I used the example of π, but this applies to other transcendental numbers as well, such as e
Kant classified statements into 4 epistemic categories based on two criteria: The Analytic/Synthetic distinction (Are statements true by definition or do we need outside information to determine their truth) and the A Priori/A Posteriori distinction (Are they independent of empirical evidence or not).
In particular he arrived at the existence of synthetic a priori truths, in opposition to Hume who believed that all statements were either analytic a priori or synthetic a posteriori.
Neither Kant nor Hume believed that analytic a posteriori truths are possible.
My question is regarding the calculation of π up to an arbitrary number of digits:
- It is a number, so presumably it contains it's own definition: Is saying "π up to 88 digits = 3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034" is an analytic a priori statement like "Two = 2"?
- But beyond a certain number of digits, nobody can come with the new digits on their own, they would have to rely on a computer to perform the calculation, so is it a analytic a posteriori (and Kant was wrong to thing analytic a posteriori truths didn't exist)?
- π isn't really a number, it is a symbol that is shorthand for a complex mathematical relation, and as such "π up to 88 digits = 3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034" is a synthetic a priori truth?
- But we can't calculate "π up to 2288 digits" without performing a mechanical procedure. So π is actually a empirical fact about the world - i.e. synthetic a posteriori. Is π then an empirical constant, similar to the gravitational constant or the charge of an electron?
How would Kant classify π? How would Hume? If π is empirical, doesn't it make theory laden, per the Quine-Duhem thesis and π would change depending on some changes in the axioms of math or geometry? What is the epistemic status of π? Given that we can never know the "true value" of π completely, is it a thing in itself, part of the noumenon?