101 Interesting Things, part thirty-two: Proof by Contradiction

Contradictions do not exist. Paradoxes exist, usually expressed in clever-sounding phrases such as, "If you want peace, then prepare for war," or, "The tree of liberty must be refreshed from time to time with the blood of patriots and tyrants." These are mere conceits of what we call "common sense," though, and all they show is that reality doesn't always conform to our expectations of "what makes sense." But a bona fide contradiction is of the form, "X both is the case and is not the case, at exactly the same time and in exactly the same respect," and such statements are categorically false. We can establish this with good old fashioned deductive logic - specifically, with implication.

A statement of implication is a conditional, an "if-then" statement, and may be phrased as, "A implies C," or "If A, then C," where A is the Antecedent and C is the Consequent. These in their turn are propositional clauses, statements which may be taken on their own and determined to be either true or false (but not both - at least not at the same time and in the same respect). The whole implication may be taken itself as a proposition, and itself has truth value. To illustrate this point, let us consider the implication, "If I encounter a pile of babies, then I will wish for a fork and knife," because I've got A Modest Proposal on the brain.

"If I encounter a pile of babies, then I will wish for a fork and knife," may be written several different ways; in addition to the foregoing, we may also write, "D encountering a pile of babies implies that D will wish for a fork and knife," or "A implies C, where A is 'D encounters a pile of babies' and C is 'D wishes for a fork and knife'." I'll spare the symbolic logic, because if you know what that stuff means then you don't need me to tell you all of this. At any rate, our A and C may each be true or false independently, and the implication as a whole may be true or false as follows:

• If I do in fact encounter a pile of babies, then A is true.
• If I do in fact then wish for a fork and knife, then C is also true and the implication is true as a whole.
• If I do not in fact then wish for a fork and knife, then C is not true and the implication is false as a whole. This is the only way for an implication to be false as a whole proposition (if A is true and C is false).
• If I do not in fact encounter a pile of babies, then A is false.
• If A is false, then it does not matter whether C is true or false, because I never actually do the antecedent thing; the implication is thus true as a whole.

This is one of the weirdest consequences of logical implication: a false antecedent validly implies any consequent. This is also why David Lewis likes modal realism, because it justifies the weirdness of implication in his mind. I just want to punch him in the face. Moving on! A professor was once asked to show how "2=1" implies that he is the pope, and according to legend, he responded as follows: "2=1, and there are the two entities 'myself' and 'the pope'; since 2=1, we two entities are in fact one and the same entity, and thus I am the pope." As I have illustrated before, this makes first premises very dangerous, because a false premise validly implies absolutely anything. As it happens, only a false premise implies absolutely anything, so if you wind up implying anything and everything, then you know you've got a false premise somewhere (common-sense version: a statement that could mean anything, in fact means nothing). But I'm getting ahead of myself here.

I'm really writing this to get at the proof that the square root of two is an irrational number, which is to my mind one of the greatest-ever marriages of mathematics and logic (aside from, oh, the entire field of geometry). We start by assuming that the square root of two is rational and seeing where things go from there. If the square root of two is rational, then it can be written as the proportion of two integers a and b (such that ^a/_b is an irreducible fraction). This implication is true, because any rational number may be so described. So we start with:

√2=^a/_b

Since we have an equivalence, we can square both sides:

2=(^a/_b)²=^a2/_b²

Now cut out that middle part and multiply both sides by b²:

2b²=a²

Interesting! Because b² can be doubled to make a², we know that a² is a multiple of two; and since a is an integer, and even integers have even squares while odd integers have odd squares, we also know that a itself must be a multiple of two. So let's say that a=2x, where x is an integer, because of all of the foregoing and the fact that even integers may be halved to create yet more integers. Going back to the start, this gives us:

√2=^2x/_b

Lather, rinse, and repeat as before:

2=(^2x/_b)²=^4x2/_b²

Multiply both sides by b² as before:

2b²=4x², and divide by 2 for b²=2x²

Wait, what?! This means that b is also an even number! Which means that a and b can both be divided by two! Which means that ^a/_b is not an irreducible fraction, and we have thus reached a contradiction. Because our initial implication is true, the antecedent (viz. "the square root of 2 is a rational number") must perforce be false for the contradiction to be resolved. Quod erat demonstrandum, motherfucker!

