For those who are still around, it was recently pointed out to me that I'd forgotten to give the solutions to the last couple of unsolved problems from my Skeptic's Circle. Since it's been a while, I'll repeat the problems here to remind you before solving them:
Lord Runolfr recently got a couple of e-mails from scammers, raising my suspicions that something big was afoot behind the scenes. I called up a few contacts, did some research, hired a few James Bond-alikes, and here's what I've figured out:
There's some evil mastermind behind the whole plan, and he's about as supervillanous as they come. This of course means that he wants to capture one of my James Bond-alikes and subject him to an intricately detailed explanation of his evil plan before killing him in a creative way. Apparently, the way he's decided to go about this revelation is through an overhead transparencies with the key points of his evil plan (you'd think he'd have better technology than my high school, but he's out to make money, so he saves it where he can).
Now of course, he's wary of this transparency falling into the wrong hands, so he's come up with a plan to keep things safe. He's figured out some method to spread the information across multiple sheets, arranging it so that if we get a hold of any two, we'll still have no idea what he's planning. So, what we need to figure out now is some possible ways he might have done this, so if we manage to get our hands on more than two, we'll know how to read them (if it's not immediately obvious). What are some possible things he could do? Remember that he's out to save money, so splitting it up to have a single word on each slide or something huge like that doesn't seem too likely.
While it's possible to split up the information in a simple way over many slides, the trick to this problem is finding a way to divide it up so that absolutely zero information is transmitted in a single slide, or even in two slides. Splitting it up into words or letters (or fragments of a letter) never accomplishes this, as the remaining pieces still give some information. To illustrate a way of transmitting zero information in a simpler case, where any one slide can be lost, I gave the following sample solution:
Pixelate the presentation, and make sure the pixels are quite large and easily distinguishable. Generate one transparency that's completely random, with each pixel being randomly either transparent or 50% opaque. Then, for the second sheet, for the places where the message is spelled out, choose either transparent or 50% opaque as necessary to make the pixel result in 50% opaque. For places where the message isn't, match the pixel to the one on the other slide. The result will be the message appearing in gray text on a mixed black and white background.
Now the trick is moving up to a case where any two slides can be lost, and together they'll give zero information. Part of what makes this problem tricky is that everyone tries to find a solution that will use the absolute minimum number of slides at first (in this case, 3). However, such a solution doesn't exist (at least that I've seen, and I tend to suspect it doesn't at all). There is, however, a solution that uses 4 slides.
Here's how to construct the 4-slide solution: As before, pixelate the message. This time, however, instead of filling the pixels with either gray or white, we'll be filling them with one of three colors of ink. This ink will be designed to absorb one third of the visible light spectrum and let the rest pass through. For instance, we could have ink that absorbs red wavelengths, allowing blue and green light through (which appears as cyan I believe), plus ink to absorb blue (appears as yellow) and ink to absorb green (appears as magenta). If we stack the three different colors on top of each other, no light can get through, and we'll have a black spot on the image. If only two colors are in the stack, it will appear as the remaining color. If just one color, we'll get a mix of the other two.
Now, for each pixel of the image, first determine whether it's part of the message or not. If it is, we want it to appear black. So, through the four slides, we'll arrange it so all three colors show up somewhere, plus one of them appearing twice. We'll randomly choose between all possible permutations that do this. Now, for pixels that aren't part of the message, we want them to not appear black, so we randomly choose one of the permutations that uses only one or two colors. The net result is the message appearing in black on a multicolored (or gray, if the pixels are small enough) background.
To see that this works, image that two slides are stolen, and look at a single pixel on both of them. There are two possibilities here: 1) both slides have the same color for that pixel and 2) the slides have a different color in that pixel. In case 1, it's possible for the remaining two slides to have the other two colors, and it's also possible they might repeat this color. For case 2, it's possible the remaining color will be on one of the other slides, but it's also possible it won't be. In the end, we can't infer anything about whether or not this pixel is part of the message.
Now, why won't this work for only three sheets? Well, let's go back to case 1 if two slides are recovered. If both slides have the same color, there's only one slide left to block more light. This third slide couldn't have both colors, so this pixel cannot be part of the message. We can't infer anything from the cases where the two slides have different colors in a pixel, but we can still gain some information by picking out some pixels we know can't be part of the message (we'll catch 1/3 of them on average, and if the pixels are small enough, we might be able to glean some of the actual message).
