Missing Things

07 October 2010

A Most Ingenious Solution

Answers to last week's conundrum!

Let's start where you probably started - with the simple probability that, with one opportunity to make the choice and four options to choose from, you have a 25% chance of picking any given option and therefore a 25% chance of picking the right one.

This falls apart as soon as you realize that there are two options marked 25%. Two correct options out of four gives a probability of 50%, so we'll pick that one...

...wait. There's only one answer of 50%, and one of four is 25%. Again.

This is probably where most people give up.


Let's think about this a little more. Three out of our four answers are now in what amounts to a superposition of states - our argument proves that each of them is both true and false. This leaves us in a position of deciding what rule to use: either "all incompletely false statements are true" or "all incompletely true statements are false".

- If we go with the second, then we have eliminated three of the four options as acceptable answers and should default to the one remaining.
- If we go with the first, then since we have just judged that three of the four answers are true we have a 75% chance of selecting a correct answer at random.

In either case we ought to choose 75% as the correct answer!

If 75% is the correct answer, we have one chance out of four of choosing it correctly at random. Gotterdammerung!


In fact, however, this is the key to solving this problem. By this chain of reasoning, we have constructed an argument that places all four answers in a superposition of states; and, depending on whether we choose "all incompletely false statements are true" or "all incompletely true statements are false", the solution is either 100% or 0%, respectively. And neither 0% or 100% is presented as a solution to choose from. This means that the best possible answer does not appear as one of our four choices, leaving us with no chance at all of choosing the correct answer! No chance at all is 0% - and so, finally, we have an answer that is permitted to be true without also being false.

Since this choice does not appear, we simply do not select any of the answers, and in so doing get the problem right.


Final thought. Amidst all these convoluted circles of paradox in such a simple question, you may have forgotten that most people give up after discovering only the first circle. Having woven our way through the rest of the problem, we now learn that giving up is the best possible answer.

This means that, of all the people who ever encounter this question, most of them will answer it correctly - despite the fact that we just proved that each of them has no chance of choosing the right answer.

I'll leave you to think on that one for a while. :D

- Thursday

30 September 2010

A Most Ingenious Paradox

I received a most excellent riddle from my deranged friend Qwip yesterday, and present it to you here in modified form:

Multiple Choice: If you were to answer this question by random guessing, what is the probability that you would be correct?

(a) 25%
(b) 50%
(c) 75%
(d) 25%


16 September 2010


Good day, there, sir, Mr. John Q. Pantisocrat! I hear you have a proposal for making the entire world a better and happier place! This sounds fantastic. I would love to be better and happier!

However, I've just spoken with a Mrs. Ida Mandonner, over there, and she says your proposal is vile and stupid and she and her compatriots will oppose it with every ounce of blood in their bodies. I don't think she likes your idea.

Oh, I see! Her "compatriots" are simply being misled by her self-serving propaganda and despotic egomania. That makes sense, lots of bad ideas have started that way. We certainly couldn't have that. How will you overcome her nefarious brainwashing to attract public support?

Educational initiatives? That sounds like a promising plan. I certainly agree that public instruction could be better than it is. Then, when everyone knows as much about the nature of things as your group does, it will be a simple matter to fix everything. If only the world agreed with you, it would be at peace!

That's funny, everyone else I've talked to says the same thing.

What if not everyone agrees with you? You wouldn't insist on coercing people against their will, would you? Mrs. Anna Domini said anyone who disagrees with her system can go to hell, which sounds mean, and Mr. Andy Quarian that everyone would be happier in his utopia whether they know it or not, which sounds worse, and Mr. Tim Spirit doesn't seem to have any plans but to kill anyone who lacks his followers' unanimity and willpower. Would you allow people who weren't sure where they were happier to come and go as they pleased?

Well, I don't think I'd call it "freedom of misery", exactly, but I see your point about brigands and freeloaders; very practical of you. Well, then, what's your solution?

Temporary freedom of misery. Hmmmm... Oh, well, that's true, most people would flock anywhere that is obviously better for them. So if that's true of your global forecast... everyone will just inevitably march towards it anyway! That's amazing!

But... well, I hate to be the bearer of bad news, but Mssr. Paul LaTariat and Miss Eva Van Detta have been saying so for a while, and now Dr. A. I. Consciousness says he can prove it in his case with science! And, well... you can't ALL be right.

So... what's so special about you?

02 September 2010

A proof about hypocrisy.

