Tuesday, February 22, 2011


I read somewhere that the theory of functions had a profound impact on Florensky's philosophy. The assumption of continuity, for him, meant that the philosophy of the modern era was doomed to see everything in purely deterministic terms. The discovery of discontinuity, of functions that do surprising and almost incomprehensible things, signaled for him a turning point. The world was full of wonder again. Learning could be spontaneous, and the mind could leap forward in faith into the unknown.

There has to be something to this. When I teach calculus, I can't help but notice that students have the hardest time processing the definition of continuity--probably because they can't imagine discontinuity. Oh, sure, maybe a function jumps a few times here and there. Or maybe a function like f(x) = 1/x has a vertical asymptote. Big deal. For the most part in my beginning calculus class we deal with functions that are continuous everywhere they're defined, aside from maybe some removable discontinuities.

The definition of continuity is simply that the local behavior of a function f(x) near x = a is the same as the value f(a). The surprising thing to students is that this might not be the case. Take a function like f(x) = x^2, and obviously finding the limit is just a matter of "plugging in" the appropriate value of x. Discontinuity is jarring. It means the function doesn't behave the way one would expect. I find it necessary to hammer this point home for students, because otherwise they'll have a hard time perceiving that continuity is really an issue at all.

There is an intimate relationship between discontinuity and freedom of the will. To construct a continuous function one simply needs a general rule, e.g. f(x) = x^5 - 5. To construct a discontinuous function, one must dissect the continuum, picking out individual points of interest and declaring that a function behave differently at those points. It is this declaration by which a mind exercises its freedom of will. Even if it is silly or strange to do so, I may simply declare that f(5) = 0, even though for every other real number x I say f(x) = 2x. It is an act of will, a bold separation of chosen points from the rest of the continuum, an assertion that my word alone defines what a function is. Its local behavior may be perfectly continuous, and it may have a limit as x approaches 5. I am by no means bound by such trends.

One who is truly skilled at dissecting the continuum can do much more terrible things than making a function with a single point of discontinuity; indeed, it is not hard to define a function which is continuous nowhere, even though it is defined everywhere. I've given my students that problem as an extra credit assignment; I'm asking them to look it up and see if they can explain it to me. They won't be able to produce the example themselves, which is not a reflection on their intelligence so much as on the nature of mathematics. There are certain things one simply cannot imagine unless one has seen.

This is why mathematics is essential to a truly liberal education, and why I find it incredible how many educated people consider math a mere specialization totally closed off even to the average intellectual. There is nothing so liberating to the mind as being able to enter a world of pure thought, a world which is simultaneously the work of our own minds and beyond the mind's control. Every field in the humanities sounds to me to be obsessed with paradigms and paradigm shifts. Mathematics has no such discussions, because there is only one paradigm: discovery through creation. Mathematics is a creative act of will in the purest sense. "Let x be," and it is. What holds most people back from the world of mathematics is their unwillingness to take that leap of faith, to make that bold assumption of creativity, to step out of continuity with the world of the concrete and "practical" (concepts which any intellectual ought to address with suspicion). More than all the scientific applications, more than all the quantitative reasoning and critical thinking skills, mathematics has this one thing to offer the world: intellectual courage.

That is, of course, if one understands it the way, for instance, Florensky did.

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