What Wifi Woo?
Sandy Shwarc reports that there's been some scare over the ill effects of all the elctromagnetic radiation going through the air, but I'm not buying it. Personally, I think this is all just an excuse to avoid having to work. Confused? Let me explain.
Take the new Ultra-Mega-Awesome Wifi tower built the other day. It had 1000 power cords going from the bottom to the top, and not one of them was initially hooked up to anything. Worse, they're all tangled in the middle so it's impossible to figure out which bottoms of wires correspond to which tops.
Now, some poor shmuck has to go and sort them all out, and the only tools he's given are a battery and a lightbulb. They somehow expect him to to hook up the battery at the bottom to a couple of wires, then go to the top and see which wires he can connect the lightbulb to to sort them out. Maybe he could pull a few tricks like tying some wires together at the bottom or top to make long wires, but it's still going to take him quite some time.
With that job ahead of him, you can see why he'd want to believe it shouldn't be done. Maybe if we could help him out and figure out the most efficient way to solve this problem, he'll be a bit more likely to accept Wifi. The tower's pretty tall and the only way up is by stairs, so he'd probably appreciate most if we could help minimize the number of trips he has to take, regardless of how much work he has to do at the top or bottom. How can we do this, and what is the minimum number of ascents and descents required?
The solution to this is pretty complicated, but it helps to look at a simpler case. Let's say we only have 3 wires. In this case, here's how you do it with just one ascent and descent: Start at the bottom. Tie two of the wires together. Mark both of these “2-_”. Mark the loose wire “1-_” (a group of 2 and a group of 1). Now ascend to the top. Hook the battery and lightbulb together, and connect one wire to one end of this assembly. Test each other wire on the other end, and count how many wires will allow the lightbulb to light up. If one wire will cause it to light up, then mark the test wire with “2-_”. If no wires will light it up, mark it with “1-_”. Repeat for all the wires.
Now, at the top you'll have two groups of wires, one of just one wire, and one of two wires. You know that these correspond to the groups you made at the bottom (and the one-wire group has the single wire properly identified). To sort between the wires in the 2-wire group, we need another step. Now, take the wire in the 1-wire group and tie it to one of the wires in the 2-wire group. Mark both of these “X-2” (where X is whatever mark was in the first digit). Mark the other wire “X-1” and leave it unconnected to anything.
Descend to the ground, and repeat what you did when you first got to the top, except this time, fill in the second digit (if it connects to zero, mark 1, if it connects to 1, mark 2). Now, at the top and bottom you'll have wires marked “1-1”, “2-1”, and “2-2”. These IDs match them all up to each other. It says nothing in your job description about untying the wires at either end, so you're done, with one ascent and one descent.
This solution type can be extended simply to any number which is a perfect triangle (1,3,6,10,15...). For instance, with 6 wires, you'd tie up one group of 3 at the bottom, one group of 2, and one group of 1. You can then identify these groups at the top. Then, you can tie up a group of 3 taking one from each bottom group, a group of 2 taking one from the 2 and 3 bottom groups, and a group of 1 from the 3 bottom group. Go back to the bottom to sort out these groups and you've identified them all.
The problem gets trickier, however, when the number you have isn't a perfect triangle. Since 1000 isn't one of these, we'll have to face this. But, it is possible to extend this solution to most non-triangular numbers. The trick is dump the extra wires into the group where the wires aren't connected to any others. Let's look at the 8-wire case to see this.
At the bottom, set up three groups. The first group (1-_) has three wires not connected to each other, and not connected to any others. The second group (2-_) has two wires connected together. The third group (3-_) has three wires all connected together. Go up to the top and identify all of these groups. Now, to set up the top groups. Set up one group which takes one wire from each of the bottom groups, and tie all of these together (X-3). Set up a second group which takes one wire from each of the bottom groups, and leave all of these tied to nothing (X-1). You'll be left with one wire from the 1-bottom group and one wire from the 3-bottom group. Tie these together in the final group (X-2). Go back to the bottom and identify these groups. You'll then have eight wires, all uniquely identified.