I mentioned this in my last post and present it for you here. I stole it from Raymond Smullyan, a puzzle enthusiast and recreational mathematician, and the author of What is the name of this book?, The Riddle of Scheherezade, To Mock a Mockingbird, Alice in Puzzle-Land, and The Lady or the Tiger?, among others. (His version, incidentally, is a lot shorter than mine is, because I'd like you to absorb this and not simply treat it as an interesting game. And also I run off on weird tangents sometimes. If I were to simply cut and paste it, I would already be done by now.)

Let's start with our definition of hypocrisy, so we all know exactly what we're talking about.

1: Any person who does not believe what he claims to believe is a hypocrite.

Not a lot to build on, I admit. We'll need a premise that we can agree is true, to use as a basis.

2: Everyone believes things.

I hope this is something we can agree on! Even without getting into onerous philosophical questions, I'm sure you have certain beliefs about, oh, the shape of world, or the usual color of plants in the spring, or what you had for dinner last night, or whether you will still be alive tomorrow morning. Anytime you say "I think", "I feel", or "I know", you're expressing a belief you have.

3: Any given belief is either true or false.

I'm losing a bit of accuracy here for the sake of clarity, because if I sincerely believe that colorless green ideas sleep furiously we can argue for a long time about whether that is true, false, poorly defined, or even meaningful (and, if it's not meaningful can I really believe it?) - so for the sake of getting on with it we'll assume that all beliefs can be clearly expressed in a way that is either true or false, and if there are some that can't we don't care about them in this argument anyway, so there. :P

4: Each of your beliefs is either true or false.

This follows as a syllogism from 2 and 3, so hopefully there's no disputing it.

5: You believe each of your beliefs is true.

Does this seem obvious? It's not. There's a whole convoluted discussion over Moore's Paradox, addressing sentences like "It's raining, but I don't believe that it's raining." The statement seems absurd, but there's no reason that the two halves of the statement can't both be true - maybe it's raining outside, but you're indoors and away from the windows and can't hear the drops. Moreover, both placing the situation in the past tense ("It rained, but I did not believe that it rained") and shifting the subject ("It is raining, but you don't believe it is raining") result in perfectly reasonable statements!

That conjunction must be playing tricks, yes? Actually, to somewhat oversimplify the situation, it's in the word "belief". Someone who does not believe that it's raining cannot honestly claim that it is raining, and vice versa. If our supposed speaker says that it is raining, and believes it, then they are lying when they say they believe it is not - and, as a person who lies about what they believe, is therefore a hypocrite. On the other hand, though, if they say it is raining but are truthfully claiming they don't believe it, then even if it really is raining they are still lying about the situation as they understand it! In either case, the statement indicates that the speaker is not trustworthy, either by malice or simple stupidity, regardless of whether the statement as a whole is true or false.*

Alas, I digress. The only real explanation is to understand that, when you make a statement, you are implicitly claiming to believe that statement is true. All those beliefs you've stored up in your head can be brought out whenever you like, and whenever you do, you're making a truth claim. We could create a list of all your beliefs, to make this an actuality rather than just a potentiality, but that would take far too much time.

6: At least one of your beliefs is false.

Here we come to a bit of a potential impasse, because it is not possible to prove this, universally, without actually systematically running through a list of all your beliefs and verifying them one by one. So this claim is supported merely by probability and psychology.

If you take that theoretical list of your uncountably huge number of beliefs and approach it without prior judgment - how likely is it that every single one of those beliefs is true? You can think of it as flipping a coin for each statement, if you like - tails for true and heads for false. Or, if you think your system of generating beliefs is a little hardier than that, roll a hundred-sided die for each statement, and only mark it false if you roll a 1. Even if you use a dice with twice as many sides as there are beliefs on your list because your judgment is just that sound, the odds only go down to 50-50 that there are no false statements on that list whatsoever!

If you can honestly tell me, after all that, that you are utterly confident in the truth of every single belief on that list - well, I have to honestly tell you, that's not quite delusions of godhood, but it's pretty close. Even the Pope only claims infallibility in religious matters, you know.

Oh, oh, oh, but wait! If I've just persuaded you that one of your beliefs is false, then by statement #5 you must believe that one of your beliefs is false...

7. You do not believe what you claim to believe.

That list of all your beliefs has an error on it somewhere.

Nevertheless, you continue to believe them all.

You hypocrite.

Want to join the club? There's always room for one more. :)

* There is a mathematical field called intuitionistic logic designed to resolve complications like this, by only using operations that preserve justifiability, rather than truth as in classical logic. As a result (and despite the name), it's actually stricter about what conclusions you can make than classical logic is! Read up, 'tis fascinating.