You can also do a little mutatis mutandis to prove that the square root of 3, if written as the fraction ^a/_b, can never be made irreducible because a and b will always both be divisible by 3 (which is impossible, thus the contradiction, thus the square root of 3 is irrational). This also works for five, and seven, and every prime number, and every non-square integer. We may even generalize this proof for any prime number p as follows, with but one quick detour:

√p=^a/_b

p=(^a/_b)²=^a2/_b²

Since p is an integer, ^a2/_b² is also an integer because it is p. No non-integer may be multiplied unto itself to create an integer (try it, I dare you!), thus ^a/_b must also be an integer. But if ^a/_b is an integer, and can be multiplied unto itself to make p, then p must not be prime after all and so we contradict ourselves: if the square root of a number p is rational, then p is not prime. Since we stipulated at the start that p is in fact prime, we have thus shown that every prime number has an irrational square root. Hey, presto!

OK, math class is over, now go outside and play!

Bullshit Pulpit: So help me, Pythagoras...

Pop Quiz! The Bible contains which two of the following three things (answered by clicking, so don't cheat):

1. Instructions on how to sell one's daughter into slavery.
2. An explicit declaration that rape is morally wrong.
3. A proclamation that God is as strong as a unicorn.

If you answered 1 and 3, congratulations! You know your Bible! Next, please rank the following three ideas in order from greatest importance to least:

1. The lives and rights of women are just as important as those of men.
2. The comparative strength of God against that of other mythical creatures.
3. People are not property.

If you ranked 2 last, congratulations! You're civilized! Finally, please reconcile your answer to the previous question with the fact of the matter of the first. If you're an atheist, or some manner of theist who acknowledges that the Bible is just some book written by primitive men with a whole mess of backward ideas, congratulations! You're rational!

Now I shall abruptly switch gears. Suppose that I hold up a math textbook in one hand, and while gesturing wildly with the other, I proclaim that I believe everything in this book is true. Yes, I know that you know where I'm going with this, just trust me for now. Suppose also that you ask me, "Well, do you believe that the quantity, 'A plus B,' squared, is equal to A squared plus two AB plus B squared?" And suppose once more that I respond, "Preposterous twaddlecock!"

Well, as it happens, (a+b)²=(a²+2ab+b²) is a quadratic equation, and these are among the foundations of algebra. Also, the book in my hand is a high-school algebra textbook, so it damn well ought to be in there. Page 382, for the sake of argument. The point here is not whether this quadratic equation is true or not, or whether my math book has any errors or not; the point is that, by pronouncing my belief in a book I have not read and do not fully understand, I have put myself in a position to look awfully silly.

Let's shift back to the original topic of this evening's symposium: Biblical and moral pop quizzes. Respondents may be grouped by their answers and I suspect that these groups will have a strong correlation with the beliefs of those respondents. Those who recognize that the Bible does not say that rape is wrong will, for the most part, turn out to be atheists (or perhaps theologians, but these will be a clear minority unless I somehow acquire a selection bias in favor of theologians). Those who suspect that the Bible does not compare God's strength to that of an imaginary creature will, by and large, turn out to be believers. I'm not sure about that first one, though - it strikes me as less preposterous than 3, but less necessary than 2 - so I'll withhold any predictions about that one for now. And there will of course be those who are neither atheist nor Christian (you know, people who are religious but don't think the Bible is true) who could reject any one of them for a whole host of competing reasons (in America, they'd be a minority).

My point is that Biblical belief, absent Biblical education, will lead to something like the following general thought process in believers: "Hmm... well, the Bible is a morally good book, the word of God in fact, and so 2 is definitely in... and 1 is pretty bad, but 3 is just stupid. If I had to guess, I'd say 3 is the missing one; and I have to guess, so I'm going to say that 3 isn't in there, and 1 & 2 are." BZZT! Wrong! Beliefs about the book formed a rational prediction, and that prediction turned out to be incorrect. Since the prediction was incorrect, we are left to examine the beliefs from which the prediction was formed. I maintain that the Bible is not a morally good book and is not the word of God, but is rather the work of many different men throughout antiquity who didn't know what atoms or germs are and probably thought that unicorns are real. Other answers are possible, though: for instance, God may have been so preoccupied over the possibility of people wearing linen and wool together that he forgot to tell the men of his chosen people that they shouldn't fuck the women of his chosen people against their will. Hey, it's cool, sometimes I forget to pay the gas bill until I get an e-mail about it.

This type of thinking leads to some fairly silly situations, though. Consider the brouhaha over the Ten Commandments, as debated in Alabama (among other places). There are people who believe, and institutions who declare, that the Ten Commandments are an advanced and wonderful set universal moral laws which were ahead of their time and are respected by all societies. Really?! What the fuck?! Do these people even know what the Ten Commandments are?!

