Saturday, August 27, 2011

Moving past that little bit of math the other day, we turn to philosophy, which is like math except much less precise, much less useful and a lot more open to discussions by laypeople in altered states of consciousness. Also a more common subject of my blogginations.

As you have no doubt cleverly deduced from the title, there is a website called, and it claims to have proof of God's existence. The even cleverer may suspect I disagree, and that an explanation of that disagreement will be in fact the main point of this post. You are entirely correct.

The format of the website is quite simple. At each stage, you are faced with a number of propositions, 4 at first and 2 at all other steps, and you have to choose one. If you choose the wrong one (as determined by the author), you are directed to a page explaining your foolishness and urging you to choose the right one. As you progressively accept the propositions deemed correct, you are guided to a final "proof" which takes these propositions as premises and God's existence as a conclusion.

It should be noted that I don't necessarily disagree with the premises. I may be hesitant on a few, but essentially all feel correct or close enough. It's just the final reasoning I object to. So while I say a thing or two about them, the really important part is just the proof. Skip to that if you want to.
Didn't skip? OK, then, we start at step 0, faced with 4 options regarding absolute truth. They are:
1) Absolute truth exists: This is the option deemed correct by the author, and mostly I agree. I wouldn't say I'm sure, but it certainly seems my best guess.

2) Absolute truth does not exist: This leads to a page titled "Absolute truth does not exist" with two options: Absolutely true and False. Both lead to the same place, a page with the title "This is not a glitch (think about it)", and shows the same 4 options that at the beginning. Of course, the author refuses to consider the possibility that the non-existence of absolute truth is relatively true.

3) I don't know if absolute truth exists: The page you get is exactly the same as if you had clicked the prior option, only with a correspondingly different title. Both options again lead to the "think about it" page. Again, the author's insistence on binary propositions leaves the obvious answers out.

4) I don't care if absolute truth exists: The followup is just a "thank you for visiting" page with an "exit" button that links to Disney. Is knowledge apathy highly correlated with being a Disney fan? Mysteries of life.

Like I said,I essentially agree here, so I moved forward to an introductory page which has some blahblahblah about how the existence of God should be obvious but anyway here's some proof and whatnot. Moving right along, we get to step 1, about logic: 
1) Laws of logic exist: While one might quibble about what "existing" means for a logical law, once again I essentially agree.
2) Laws of logic do not exist: The page claims that either you arrived at this conclusion using logic, or it was an arbitrary decision and you might choose otherwise next time. Once again, this is not strictly exhaustive, one might come to decisions using non-logical but time-invariant reasoning. People do not, in fact, use logic alone to come to most decisions, but it's not really a point worth arguing.
Onwards to step 2 (electric boogaloo), which deals with the subject of mathematics:
1) Laws of mathematics exist: This is actually equivalent to step one. Math is just logic, wherein you say that, given some axioms, something is necessarily true or necessarily false. Accepting logic is accepting math, and viceversa. So I again agreed.
2) Laws of mathematics do not exist: This point essentially says that you use math all the time, so you can't deny it and be consistent. I agree, though one might argue about the subtleties of using math vs declaring math to be law, etc.

Which leads us to step 3, which leaves the purely logical domain and enters the empirical one:
1) Laws of science exist: Agreed, with the usual provisions on what it means for a law to exist, etc. But, yes, I do believe laws of science are real and useful, otherwise I wouldn't be trying to become a scientist.
2) Laws of science don't exist: It redirects to a page with the same argument as above, except for science instead of math

And then step 4 which takes us to the complex field of ethics:
1) Absolute moral laws exist: This one is tricky. I'm a moral realist, i.e. I consider right and wrong to be objective concepts. Results of quirks of the evolution of humans as sapient social animals, ultimately, but nonetheless real. What I am not is a deontologist, i.e. someone who believes that morality is a set of rules that must be obeyed absolutely, which is what one usually thinks when talking about moral laws. But nonetheless there are laws concerning right and wrong, or at least one law, which says (roughly) "calculate the possible consequences of your actions, weigh them according to what you value, and do the best thing possible". So I agree or disagree depending on what exactly is meant. This is ultimately of little relevance to the final proof, though.
2) Absolute moral laws do not exist: This leads to a second set options, asking whether raping children for fun is absolutely wrong or not. I actually can conceive of situations where raping children for fun would, in fact, be the right thing to do. They are convoluted and ridiculously improbable, of course, but still. If you pick "not", the next page goes on about how moral subjectivism is bad and blah. I could respond to it (I used to be a moral subjectivist, and had seen similar arguments before), but it would get long and boring.
Steps 5, 6, and 7 are on the nature of all these laws previously mentioned. Step 5 deals specifically with their materialness :
1) Laws of logic, mathematics, science and absolute morality are immaterial: I prefer "abstract" to "immaterial", since the latter implies they are made of something, which just isn't so. But in any case, sure, they certainly aren't objects made of matter. They are things that hold true about things that are made of matter, (or about other abstract entities).
2) Laws of logic, mathematics, science and morality are material: Leads to a page asking you to say where in nature you can actually find them and blah.
Step 6 deals with universality:
1) Laws of (etc.) are universal: I don't really consider something to be a law (in the sense used here, not the legal one) unless it is, in fact, universal. A law which only works in some circumstances is not a law, just a special case of a greater, actual law. So yeah, sure.
2) Laws of (etc.) are individual: Once again our friend ignores a vast number of possibilities. There are a number of different scales between universal and individual. But anyway, leads to a page using the same "you assume X to live your daily life so you can't deny it".
Step 7 is highly similar to its direct predecessor, but this time it's about change:
1) Laws of etc. are unchanging: Again, if laws change over time they are just special cases of laws that dictate their behaviour at each time.
2) Laws of etc. are changing: Leads to the typical argument.

