"Polish hand magic", and satisfying answers

[steuard]

A recent SMBC comic featured "Polish hand magic", a rather remarkable mathematical trick for multiplying on your fingers. I want to talk a bit about the trick, and maybe a little bit about the broader philosophical idea involved. So go read the comic, and then I'll babble a little.

I'm a big fan of "But... why?" questions, and I like to think that a lot of time they do have satisfying answers. Or, well, at least that you can upgrade to a more prestigious class of confusion. So here are some thoughts on Polish hand magic, reduced to a sequence of oversimplified bullet points.

General observations:

• Fun fact: This only works because we happen to use a number system whose base is twice the number of fingers on one hand. (The proof works if you replace "10" with any number, as long as you simultaneously replace "5" with half of that number.)
• The point of this trick is to make multiplying large one-digit numbers as easy as multiplying small one-digit numbers. (It works more generally, but that's the only case where it's useful.)

Understanding the ones place:

• You can swap large one-digit numbers with small ones using 10-x. (The number encoding is designed so "fingers down" gives this.)
• (10-x) * (10-y) automatically has the same ones place as x*y, since all the other terms are multiplied by at least one 10.

Understanding the tens place: (Ha! This part only helps a little.)

• The difference between our goal "x*y" and our ones place trick "(10-x)*(10-y)" from above is "-100 + 10*(x+y)". Writing this as a contribution to the tens place: "10 * (x + y - 10)".
• We can split that "-10" to write the tens place as "(x-5) + (y-5)".
• By dumb luck, "x-5" is the number of fingers up. (This is the step that wouldn't work if base 10 weren't two times the fingers on one hand. You can pronounce it "history" instead of "dumb luck" if you'd like.)

I'll readily admit that none of that really gets at the "why" deep down: I understand the math better now that I've thought through it and there are some neat things there, but the explanation isn't as satisfying as the trick seems to demand. But from the looks of it, that was to be expected in this case because the trick relies on a mathematical coincidence about our fingers and our number system. I suppose that's satisfying in its own way: to know when something really is just a surprise.

The broader issue that this touches on is our scientific desire for a satisfying explanation of the workings of the universe. I've always hoped that once we really understand the foundations of physics, we'll know the reasons behind all of the seemingly random patterns in particle physics and astronomy and cosmology. ("Why are there four fundamental forces? Why are some of them so much stronger than others? Why are there three copies of the fundamental particle multiplet, with such different masses?" And so on.) It would seem almost cruel if there weren't some deeply satisfying structure beneath it all, and one big hope for string theory has been that it will provide those answers.

Or at least, it was. These days, people have come to realize that no matter how unique the basic structure of string theory may be, the connection between those immutable laws and the particle physics we actually observe depends on many details of how the universe happens to be shaped here where we live. I didn't want to accept that at first, but it wouldn't be the first time science turned out that way. Kepler's early attempts to explain the orbits of the planets in terms of nested Platonic solids seem almost laughable now that we know the true history of the solar system: at this point, asking for a fundamental reason why we have the planets we do doesn't even make sense. So while there's still some hope that string theory will pick out our particular universe as uniquely preferred, it doesn't have to be that way.

So there's the question: When is it reasonable to hope for a deeply satisfying answer, and when should we expect that much of even a beautiful pattern is just due to random chance? Is there any way to guess in advance?

Comments

[beth_leonard]

When is it reasonable to hope for a deeply satisfying answer, and when should we expect that much of even a beautiful pattern is just due to random chance? Is there any way to guess in advance?

I'm reading the book, "Why Faith Matters" for a class at church and I haven't yet gotten to the chapter on "Does Science Disprove Religion?" but we already discussed it in class. I have trouble putting my answer to your question into words, but the sense of Wonder and Mystery are significant. I think humans do hope for a deeply satisfying answer. Science is science, religion is different. Asking one to prove or disprove the other means you're asking the wrong questions.

--Beth

[steuard.livejournal.com]

It probably says something about my worldview these days that it hadn't even occurred to me that this idea touched on issues of science and religion, but of course it does. One of the big ideas associated with religion (perhaps the biggest) is that there is a deep, satisfying reason for everything that happens in the world. Even if we mere mortals aren't privy to the details, we can have faith that the universe will eternally play out according to God's plan. (Well, okay, some of that may be specific to monotheistic religion or even to the Judeo-Christian-Muslim tradition. Maybe.)

My friend Sean Carroll once gave a talk entitled "Why (Almost All) Cosmologists Are Athiests". I haven't thought deeply enough about his arguments or conclusions to decide how much I agree, but the issues he raises are certainly things that I would like to see addressed in any serious discussion of the topic. I don't know if you'd find it relevant or interesting (I'm optimistic that it's at least not insulting), but you might have a look.