However, it turns out there's a problem for some numbers of wires, such as 5 and 9. If you try to do it this way, you'll have too many wires in the group where they aren't connected to anything. I won't go into all the details here, but your best solution with this method is to have the extra wires in the unconnected group on both trips. You'll end up with two wires marked 1-1. Then, connect one of those to another wire when you're at the bottom (1-1a) and leave the other unconnected (1-1b). Go back to the top, and figure out which of the wires is 1-1a by testing its circuit with the one you tied the bottom end to, and the other is then 1-1b. You'll then have figured them all out with just one extra ascent (you also have to descent to go home, but that's just a technicality).
It turns out that this particular problem comes into play when the number of wires is one less than some triangular number. So, you get problems with 2, 5, 9, 14, and so on. The 2-wire case allows a special solution with only one ascent (attach both wires to the battery at the bottom, marking them with the pole they're attached to. Then go to the top and see which way you have to orient the lightbulb so it'll go on, and match up the poles), but the others will all require 2 ascents and 1 descent with this method to figure it out. Any other number can be done in 1 ascent and 1 descent with this method (3 wires can actually be done in a single ascent with a simple extrapolation from the 2-wire case). Since 1001 isn't a triangular number, our 1000-wire case can be done in just one ascent and one descent using our method.
Side note: There is another, much more complicated method which will allow you to solve this problem in just one ascent and one descent regardless of the number of wires. However, the average time this method takes is proportional to the number of wires to the fourth power; while the method I've given takes time simply proportional to the number of wires squared. For large numbers of wires, the extra time the other method takes at the top or bottom would easily outweigh the time of a single ascent of descent. However, for 5 or 9 wires, it might be worth it. I won't go into it here, though.
Friday, November 30, 2007
For those who are still around, it was recently pointed out to me that I'd forgotten to give the solutions to the last couple of unsolved problems from my Skeptic's Circle. Since it's been a while, I'll repeat the problems here to remind you before solving them:
Friday, November 23, 2007
This just in, apparently the top bishop in Quebec has gone and apologized for pretty much everything the church has done wrong. Everything he could think of, at the least. First of all, it definitely is nice that someone in a prominent position there is sorry, and I believe this is the first time I've heard any of them apologize for the child abuse that went on. However, I do have a few gripes.
The first is motive. This bishop wasn't apologizing just because he was sorry about all that had happened. He apologized because he thought it would help draw people back into Catholicism. This doesn't mean he isn't actually sorry, and I don't doubt that this man in particular probably is, but it does mean that it isn't as big a reason for him as attracting people to the religion.
My second problem is that although an apology is fine, what I really want is a promise to try to improve. Weeding out the bad ideas you know about is a good first step, but other bad ideas will keep popping up. You have to get down to the root causes and pluck them out. Unfortunately, this is something I expect the church never to do, as it would involve using logic instead of faith, relying on evidence instead of divine revelation, etc. In short, it would take all the religion out of religion. Not that I'd mind seeing that happen, but it's not going to.
Monday, November 19, 2007
I'd like to make a few comments about my recent post, Intelligent Planting. I left the narrative without any comments there, as I figured this was an allegory that worked well enough on its own. I set up a parody of Intelligent Design to better illustrate all the leaps in logic design proponents expect people to make. It starts with jumping from "it doesn't look random" to "it was designed." Then it jumps from "it was designed" to "God/Pete designed it, and he also did all these other things recorded in the Bible." Now, of course, design proponents are all about hiding their religious affiliation, but it's there, and it is their ultimate goal, whether they'll admit it or not. I also then threw in some of the other doggerel they use to justify this for good measure, particularly mocking the appeal to faith.
That all being said, you might be somewhat surprised to learned that Intelligent Design wasn't my initial impetus for writing this story. Instead, this comes from a different argument for God which makes much the same leap in logic (from saying there was someone or something to saying it was God). This was what's known as the Cosmological Argument. When boiled down, it essentially becomes, "There was an ultimate cause for everything, therefore God."
You might want to take a moment to read through the Wikipedia article on this argument, linked above. What I'd like to call to your attention is the simple fact of how many variations on this argument there are. The Cosmological Argument is often presented as being strictly logical, but if that were so, then you wouldn't expect these variations on exactly how the Prime Mover/Uncaused Cause/God started things. Therefore, it would seem that most or all of these arguments are likely making some assumptions behind the scenes (or are just fallacious).