26 August 2010

I am a hypocrite.

Okay, okay, the title makes it seem almost as if I forgot everything that I wrote in my last lengthy tirade during the month that I mysteriously vanished. I am hypocritical. Happy now?

Maybe not. Allow me to elaborate. This isn't about how I vowed not to be one of the people who get a blog, update it a few times, and then leave it to rot - even though it feels like that's what I've been doing, technically that just makes me an oathbreaker. (Aaaaaand that makes me kind of depressed. But I'm back now, and as long as I keep coming up with unusual information that ought to affect the philosophies we live by, I'll keep coming back. Right. Moving on...)

Hypocrisy can be briefly summarized as the "do as I say, not as I do" principle. There is a weak use of the term that can be applied any time someone says one thing and ends up doing another - for someone to promote wearing seat-belts, but to be so absent-minded that he forgets to don one whenever he actually gets in a car, for instance, is certainly saying one thing but doing another, and by this general definition he is certainly a hypocrite.

This use, however, broadens the term to the extent that it encompasses any failure whatsoever - an athlete who says he will become an Olympian but does not perform to standards, an advocate of celibacy who nevertheless succumbs to lust, a pathological kleptomaniac who nonetheless acknowledges that theft is wrong. I would be hard-pressed to place such people in the same category as, say, Anthony Comstock*, who happily drove fifteen people to suicide in the name of the morals of the young and innocent. Human failures are ubiquitous (I think I shall make a proof of this in my next post); by so broadening the definition of the term until it is all-encompassing, we make it impossible to discuss.

* For the sake of fair hearing, the link refers to an article with a mixture of praise and criticism, and excellent depth.

So - rather than discrepancy between a person's claims and actions, let us talk of hypocrisy as the discrepancy between a person's claims and beliefs - not only saying one thing and doing another, but believing one thing and saying another. This must be considerably trickier to discern - how else are we to know what's going on in a person's head unless he says?

"Out of the abundance of the heart..."

Oh, even better: "By their fruits you shall know them." I think the usual way this verse is expressed (it's Matthew 7:16-17, by the way) can lead us to miss something key. "A good tree produces good fruit, and a bad tree produces bad fruit." Think about it. I don't know how it reads in the original Greek, but simply with regards to the agricultural analogy it seems it would be more accurate to say "A good tree is one that produces good fruit, and a bad tree is one that produces bad fruit." It agrees more with verse 16, too. Anyway, main point - regardless of what someone says, their actions will demonstrate where their true thoughts are.

And this - this is central.

I say I believe in free will. Very strongly (both the saying and the believing). Because it is inconceivable to me on the one hand that the thinking, perceiving, core that is me inside my head is just so much exhaust from the overcomplicated engine of my body that I can't in any way rely on what it tells me about the universe (as I conclude from various atheists), and simultaneously on the other hand that, awesome as it would be, a cosmos-sized Robinson/Goldberg machine is better suited than irreplicable free wills to glorify God (as I understand certain predeterminists).*

* I can build a Robinson/Goldberg machine - actually, I have built one. Building a cosmos-sized one would just require a cosmos-sized me, maybe not even that. Omnipotent God is infinitely more creative than a cosmos-sized me.

On the action end of thing... I have issues. I'd prefer not to elaborate, thanks very much, but suffice to say they are mainly psychological and neurological in nature, and I've made myself serious impediments to quite a lot of basic life, since graduation. I take medication, and it helps, to a noticeable extent, but quite a lot of it is simply bad habits. (Where did the good habits I used to have go? Excellent question. No idea.) And what I've done as regards this matter is... well, let's say that if we were to categorize my failures in terms of the cardinal sins, Sloth would be at the top of the list. And, this is the key part, basically all of my attempts to deal with these problems have involved modifications to my environment - working in a more conducive location, taking instructions from different people, spending less time on the computer, increasing my medication beyond its effectiveness, et cetera.

Since when is psychological behaviorism compatible with free will

(Notice my clever use of an interrobang there? I can't believe I finally remembered to do that!)

I feel like I'm going out of my way to validate the idea that the cognitive mind has little to no effect on the physical engine, which is not what I had in mind when I sought out this post.

I'm going to go disturb the universe now, 'kay? Back in a minute.

10 June 2010

The Mousetrap

Because I have the end-of-quarter approaching, this post is brief and noticeably different from what passes for "the usual" on this blog.

