Mathematical simplicity and a priori probability
In the previous sections I have argued that neither the apparently nessicary physical consequences of mathematics nor the role of mathematical objects in our best physical theories give us reason to reject the practice-projection view of mathematics which I advocate. But a further mystery remains, one which, so far as I know faces all present philosophies of math at about equal strength. This is what you might call the ‘god is nice’ problem: why is it that what appear to be the fundamental laws of the universe can be stated, and stated simply in the language of mathematics?
The answer I propose starts with a familiar story about induction. As Nelson Goodman pointed out doing induction involves a kind of choice or preference ordering between different hypotheses/predicates which goes beyond the requirements of logic. When we see the sun rise 100 times this convinces us that it will rise the next morning as well, though our actual observations are equally consistent with the theory that every 101st time the sun will take a break, or that the sun will rise every day until tomorrow after which it will never rise. So human beings have a kind of shared taste in doing induction, preferring certain kinds of hypotheses to others or (to put the same thing differently) we largely share the same a priori probability measure on the space of possible worlds, one which assigns a large probability space to the worlds which follow laws like the sun will rise every day, and low probability to laws like the sun will rise every day but next Tuesday.
This suggests the question: how did it come to be that the laws which people think are likely tend (at least in the past) to have actually been confirmed? How is it that we seem to be built to perform induction in (largely) the right way? The task here is one of naturalized epistemology. Hume and Goodman have provided irresistible arguments that no non-question begging justification for induction (or, more precisely, doing the way we do it) can be given. But assuming, as we can’t help but do, that our way of doing induction is right there remains the task of using our full faculties including induction to account for the curious match between what people tend to expect when they see whether a will be followed by b the 101st time, and what actually happens. Surely we don’t have some kind of a priori probability sensors?
The standard answer to this question appeals to evolution. We can imagine creatures (we might not want to call them believers) whose behavior corresponded a number of different a priori probability functions. There could be counter-inductive creatures, for which seeing A followed by B many times would incline them more and more to prepare for A not to be followed by B the next time. Or there could be creatures which acted as though they expected time (or space) to be divided into epochs with radically different laws, so that observing patterns in one place would have no effect on their behavior in another place or time. But, if we are right about the earth’s past then all these creatures would have a bad time of things. A creature that ate more or a certain kind of mushroom because the last mushroom of that type it ate had made it ill, or one which applied none of what it had learned about preditors or food in any one region in its migration pattern to any another would not be likely have many decendents. So we can explain some of the mysterious match between the kinds of laws we expect to find and the ones whose predictions do turn out to be confirmed by seeing how evolution would have selected creatures which tend to expect the kinds of patterns which our world actually evinces.
Another bit of support might come from the fact that our brains, which realize this tendency to do induction one way rather than another are part of and built out of the same materials as the world around us. In our world it is much easier to build a machine whose beliefs change in a standard inductive fashion (what we would call, expecting the same things to happen in the future as it has observed in the past) than one which did induction in a grue-theory like manor (expecting all the basic properties of things to do what we would consider as changing after a certain day: expecting emeralds dug up later to be blue, and cups dropped later to float etc) or as if there were temporal epochs with different laws. You would have to make a clock to tell grue-like machine when to switch its behavior on the special day, and one to tell the third machine when a new epoch had started so it could swap out its memory bank for another. In contrast, in a world with temporal epochs each of which had radically different laws it would be very hard to build a machine which ‘expected’ the same kinds of things to happen in one epoch as in the next, since the materials out of which that machine was built would themselves change. This is not to say that it will always be easy to build a machine in any world that does induction ‘right’ or ‘well’ for that world: there could be a world whose laws were epoch-like for gems and normal for brains and most other things. Nevertheless, in out world there seems to be a match between the way that brain-materials behave and the way that the external objects whose behavior brains predict behave, so that creatures which do induction ‘the right way’ are simpler and easier to build/more likely to be produced randomly than those which do it very differently.
Now this evolutionary story is so far just about scientific induction, and we haven’t said anything about math. But suppose it is right, and evolution has given rise to creatures whose behavior embodies a certain a priori probability measure over different possible worlds or theories, and that this way of assigning prior probabilities is right for our world (it mostly leads one from the evidence one encounters to expect what will actually occur). Pre-human creatures have, then, embodied in their minds, a general shared way of going from finite actual observations to a law which encompases those observations and predicts further observations (namely, the law which the shared probability measure assigns highest probability to conditional on those observations). And if the creatures then come to know (or in any case act on) this information about each other, it can be very useful. If I want you to believe that things tend to slip off the rock, so you shouldn’t put our eggs there, I can get you to believe this by putting some other things on the rock, and having you watch as they fall off. I know this is the kind of thing which would lead me to expect that eggs would also fall off the rock, and that we tend to go from observations to future predictions in the same way, so I can infer that this will also cause you to form that belief. Similarly, if I want you to know when I am going off to sleep rather than just wandering away for a short while, I might wave my hands around a lot each time before I go to sleep and not under other circumstances. For I know that our shared manner of doing induction will suggest, on the basis of this evidence, the rule that I always shake my hands around before going to bed. And note further, that once a pattern like this is established I can use it to trick you, by waving my hands around before I go off to sleep with your wife, or if one night I slump down in exhaustion without waving my hands first you can see that I have made a mistake, or in any case are violating my regular pattern.
Thus, there may be a sort of bridge between descriptive laws and normative linguistic rules. Our original capacity to go from observations to a law which makes future predictions, may also be exploited to give us a way of going from finite examples to a general rule or pattern, and since this ‘way’ is shared, a way of communicating and making claims. Our idea of what it would take for a person who has given us a few examples of how to use the word gouche or apply the successor operation to continue on in the same way (as opposed to making a mistake) may be a sort of simplifying off shoot of our idea of our inbuilt irresistible tendency to believe that nature will go on ‘in the same way’. This would explain why the kind of laws which we are inclined to think a priori likely also tend to be simple to state in our language.
And if we put this story about language together with the earlier point about evolution, we get the following theory about what mathematics (on the logical/practice-projecting view) should match well with, and be convenient for stating the kinds of patterns we find in the world. Mathematics is a matter of continuing patterns in the way that humans do, together with logic. The this linguistic or normative sense or what rule a bunch of examples embody, or what it would amount to go on in the same way, originally sprung from our inbuilt sense of what it would take for nature to go on in the same way, our inductive expectation that the world will embody certain kinds of patterns. This expectation itself, was molded by evolution to match the kinds of patterns which there actually are in the world.