For example, let's take the argument of Thomas Aquinas, one of the more complete versions. From the Wikipedia summary:
1. Every finite and contingent being has a cause.
2. Nothing finite and dependent (contingent) can cause itself.
3. A causal chain cannot be of infinite length.
4. Therefore, there must be a first cause; or, there must be something that is not an effect.
Few would argue with points 1 and 2, but let's take a look at point 3. Why is it that a causal chain cannot be of infinite length? Presumably this is simply stated because the thought of it seems absurd, but is it really? Let's extend things into the future. Under most modern models for the universe and many religious models as well, time will go on infinitely into the future. This means that as long as it keeps going, we'll keep on having a causal chain. Thus, the causal chain will extend infinitely into the future. Ipso, a causal chain of infinite length. (Man, have I been itching to properly use "ipso" in a sentence...)
So, if it can extend infinitely into the future, what's wrong with having a causal chain extend infinitely into the past? It's at this point that it seems a bit more absurd instinctively, but logically it doesn't have to be. All laws of physics we know of are time-reversible, with a single exception that allows us to see order in time, the collapse of a wavefunction. If you compensate for collapse and run things backwards in time, you can see the same theme of causation occurring. Instead of a sperm and egg causing a zygote, you get a zygote causing a sperm and an egg, for instance. Running things this way, it doesn't seem so absurd that things might go on forever.
So here we have the problem with this particular argument: a false premise. The argument may still be technically valid (the conclusion can't be false if all the premises are true), but with a false premise, it's unsound, and we have no reason to believe the conclusion given this argument. Now, this doesn't mean that there wasn't actually some first cause, it only means that this argument doesn't prove it. So, let's entertain the idea that there was a first cause now, for completeness' sake.
What can we say about this first cause? Well, nothing, really. We can't claim it must have been intelligent, or even complex in any way, as it's easily possible for intelligence and complexity to arise from unintelligent, simple conditions, driven by a little randomness. However, we have a lot of religious people pointing to their own god and saying it fits the bill of a first cause. The argument for a first cause, even if it were valid, doesn't give us any reason to believe that the first cause is anything like their god, but that's not necessarily a problem.
What we can do is treat their god as a hypothesis to explain the first cause. A tactic like this is often done by scientists; we have a problem, so we hypothesize something to explain it which is a bit beyond what we know. Since it's beyond what we know, it often comes with the ability to predict other phenomena we haven't tested for yet. So, we then go and test for those phenomena. If they exist, we have evidence that this hypothesis is true. We can do roughly the same thing for the hypothesis that a god was the first cause.
The immediate problem is that invoking a god here is a gross violation of Occam's Razor. An explanation that invokes a particular brings in many interrelated claims, and has many, many predictions beyond the simple creation of the universe. This doesn't mean it isn't true, however; it just means that we're going to need a lot of evidence to support it. Otherwise, a simpler explanation (or less precise god) will be much preferred.
Now, there are many deities we can choose from, so I'll only use a couple examples here, positioned at extremes. Most other deities will fall on the continuum somewhere between these, and a mix of the applicable arguments will apply.
First up, I'm going to take the god believed in by many evangelical Christians in the US. This is the god discussed in the Bible, who did all the things claimed there. He's omnipotent, omniscient, omnipresent, and just generally omni. He also takes a role in our day-to-day lives. He listens to and answers prayers. He smites those who displease him in any way. When people die, their souls are judged by him. If they've led a ridiculously devout life, free from even the slightest pleasure (shadenfreude over thinking about sinners going to hell excepted), they go into heaven, a place of eternal bliss. If they're even slightly off, or believe in a slightly different god, they go to hell, a place of eternal torment. (Aside: I don't particularly care if anyone believes in exactly this god; I'm just using it as an extreme example.)
This the hypothesis to be tested. On the other end, we'll have the null hypothesis, which we'll be comparing this to. At the end, we hope to be able to reject either this or the null hypothesis. In this case, we can use the null hypothesis, "No god exists." If we find sufficient evidence for this god, we'll be able to reject this null hypothesis.
So, what of this is testable? If you say "none of it," scroll down a bit. I've got your untestable god there. This is a god who interferes with the world. If there are natural effects of supernatural causes, they can be tested for. Anyways, there are two big points here that we can test: Intercessory prayer and smiting the heathens. Let's start with prayer. This is something that actually has been scientifically tested. Repeatedly. And then some more. And again, because every time, the results weren't satisfactory. Whey weren't they satisfactory? Because the tests were either poorly done, or they didn't show any effect to prayer. Even if you don't agree that the ones who showed an effect were poorly done, it's still only a marginal improvement. It's nothing compared to the effect you'd think an omnipotent god like this could have.