I finished reading Agatha Christie's famous play, The Mousetrap, yesterday, and I'm not sure that it was worth the hype derived from its well-known resistance to exposing the ending. However, in the interest of furthering the joke, I will note that the following concerns the denouement and, thus, contains spoilers.


The female lead gets a nice hat, and the pie burns.

03 June 2010

Use the force

Take a slinky and stretch it out so that it's fairly taut - across a room, say. Wiggle one of the ends up and down. Notice how whenever you disturb part of it, the disturbance moves away down the length of the slinky?

This is what in physics we call a wave - a disturbance that moves. There's a whole bunch of interesting stuff that happens with it that you can play with (what happens when the disturbance reaches the end of the slinky? is a good place to start), but there's one thing in particular I want to draw your attention to.

Disturb the slinky again and pay close attention to the little loops that compose the spring. What you should notice is that while the disturbance as a whole travels away, the individual pieces of matter basically only move back and forth in line with however you disturbed it; up and down, side to side, whatever. This shouldn't be a huge surprise, right? It's not like you grabbed part of the slinky and threw it across the room or something. The particular loop of the slinky you disturbed just pulls on the neighboring loops, which pull on their neighbors, and so on; it's a function of the fact that the slinky is springy and tries to return to its original shape due to tension forces. The "wave" isn't an actual object, it's just a description of the process as a whole.

So then what happens if you set up something, like a domino chain or something, next to the far end of the slinky and then disturb the slinky sideways? You can try it if you like, though what happens is essentially what you'd expect - the wave propagates down the slinky and runs into the dominoes and knocks them over.

But we just said that the wave isn't a physical object. It's just a little disturbance in the force (heh heh); you just accomplished the same thing as if you'd thrown a baseball down to the other end of the room to knock the dominoes over, only the only thing that moved across the room is a mathematical description of the force and energy transference taking place in the slinky. No actual, physical object crossed the room at any point.

Of course, you could argue that the same thing happens if you just stretched out a chain of dominoes across the room to reach the domino on the other side, but each individual falling domino moves slightly towards the next one. There's a net motion involved in the right direction. But with the slinky, each little bit of matter only moves from side to side (if you were careful). Not only does no object cross the room, no object even moves in the right direction to cross the room.

Congratulations. You just affected an object a whole room away from you with nothing but the power of math.

20 May 2010

Worldbuilding Worldview

I like to write stuff.

Actually, strictly speaking, I don't like the actual act of putting words on paper so much as I like coming up with the things that can be turned into words on paper, but that's part of the reason I got this blog in the first place.

Anyway, in either case, this is probably not a huge surprise.

What I was doing recently, while I was writing - or, as the case may be, not writing - that's a well-worn joke by now, I'm sure - was designing a universe. Not from scratch, that's rather beyond me, but starting with our universe and... tweaking things about it. Adding new and interesting particle sets that act in unusual ways, making gravitational fields act perpendicularly to the movement of electrical charges, that sort of thing. But, since none of these worlds are particularly interesting without people in them, I get to worldbuild the people too - and, even more interestingly, how they think. This means some primitive sociology, since to understand how my fictional societies think I need to figure out how normal people think first. Tricky.

The modern world operates on what might simplistically be termed the scientific method - we try different approaches and use the ones that seem to work. Why is that, exactly? Well... we've tried it before, and it seemed to work, so we keep using it.

That's very intuitive, yes?

That's when it occurred to me that this is rather similar to the charge thrown at various religions by ardent scientific humanists - you know the one, that "you only believe that book because it tells you to!" This, in turn, led me to wonder what other kinds of worldviews I can self-justify in this way. Here's my list...

(*NOTE: This is a thinking exercise more than anything else, so it's okay if I'm reductive to absurd lengths.)

- Scientific: Try everything and keep what works. We've done this before, and it seems to work.
- Post-modern: We took a poll among everybody, and we all agreed that we're mostly sane and have an accurate perception of reality. So if we ask each other, we've got a good chance of getting the right answer.
- Separatist: No need to bring in other people. I'M more usually right than wrong. Right? Right.
- Authoritarian: This guy here is really smart/enlightened/powerful/inspired, and he's right most of the time, which indicates a strong connection with the truth. So when he says we can trust him, it makes sense to believe him.
- Intuitive: My very existence can be described as "I am true". I have this in common with everything else that's true, a connection I should notice. The things I know best are those which are most familiar to me, so the things that feel most familiar are most likely to be true - actual familiarity as well as gut instincts, intuitions, and feelings of deja vu. It gets trickier if things don't seem familiar, so I should keep finding more information until it does connect with something familiar.
- Gnostic: Everything I encounter, true or false, has in common that I have encountered it, and is therefore connected to everything else I've encountered. By exploring this inner existence and finding what is true about myself, and following the connections from myself to the world, I can determine truths about the world. Deep inside, you know I'm right.

