Missing Things

21 February 2010

Kurt Gödel

I've been really bad about updating this - last week I was sick, but yesterday I was just lazy. Shame on me. But - I'm sworn not to become one of those people who starts a blog and never updates it, so.

Time to start talking about the other half of the title.

The Library of Babel, if you'll remember, is a metaphor for the inherent vagueness of truth and falsehood - you cannot know that a statement is true simply by looking at it. You have to compare it to itself, and to other statements that you know to be true... which you don't necessarily really know to be true, either.

So doubt everything. Test every book you read. Can it describe itself? Or does it rely on another book that can, or another book that relies on another book that relies on an entire series that can? If it can survive your earnest flame - and you must be in earnest to avoid deceiving yourself - you may trust it with your life. If not, it was not worth keeping to begin with, as useful to you as the belief that you can live without breathing.

And if you think that's no way to live... I have a bridge to sell you.

There's this construct called mathematics, which is basically the above taken to extremes. It's not just for numbers - that would be the subject known as arithmetic - but the rigorous calculation of fact. Set theory is probably one of the most general sub-categories, dealing with absolutely anything that can be grouped; but also predicate logic, with begins with tautologies like "if A is true, then A is true" and "either A is true or A is not true", and builds from there. Because of this, mathematical proofs are absolutely reliable... and this leads to problems.

Let's talk set theory. Any group that can be described constitutes a set - the set of all rational numbers, for instance, or its subset that contains only the numbers 4, 18, and 6... or the set of all the books in my library, or the set of everyone who has ever had the name "Julius Caesar". You can even define a set whose elements are {white, 14, Literacy, [you]}, as long as you don't include yourself (or literacy, or white, or 14) more than once.

There are also sets whose elements are other sets - the set of {white, 14, Literacy, the set of all the books in my library}, for instance. This is where a few important distinctions come in:

(1) 14 is not {14}. A set containing a single element is not the same as that element; saying {14} + {7} = {21} is like saying that {apple} + {orange} = {some bizarre sum equal to apple+orange}.

(2) {14} is a subset of {14, white} because all the elements of {14} are also elements of {14, white}. 14 is not a subset of {14, white} because it is not a set. More weirdly, 14 is an element of {14, {14, white}} and {14} is a subset, but {14} is an element of {{14}, {14, white}} but not a subset. The brackets are important!

(3) {14, white} and {white, 14} contain exactly the same elements (i.e. they're subsets of each other), which means they must be the same set. Order doesn't matter.

Confused yet? If you are, ask me and I'll try to explain better.

18 February 2010

A thought about enlightened monarchy

An enlightened monarch is one who embraces the principles of the Enlightenment and uses them to support his own reign; i.e. a ruler who does things like patronize the arts, improve his subjects' standard of living, etc, because if he fulfills his responsibilities and makes his people safe, free, and happy, they'll tend to prefer that he stay in power rather than rebel.

With that in mind, that golden rule of compassion and self-giving, "Do unto others as you would have others do unto you", seems more like enlightened selfishness.

If that's the case, what does selflessness look like?

13 February 2010

Socrates meets Malacoda

Inspired by Peter Kreeft's Socrates Meets... series, and Dante's Inferno.


[Setting: A nondescript beach sloping down to the ocean on the right and steeply rising into a high row of rocks on the right. Between them, unhurriedly, paces Socrates, head in the clouds and lost in thought, wandering in no particular direction other than simply down the beach. In front of him and on top of an especially flat part of the rock wall crouches Malacoda, watching and smirking fiendishly. Socrates sees him, appears puzzled, and speaks.]

Socrates: Good day, sir. Where are we… and who are you?

Malacoda: That first is complicated. As for the second, you may call me Malacoda.

Socrates: That is your name?

Malacoda: Indeed.

Socrates: But who are you? That does not actually answer my question.

Malacoda: Hmm. I am generally considered a demon, but not in the way that you understand the term. Consider me one of a number of gods of the afterlife, if you would. A devil.

Socrates: Ah. So I am, in fact, dead, then. I had wondered. This place is part of the afterlife, then?

Malacoda: Quite true. In fact it is the entirety of the afterlife, and a bit more besides.

Socrates: I’m sorry? Could you possibly elaborate?