But wait! "Absence of evidence isn't evidence of absence!" I hear you cry. Ah, well there's a big exception to that. Absence of evidence can indeed be evidence of absence when you've properly and thoroughly looked for evidence. I call it the Modus Tollens Exception, as you can phrase it in the following logical form:
P1: If A exists and we use method M to search for evidence, we will find evidence E.
P2: We used method M to search for evidence, and did not find evidence E.
C: Therefore, A does not exist.
Boiling it down to the simple logic, this is the valid structure:
P1: If A and M, then E.
P2: M and not E.
C: Not A.
This is essentially the Modus Tollens argument form, with a small complication of an extra requirement in the premise. Since we're using a logical form here, my argument that absence of evidence is in this case evidence of absence is also valid.
It's a bit harder to test the prediction that this god will go around smiting heathens, as we can't really control things here. However, if you look at what happens in the world, there isn't good evidence that this takes place. For instance, many evangelicals claimed that Hurricane Katrina was their god smiting New Orleans for all the debauchery that takes place there. The problem was that the French Quarter, which was where most of the debauchery took place, was one of the least-damaged areas. This particular claim, at least, doesn't hold water. (Ugh... I swear that pun was unintentional.)
In any case, testing prayer alone is sufficient here. We can, of course, add to the evidence for all the other claims about this god, though it's not necessary for the time being. Since we have evidence that prayer to this god doesn't work, we can reject the hypothesis that this god exists. And no, we don't go and accept the null hypothesis that no god exists; we just say that we fail to reject it. It could well be true, but we haven't shown that here.
Alright, now let's switch to the other extreme. Let's take a Deist god. This god created the universe, and then just kind of sat back and watched. Or maybe he went off to create another universe, or just took the next few billion years off to slouch around and watch TV. Or maybe he's a "she," or an "it," or some other gender we don't have a pronoun for. Not much is claimed about this god. In fact, aside from that he created the universe, not anything is claimed about this god. In contrast to the previous case where so much was claimed about the god that it was easy to find evidence against him, here, we don't have anything claimed at all beyond what we know happened. With this, we can't make any testable predictions about him, so we can't scientifically test for his existence.
This is still a hypothesis, though. The problem is that it's a completely useless hypothesis. There's nothing we can do to improve upon it, or get any further evidence that it might be true. There isn't even anything we can do to differentiate it from similar hypotheses, such as saying that instead of a god, a Flying Spaghetti Monster created the universe. Or we could say that the universe spawned yesterday from primordial slood with all the particles in just the right positions and velocities for us to be here with all this evidence for a past and memories of it.
In the end, there's no way to reject this hypothesis. But coupled with that is the fact that there's no way to get evidence for it. There's no reason for you to actually believe in it. The fact that it can't be disproven is no reason to believe, as following that logic would lead you to believe a million contradictory explanations for the beginning of the universe. This is why science doesn't do anything with untestable hypotheses; they're utterly useless.
In the real world, most gods people believe in fall somewhere between these extremes. They try to balance out not contradicting reality with having enough predictions to be useful. However, this doesn't really solve any problems. In order for their to be evidence of a god's existence, it has to make some predictions that later turn out to be true (and of course, can't be adequately explained without him. A god predicting gravity isn't a big deal). Simply throwing away disproven predictions and holding onto untestable ones still doesn't give anyone a reason to believe in this god.
However, I could well be wrong. If I am, if there's some god out there with good evidence for his existence, I would in fact quite like to know about it. I'd expect to have heard of it by now, but you never know. Maybe the right study just hasn't been performed yet. In which case, I challenge any believer who believes they have a testable prediction about their god to go out and perform a study to test it. Perform it well enough, and a positive result could be just what you need to convince me. Until then, I'm happy living my life accepting the null hypothesis as most plausible.
Saturday, November 10, 2007
Bah, I've been remiss about this. I missed out on linking to the 72nd Skeptic's Circle and the 73rd Skeptic's Circle when they came around, so there ya go.
In other news, if you're going to conduct a demonstration claiming that a snake not biting you is evidence of your faith and God's existence, you have to prepared to take the fact that it instead bit and killed you as evidence against this.