What this exercise is demonstrating to me, more than anything else, is the importance of paying attention, not to where your theory is right, but to where it is wrong. You can prove anything you like; but you can't disprove the truth, and if you succeed you obviously did something wrong.

06 May 2010

Something quick to catch up

Let's suppose you're thirsty, and the only convenient source of drink nearby is a vending machine, dispensing $1.25 drinks.

You have only $2.00 on you, in bills. The vending machine doesn't give change.

I happen to be walking by, though, and I have a couple of quarters on me. I'll give you my two quarters if you'll give me a dollar.

Alternatively, you could just put $2.00 in the machine and let it keep the change, which would cost you an extra $.25 than if you bought my quarters.

Or you could go without a drink, but you're really thirsty and don't want to do that.

So... How did my $.50 suddenly double in value?

29 April 2010


I am not a Christian.

There are lots of people who think that this term is used far too often, by people who call themselves this simply by habit, or preference, without actually understanding what it means and living it. These people tend to prefer to call themselves "believers" or "followers of Christ" or something equally ethereal and pleasant-sounding, to distinguish themselves from the petty heathen who have mistakenly confused their own beliefs for the perfect and life-changing doctrines espoused by our Lord Jesus Christ.

Unfortunately for the proliferation of these terms, the people who use them almost never agree on the manner in which one's life should be changed, and consequently are as disunited amongst themselves as the remainder of the church that they criticize.

I am not a believer or follower of Christ.

In the common parlance, people tend to refer to the kind of people who believe that their beliefs are right and everyone else's are wrong as "fundamentalists" (typically described as "taking the Bible literally"), because that is so mean to all the people who are wrong to actually tell them so, and it is wrong of you to do that! (And fundamentalists advocate, like, killing people who disagree with them, but we only mock them, so we must be better people than they are!)

I am not a fundamentalist.

In lieu of continuing this ridiculous pattern further than necessary, I will also proclaim the following: I am not an American. I am not a Caucasian. I am not a brunette. I am not a gentile. I am not a logician. I am not a scientist. I am not an artist. I am not a philosopher. I am not a body. I am not a soul. I am not a mind. I am not a human. I am not a person.

At least, not in the way you are probably thinking of any of those things.

The thing is, those things are all categories. Categorizing is a human and, you may be surprised to hear me say, good thing to do. We have to stick labels on things in order to make sense of the universe, and just because we label it doesn't mean that the label is arbitrary or wrong, like so many nominalists would have you believe.
But, when we do this, we tend to assume that labeling something is like placing it into a file instead of a pasting a sticker on it. Truth is, if everything exists outside your own head, it has to be the latter.

If we describe something as a door, what we're really saying is that this thing can be opened and closed, and when it's open you can walk through it.

If we say something is an electron, we mean it's tiny thing possessing a certain amount of charge, and a certain amount of mass energy, and behaves like this in these situations.

If we say something is a fact, we mean that it's a statement about the universe (and usually, that it happens to be empirically true).

If we say it's a number, we mean it's a abstract object that can be counted or measured.

In fact, the only real noun in English, or any other language, is "thing". Every other noun just means "thing with descriptions X, Y, Z, etc", "thing with adjectives". Some of these properties are extremely complex, like for words like "human", but it can still come down to something like "thing that is alive, and as an adult is able to walk on two legs (things used for walking), manipulate things with two hands (things used for manipulating, that have smaller things on them that can grip other things), and talk, see, taste, smell, and hear with a head (thing attached to another thing that is able to react automatically and contemplatively in a lot of different ways to things around it) and performs abstract and conceptual thought and communication, OR is descended from another human".

All I'm saying is that you have an independent existence from any of these stickers, and if you and the sticker don't match, the sticker is wrong. Object-oriented linguistics, if you will. I think this is important, because being capable of abstract and conceptual thought means that we end up fixing or adjusting what each sticker means constantly, and a lot of the time something that the sticker used to accurately describe no longer does. Especially when we start referring to the collection of all objects with a particular sticker on it as part of some uniform whole, as if the sticker was there first. No, I'm not being a pedant about this whole thing.

