic idealism suggested by today's xkcd comic titled Certainty (click on the image to see a bigger version so you can actually read the words):



It is, unfortunately, untrue.

Now, in practical terms, math as a school subject has a lot less uncertainty associated with it than, say, literary criticism. If you're asked on an exam to solve a given problem, chances are there is a specific answer which the instructor has in mind. However, even at this basic level, issues of interpretation come into play, as illustrated by what happened to Heath Raftery on a probability test. (It's particularly funny given how many people commented on that post to try to explain to Heath why he was wrong, when in fact he was absolutely correct.)

Even on a foundational level, mathematical truth is not as absolute as it is often assumed to be. Ever since Kurt Gödel's famous incompleteness theorems, it's been known that any sufficiently powerful axiomatic system cannot be both consistent and complete. But it is not necessary to invoke such a deep result to understand the potential limitations of mathematics. The rules of inference used to generate theorems are themselves crafted from empirical observation of the world. We learn from experience that if, say, A implies B and B implies C, then A implies C. These rules seem so natural that it can be hard to imagine that they are not of necessity true. The truth is that the rules of inference must be assumed to be true, along with any other axioms we are using. And not all mathematicians agree on which rules of inference should be included; for example the field of constructive mathematics rejects certain types of inferences used in classical mathematics.

The nicest explication of this idea that I've come across is an essay written in 1895 by Lewis Carroll, titled What the Tortoise said to Achilles. This dialog between the fictional Tortoise and Achilles was also included in Douglas Hofstadter's book Godel, Escher, Bach: An Eternal Golden Braid and served as the jumping-off point for Hofstadter's own set of dialogs. (If you haven't seen it before, I strongly urge you to read Carroll's essay; it's short, it's funny, and as they say, it contains no mathematics!)