No, as it turns out, they don't. Ask a person on the street, or a Christian in church, to name all ten and you will get confused looks at least 90% of the time, and complete answers from perhaps two or three percent (again, unless you've got a selection bias in favor of theologians). Just about everyone will be able to give you "don't kill" and "don't steal." Other popular ones include "honor your parents" and "remember the Sabbath." Let's see if I can get all ten from my desk at work with no Bible (I'm not using SAB to cheat, I promise!):

1. I am the Lord thy God; thou shalt have no other gods before me.
2. Thou shalt not make unto thee any molten gods.
3. Honor thy father and mother.
4. Thou shalt not kill.
5. Thou shalt not steal.
6. Remember the Sabbath and keep it holy.
7. Thou shalt not bear false witness against thy neighbor.
8. Thou shalt not commit adultery.
9. Thou shalt not covet thy neighbor's wife, nor his goods, nor his ass, nor his house.
10. Thou... umm... something about yeast?

Shit! Only nine! OK, let's see how I did:

From Exodus 20:
1. Thou shalt have no other gods before me.
2. Thou shalt not make unto thee any graven image.
3. Thou shalt not take the name of the LORD thy God in vain.
4. Remember the sabbath day, to keep it holy.
5. Honour thy father and thy mother.
6. Thou shalt not kill.
7. Thou shalt not commit adultery.
8. Thou shalt not steal.
9. Thou shalt not bear false witness against thy neighbour.
10. Thou shalt not covet thy neighbour's house, thou shalt not covet thy neighbour's wife, nor his manservant, nor his maidservant, nor his ox, nor his ass, nor any thing that is thy neighbour's.

Argh! So close! OH, WAIT. NO, I AM NOT EVEN CLOSE. You know why? Because those aren't the Ten Commandments. I can prove it right now, too. Just grab your fuckin' Bible and follow along with me, no sleight of hand or shenanigans here at all.

Got your Bible? OK, open up to Exodus 19:20. This is after the Israelites have left their made-up captivity at the hands of the Egyptians (the latest archaeology shows that the Israelites, as they came to be known, were probably the lower classes of extant Canaanite civilizations who rose against, broke stuff, and then ran to the hills, using Egypt as a stand-in for their faceless oppressors), but before the milk and honey hijinks. In chapters 19 and 20, Moses goes on up to Mount Sinai and has a chat with God - no tablets, no commandments, they're just talking - and then Moses comes down. The End. Later on, in Exodus 31, Moses goes back up to Mount Sinai. This time, there is writing, but the Bible itself simply states that God wrote what he and Moses talked about - so it ought to be the same, right? Still no mention of any commandments, though. Well, Moses goes back on down with his two stone liqui-gels tablets, but he sees that Aaron has got them all worshipping a golden calf, so he throws a tantrum the tablets to the ground... no, wait, he throws a tantrum, too... and the tablets break. Then Moses has to talk God out of killing every last fucking Israelite stone-dead (no joke: read Exodus 32:9-14).

Nowhere in this part of the story is the writing on those tablets explicitly reviewed, and neither is this piece of genius referred to at any point thus far as "The Ten Commandments." You have to go back to chapter 20 to get the words themselves which are today called the Ten Commandments, and you have to go forward to Exodus 34:28 to get the word "commandments," and this is a problem. In Exodus 34, it says that Moses goes up Mount Sinai a third time, and this time he writes what was written on the tablets before, which writings bear no apparent relationship to what they had talked about the first time, and only now are they called commandments. So far, so good... except the commandments are entirely different, viz:

1. No other gods.
2. No molten gods. (So far, so good.)
2. Keep the feast of unleavened bread. (What?)
2. Work six days, rest on the seventh. (Oh, OK, back to normal...)
2. Ovserve the feast of weeks, the firstfruits of the wheat harvest, and the feast of ingathering at the year's end. (That's quite a few parties...)
2. All the male children should appear before God three times a year. (Double-what?!)
2. Don't offer a blood sacrifice with leavening.
2. Don't leave the passover sacrifice until morning.
2. The first of the firstfruits of the land are God's.
2. Don't boil a baby goat in its mother's milk.

Really? I mean, really?! Those are the Ten Commandments? You're goddamned right, they are. That is, until you get to Deuteronomy 5, where we're back to that first conversation that Moses had with God*.

