All philosophers of math will agree that people do something they call "math", and that this activity is practically helpful, in a certain sense. This is often put pretty loosely by saying `Math helps us build bridges that stand up'. But I think we can say something much clearer than that. Here goes:
Our grasp of math (such as it is) has at least three aspects:
- We can follow proofs. You will accept certain kinds of transitions from one mathematical sentence to another, (or between mathematical sentences and non-mathematical ones) when these are suggested to you.
- We can come up with proofs. You have a certain probability of coming up with chains of inference like this on your own.
- Proofs can create expectations in us. Accepting certain sentences makes you disposed to react with surprise and dismay should you come to accept other sentences. e.g. if you accept "n is prime" you will react with surprise and dismay to a situation where you are also inclined to accept "n has p, q, and r as factors".
Now, the sense in which our mathematical practices are helpful is this:
First, our reasoning about math fits into our overall web of beliefs in such a way as to create additional expectations. Here's what I have in mind: Fix a situation. People in that situation who realize their dispositions to make/accept mathematical inferences arrive in a state where they will be surprised by more things than those in the same situation who don't.
For example, plonk a bunch of people down in front of a bowl of red and yellow lentils. Make each person count the red lentils and the yellow lentils. Now give them some tasty sandwiches and half an hour. Some of the people will add the two numbers. Others will just eat their sandwitches. Now, note that the people ho did the math have formed extra expectations, in the following sense. If we now have our subjects count the lentils all together, the people who did the sum will be surprised if they get anything but one particular number, whereas those who didn't do the math will only be surprised if they get anything outside of a certain given range.
Secondly, the extra expectations raised by doing math are very very often correct. When doing mathematical reasoning about your situation puts you in a state where (now) you'd be surprised if a certain observation/reasoning yields anything but P, applying this process tends to yield P. (This is especially true if we weight the satisfaction/dissatisfaction of strong expectations more heavily). Thus, composing a process of mathematical reasoning M with some other reasoning processes A yields nearly always yields correct expectations about the result of going through a different process B, if it yields any expectations at all.
And finally, this is (potentially) helpful, because it means not only do we acquire the disposition to be surprised if B yields something different, but any further inferences/actions which would get triggered by doing B happen immediately after doing A and M without having to wait for B to take place. For example, in the case from the previous post: if we imagine that all of our sample population have inductively associated counting 1567 lentils in total with having enough to make soup, the people who did the addition after counting the lentils separately, start cooking earlier than those who did something else instead.
Doing math is practically helpful in the sense that spending time doing math raises extra expectations (relative to spending that time eating sandwiches) about the results of certain other processes, and these expectations are generally correct. Thus, mathematical reasoning constitutes a reliable shortcut, leading us to take whatever actions would be triggered by going through some other process B without actually going through B.
NOTE: I don't mean to suggest that this is all there is to math, or that math is somehow *merely* instrumental. I'm just trying to concretely state some data about the successful "applications" of math, which I think everyone will agree to.