Followup to: Beautiful Math

If you looked at the sequence {1, 4, 9, 16, 25, ...} and didn't recognize the square numbers, you might still essay an excellent-seeming prediction of future items in the sequence by noticing that the table of first differences is {3, 5, 7, 9, ...}. Indeed, your prediction would be perfect, though you have no way of knowing this without peeking at the generator. The correspondence can be shown algebraically or even geometrically (see yesterday's post). It's really rather elegant.

Whatever people praise, they tend to praise too much; and there are skeptics who think that the pursuit of elegance is like unto a disease, which produces neat mathematics in opposition to the messiness of the real world. "You got lucky," they say, "and you won't always be lucky. If you expect that kind of elegance, you'll distort the world to match your expectations - chop off all the parts of Life that don't fit into your nice little pattern."

I mean, suppose Life hands you the sequence {1, 8, 27, 64, 125, ...}. When you take the first differences, you get {7, 19, 37, 61, ...}. All these numbers have in common is that they're primes, and they aren't even sequential primes. Clearly, there isn't the neat order here that we saw in the squares.

You might try to impose order, by insisting that the first differences must be evenly spaced, and any deviations are experimental errors - or better yet, we just won't think about them. "You will say," says the skeptic, "that 'The first differences are spaced around 20 apart and land on prime numbers, so that the next difference is probably 83, which makes the next number 208.' But reality comes back and says 216."

Serves you right, expecting neatness and elegance when there isn't any there. You were too addicted to absolutes; you had too much need for closure. Behold the perils of - gasp! - DUN DUN DUN - reductionism.

You can guess, from the example I chose, that I don't think this is the best way to look at the problem. Because, in the example I chose, it's not that no order exists, but that you have to look a little deeper to find it. The sequence {7, 19, 37, 61, ...} doesn't leap out at you - you might not recognize it, if you met it on the street - but take the second differences and you find {12, 18, 24, ...}. Take the third differences and you find {6, 6, ...}.

You had to dig deeper to find the stable level, but it was still there - in the example I chose.

Someone who grasped too quickly at order, who demanded closure right now, who forced the pattern, might never find the stable level. If you tweak the table of first differences to make them "more even", fit your own conception of aesthetics before you found the math's own rhythm, then the second differences and third differences will come out wrong. Maybe you won't even bother to take the second differences and third differences. Since, once you've forced the first differences to conform to your own sense of aesthetics, you'll be happy - or you'll insist in a loud voice that you're happy.

None of this says a word against - gasp! - reductionism. The order is there, it's just better-hidden. Is the moral of the tale (as I told it) to forsake the search for beauty? Is the moral to take pride in the glorious cosmopolitan sophistication of confessing, "It is ugly"? No, the moral is to reduce at the right time, to wait for an opening before you slice, to not prematurely terminate the search for beauty. So long as you can refuse to see beauty that isn't there, you have already taken the needful precaution if it all turns out ugly.

But doesn't it take - gasp! - faith to search for a beauty you haven't found yet?

As I recently remarked, if you say, "Many times I have witnessed the turning of the seasons, and tomorrow I expect the Sun will rise in its appointed place in the east," then that is not a certainty. And if you say, "I expect a purple polka-dot fairy to come out of my nose and give me a bag of money," that is not a certainty. But they are not the same shade of uncertainty, and it seems insufficiently narrow to call them both "faith".

Looking for mathematical beauty you haven't found yet, is not so sure as expecting the Sun to rise in the east. But neither does it seem like the same shade of uncertainty as expecting a purple polka-dot fairy - not after you ponder the last fifty-seven thousand cases where humanity found hidden order.

And yet in mathematics the premises and axioms are closed systems - can we expect the messy real world to reveal hidden beauty? Tune in next time on Overcoming Bias to find out!