In my last post, I proposed that that our methods of reasoning about math are "practically helpful", in (at least) the sense that they act as reliable shortcuts. Mathematical reasoning leads us to form correct expectations about (and hence potentially act on) the results of various processes of observation and/or inference, without going through these processes.
Now I'm going to give some more interesting examples of (our methods of reasoning about) mathematics being practically helpful to us in this way.
The general structure in all these is the same: Composing a process of mathematical reasoning M with some other reasoning processes A yields a result that's (nearly always) the same one you'd get by going through a different process B.
1. Observe computer (wiring looks solid, seems to be running program p etc.), derive that program it's running doesn't halt, expect it to still be running after first 1/2hour <--> observe computer after 1/2 hour
2. Observe cannonballs, form general belief about trajectory of ball launched at various angles, observe angle of launch, derive where trajectory lands <---> measure where this ball does land.
3. Prove a general statement, expect 177 not to be a counterexample <---> (directly) check whether 177 is a counterexample.
4. Conclude that some system formalizes valid reasoning about some math truths, expect that you aren't looking at an inscription of a proof of ``0=1'' in that system <---> check what you have to see if it's an inscription a proof in the system, if it ends in ``0=1''.
5. Count male rhymes in poem, count female rhymes, then add <---> Count total rhymes
[Special Case Study: Number Theory
If we focus on the case of reasoning about the numbers, we can see that there's a nice structure of mathematics creating correct expectations about mathematics which creates correct expectations about mathematics, which creates correct expectations about the word.
- general reasoning about the numbers: Ax Ay Az ((x+y)+z) = (x+(y+z))
- calculations of particular sums: 22+23=45
- assertions of modal intuition: whenever there are 2 apples and 2 oranges the must are 4 fruit
- counting procedures: there are two ``e"s in ``there''
Note that each of the above procedures allows us to correctly anticipate certain results of applying the procedure below it. ]