Friday, June 5, 2009

101 Interesting Things, part sixteen: Russell's Paradox

My last two philosophically-oriented posts have chiefly concerned the limitations of language as a system of categorization in a bottom-up (or "designoid") Universe. This is as opposed to a top-down (or "designed") Universe, in which things in the world are designed intelligently to fit into preconceived categories, which would give language metaphysical primacy over reality. I want to take a little bit of time today to show how this quickly causes a worldview to degenerate into absurdity if we insist that language really does stick to the world and that our categories are somehow "real" instead of "imaginary." This dovetails perfectly with my planned entry on Russell's Paradox as part of my 101 Interesting Things series, so here I go!

Words are used to label and categorize things in the world, but words themselves may also be labelled and categorized as nouns, verbs, adverbs, adjectives, prepositions, and so on. We can also create categories arbitrarily, such as "seventeen-lettered," which refers to those words which have seventeen letters. Every word is a member of the category, "words." So far, so good.

There's a rather interesting way of categorizing words, specifically: according to whether or not they describe themselves. "Word," for instance, is itself a word. But "words" is not more than one word, so it doesn't fit into the "words" category, if you want to be a stickler about quantifiers. "Seventeen-lettered," however, is a seventeen-lettered word, and so describes itself. Such words are called "autological," because they refer to themselves and so are metaphorically contained in their own boxes. If a word does not refer to itself, then it is "heterological." The word, "verb," is not itself a verb, and so is a heterological word.

Logically speaking, these categories ought to be exhaustive: i.e. one or the other of them should apply to every single word. After all, it seems intuitively obvious to an almost painful degree that a word should either refer to itself or not. "A or not-A" is a true disjunction, after all, and the proposition "A word is autological or it is not autological" is surely of that form. Defining "heterological" as "a non-autological word" seems to complete the disjunction and make it so that every word goes into one or the other of these boxes.

Here's the question: into which box should the word "autological" be placed? If "autological" is, in fact, an autological word, then it's an autological word and it goes into the autological box. But! If autological does not refer to itself, then it does not, and so it would go into the heterological box. This seems simple enough - so how do we decide the question? What can we do to test whether the word "autological" is itself autological or heterological?

Umm... oops! As it turns out, there is no way to decide the question. Sure, we can arbitrarily stipulate that it's one way or the other, but we can't come up with any justification for putting the word "autological" (which, as a word, clearly belongs in one or the other box but not both), into one or the other box but not both. Crap!

But wait, there's more! How do we classify "heterological?" If "heterological" is a heterological word, then it goes into the heterological box - but then it goes into its own box, and so it's an autological word, and goes into the autological box - but then it doesn't go into its own box, and so it's a heterological word, and goes into the heterological box - but then it goes into its own box, and so it's an autological word, and goes into the autological box... and so on ad infinitum. If it goes into one box, then it doesn't belong there for one reason, but if it goes into the other box, then it doesn't belong there for another reason. This is because "whether a word refers to itself" and "whether it goes into its own box" are "supposed" to mesh every single time, but with the word "heterological," they are necessarily opposed: if heterological refers to itself, then it goes into its own box, which makes it autological. But its own box is reserved for words which do not refer to themselves, so it can't go in there if it's autological! But that's the only way it can go into its own box, and so on and so forth. Oops!

Now, "autological" and "heterological" are intuitively coherent categories - we can make sense of them - but it's clear that they break down because of themselves. What's wrong with this picture? Is it the things we're trying to categorize to blame, or is it the way we're trying to categorize them, or is it categorization itself that's messing us up? Hmm... interesting question. If only there were some stalwart hero of logic to come to the rescue and show us what's up...

OK, so there was this great guy by the name of Bertrand Russell, and he was an outstanding logician, and he's my hero and I want to have his babies but he's dead now, and he had this great insight into what is known formally as "set theory," which is really just a fancy hat that philosophers put on "categorization" so it looks like it belongs in the ivory tower. This guy, Gottlob Frege, was going around running his mouth about how there's a "set of all sets," which was his way of saying that "there's a category that contains all categories: the category of categories." Like being the King of Kings, you're still a King, but you also rule over all other Kings, so the set of all sets is the set that contains all other sets. Components of a set, the things that make it up or go into its box, are called "elements" of that set, so the set of all sets has itself as an element of itself, which is kind of neat. Then Bertrand Russell arrived on the scene and said, "Wait a minute! What about 'the set of all sets which are not elements of themselves?' Is that in your set of all sets?" And Frege said, "Sure! After all, it's a set!"

But what Frege didn't realize was that "the set of all sets which are not elements of themselves" is to set theory what the word "heterological" is to language. Bertrand Russell knew this, because he knows everything and is awesome, so he told Frege and Frege was like, "Yeah, well, whatever." But then Bertrand Russell said, "Hold on! The set of all sets which are not elements of themselves results in a contradiction when we try to determine whether or not it is in fact an element of itself. But it is a legitimate set nonetheless, as you say, because we can coherently state the criteria for whether something is or is not in that set, and that's what defines a set. So the statement, 'It is true that there is a set of all sets,' results in a contradiction when we try to resolve one of its entailed implications - namely, whether or not 'the set of all sets which are not elements of themselves' is an element of itself - and in formal logic, this means that our starting premise is in fact false! Therefore there is no 'set of all sets,' quod erat demonstrandum, motherfucker!" Then Bertrand Russell folded his arms across his chest, smiled smugly, and flew off in a rocket ship to take tea from his Celestial Teapot. True story.

Now here's the really interesting part: we can mutatis mutandis the above all the way to "It is logically necessary that 'category' is not itself a category, because all categories would go into it, including the category 'heterological,' which results in a logical contradiction." But category is a category, conventionally speaking, because it meets the criteria we've set forth for "being a category." Or, in other words, we talk as if there are categories all the time without our brains exploding, so we can see the conventional/metaphysical split rather clearly. Take it one level higher, and we see that language, while capable of formulating conventionally true statements, can never attain metaphysical truth because it entails that there are categories as a category, which entails a contradiction. But that's the only way we can talk about things: as categories, such as "things," and by using language which is intrinsically imprecise and not "sticky." So, to answer our earlier question, categorization itself is to blame.


Categories are imaginary. Language is all in our heads. Top-down Universes might even be logically impossible, though I can't think of how to prove it at the moment. What say you?

Quick End Note: One practical implication of this is that the micro/macroevolution distinction is purely imaginary, which anyone with half a brain can tell you, but now has been conclusively proven. Arguments citing this as a premise in support of ID are thus made categorically invalid beause this supporting premise is tautologically false - the worst kind of false. Rock on.

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