Tuesday, November 06, 2007
One day, two friends, Jim and Mike, were out hiking through the wilderness. They came upon a beautiful patch of wildflowers, and while admiring it, they struck up the following conversation:
Jim: Amazing, isn't it?
Mike: Yeah. That Pete sure is something.
Jim: Pete? Who's Pete?
Mike: He's the guy who planted all of these flowers. You didn't think they'd be arranged so beautifully on their own, did you?
Jim: Well, actually, I did. Flowers evolved to be beautiful; it's part of how they attract bees to pollinate them or something. We can ask my friend Rick when we get back to town if you want the whole story; he's a botanist, so he should know. It's not too surprising they'd be beautiful.
Mike: Meh. I don't buy it. Even with that, the way they're arranged is too pretty.
Jim: I don't know, it seems pretty random to me. The human mind is good at picking out patterns, though, so the few that form randomly stand out to us and make us think it's beautiful.
Mike: Trust me. I know design when I see it, and that patch has too many nice patterns in it to be random. Someone must have purposefully planted them that way.
Jim: And that someone is Pete?
Jim: ... I think I'm missing a step here. I can see how your logic and way of thinking would lead you to believe someone must have planted those flowers – even if I don't agree with it – but how does that extend to it being a specific person? Do you know Pete or something?
Mike: Well, no.
Jim: Do you know anyone who's seen him? Or, maybe any newspaper articles about him planting wildflowers in this area?
Mike: Not exactly, no.
Jim: Then why do jump from “someone” to “Pete”?
Mike: Well, you see, when I was a kid, my mother read this book to me, all about Pete and his work. I've grown up believing in Pete ever since, and I make an effort to spread the word about him when I can. His work shouldn't go uncredited.
Jim: Okay, I guess that's better than nothing. I'd like to see this book sometime, though. See what all the fuss is about.
Mike: Oh, well you're in luck. I always bring a copy of it along when I go hiking.
(Mike searches through his bag, picks out a book, and hands it to Jim. Jim starts reading through it.)
Jim: Huh. Well I can see where you got some of your ideas from, but this just isn't too convincing to me.
Mike: Why not?
Jim: Well, for one thing, it isn't very consistent. For one thing, his last name changes spelling over the course of the book. It starts off as “Gardener” – a name whose appropriateness makes me suspect this started off as a simply children's book, but I digress – but in the last chapter it becomes “Gardner.”
Mike: Oh, well that's just a typo.
Jim: All seven times?
Mike: Okay, maybe not. I think different chapters might have been written by different people, so that would explain it.
Jim: Ah, I can see that. Especially if this just started as a children's story which got misinterpreted.
Mike: Hey! Don't disrespect my beliefs like that!
Jim: Sorry, but as your friend, I feel it's my obligation to tell you that what you're saying isn't that convincing.
Mike: What do you mean? Isn't the design of these flowers along with this book proof enough for you?
Jim: Well, for all I know, this book could have simply been intended as fiction. The design of these flowers – which I still don't agree to, mind you – is the only real evidence you've presented. Besides, I've heard of a few different stories about this type of thing. I think I recall one similar book, except it was a guy named Phil, not Pete.
Mike: Yeah, we get into fights with the Phil-believers all the time about this.
Jim: That's just the point. From your flimsy evidence, all you can argue for as that someone must have planted the flowers. There's no reason I should prefer Pete over Phil or maybe Bruce, Doug, or Pat!
Mike: That's where this book comes in. We've got evidence that there has to be someone planting these flowers, and we've got a story about Pete doing just that. Isn't that enough to believe in Pete?
Jim: Not really. I mean, a book like that isn't very good evidence. It's cobbled together by multiple authors, contradicts itself in many places, and doesn't even provide any way to verify any of it. And I'll remind you that I still think the layout of these flowers is simply random.
Mike: Well you see, that's where faith comes in. I mean, logically, it might not seem to all fit together. But you have to trust in Pete, and know that he really does exist, and somehow this all makes sense.
Jim: You say that as if blind trust is a virtue.
Mike: Pete says it is.
Jim: But you can't know that Pete's right unless you already accept that he exists.
Mike: Which I do, because of my faith.
Jim: You ever get the feeling we're arguing in circles?
Mike: Pete says the circle is perfect, and therefore circular logic is also a thing of beauty.
Jim: I don't think I want to be friends with you anymore.