Malacoda: Certainly. This land, from the shore of the Dirac Sea beside you to infinity behind me is the space where things are kept that do not exist. It is the home of the library of unwritten books, it is where we keep Theseus’ ship now that he has replaced every plank, it is where we hide the square root of infinity, and it is where [picks up a candlestick and ignites it with his breath] the flame of a candle goes after it has been blown out. Some call it Tumbolia, some call it Super-Sargasso Space, but properly it is known as Potentia. Naturally, now that you have ceased to exist, you have arrived here.

Socrates: You have said something very interesting to me. It is my understanding that at death, the body and soul are separated and each goes its separate way. Am I correct so far?

Malacoda: Yesss…

Socrates: And the body slowly returns to the material world.

Malacoda: Yes.

Socrates: Now, it is my understanding that the soul, the essential I, moves onto the spiritual world. But you have just said that I cease to exist and therefore comes here.

Malacoda: You have said it, Socrates.

Socrates: I find there is a rather stark difference between these two statements.

Malacoda: Not really. Since this spiritual world of yours does not exist, either, your essential “you” does in fact go there – it is part of Potentia.

Socrates: I think you’re lying.

Malacoda: That’s possible. We devils do so frequently. Wherein lies the falsehood?

Socrates: If nothing here exists, then what of yourself?

Malacoda: Oh, I do not exist either. I was dreamed up by Dante Alighieri. I am a construct of his mind alone, I never have been real, and I never will be. Thus, I have been made responsible for supervising Potentia on Hell’s behalf.

Socrates: Literary characters are here, too?

Malacoda: Didn’t you know? You yourself are a character of Plato’s invention and have no proper existence either.

Socrates: Don’t be absurd. Of course I exist.

Malacoda: Yes, you exist in Potentia.

Socrates: Very funny. You give this place a sort of twisted sense but it so flagrantly violates basic logic it’s impossible to believe you.

Malacoda: I suppose you cannot really think through the entire thing clearly yet. What can I do to prove it to you?

Socrates: You are going to try to prove empirically that our current location is not empirical?

Malacoda: Very sensible for such a place, isn’t it?

Socrates: Not if the place itself is not very sensible.

Creon: That is the point, I think.

Socrates: Who are you?

Malacoda: My empirical evidence; allow me to present the fictitious Creon of Thebes.

Socrates: Where did you come from? How do you know what we are talking about?

Malacoda: Has he not been here all along?

Socrates: No!

Malacoda: Precisely.

Creon: I have to agree with Socrates.

Malacoda: What?

Creon: You’ve been talking solely about terms so far. That is fine as far as logic goes, but as you both have come close to pointing out, logic is not very practical for talking about inherently self-contradictory things. Well, as far as empirical evidence goes, I can tell that I am currently standing on this beach, and therefore both the beach and I exist.

Malacoda: But I can provide any number of objects that do not exist or cannot possibly exist. Would you like to see an anti-knot?

Creon: I could not say. What is it?

Malacoda: If you tie a knot in a rope, the anti-knot is the knot you'd tie to cancel out all the twists of the first without actually untying it - and, since neither of them exists after they cancel each other out, they end up here. Alternatively, look at this rope with three ends, or this seven-sided cube.

Creon: This is irrelevant. I’m holding it right now. As far as I’m concerned, it must exist and you’re just wrong about it being impossible.

Socrates: I don’t suppose I could trouble you with a few questions?

Malacoda: Oh, not at all. Be my guest.

Socrates: All right. Could I confirm something with you first? I just want to be sure that when I am speaking to you I am speaking.

Malacoda: …what?

Socrates: I am making statements, yes?

Malacoda: That seems to be trivially, even tautologically, the case.

Socrates: And these statements, they are meaningful?

Malacoda: I suppose. Although not very.

Socrates: But they do, in fact, possess meaning.

Malacoda: I am responding to you, am I not?

Socrates: Is that a yes or a no? If you do not exist then some would say you cannot respond.

Malacoda: Clever! But no, I may not exist, but my responses do.

Socrates: How is that possible?

Malacoda: Because someone who exists is recording them.

Socrates: …Oh. Well then, this place Potentia. It is where things are kept that do not exist?

Malacoda: Indeed.

Socrates: And only things that do not exist?

Malacoda: …yes…

Socrates: Do some things exist?

Malacoda: Excellent question. I shall not say.

Socrates: Well. Does Potentia exist?

Malacoda: Umm. Suppose I say yes?

Socrates: Then I ask you how a place can exist when none of its features do.

Malacoda: Then I say no.

Creon: So this place is a part of itself?

