Temper

I was very unphilosophic recently, and I'm just noting why.  Our next door neighbor has been having lots of work done on his house lately, and the construction workers have been parking in our driveway.  In an effort to be neighborly, we didn't object, although it would seem to make more sense to park on the street.  Yesterday I was backing out of our garage, and all I could see in my rear view mirror was the truck parked at the bottom of our driveway.  As I tried to get around it, I turned too sharply and ran into the trees lining the driveway.  One of the workmen was standing ten feet away talking on his cell phone.  I was furious, although turning too sharply was my mistake.  He could have said, "Is my truck blocking you?  Let me move it."  But he didn't.  He did yell to warn me as I hit the tree, but it was too late.

I was so mad that I cussed him out, which is somewhat unlike me.  So much for being neighborly.

Plato's Cave of Shadows

After writing the previous post, something called my attention to Plato's allegory of the cave of shadows.  It make me think how similar his example of shadows was to the two-dimensional world visualized in the Scientific American article, although he was interested in a completely different issue. The Scientific American article was concerned with the external, physical world, while Plato was concerned with the internal, mental world.  For Plato, the two-dimensional world of shadows represented the unenlightened world in which most men live.  As they improve themselves they exit the cave of shadows and see the bright, three-dimensional world, but then, I think, Plato argues that after seeing the bright world, leaders have to re-enter the cave of shadows to lead those poor souls with the knowledge they have gained from the outside world.  If they stay in the light, they are no benefit to their fellow men.

Although their points are different, it's interesting that thinkers thousands of years apart resorted to the same type of thought experiment, although in one case (Plato's) what is learned in the three-dimensional world benefits the two-dimensional, while in the other (the Scientific American's) what is learned in the two-dimensional world helps explain the three-dimensional.  One is concerned with the physical world, while the other is concerned with the metaphysical.

Two Dimensional Gravity

The April Scientific American has an article on two-dimensional gravity, "Quantum Gravity in Flatland."  It shows how you can simulate gravity to some extent in two dimensions by changing the topology of the two dimensions. 

For me, the main interest is in seeing how an n-1 environment can help us understand an environment with n dimensions.  We think that we live in an environment of three spacial dimensions, plus time.  What if we live in an environment of more dimensions, which we cannot sense.  We have five senses: sight, touch, hearing, smell, and taste.  Most of what we know about our universe comes from just two of those, sight and touch.  Microscopes, telescopes, machines and tools help us extend these senses to very large and very small things, but we are still basically depending on sight and touch.  What if the things we sense are only part of the universe; they might be just part of, or effects of, things which we cannot sense directly. 

That is what is interesting to me about two-dimensional simulations of a three dimensional world.  Suppose you lived in a two dimensional world where you could only see the flat shadows of three-dimensional objects.  As the sun rose and set, shadows would go from being infinitely long to small, and back to being infinitely long again.  What could you figure out about the object; would you think it changed size or shape, when only the light source was moving? 

Or what if there is some other dimension, like time, that is not a physical dimension.  Could we even conceive of what it is?  Basically what makes a dimension is our ability to measure it.  We measure time with clock.  Einstein found that the measurable dimensions are not absolute, but vary relative to each other.  Could it be that we are looking at the moving "shadows" and that there is something out there that is constant? 

String Theory

The reports of neutrinos moving faster than the speed of light and the entanglement of electrons lead to the possibility of additional dimensions. Extra dimensions could also help explain the possibility of dark matter and dark energy, which seem to be necessary to explain the makeup of the universe, although there is no physical evidence of them.  One response to this possibility is string theory. 

To see the possibility of additional dimensions, think about trying to figure out our three dimensional universe by looking at two dimensional shadows.  By looking only at the shadows, it would be very difficult to figure out exactly what the three dimensional shadows casting the shadows looked like.  A sphere, for example would cast a circular shadow, but so would a cylinder, if it was held the right way.  An egg could cast a circular shadow or an oblong shadow, and so on. 

For string theory, consider drawing a line on a sheet of paper.  If you look at the edge of the paper, you would only see a dot, where the line started, but if you could look down on the paper, you might see an infinitely long line, if the paper was infinitely long.  Or, you could draw a line that started at one point on the edge of the paper, then then went in crazy curves all over the paper, before coming back to the edge.  If you looked only at the edge, you would see only two points, one where the line started and one where it ended.  You would think these were separate points, but in fact they would be part of the same line.  This concept could explain the idea of entanglement, which says that if you know the state of one electron in a pair, you instantly know the state of the other.  In the line case, you would think you are looking at two separate points, but in fact you are looking at two ends of the same line.  Therefore, if one point was blue, for example, you would know that the other point would also be blue, because they are part of the same line.  But from looking only at the two points on the edge of the paper, it would be impossible to know what the line did between those two points, whether it went in a relatively smooth curve from one to the other, or went for miles and miles in all sorts of crazy curlicues between the two points. 

NYT Likes Carmelo Anthony

The New York Times ran a complementary article about Denver's Carmelo Anthony today. He is apparently one of the stars in a documentary of one of the first professional black basketball teams.

NY Times Math Blog

Finally a blog that it makes sense to list in the Colorado Athenian: a New York Times blog on math.

Rein in Goldman

One problem with Goldman Sach's huge earnings is that it shows that they are taking too many risks. The various articles about it say that they continue to take huge risks, although they are clearly winning. The WSJ said that they had reduced their leverage, but they still made too much money to be trading conservatively. It illustrates the need for stronger regulation. Goldman can't control itself. Arguably its traders are smart enough to beat the market on a regular basis, but unfortunately for our financial security if Goldman can trade wildly, so can everyone else, which includes whoever will replace Lehman, Merrill Lynch, etc. Goldman is still doing what Wall Street firms were doing a year ago when they almost destroyed the global financial system. They shouldn't be allowed to continue to do that. Government has to cut back their margins, leverage, portfolio size, something, so that they are not too big to fail, and also not big enough to destroy the world.

Another problem, illustrated particularly well by Matt Taibbi in Rolling Stone, is that while playing the market has been good for Goldman, it has not been good for America. He says Goldman is responsible for many of the recent financial bubbles, including the oil bubble that ran gas over $4 per gallon last year.

An additional problem is the excessive compensation. It's not good to have such huge differences in wealth. It makes America look and act like a banana republic. The rich guys -- capital -- control everything, including the government, while the poor guys -- labor -- suffer for the mistakes of the rich guys, e.g., in almost destroying the global financial system. Maybe it should be okay for individuals to trade like Goldman does with their own money, but a bank should not be able to. That would somewhat limit the number of fantastically rich people. But also a more progressive income tax would cut down the gap. It looks to me like Reagan destroyed the wonderful America that existed for about 30 years after World War II when he significantly reduced taxes on the rich. Obama's election may show how uncomfortable a slight majority of Americans has become with this situation.