But perhaps I'm being a little pedantic.

22 April 2010

A puzzle

You have a rectangular field containing an irregularly-shaped pond whose area you wish to know.

The tools you have at hand are the property deed for the field, a cannon, and an infinite number of cannonballs.

How do you find the area of the pond?

15 April 2010

Test Yourself

Please choose the best answer for each question.

Favorite Color:

1) What is nescience?
(a) Orange
(b) True
(c) False
(d) All of the above
(e) I don't know

2) What is veracity?
(a) Silver
(b) True
(c) False
(d) All of the above
(e) I don't know

3) What is your favorite color?
(a) Blue
(b) True
(c) False
(d) All of the above
(e) None of the above

4) What is melange?
(a) Grey
(b) True
(c) False
(d) All of the above
(e) None of the above

5) What is contradictory?
(a) Vermilion
(b) True
(c) False
(d) All of the above
(e) None of the above

6) What is the capital of Assyria?
(a) Ashur
(b) Calah
(c) Khorsabad
(d) Nineveh
(e) None of the above

7) What is the maximum airborne velocity of an unladen swallow?
(a) 1 m/s
(b) 5 m/s
(c) 11 m/s
(d) 200 m/s
(e) 42

8) Is this a question where one of the possible answers goes "blam"?
(a) No.
(b) BLAM
(c) BLAM
(d) BLAM
(e) Actually, I think several of your possible answer make that noise.

9) Cake. - True/False

10) Nobody expects the Spanish Inquisition! - True/False

11) This sentence no verb. - True/False

12) Think of a number, any number.

13) Where's Waldo?

14) Fill in the blank: __________

11 April 2010

It's not Thursday yet, but this is just a tech update

I've added the feedback bar thing. I had wanted a nice number of dichotomies so you could tell me that your impressions are that what I write is kind or cruel, true or false, beautiful or ugly, confusing or clear, justified or unjustified.

Unfortunately, I can fit all those into the little box but they'll only put up a few anyway. Foul technology, curse your surprising but inevitable betrayal!

So now there's just three options - good if you think I was persuasive, bad if you think I was unkind, and ugly if it's just too confusing to tell what just happened. (I expect this one to be quite popular.)

Also, I've gone back and added a bunch of links into Socrates Meets Malacoda for explaining what I had in mind with all the various philosophy/literary/science-y geeky references I made - maybe this will clear things up a bit?

08 April 2010

Sparkling angels, come and see

The idea of angels probably irritates me more than the idea of gods does. They're practically identical for all we know about them - they're supernatural and a Lot Better Than Mere Mortals, For Sure, but angels serve even higher beings and gods generally don't... but angels take the lead due to the outstanding problem that the book I've sworn to pledge as the truth, the important truth, and nothing but the truth, claims they exist and gives them fairly prominent roles in the story, and yet still manages to skimp on the details of what, exactly, they are, and how, exactly, they matter.

The details it does provide, however, are fairly clearly in direct contradiction to the average person's conception of them. If you manage to get a clear answer out of this hypothetical person on "such a religious question", she is likely to give you three images of what angels look like:

(a) A divinely cute infant or young child, but with wings.
(b) A divinely good-looking woman, but with wings.
(c) A divinely powerful knight, but with wings.
I suppose there's also (d) some combination of the above, usually (b) and (c) because both possible combinations ("female" plus "powerful", and "knight" plus "good-looking") are apparently synergistically attractive, and warrior children don't really seem like a very divine idea.

You may notice a couple of recurring elements, and oddly they're the only ones that come close to approximating biblical descriptions (which tends to multiply both the "divine" and "winged" aspects to redundancy and beyond). I can think of three good reasons the others are wrong:

First, a fact that I've heard circulated frequently and so I presume is fairly well-known, there are very few angels in the Bible who open their messages with a preamble of "Hey there!" It's generally closer to "Fear not". This would seem to rule out (a), for whom this would be unnecessary unless every biblical hero and heroine happened to have a horrible phobia of children with wings*, and (b), whom studies show should worry more about other emotions than fear getting in the way of the message. Similarly, I can't think of any explicitly described manifestations with more than a passing resemblance to a human being, barring when Raphael goes incognito for most of the Book of Tobit, which would seem to eliminate (c) and (d) as well.

* This isn't necessarily a bad assumption to make, but you do have to consider that among such heroines is Mary, who had very explicitly never had children before.

