[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.