[jules]

Pi isn't defined by how we measure it. It's defined mathematically. Predicting how a measurement of the ratio between the perimeter and the diameter of a circle comes out is just one of the predictions we can make having calculated pi inside mathematics.

[abrezas]

But if at some point further decimals cannot be experimentally proven to be right, at best you can say "pi is such and such for such precision" and any definition that gives the same number up to that precision must be accordingly accepted. Otherwise you need to demonstrate an experiment that uses further precision.

But we don't do that, and that's why mathematics don't have to do with experience, they are an entirely different tool that also happens to be useful in experimental science.

[jules]

If mathematics doesn't have to do with experience, how do you explain the fact that when we measure pi it's 3.14?

Pi is just pi defined in a mathematical way. It's true that there are other numbers or objects which can make a prediction about the measured ratio of the perimeter to the diameter, but those are not pi. Whether those other objects may be equally valid as pi for making that prediction depends on the details. There may be reasons to prefer one to the other even if they make the same predictions up to measurement precision. We usually prefer the simpler explanation for example. This is equally true in physics and other subjects. Note that as a device for predicting the ratio of the perimeter to the diameter, pi is not perfect. Our space is curved, so for large circles the ratio will deviate from pi, and you have to use a more complicated method based on Riemannian manifolds.

[hyperpape]

"why mathematics don't have to do with experience" is too strong. Nothing you just said implies that math isn't intimately connected to empirical predictions in some way. What it shows is that contrary to Jules' earlier statements, there's no statement by statement correspondence--for any given mathematical statement, you can't find an interesting empirical prediction you associate with it.

[abrezas]

I don't believe they are entirely independent, but there's nothing you can look outside your window that will change any mathematical theorem.

[jules]

I agree with this statement (depending on how you interpret "change": if the world was different we may have been lead to prove different theorems), and nothing I've said previously contradicts it.

[jules]

I never claimed that, in fact I said precisely the opposite:

> Of course this isn't always the most interesting test

[hyperpape]

Substitute non-trivial for interesting. For instance, a calculation of pi to 50 digits will map to the same test as a calculation of pi to 51 digits. But the claims have different content so they should map to different tests!

[jules]

So?

(By the way, they do map to different tests if you view it as a statement in constructive logic, rather than geometry.)

[hyperpape]

They don't map to distinct empirical tests.

And that's the basic requirement for a semantic theory: map distinct statements to distinct contents.