Malacoda: Yes. Is this somehow impossible? The place you call Earth contains the entirety of the Earth, doesn't it? Besides, even if you think that there is something inherently impossible with a space containing itself, being impossible is part of the nature of this particular space.

Socrates: Can you show me something that is both nonexistent and indescribable?

Malacoda: [produces such an object] Voila.

Socrates: Impressive.

Malacoda: Thank you.

Socrates: But unsuccessful.

Malacoda: What?

Socrates: That thing there can be described as “something both nonexistent and indescribable”. Since I just described it, it is not indescribable. Moreover, I can say that about anything that “only exists in Potentia” – so if it is indescribable, it cannot not exist.

Malacoda: So?

Socrates: If it does not not exist, then it must exist.

Malacoda: Well! I concede that some things exist, then.

Socrates: But by definition, something both nonexistent and indescribable cannot possibly exist. So now we have an object that neither exists, nor does not exist – a logical impossibility.

Creon: These are just word games. If he cannot produce it for you, then obviously it does not exist, and that’s the end of it.

Socrates: I don’t believe these he’s contributing anything to the discussion. Can we be rid of him?

Malacoda: Of course.

Socrates: How do you do that? Did you make him start existing?

Malacoda: Well, what if I did?

Socrates: Then you’re a creative being and certainly no devil, if as you said a devil is a god of death. It’s as I thought. You have been lying about this entire situation. Potentia cannot exist as such, and neither can any of the things – or people – you have attempted to use as proof. Well, this has been a fascinating diversion, but I have an afterlife to investigate, so if you’ll excuse me…

Malacoda: Sssss! Objective unreality defies your wordplay! Tell me. Did you ever meet Euthyphro?

Socrates: I… hmm. I do seem to recall conversing with him, but I could have sworn he was dead…

Malacoda: Exactly! You remember a fiction Plato wrote about you. You’re sufficiently nonexistent for me to keep you stuck here!

Socrates: Do you mean to say I’m a prisoner of a nonexistent being in an impossible place for no good reason?

Malacoda: Precisely!

Socrates: And, to be clear on this point, there is absolutely no way to escape?

Malacoda: None whatsoever!

Socrates: Ah. Well, if a way to escape doesn't exist, then I'll just go find it, won't I?

Malacoda: Wait, what? That's... umm...

Socrates: Goodbye.

"The universe, which others call the Library..."

The Library of Babel is a short story by Jorge Luis Borges describing its own title. The Library is an entire world completely filled by hexagonal rooms of bookshelves, containing all the possible permutations of 25 symbols - 22 letters, plus the period, comma, and space - that could be contained in books that are precisely 410 pages long, 40 lines per page, 80 symbols per line. Every possible book that fits these parameters is somewhere in the Library, exactly once. Somewhere, there is a book containing nothing but four hundred and ten pages of MCVMCVMCV, but also the Encyclopedia Britannica, Shakespeare's First Folio, and a book that describes how to construct a perpetual-motion machine.

And, of course, a book whose title page is from the First Folio (by MCV), but the rest of which is from a faulty version of the Britannica that contains a description of a perpetual-motion machine. Thus, the Library contains all truth - but also all falsehood.

Obviously, somewhere, there is an index - a catalogue, explaining where all the books containing truth can be found, and one in every language, no less. But of course there are also a countless number of flawed indices, many of which have simply misplaced a period or substituted a word, but many of which are seemingly flawless except that where they say you should be able to find your own biography there is actually a copy of Finnegan's Wake. How, then, can you tell the true from the false?

You can start by picking an apparent index and looking it up in itself. If it's not there, you obviously can't trust it, because if it is true it should be listed in the book! Moreover, if it gives you a wrong location, you can't trust it, either. This is the easiest thing to begin with - any index which cannot account for itself is not a reliable index.

When you have an index that satisfies this basic condition, you can check the reliability of the rest of the list to make sure that all the books it contains are where it says they are. Then you must test the reliability of those books to make sure that they actually contain what they say they contain in the title; if any of these books are themselves indices (more than likely, for a library of this scope), this means testing a number of their contents for accuracy as well, ad infinitum if it goes that far.

Difficult, you say? Yes. Impossible? Necessary, if you wish to believe you know anything.

I hate hitting people with frying pans, but... if you haven't realized what this entire entry is a metaphor for yet, well, the majority of the rest of this blog is probably not for you.

As for Godel... he can wait until later.