How could it be that mathematics formulated in a priori necessity in the armchairs and heads of mathematicians, applies to the messy, far-off, contingent, natural world?

Here's an analogy that I think suggests an explanation.

Have you ever played a rules-enforced online game? A rules-enforced online game is great for learning if you don't know the rules of a game. That's because if you make an illegal move, the game reacts by pushing your piece back to the starting point, and making a rude noise. Now perhaps we're given a game like chess, but with different rules. We want to figure out what those rules are. We can try to find out by testing sequences of moves (let's suppose we're playing against a computer). And as the game goes on, we'll see that some moves are allowed while others are not.

Now imagine that we have a book of games. In this book are all the kinds of games that are like chess, only different. The book not only lists the rules of each game, but also the ways that it's possible for the board to look given those rules, and all sorts of other true things about the game as defined by those rules. For instance, some of those games will be winnable within five moves. Others will not. And of course, the fact about which ones will is dependent on the initial rules. The fact about which ones will is implied by those rules. This book will have that information about each game.

We may imagine that this book was written by some obsessed monk, with no gaming experience, hard-set on cataloging as many possible chess-like games as she could. Or, better yet, an old-fashioned computational device (with no experience at all) put to the task. Either way, the process is: define a set of rules for each game, and work out as many truths as possible about the game (trying one's best to focus on the general ones) by asking what those rules imply about possible game states and how to move from one possible game state to other possible game states.

Consider just how helpful this book will be to our project. We want to find out which game we're playing. We want to find out all the rules exactly. We can bump around and use trial and error. Perhaps we discover, in our bumping around, that the game is winnable in under five moves. That means that the mystery game may be described in the part of the book that chronicles those games that are winnable in under five moves, and it isn't in that part of the book that deals with those not winnable that way. Now it seems we've got some knowledge for free. We've learned to test the other kinds of general properties that our book tells us we should expect from any game winnable in five moves.

Now we try something else. We list all the possible first moves, as given to us by trial and error against the rule enforcing game engine. We then look up all those games that allow that collection of first moves in our book. And when we do, we'll most likely find some knowledge about the game that we didn't before. Perhaps all games that have this set of allowable first moves include a rule in their rulebook that states "you may move at most 10 squares on your first turn." Notice that this is not necessarily an implication of our discovery about possible first moves, a property that all games that have these rules share. Instead, the book, and its catalog of games and truths about those games, allows us to make this deduction. It gives us more knowledge.

And as we play around with our game and our book, we get a sharper and sharper idea of what the rules might be.

Now let's talk about the analogy. The mystery game plays the role of our cosmos. The game's rules are the laws of nature. The book of games is mathematics. The games within are mathematical structures. The rules of each game are the axioms and rules that, taken together, entail those mathematical structures.

Notice that there isn't really a problem of application in the case of games. There isn't a problem of unreasonable effectiveness of our gamebook in helping us explain how our mystery game works. The explanation for application is that whatever the mystery game is, it operates like any other game: there are rules, and the possible board states arise from those rules. And the book of games is simply full of explorations of the implications of possible sets of rules. Assuming it's lengthy enough, or we're lucky enough, it wouldn’t be such an unexplainable event that the book tells us some things about the game we're trying to figure out. Even though the author of that book never had any experience or prior knowledge of the mystery game.

This is easy to see in a simplified situation of games, but harder to see in the situation of mathematics and the natural world. Surely the analogy comes apart from the reality in two places. First, the natural world, and structures within it, tend to be far more complex than any chess-like game. And second, there is no great book of mathematics that has come close to anything like completion.

And yet, these differences need not nullify the analogy. Increase the complexity of the game, and perhaps the book of games does not talk about games, but types of games that share certain qualities as defined by a set of rules. Now, these rules may be approximations of the actual rules, but they may still yield implications about what is possible in terms of moves and game states: implications that we can go out and test in our very complex mystery game. And it may be that the book of games in the case of mathematics is not only approximate but also (to put it mildly) incomplete. That's ok for our analogy too. We'd still prefer to have the book alongside, and we'd use it in much the same way. We'd use it as a kind of library of implications. A partial library of how partial structures arise from rules.

Now here's a point that would, if it weren't true of our cosmos, absolutely scuttle this analogy. The mystery game, no matter how complex it turns out to be, is regular. That is, the rules don't change and always apply. This is especially important when we attempt a wrong move, and the game pushes our piece back and makes a rude noise, every time. If our game didn't do that, our book would be useless. But then our task would be meaningless as well--for there wouldn't be any rules to discover. And it seems that our cosmos is this way too. It appears to be regular in imposing its rules. If it weren't, the big book of mathematics would useless to science. But then again, the task of science would be meaningless as well--for there wouldn't be any rules to discover (much less any scientists to discover them).

So here's a kind of anthropological explanation for the effectiveness of mathematics to the natural sciences. Of course our cosmos yields to the great book of mathematics, because a cosmos that didn't wouldn’t have us in it. In short, only a regular universe can harbor intelligence, and a regular universe is mathematically describable.