And finally, we arrive at step 8, the part that is the purported proof rather than just the building up the premises.

The argument begins as follows, and I quote:
To reach this page you had to acknowledge that immaterial, universal, unchanging laws of logic, mathematics, science, and absolute morality exist. Universal, immaterial, unchanging laws are necessary for rational thinking to be possible. Universal, immaterial, unchanging laws cannot be accounted for if the universe was random or only material in nature.
A random universe can very well be lawfully random. The Copenhagen interpretation comes to mind. And I don't see why a purely material universe can't have statements about it that are true yet not material themselves, I mean that's what most materialists mean by material universe, myself included. So this "cannot be accounted for" thing is just glaringly lacking in justification. When you consider the long essays over ultimately inconsequential things, this is quite annoying (I say ultimately inconsequential, because you don't really need to agree to all the laws put forth, it's enough to agree with one set of them for the purposes of the argument)

It continues to say:
The Bible teaches us that there are 2 types of people in this world, those who profess the truth of God's existence and those who suppress the truth of God's existence. The options of 'seeking' God, or not believing in God are unavailable. The Bible never attempts to prove the existence of God as it declares that the existence of God is so obvious that we are without excuse for not believing in Him.
Because obviously, someone who doesn't already believe God exists gives half a shit about whether the Bible allows us not to believe, or whether it considers God to be evident. I mean, the Revelation of Ungod teaches us that gods don't exist, but I don't expect that to sway the author of this piece.

Anyway, that is followed by some corresponding Bible verses (Romans 1:18-21, if you're interested), and then the remark:

The God of Christianity is the necessary starting point to make sense of universal, abstract, invariant laws by the impossibility of the contrary. These laws are necessary to prove ANYTHING. Therefore...
Whoa, mate. How the fuck d'you figure that? Even accepting your premise that a material universe isn't enough, why does the starting point need be a god, let alone your god? This claims to be a logical proof, so what possible chain of reasoning based on the premises so far results in "The God of Christianity", a complex proposition not mentioned once in any premise? Someone doesn't understand what "logical proof" is, methinks. New elements don't just jump out of thin air.

The final statement of the proof is "The Proof that God exists is that without Him you couldn't prove anything." It gets its own page and all. And while I could argue the finer points of whether you actually can prove anything at all, that's not really where the proof fails. At each step there were subtle points that can be argued to death, but it's useless to get bogged down on that when there's that giant gaping hole in the argument.

So putting aside my hesitation at some of the premises and whatnot, my refutation of this "proof" is short and simple: There's no reason given why a material universe doesn't account for universal law, and there's even less reason to assume the only thing that does is one particular God. Anything else would be nitpicking.

It's disappointing. With all the build-up and extensive arguing for each mostly obvious premise, the final steps of the proof, the most vital and the most controversial, are stated outright with no reasoning behind them.

Tuesday, August 23, 2011

Slicin' spheres

Illustrative MS Paint diagram 1
Imagine a spherical structure of radius r, which has within it multiple circular floors (see illustrative MS Paint diagram 1). You want to know the total area of those floors. The area of a circle is pi*radius^2, so you can easily calculate the surface of a floor located in the exact centre, since its radius is the same as that of the sphere, but the other floors are trickier. Let's say that all the other floors are separated by a distance h. How do we find the total area?

The most obvious approach is to find a formula for the radius of each floor, plug it into the pi*r^2 formula, and repeat for each floor. Let's try that, first.

Illustrative MS Paint diagram 2
Take a look at our second MS Paint diagram, with coloured lines labelled x, r, and h. The green r length and red h length we already know (and are, in fact, r and h, i.e. the radius of the sphere and distance between floors). You can see that the blue x line is the one whose length we want to know, the radius of a circle located at a distance h from the centre. X, r, and h form a right triangle, so by Pythagoras' theorem, we know that x^2 + h^2 = r^2. From which we can deduce that x^2 = r^2 - h^2. We could take the square root to find out the value of r, but we don't need to. Why? Because all we want x for is to replace it in the circle formula pi*radius^2, where it is already squared.