Second, whatever else angels may be, the Bible is fairly specific that humans alone are made in the image of God. If you believe this refers to physical structure, this should suffice; if a reference to God's triunity as mind, body, spirit - well, angels are obviously spiritual beings and occasionally rebellious, which would leave the body as the key difference.

Finally, outside of their allegiances demons and angels are fundamentally identical. There's no support for the notion that good is pretty and evil disgusting (or vice versa) other than peoples' astonishing tendency to merge goodness, truth, and beauty into a single axis; quite the contrary, metaphors involving demonic activity while "disguised as an angel of light" are common. If demons can be light and still evil, angels can be dark and still good.

This is all just a roundabout excuse for me to cite C. S. Lewis, who deals with this quite a lot in The Space Trilogy (another of the world's best stories nobody has ever read), especially because several of his "angels" are the personifications of planets rather than divine messengers. When two manifest near the end of Perelandra it takes them some time to figure out an appearance that won't cause the human protagonist to be physically ill; before they finally settle on sort of metallic colossi, some of their attempts are closer to vertiginous spaces swirling with geometric shapes than any real object. Lewis justified this elsewhere simply by pointing out that everyone expects the forces of evil to be terrible and threatening, and that's all very well, because you can always have faith that the forces of good will swoop in and save you. But when the same applies to the ones in which you trust for salvation... one way or another, you will have to undergo a very radical change of mind.

And when I am explicitly imperfect by nature, and wisdom begins with the perfect fear of an omnipotent God and ends with it being dispelled by his perfect love, that sounds far more real to me than any pretty fledged human.

And now for something hopefully more comprehensible

I don't understand gods. When someone talks about advances in technology making humans as gods, or some fictional character believes that by doing some particular thing he or she will become godlike, I literally fail to see what the big deal is, and I'm convinced this is a terminology problem.

My beliefs are fairly straightforwardly monotheistic - I'm convinced that there is precisely one god. (Usually I'm going to capitalize this, as in God, not specifically for reverence - I can kind of see how "I consider God so important I'm willing to violate my otherwise-rigorous use of basic grammatical correctness for Him" makes sense, but considering how often "otherwise-rigorous" is a blatant falsehood I don't really buy it - but so that there's at least one visible difference between my use of the term and anyone else's.) More specifically, I believe in a God with the triple characteristics of omniscience, omnipotence, and omnibenevolence, which isn't really a word but I'm going to violate my otherwise-rigorous use of etymological yes I'm kind of being a hypocrite but it makes sense here, okay?

I think this means I use the same basic definition as the average atheistic person, and we just need to figure out whether such a God exists or not. That takes care of the first entry in Merriam-Webster. (I'm not sure what they mean by "supreme or ultimate reality" - it seems to imply that God is somehow more real than anything else, and I don't think I buy the concept of one person or object being more real than another one... is it possible to cite a dictionary for lacking rigor?)

Ignoring the mystical and arbitrary definitions, that leaves (2) - a being or object believed to have more than natural attributes and powers and to require human worship; specifically : one controlling a particular aspect or part of reality. This is the definition that seems to permit polytheism (and any forms of monotheism that don't believe in an omnipotent god), and my problem with it is that it's merely descriptive - not explanatory, and not compelling. My reasons for this are based on the key lines "more than natural attributes and powers", and "requires human worship":

What do we mean by natural? In the past powers that would have been considered unnatural would include things like, oh, sending a message to the other side of the ocean instantly, or throwing rocks into the air and not having them fall immediately back down. This is where Clarke's Third Law - "Any sufficiently advanced technology is indistinguishable from magic" - comes from. Taking that to its logical conclusion, if we're presented with some claimant to the title of a god we never have any good reason to believe any particular demonstration of powers is "more than natural"; sure, it looks like magic, but that's just because we don't know how to duplicate it. And since there's no real reason in theory that human understanding can't increase indefinitely (refer to the conclusion from my last post, that there's always more knowledge to know!), any power that's less than omnipotent can, and, probably, will be surpassed.

It doesn't make any more sense to worship Zeus than it does to worship your R&D department.

Now, there is a counterpoint to this, which is that it is right to pay more respect to more powerful people (which presumably means more respect for Zeus than for R&D, at least until the R&D chair is able to beat Zeus up). The problem with this is that it collapses into some form of might-makes-right (possibly bright-makes-right) scheme, which offers no grounds to argue with Zeus when he says to stop funneling so much of your budget into research and development, unless you resort to citing a higher authority than Zeus, and every possible higher authority can be answered this way... until you decide to cite an all-powerful, all-knowing, all-loving authority, who, funny thing, either exists or doesn't.

