It’s a good thing I left off where I did, because I realized that I can’t come up with any convincing evidence that reality is continuous and not just a large number of small pixels. I tried to resort to some mathematical stuff, but I’m just not very good at this metaphysical stuff. To give you an idea, it took me around 3 years to convince myself that the natural numbers and the even numbers have the same cardinality.
Yeah, now you know why I mostly stick to occipital-style intuition.
My first plan of attack was to show that pi is a transcendental number*, and pi is a real thing, therefore reality is continuous. This was flawed, naturally, because I have no idea whether pi is a real thing or merely an idealization that gives us really useful estimates. This phenomena-noumena stuff really fucks with my head, although I’m starting to get it (I think).
My second plan of attack was to try comparing noumena to phenomena because if reality is pixelated an error term should pop up. Maybe draw increasingly perfect circles and measure pi. Well for one, it’s hard to measure stuff without using material instruments that are subject to larger-than-or-equal errors than the ones I’m seeking. I lack the reasoning skills to prove that such an experiment is impossible, and I lack the imagination to dream one up. Bother.
Therefore, I’ve been unable to conclude that there is such a thing as a continuous probability density function in reality. Poop.
*It would be enough to show that an uncountable number of rational numbers exists even though that leaves holes in the real number line, because the definition of a limit only requires that there is at least one number between two other numbers.