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.

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*Related*

http://koanicsoul.com/forum/viewtopic.php?f=18&t=142&p=5163#p5135

Oops, I think I missed that back in the days before I discovered the “View Unread Posts” button.

Fantastic. I’d actually had the idea a few months ago that information could be stored merely by pointing at positions on the number line. I really, really wish I could keep up with you so we could have a conversation.

http://arxiv.org/pdf/math/0404335.pdf

“It always bothers me that, according to the laws as we understand them today, it takes a computing machine an infinite number of logical operations to figure out what goes on in no matter how tiny a region of space, and no matter how tiny a region of time. How can all that be going on in that tiny space? Why should it take an infinite amount of logic to figure out what one tiny piece of space/time is going to do? So I have often made the hypothesis that ultimately physics will not require a mathematical statement, that in the end the machinery will be revealed, and the laws will turn out to be simple like the chequer board…”—Feynman

We’ll get to this below.

“I call this theoretical theoretical physics, because even if these models do not apply to

this particular world, they are interesting possible worlds!”

“Through the years I’ve noticed many times, as an armchair physicist, places where physical calculations diverge to infinity at extremely small distances. Physicists are adept at not asking the wrong question, one that gives an infinite answer. But I’m a mathematician, and each time I would wonder if Nature wasn’t really trying to tell something, namely that real numbers and continuity are a sham, and that infinitesimally small distances do not exist!

Two Examples: the infinite amount of energy stored in the field around a point electron according to Maxwell’s theory of electromagnetism, and the infinite energy content of the vacuum according to quantum field theory.

In fact, according to quantum mechanics, infinitely precise measurements require infinite energies (infinitely big and expensive atom smashers), but long before that you get gravitational collapse into a black hole, if you believe in general relativity theory.”

I just choose not to believe in point particles.

“The paradox of the whole being equivalent to one of its parts, may have deterred Galileo, but Cantor and Dedekind took it entirely in stride. It did not deter them at all. In fact, Dedekind even put it to work for him, he used it. Dedekind defined an infinite set to be one having the property that a proper subset of it is just as numerous as it is! In other words, according to Dedekind, a set is infinite if and only if it can be put in a one-to-one correspondence with a part of itself, one that excludes some of the elements of the original set!”

To respond to Feynman, if we can’t store an infinite amount of information in a finite amount of space and time, then how can we store enough information in our brains to have an accurate picture of the universe?

Information has to be unphysical if our understanding of the physical world isn’t to be flawed.

“Borel pointed out that if you really believe in the notion of a real number as an infinite sequence of digits 3.1415926…, then you could put all of human knowledge into a single real number. Well, that’s not too difficult to do, that’s only a finite amount of information. You just take your favorite encyclopedia, for example, the Encyclopedia Britannica, which I used to use when I was in high-school—we had a nice library at the Bronx High School of Science—and you digitize it, you convert it into binary, and you use that binary as the base-two expansion of a real number in the unit interval between zero and one!

So that’s pretty straight-forward, especially now that most information,including books, is prepared in digital form before being printed.

But what’s more amazing is that there’s nothing to stop us from putting an infinite amount of information into a real number. In fact, there’s a single real number, I’ll call it Borel’s number, since he imagined it, in 1927, that can serve as an oracle and answer any yes/no question that we could ever pose to it. How? Well, you just number all the possible questions, and then the Nth digit or Nth bit of Borel’s number tells you whether the answer is yes or no!”

“You see, some mathematicians have what’s called a “constructive” attitude. This means that they only believe in mathematical objects that can be constructed, that, given enough time, in theory one could actually calculate. They think that there ought to be some way to calculate a real number, to calculate it digit by digit, otherwise in what sense can it be said to have some kind of mathematical existence?”

In contrast, I’m a non-constructive Platonist/Intuitionist(I don’t think there is a meaningful difference), and we all know what kind of circles Brouwer moved in.