I once had a professor tell me that, convoluted as the Bible was, it does not contain any contradictions because a contradiction is of the form "X is the case, and X is not the case." Leaving aside simple paradoxes (such as "People are good, people are evil," where "evil" means "not good" and therefore "people are evil" means "people are not good"), this is still false because we have here the statement, "The Ten Commandments are A, such that any B which is not A is not the Ten Commandments" (for any statement of A=C, it is implied that there is a B which is not C and is therefore also not A, otherwise A=C is an uninteresting tautology). And then later on we have the statement, "The Ten Commandments are B." So check it out in premise format:

1. A. (Where A is the Ten Commandments in Exodus 34 described as the Ten Commandments).
2. A implies ~B. ("~" means "not," or in this case, "B is not the case," where B is any non-A set of the Ten Commandments.)
3. ~B. (From 1 and 2, above, with modus ponens.)
4. B. (Where B is the Ten Commandments in Deuteronomy 5 or Exodus 20, which is also a non-A set of the Ten Commandments - it's a "true B," in other words.)

So we have here A, A implies ~B, therefore ~B, and B. We have here a contradiction: ~B and B.

Something is wrong here. Options include, but are not limited to:

1. There are actually Twenty Commandments, with some duplicates.
2. Moses wrote it wrong (or dictated it wrong, or was transcribed wrong) in Exodus.
3. Deuteronomy is wrong with respect to the Big Ten, as well as parts of Exodus.
4. God changed his mind.
5. There has been a change which would be rather difficult to trace today and which has resulted in different accounts of what was supposed to be the same mythical (not literal) event.
6. The whole thing's hogwash.

As it happens, 6 is true - but so is 5! (Happily, they're not mutually exclusive like B and ~B, so they can both be true at the same time without contradiction.) You see, some people (who apparently had nothing better to do) have determined that the Old Testament is actually an amalgamation of no less than five distinct sets of writings. One of those sets of writings, the latest one (omitted by the Wikipedia article, it's called "R" for "redactor"), is the result of an ongoing attempt to work the other four into one continuous narrative, and this is why we have the Bible as we have it today (well, that and the third Council of Carthage in AD397). To accomplish this task, many techniques were employed under the general heading of midrash, the important parts of which (for our purposes) amount to the Biblical equivalent of DC re-telling Superman's origin over and over again, or of Nintendo's continuous re-telling of the Legend of Zelda: it is a regular sexing-up of otherwise outdated material for a new generation of audience members. One of these techniques is to take different accounts of the same event, like Moses' trip up Mt. Sinai, and to say that all versions happened in sequence (as opposed to instead of one another).

This professor of mine tried to explain away the "B and ~B" contradiction outlined above - yes, that very one - by saying that since the current version is just a re-telling of several other versions rolled up into one, it's not "really" a contradiction. That's like saying, "Well, you see, the original Superman was named Clark and raised by John and Martha Kent, but the next Superman is named Arthur and raised by John and Martha Dent, and so even though the newest Superman literature says in one place that he's always been named Clark and raised by the Kents, but in another place that he's always been named Arthur and raised by the Dents, it's not really a contradiction because it's just a re-telling." Oh, and don't forget that a whole lot of people worship Superman in this hypothetical world, insist that his comic-book history is the inspired word of Superman (who is perfect and therefore never lies) which is without flaw or error, and also that when people die they will be either resurrected by Superman to live in the Fortress of Solitude with him (for serious!) or they will be imprisoned in the Phantom Zone forever.

That's stupid.

Anyway, I really think that if more people read their damn Bibles, we'd have a lot fewer believers. Or at least a lot fewer literalists (though there would still be some). Also, I want to swear on math textbooks in a courtroom now. "Do you solemnly swear to tell the truth, the whole truth, and nothing but the truth, so help you Pythagoras?" "I do, so help me Pythagoras."

- - -

* - Extra Credit: OK, so there are a few other things going on here as well. First of all, beyond Moses arguing with God (that's how you know he's a true Jew), you've also got the part in Exodus 34 where God say's he'll do the writing at the start of the chapter, but then he pussies out and has Moses doing the writing by the end of the chapter. Also, in Deuteronomy 5, it says that God made the covenant with Moses at Horeb, whereas Exodus places all covenantry clearly at Sinai. And do keep in mind that the only place in the Bible that the Big Ten are called the Big Ten - that is, the only place in the whole fuckin' Bible where the words "The Ten Commandments" are used thus - is in Exodus 34. What do you think ought to be the official ruling on this one?

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.

Poof.

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.