Illustrative MS Paint diagram 3
Replacing, we know that, for a circular section of a sphere at a distance h from the centre, the area is pi*(r^2-h^2). That's the floor directly above the centre, at any rate. It's also the floor directly below the centre, since they are the same size. But we want to know the total surface, so we need to know all floors. So what we do is we name each floor by the scheme in diagram 3, and amend the formula to be pi*[r^2-(n*h)^2], where n is the floor number. That way, we are in effect doing the same thing that in the paragraph above, but replacing each h with n times h. So, for floor 1, the formula remains unchanged. For floor 2, we use twice h rather than just h, because it's twice as high. Notice that, cleverly enough, for floor 0 the formula is just the old pi*r^2. Also notice that I gave some of the floors negative numbers, but I could just as well have made them positive. It doesn't make a difference, because n^2=(-n)^2.

So there we have our complete formula,, for every n floor. Now it's a relatively simple matter of replacing every floor in the formula, and adding them all up. Easy if you have 3 floors, to be sure, but what if you have more?

Well, in the original problem I considered, the radius was 150m and the distance was 3m. That is to say, ~50 floors in each direction. Admittedly, at that point I declared myself too lazy, left the person who was asking me the question with the general formula, and told them that if they wanted it they could do it themselves. (They didn't, but I remembered what little I have of skill with Python and wrote a short script to do it for me). But I then thought of this simple trick to compute an approximate answer. It's useful if you don't need an exact answer and can't ask a computer or another person to do it for you. In fact, it's because I was surprised the trick worked so well that I ended up writing this blog post.

Image stolen from Wikipedia
The reason I thought of this trick is because the idea of slicing up a sphere in lots of smaller pieces and adding them up reminded of the concept of an integral. Very rough explanation: An integral is something you can use to calculate the area below a curve. The idea is that you divide the area into small rectangles, whose surface is easy to find. The more rectangles, the closer you get to the exact value. See diagram to the right, which this time I didn't make myself in MS Paint but rather took from Wikipedia*. You're welcome. Anyway, the same principle can be used for volumes (and happens to be something I've been doing in class a lot, lately). The trick we need for today's problem is to go backwards from an integral. I'll explain how that works in a moment.

Illustrative MS Paint diagram 4
So let's go back to the beginning. You've got a sphere and many parallel slices, whose area you want to know. You already know the radius of the sphere , so you can easily find its volume as 4/3*pi*r^3. And then you think: "I have all these parallel slices at the same distance of each other. I bet that if I took each slice, made it into a cylinder, and stacked them all on top of each other I'd get something very similar to the sphere". You can imagine something like our 4th illustrative MS Paint diagram. The red is the stacked cylinders, black is the actual sphere (I know they look like rectangles on a circle, I don't feel like adding 3d effects and the idea is the same). Anyway, you can see that is a reasonable approximation, though of course not perfect. The more slices, the better the approximation, so for the ~100 slices of my original problem this is more than good enough.

So, you add the volumes of the cylinders and get a result that is close to the volume of the sphere. But what are you doing when you are adding the volume of the cylinders? Well, a cylinder's volume is height times area of the base. So we have base area of the first cylinder (ba1) times its height (h1), plus those of the second (ba2*h2), plus those of the third (ba3*h3)... But wait, all those h1, h2, h3 are the same, so we can just call it h. So we have: Volume of the sphere ~= ba1*h + ba2*h +ba3*h +.... The symbol ~= means "approximately equal".

With me so far? We have everything multiplied by h, so we can just write it as volume of the sphere ~= h*(ba1 + ba2 + ba3 + ba4 + ...). But what is this sum of ba's? The sum of the base areas of all the cylinders. And what is the base area of those cylinders? It happens to be the area of the circular sections we were trying to calculate in the first place!

So we just divide both sides of the equation by h, and we have that (volume of the sphere)/h ~= sum of the areas of the circular sections. And the volume of the sphere, I remind you, we already know from its radius. So this long explanation adds up to a very simple formula, which is the biggest advantage of this method: It's ridiculously easy to use. Just take r and h, and plug them in here:

As for that "approximately", how approximate is it? In my original problem, radius = 150 m  and h = 3 m, the result I got from my Python script is 4,711,917.9068 (pi to ten decimal places, just because that's as much as I know from memory). Using the formula above, I get 4,712,388.9804. That's an error of about 0.01%. So, y'know, pretty good. Considering the result I reported to the person who asked me was "about 4,700,000 m^2", far more accurate than I needed.

To my imaginary readers, I promise I will get back to writing fiction rather than math and shitty .bmps. Eventually.

*The image's copyright holder is Wikipedia user KSmrq. Since it's under a Creative Commons Attribution-Share Alike license, so is this individual post.