Charlie

Thanks, that makes a lot more sense than how I'd seen it before. That also explains why it's discussed in the textbook in a chapter on the calculus of variations. One question though: does that mean it's possible for light to traverse several paths to the same point, if each of those paths is a local extremum? And don't worry about the long delay. I completely understand that you would have other things on your mind at the time.

This is something I've been having trouble understanding. According to my textbook, Fermat's principle is this: Light travels by the path that takes the least amount of time. But this seems contrary to my day to day experiences with light. For example, if you shine a flashlight at a block of some medium with a really high index of refraction, the light will go through the block, but very slowly. Even though It would be faster for light to go around the block (by bending up and over it), that's not what happens. But isn't that what Fermat's principle predicts would happen?

Nuclear power has always interested me. It's more efficiant than any other method we have, it's extremely safe, and it doesn't pollute. With all these advantages, I would think that it would be the dominant source of energy. However, ever since three mile island, no new nuclear plants have been built, which I understand is mostly due to rediculously strict government regulations on new plants. I was wondering if it would be a good idea to invest in a company that is involved with nuclear power production. Even if no new plants get built, there seems to be a lot of potential for profit in the industry. Also, other countries such as Japan, China, and France seem much more leniant. If a domestic company is a bad investment, would it be a good idea to invest in an overseas nuclear energy company?

I know this one because it was the topic of my essay for the ARI essay contest, so I read that part about 20 times this summer. He says that John Galt found the fountain of youth. He searched everywhere, and suffered horribly before he finally found it on top of a mountain. But he didn't bring it back because it couldn't be brought down.

That building is absolutely beautiful

It seems like you're trying to deduce that deduction isn't important. You make some clever arguments, but I'm not really sure what you're trying to prove. Also, your "case for induction" relies upon deduction. Simply knowing that Magellan went around the world isn't enough to use induction to prove that the world is round. You use these premises (arrived at through induction): 1) All round things do not have edges 2) All flat things do have edges 3) Magellan went around the world without reaching an edge to DEDUCE that the world is round.

I'll have to think about this some more. I'm not sure if I accept the idea that "holes" are logically impossible, but Alex S........'s essay had a very good explanation of why unbounded doesn't necessarily mean infinite.

I'm aware it's not entirely reliable. but in this case it seemed to be the simplest way to back up my argument that the big bang theory does not require a creation of the universe. I doubt it would contain any major factual error for very long (I searched the Ayn Rand for dondigitalia's claim that it described all romantic relationshiops in her novels as involving adultery, and could not find any such claim).

In response to Stephen Speicher's comments: There's no need for space to come from nothing- it's always not been there.

Notice it doesn't say anything about this actually being the beginning of the universe, or about matter being created from nothing. It specifically says that we cannot make predictions about anything before the Planck time. Any talk about what happened before then is mere speculation. Stephen Speicher brought up many examples of this sort of speculation by famous physicists in what he wrote, but none of those people actually show how the Big Bang theory inevitably leads to a creation of space, time, or matter. The quote by Stephen Hawking was the only one which even mentioned proof that an idea of physics meant there was a beginning in time- and he was using general relativity, not the big bang theory. I've noticed that a lot of incredibly brilliant physicists and mathematicians make absurd philosophical statements, and I think this might be one of those instances.

I don't agree with this. It's possible to determine the shape of an object without going outside of it. For example, I can determing that the room I am in is basically a cube without needing to go outside of it. The universe just happens to be a lot bigger. If there was in fact a big bang, then light (and other radiation) would have been racing outward from it ever since then. Nothing can go faster than light, so light would be what is furthest from the origin. So the "boundaries" of the universe would simply be the outer periphery of light. Light, of course travels at a constant speed in a vacuum, so it would have traveled equally far from the origin on any side. In other words, the shape of the universe is a sphere.

Hi Klaus. I was wondering if you had any pictures that people you'd tutored in this way had done. Some "before and after" pictures would be great. Also, can you teach Paint Shop Pro, or just Photoshop?

I really liked the visuals in this movie, but that was the only part I liked. The plot and the characters were just... uninteresting.

I just wanted to say that I highly recommend Dream Theater as well. They completely break the stereotype of the dumb rock band playing noise as loudly as possible. Three of them trained at the Berklee school of music (a rather famous music school) and one of them trained at Juliard (which is pretty much the most famous music school of all). All of their songs show an amazing degree of proficiency, and what's really remarkable is that they don't confine themselves to any one style- pretty much any musical style you could ask for has been included in at least one of their songs. That being said, however, I think that sometimes they sacrifice musicality for the sake of showing off what they can do.

I think Carlos makes a good point that some things are simply too potentially dangerous to allow a private citizen to possess(and, as David pointed out, the real question is whether or not they should be allowed to possess these things, not whether they should be allowed to import them). Nuclear weapons are a good example, since there's really no legitimate use a private citizen could have for them, and one falling into the wrong hands could have devastating consquences. I'm not sure where you'd draw the line as to what things people should be allowed to possess... that's a dicey issue. But as to the question of tariffs, I'd say you make a good point. Only, I wouldn't call them tariffs, because as I understand it that's a tax on the value of the goods being imported... the cost of the actual importation has nothing to do with it. This would be just a fee to cover the necessary cost of inspection. But, on the other hand... wouldn't a fee (even a low one) serve as an incentive to make people try and smuggle goods into the country, rather than paying to have them inspected?