I thought the third part of the series was the strongest in at least one sense — this is the part that seemed to most fully and most compellingly do what it set out to do, which was to show how numbers are interwoven into the world all around us. Specifically they examine music, architecture, the universe (in terms both cosmological and spiritual), and the future.
The discussion, which I applaud everyone involved for taking the time necessary to do properly, was a beautiful beautiful thing. Numbers are built into the human musical impulst in a deeply primal way. Simultaneously eerie and true. Whatever the reason, whatever the explanation, aesthetically pleasing harmonies correspond to ratios (of frequencies, of string lengths, etc.) which have very good approximations by fractions with very small numerator and denominator. And that’s been true for a lot longer than humanity has had knowledge of that fact or the sophistication of language to verbalize it. But how can that be? What is responsible for that? Does the string know about rational approximation theory (a sophsticated subject in its own right)? Is there a little mathematician in my ear who works out the frequency ratios and tells me how pleasant I think the chord is? Spooky.
(Incidentally, the experiment described with a plucked string is one you should all actually do. Strings are visibly, discernibly, dramatically “happier” to vibrate in ways corresponding to simple fractions.)
The remarks about contradictions and multiple universes and truths in connection with the tango are mathematically deep, maybe deeper than anything else in the program. This distinction between truth as used in mathematics, in metamathematics, and in common discourse, deserves a lot more attention. This may be the piece of the core of mathematics most misunderstood by nonmathematicians. (From now on, I will think about this as the Tango Principle of mathematics.) You should all listen to this program just for that clip, even if you haven’t listened to any of the other parts.
I’m glad that so much was made on zero and the nature of zero. Zero is a deep and powerful idea, and its discovery/invention was a major milestone in numnber systems. It is absolutely essential to any kind of place-value-based system, which as I’ve already said comprise an important innovation. The digression into Hindu and Buddhism conceptions of being and nothingness might have seemed out of place, but for once I don’t think so. A very large part of making sense of mathematics is making sense of the difference between zero and nothing, of understanding the nature of identity operators. I never thought of zero objects as corresponding to the “full void” notion in Buddhism, but there is something there.
I have mixed feelings about the concluding paragraph, which was excerpted from a book by the presenter Cecil Balmond, and which, it is obvious from the emotion in his voice as he read it, carries great significance and weight for him personally. I’ll just say that 1) I didn’t get it and 2) it’s not the kind of note I would have chosen to end on. It felt like a psalm from some numbery religion. The reason I would not have ended that way is this: the average person finds math inscrutable enough, finds statements about the interestingness of numbers headache-inducing enough, that adding a layer of mystification seems unhelpful to me. Maybe that’s why I am an educator and he has a show on the BBC (and the mathematician he spoke to about the Google age of numerocrats, Keith Devlin, is the official mathematician of NPR). That said, to the extent I understand his mystical language, I agree with what he says. Numbers and patterns describe our thoughts and our ideas and our lives, and simultaneously they transcend all these. Numbers infuse our mental, practical, and spiritual lives in ways we understand and in ways we don’t understand.