/lit/ - Literature

Principia Mathematica exposed math as being the enigmatic bullshit that it really is. We aren't even sure what numbers are yet. For the sake of argument, consider the below hypothesis.

There are an infinite number of numbers.

Therefore there are an infinite number of possible values of 1+1. Therefore the probability that 1+1=2 equals zero. We express this identity as [math]P^1_1(2)=0[/math]. Indeed [math]P^1_1(x)=0[/math] for all x. However in the "real" numbers we must have [math]\int_{-\infty}^\infty P^1_1(x)\,dx=1[/math], a contradiction by the Dominated Convergence Theorem.

The obvious resolution is to introduce enough infinite and infinitesimal numbers to the so-called "real" numbers that the line ceases to be second countable. However the simplest and most elegant solution is simply to notice that infinity does not exist: that there is at most a potential (rather than completed) infinitude of real numbers, and the value of 1+1 is an unknown, possibly changing, value among them.

Hey /lit/, I am writing a paper on the validity of synthetic, a priori principles. Most of what I know comes from Kant's "Critique of Pure Reason" and "Prolegomena."

I critique Kants' proofs from a logical viewpoint; mainly via the framework used in Oliver A. Johnson's "Denial of the Synthetic, A Priori."

I will need to field questions from my class and then in a one-on-one with my professor. I was hoping some of you well-versed in the subject could toss up possible problems/challenges/interesting implications/general discussion regarding my argument or the topic/philosophical works in general.

I will update the string of my argument in separate posts, accompanied with the diagrams/proofs I'll be writing on the board during my presentation.