Followup toBeautiful 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!

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Dominic: How many bits does it take you to communicated the formula you discussed? How many for just the last term of that pattern?

When you wrote "But neither does it seem like the same shade of uncertainty" I suppose you mean that it doesn't seem that way, to you. Nor does it to me. But before, as a thinking person, I suggest that the difference is meaningful, I need a context or a reason. You haven't provided one, and that's why your argument has the flavor of religion, to my palette.

I'd love to see your answer to the actual skeptical argument, rather than the straw man you offer, here. Here you are doing the equivalent of announcing "I'm thinking of a number!..... 5!...... I'm right again! My quest for order is rewarded!"

If you use mathematics to find order in the messy world, and you succeed, does that amount to a proof that the order you found is the actual order? Kepler would have argued yes! So would have Newton. Both were wrong. We know they were wrong. Wrong but their ideas are enduringly useful, as far as we know... so far... The skeptical position is not one of denying the value of ideas, but rather that of continuing the inquiry.

When my inquiry ceases, my beliefs become hardened premises that define my world and prevents me from benefiting from ideas of people with different premises. That's fine in a simple world. A gamer's world. I've become convinced that there is no simple world, except in our fantasies. Overcoming bias is about finding our center in a messy world. It's about overcoming fantasy.

Isn't the last value in the sequence definitionally zero for the first five terms, reducing the entire first term to zero as well and leaving only the n^2?

Does this mean that if we gave you (1, 4, 9, 16, 25, 36...) you would claim the next value in the sequence was 6! plus 49??

Michael: do you think we should decide that the simplest formula is the best?

But then how do we define simple? What do you mean by 'communicating' and 'bits'? Do we assign arbitrary complexity points to the operators? What would be the relative complexity of a power operation as compared to a multiplication? And what of my pet operator I just invented that lets me replace "(n - 1) (n - 2) (n - 3) (n - 4) (n - 5)" with "5##" or something similarly silly?

Ask yourself, how can we be sure we have the simplest explanation? What is the simplest formula for the sequence {1, 2, ...}? Is it the powers of two or the natural numbers? What about the sequence {1, ...}? Is it really sensible to ask such questions?

I think you could use Kolgomorov complexity to define simple, for these purposes. That way replacing your formula with "5##" wouldn't make it any simpler, because the machine would still have to execute all those multiplicative operations.

How can we be sure we have the simplest explanation? We can't be sure, because new data could come in to make us change our minds. But given a finite amount like {1, 2, ...} we can still weight possible formulae by Kogomorov complexity and prefer the simpler hypothesis.

I think natural numbers is simpler in this case, because n is simpler to calculate than 2^n. As for {1, ...} I don't think we have enough information to locate a hypothesis.

I'm uncertain about what I've said, so please correct me if I'm wrong about anything.

Given the set: {1, 4, 9, 16, 25, ...} And asked to identify the next number, the answer is: 156 For the sequence is obviously generated by the following formula: ((n - 1) (n - 2) (n - 3) (n - 4) (n - 5)) + (n ^ 2)

It is left to the reader to manipulate the formula into an unreadable form, so that it's hard to see how it works. Especially fun is adding an irrational multiplier to the '(n - 1) ... (n - 5)' part.

And notice this method works for any sequence given to a finite number of elements, for, indeed, there are an infinite number of fully-specified sequences that fit.

That's what I call beauty.