In the first case, starting with p such that the highest power of 2 that divides p is an integer power of 2 (2^k for some integer k); then the highest power of 2 that divides p² is 2^2k; then the highest power of 2 that divides 2q² is also 2^2k; then the highest power of 2 that divides q is 2^(2k-1); therefore q must be a multiple of 2^(k-0.5); a noninteger power of 2.

This implies that there is a number 2^(0.5). It makes no claims as to whether or not this number is rational, or integer; it merely claims that such a number must exist. (Consider: if I had started instead with the equation x²-4=0, I would have ended up showing that a number of the form 4^(0.5) must exist - that number is rational, is indeed an integer).

Now, I think I can prove that an integer q which is a multiple of 2^(k-0.5) but which is not a multiple of 2^k, for integer k, does not exist; but I can only complete that proof by knowing in advance that 2^0.5 is irrational, so I can't use it to prove the irrationality of 2^0.5. I can easily prove that a rational number of the form 4^(k-0.5) for integer k does exist; indeed, an infinite number of such numbers exist (examples include 2, 8, 32).

No matter how forcefully that first passage conveys the irrationality of √2, it does not prove it.

At the Princeton graduate school, the physics department and the math department shared a common lounge, and every day at four o'clock we would have tea. It was a way of relaxing in the afternoon, in addition to imitating an English college. People would sit around playing Go, or discussing theorems. In those days topology was the big thing.

I still remember a guy sitting on the couch, thinking very hard, and another guy standing in front of him saying, "And therefore such-and-such is true.

"Why is that?" the guy on the couch asks.

"It's trivial! It's trivial!" the standing guy says, and he rapidly reels off a series of logical steps: "First you assume thus-and-so, then we have Kerchoff's this-and-that, then there's Waffenstoffer's Theorem, and we substitute this and construct that. Now you put the vector which goes around here and then thus-and-so . . ." The guy on the couch is struggling to understand all this stuff, which goes on at high speed for about fifteen minutes!

Finally the standing guy comes out the other end, and the guy on the couch says, "Yeah, yeah. It's trivial."

We physicists were laughing, trying to figure them out. We decided that "trivial" means "proved." So we joked with the mathematicians: "We have a new theorem -- that mathematicians can only prove trivial theorems, because every theorem that's proved is trivial."

The mathematicians didn't like that theorem, and I teased them about it. I said there are never any surprises -- that the mathematicians only prove things that are obvious.

From "Surely You're Joking, Mr. Feynman!": Adventures of a Curious Character

After I spoke at the 2005 "Mathematics and Narrative" conference in Mykonos, a suggestion was made that proofs by contradiction are the mathematician's version of irony. I'm not sure I agree with that: when we give a proof by contradiction, we make it very clear that we are discussing a counterfactual, so our words are intended to be taken at face value. But perhaps this is not necessary. Consider the following passage.

There are those who would believe that every polynomial equation with integer coefficients has a rational solution, a view that leads to some intriguing new ideas. For example, take the equation x² - 2 = 0. Let p/q be a rational solution. Then (p/q)² - 2 = 0, from which it follows that p² = 2q². The highest power of 2 that divides p² is obviously an even power, since if 2^k is the highest power of 2 that divides p, then 2^2k is the highest power of 2 that divides p². Similarly, the highest power of 2 that divides 2q² is an odd power, since it is greater by 1 than the highest power that divides q². Since p² and 2q² are equal, there must exist a positive integer that is both even and odd. Integers with this remarkable property are quite unlike the integers we are familiar with: as such, they are surely worthy of further study.

I find that it conveys the irrationality of √2 rather forcefully. But could mathematicians afford to use this literary device? How would a reader be able to tell the difference in intent between what I have just written and the following superficially similar passage?

There are those who would believe that every polynomial equation has a solution, a view that leads to some intriguing new ideas. For example, take the equation x² + 1 = 0. Let i be a solution of this equation. Then i² + 1 = 0, from which it follows that i² = -1. We know that i cannot be positive, since then i² would be positive. Similarly, i cannot be negative, since i² would again be positive (because the product of two negative numbers is always positive). And i cannot be 0, since 0² = 0. It follows that we have found a number that is not positive, not negative, and not zero. Numbers with this remarkable property are quite unlike the numbers we are familiar with: as such, they are surely worthy of further study.

• Timothy Gowers, Vividness in Mathematics and Narrative, in Circles Disturbed: The Interplay of Mathematics and Narrative

Only on LessWrong would a statement with that much insight be downvoted because it could be taken to signal something vaguely positive about religion.

Thinking is an act, feeling is a fact. Don't bother comprehending, living is far beyond any comprehension...

-- Clarice Lispector.

Original in Brazilian portuguese:

Pensar é um ato, sentir é um fato. Não se preocupe em entender, viver ultrapassa qualquer entendimento...

Before you downvote: I think this has a lot to do with rationality because we tend to be caught up in thinking and the models about the world we create in our minds, actually science is about this. But those models have limitations and are often wrong as the history of science shows time and again.

EDIT: added the originals. Fixed typo.

"Given the nature of the multiverse, everything that can possibly happen will happen. This includes works of fiction: anything that can be imagined and written about, will be imagined and written about. If every story is being written, then someone, somewhere in the multiverse is writing your story. To them, you are a fictional character. What that means is that the barrier which separates the dimensions from each other is in fact the Fourth Wall."

-- In Flight Gaiden: Playing with Tropes

(Conversely, many fictions are instantiated somewhere, in some infinitesimal measure. However, I deliberately included logical impossibilities into HPMOR, such as tiling a corridor in pentagons and having the objects in Dumbledore's room change number without any being added or subtracted, to avoid the story being real anywhere.)

It is absurd to divide people into good and bad. People are either charming or tedious.

-- Oscar Wilde

I argue that my brain right now contains a lossless copy of itself and itself two words ago!

I'd argue that your brain doesn't even contain a lossless copy of itself. It is a lossless copy of itself, but your knowledge of yourself is limited. So I think that Nick Szabo's point about the limits of being able to model other people applies just as strongly to modelling oneself. I don't, and cannot, know all about myself -- past, current, or future, and that must have substantial implications about something or other that this lunch hour is too small to contain.

How much knowledge of itself can an artificial system have? There is probably some interesting mathematics to be done -- for example, it is possible to write a program that prints out an exact copy of itself (without having access to the file that contains it), the proof of Gödel's theorem involves constructing a proposition that talks about itself, and TDT depends on agents being able to reason about their own and other agents' source codes. Are there mathematical limits to this?

a lossless copy of itself and itself two words ago

But our memories discard huge amounts of information all the time. Surely there's been at least a little degradation in the space of two words, or we'd never forget anything.

Still, I don't think you could compress the content of 1000 brains into one. (And I'm not sure about two brains, either. Maybe the brains of two six-year-olds into that of a 25-year-old.)