Here comes another exercise in stating the obvious just because I happened to be thinking about it.

About a year ago, on a whim, I bought a nice “fifteen puzzle” (the boring granddaddy of all sliding block puzzles). I can’t say I play with it all that often (it’s the sort of thing that benefits from infrequency), but the other day I had it with me on the subway and was fooling around with it. The little booklet that comes with it is written in a sort of retro/quaint mode and, alongside the various challenges (1-15 vertically! 1-15 backward! Evens on top! Odds on top!) it encourages you to make up your own patterns and then see whether they’re possible or not. “There’s no way of knowing until you try!” it says.* Because, you see, not every conceivable pattern of 1-15 can be created with one of these things. Some of them are logically impossible to reach.

This was the crux of Sam Loyd’s infamous 1870s “14-15 puzzle” hoax-contest, wherein a $1000 prize was offered to anyone who could set the numbers in order, from an initial position where the 14 and 15 were swapped. It’s impossible.**

The principle, to the degree that I understand it, is that only positions that entail an even number of tile-swaps can be reached (unlike the odd-numbered single swap of 14-15). There are several ways of demonstrating this, some simpler than others, but, and this is the important thing, they all involve some sort of constructed argument, a la “since each move entails a shift of one blah blah, we can reason that only an even number of shifts blah blah meaning that the parity blah blah.” They lead us down a logical path away from our actual experience of the thing, toward a law that is perfectly correct and absolutely abstract, a law that I can just barely comprehend, much less hold in my head in any way that will affect my experience of the puzzle. The only real way to incorporate this law into my experience of the puzzle is to know that it exists but is, for my purposes, more or less inscrutable.

This is why the booklet cheerily tells us that “there’s no way of knowing until you try!” Because, though this is untrue, it is, unavoidably, the way we experience the puzzle. If you showed me a pattern and asked me if it was solvable, I would have to say that while math could answer that question, I personally could not. I would have to take recourse to wielding math.

This is not a problem. The problem is that, while the underlying law that governs the 15-puzzle is beyond what any person can intuit, the fact that there is an underlying law is absolutely clear. When you’re moving those damned numbers around trying to get two of them to switch places, you begin to feel with some certainty that some higher order is working against you. It is intuitively clear in an immediate, non-abstract way, that some kind of all-powerful law blocks your way. And – and this is the reason people don’t like math – nobody wants to believe that the immovable force in their way is a concept too abstract for the human mind to ever really process. It’s upsetting and unsatisfying, in exactly the sense that actually solving such a puzzle is satisfying.

Or take Nim, which I just played for a long time, getting creamed again and again and again by this stupid computer until I cheated and looked up how to win.

Play it for a bit. You know that something is going on here, some kind of principle – you can feel it! – and you can also see how self-contained and simple the whole thing is. Your gut tells you that soon you’ll figure out the strategy, like you eventually did for tic-tac-toe, and start winning. But you don’t. Why? Because you’re looking for something that has to do with moving the little circles, when in fact the real underlying principle involves bitwise XOR. That’s right, there is no simpler way to explain the underlying order of the game than to express the game state as an “exclusive or” operation applied to the positions of the circles converted into numerical values in binary. Once you view it this way, in fact, it’s not a game at all; winning becomes completely deterministic. Also, once you view it this way, you’ve stopped playing the game and started doing math.

The great horrible revelation that math offers is that the order underlying the universe is both knowable and totally unsympathetic. You can work it out on paper, but it will never fit in your head. A lot of people HATE hearing that. It’s so unsatisfying as to feel almost offensive. You’re telling me that when I felt like I was matching wits with some clever opponent, doing this 15-puzzle, I was actually matching wits with some kind of faceless, alien sort of truth that can only be hinted at in complete abstraction, with a bunch of numbers? You’re telling me that I’m supposed to shake hands with this creepy thing that can’t see, touch, or even think about directly?

The long history of bad early science is all about people making up theories that felt like they might be right – like, that maybe the world was made of, uh, heavy stuff, and wet stuff, and airy stuff, and I guess fire. That seems like it covers pretty much everything. Good guess, guys, but the correct answer would be that the world is made up of… well, actually the truth is so alien that it can’t really be described in normal language, but if you study for several years I can begin to explain it to you. Numbers are your best bet. Whatever that crazy stuff is, for now you should just know this: it’s indifferent to you and me, and it’s the correct answer. I bet it’s a relief to finally know the truth, huh?

The scientifically/mathematically-minded want to believe that the manifest CORRECTNESS of math and science are enough to win anybody over. But they are overlooking the fact that it is unsatisfying to the point of being disturbing, for many people, to think that the path to truth is the path away from actual personal understanding. This is why religions that offer no correct answers and no justifications can still flourish – because the wrong answers they offer are at least humanly assimilable. The human desire to make up superstitions is never going to be squelched – given the choice between believing that you might get cancer if you step on a crack, or believing that you might get cancer if the combinatoric matrix of all the molecules in the universe align in ways more complicated than anyone will EVER be able to express or comprehend, there’s really no contest. One might be correct but it is fundamentally impossible to believe it. Only on paper.

So: that’s why people don’t like math. The kid says to the math teacher, “I don’t want to do these integrals. I can’t relate to them,” and the teacher says, encouragingly, “Oh don’t worry, nobody can relate to them! They’re truth!” People don’t like math because it makes them feel hopeless and insignificant, and reminds them of their own mortality. No, I’m serious!

My point is that math and science educators need to accept that theirs will always be an uphill battle against human nature, that they are denying us the things that make us feel secure, and that real math and science education needs to offer some kind of philosophical perspective to help cushion that fall. I’m all for the fall, though.

Um, so here’s a little list of some of the good puzzle links.

LogicMazes
ClickMazes
PuzzleBeast
MathPuzzle (heavy duty!)
The Sliding Block Puzzle Page

Let me close by saying that I, for one, always kind of liked math. But I have come to believe that this was a reflection of my personal way of dealing with the universe; as a child, I rarely felt the need to forge any kind of personal relationship to information to feel that I had assimilated it. People like me took math in stride – any kind of abstract manipulation just was what it was. But that tied in, I think, to my tendency not to try to make any sense beyond the surface of books or movies, just to accept them for whatever they were. I basically led a low-empathy existence. That kind of mental philosophy has its benefits and its shortcomings, and at any rate I’m in a rather different place now. But the people for whom empathy and understanding are linked, and there are many such people, can’t just dive into all those numbers; they’re frightening. Math teachers gotta understand that; that’s all.

* Or something like that. I don’t have it on me right now.

** And also, Sam Loyd didn’t actually invent it.