How is the sense in which you say that “math doesn’t directly make predictions about the world” in the paragraph quoted below not equally well a sense in which astronomy “doesn’t directly make predictions about the world”?

Sure, in one sense this is true but that isn’t in conflict with my point. I’m exactly saying that there isn’t anything special about math+bridge laws as opposed to astronomy + bridge laws.

What I’m saying is that many of the apparent problems about mathematical knowledge disappear if we are sufficiently precise. There is no problem about why mathematics is useful because mathematics itself is not useful. There is a problem about why math+bridge laws is a useful theory but math+bridge laws is not fundamentally different in kind from astronomy=bridge laws so it doesn’t call out for any special explanation over any above a general explanation of how we gain scientific knowledge.

I mean imagine someone said, “Isn’t it amazing that English sentences can be used to make predictions about the stars. How could it possibly be that these sentences could tell us useful things about the stars?” The correct response would not be to further his misunderstanding and assign some deep theory to English sentences but rather point out that English sentences themselves make no predictions about astronomy at all. Rather it is the sentences in astronomy textbooks/papers/whatever in combination with certain methods and rules (bridge laws) that together make predictions about the stars.

Now I agree it is a deep puzzling question as to why there is a simple useful theory about stars (or object persistance) but it’s simply misguided to think that there is a deep quesiton about why english sentences seem so useful in make astronomical predictions. Given any useful theory we can always point out properties it has (being written in english) but just noticing that our theory has some property (written in english) doesn’t raise any puzzling questions. Of course if you found out that all possible ways of creating a useful predictive theory of the stars had to be formulated in English that would call out for an explanation.

Is it the case that mathematics is necessary to create useful predictive theories? If mathematics means something that human mathematicians would recognize and understand as mathematics I’m unconvinced. While it may be true that humans all use recognizably similar ‘mathematical’ processes to reach scientific predictions this is no more profound than the fact that all human languages share certain regularities but why should we believe that all aliens, or more generally most succesful beings in most possible worlds use something we would recognize as math to make their predictions? Ironically, it would seem the only time we would be justified in demanding an explanation of why math is so useful is if we already had an answer in the form of an argument showing that math is uniquely useful in making predictions about the world.

Of course you might try and avoid this issue by defining math much more generally. Mathematics, one might say, is some observation independent way of moving from one set of beliefs (understood here as dispositions to respond to external stimuli) to another set of beliefs. However, the need for a system like this is answered by my point about the indexing theorem in recursion theory. Given any way computable way of representing sufficiently complex descriptions of the world there will always be multiple ways to refer to the same state and some computably enumerable means to search for equivalences amoung these descriptions. If that’s all it means to be mathematics then we already know why it’s useful.

Basically my question is: why doesn’t everything you are saying about math not making direct observable predictions apply to astronomy?

More or less it does. That’s the point. Just as there are no substantial problems in the philosophy of astronomy over and above those in the philosophy of science in general there are also no special problems in the philosophy of math.

How is the sense in which you say that “math doesn’t directly make predictions about the world” in the paragraph quoted below not equally well a sense in which astronomy “doesn’t directly make predictions about the world”? Astronomy never tells you what you will see directly, only via bridge laws about your eyes and telescopes working right. You could always say that all there is in the sky is a colossal floating pink elephant, which we could see if we looked, if only our eyes didn’t always go in the fritz.

“While it may appear that mathematics directly makes predictions about the world (if I have two apples in my bag and place another two apples into my bag I have four apples in my bag) in fact it’s only the combination of mathematical theorems with contingent bridge laws that makes these predictions (apples don’t appear or disappear when I place more of them together). One might try and minimize the significance of these bridge laws by saying something like “So long as apples don’t appear or disappear the number of apples in my bag is the number of apples I added minus the number I removed.” However, this merely begs the question by working in our expectation that the plus operation on the natural numbers describes how objects behave into the definition of appear or disappear. I could equally well claim that apples were disappearing and reappearing all the time but if they didn’t do so we could see that adding n apples to a bag with m apples in it results in a bag with n x m apples in it”

The sense in which “if there are exactly 2 red apples in the bag and exactly 2 green apples in the bag at 12:00 and no red apples are green apples then there are 4 red-or-green apples” and “the earth spins at such and such a rate as it goes around the sun” predicts stuff about the world correctly is not that one can reconcile these claims with any recalcitrant experience. As Quine pointed out one can reconcile any theory with observation, provided one is willing to make suitable adjustments to the rest of one’s theory (e.g. our telescopes systematically fail when we look at the sun, or it’s possible to steal apples through a bag without damaging the bag, and there’s an apple stealer who is only interested in stealing apples when there are being counted).

Rather, astronomy and concrete arithmetic ‘make correct predictions about the world’ (to the extent that any single theory – rather than a whole web of belief – can) in the sense that combining them with everything else we have reason to believe results in novel predictions that match up with what we actually observe. That is when combined with our other beliefs about the world, 2+2=4 yields the right thing about what you will get when you count the apples right off the bat! It says: expect to find 4 apples if the bag is still intact, less than 4 there’s a whole in the bag, maybe 0 if there’s a hole and you have been walking through a bad neighborhood. 2+2=5 can only be made to yield the right predictions if we give up a lot else that we thought we knew about apples, physics etc.

Basically my question is: why doesn’t everything you are saying about math not making direct observable predictions apply to astronomy?