I have no issue in general with the existence of any sort of less-than-all-powerful being. But it's also entirely irrelevant to anything of more than temporary importance.

29 March 2010

Waiting for Gödel


This is the part where things start getting complicated, because set theory can, in fact, be defined in terms of itself. You see, any mathematical theory can be defined as a set of statements - some of which are unproven and assumed, and called axioms, and others of which are called theorems because they are implied by the axioms.

For example: The axioms of the basic arithmetic we all learn in school are as follows.
- Every number is equal to itself. If one number is equal to a second number, that number must be equal to the first. If a third number is equal to the second, it must also be equal to the first. But only a number can be equal to a number. (Equality defined!)
- 0 is a "natural" number, and adding one to any natural number yields another natural number. (Addition permitted!)
- If you have two numbers and add one to both of them, and get the same result, the two numbers must themselves be equal. (Subtraction permitted!)
- If you define some set of numbers such that if it contains some natural number then it must also contain that number plus one, and define it to contain 0, then that set contains all natural numbers. (The principle of induction!)

Arithmetic theorems include 1+1=2; 1+2=3; 2+2=4; and so on. It should be clear that there are an infinite number of these, even though there are some distinct limits - this arithmetic system doesn't even allow for multiplication yet, let alone fractions, irrational numbers, negatives, and more complicated ideas! But such concepts can be incorporated, by adding more axioms to get a larger system that still contains arithmetic.

For centuries, mathematicians have held two dreams. On the one hand, that it would be eventually possible to extend arithmetic logic - that is, the logic that underlies how we experience the universe on an everyday basis - to the point that it would contain all possible true statements. The day was dreamed of when two philosophers in a dispute would, instead of saying "let us argue", would say "let us calculate"; and then sit down and work out the equations underlying the properties of the soul, or the top quark, or God - a system both eminently powerful, and eminently practical.

On the other hand, there was the fear that mathematics is itself broken in some way, that one day someone would calculate some grotesque and immense equation, prove it true beyond all possible doubt or inherent limitation, and then realize that you could apply it in such away that 0=1. A waste of millennia of research and imagination and rigor, and the seed of a potential existential crisis among the entirety of the sciences and philosophies, which rely so heavily on the idea that human reason can understand the cosmos to begin with.

The dream and the nightmare are fundamentally in opposition to each other, and either would precipitate a tremendous shift in human understanding.
And Kurt Godel was the man who proved both were impossible.

The First Incompleteness Theorem: Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete.

In simpler terms, any theory that includes the axioms discussed above - which we know to be true - either contains false statements, or it excludes true statements.

Let me repeat that. If a theory can be made to contain all truth, then it also contains at least one statement that's false. If you can fix it to kick out that one statement, you also have to kick out at least one statement that's true.

This is the strongest argument for agnosticism that could ever possibly be made, and it is provably true based on everything we understand about logic and mathematics. And if you think that's bad, it gets worse.

The Second Incompleteness Theorem: For any formal effectively generated theory T including basic arithmetical truths, and also certain truths about formal provability, T includes a statement of its own consistency if and only if T is inconsistent.

That is, one of those statements you can never prove is that you're right. And if your theory can prove that all the statements it implies are true, they aren't.

Funny thing is, we can still define the "universal set" of all statements that are true. There's just no theory that can tell us what all those statements are. Sure, we can identify a "Godelian statement" for every theory we come up with, and add an axiom to the theory to account for it. We can splice a couple of mutually-cohesive theories together to cover even more ground. But these just make a bigger theory, and we already proved every consistent theory has a true statement outside it. We can make infinite theories, but the thing about eternity is, there's always more of it.

We can mark off a section of the Library of Babel that contains all the books we can prove to be true, but there's always another secret just beyond the horizon and we'll have to fight through a bunch of lies to get to it - and if you should find a lie in your own head, well, you'll have to throw it out, with all the ones relying on it that you thought you'd already proven.

Sound perilous? If you care about what you think, it's the only choice you've got. The quest for knowledge is an unending struggle against infinite odds. Besides, they say an inconvenience is only an adventure wrongly considered.

Bring me that horizon.

25 March 2010

I am being lax

At this rate, I might as well change my name to "Once a Month".

Not done explaining Godel yet, sorry. But I highly encourage you to go check out this essay, as it is pretty awesome.

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.