This is where I think "common sense" messes us up. If one thinks "of course space is flat!", well, you have to ask yourself why you think space is flat. My guess is it would be a combination of (a) our sense that we can look in straight lines in all directions if nothing is blocking the view, (b) the way we usually associate curvature with some thing, and empty space is nothing, so it's hard to imagine nothing being curved, (c) maybe we remember seeing x-y-z cartesian axes at some point, and those are straight lines, and (d) we all either inherit or intuit an aristotalean image of what "empty space" is, and it's a euclidean (flat) geometry.

The key is that all these images/concepts are rooted in our experience, and modern physics has shown that when you get into regimes outside our regular experience, common sense doesn't help us, in fact, it actually hinders us. How we think of things behaving is based on observations of conglomerations of 10²³ atoms or more; when you're dealing with just one atom, you will find it behaves quite differently than common sense would indicate.

Here's an example that's not quite as esoteric: we're talking about whether space can be curved. Part of the problem is that we generally (unless you've had a fair amount of math) don't properly generalize those words: space, curved, straight, etc. So what is "space"? I'm sure mathematicians can give you a better definition, but I think it will be sufficient to say that space is an abstract collection of points with some arbitrarily small distance between them. We say the space is n-dimensional if we need n numbers to characterize those distances. Whether space is flat or curved depends on how you can characterize that distance. In a flat space, the Pythagorean theorem (extended to n dimensions) will adequately describe the distance. If you want to define a finite extension within that space, you need to add up (i.e. integrate) the tiny distances between all the points in the space along the extension. The surface of the earth (if you ignore mountains and valleys) can be approximated as a two-dimensional curved space. What does that mean? Well, in technical terms, it means that the metric is no longer ds² = x²+y². That is, all the rules of euclidian geometry no longer apply: the interior angles of a triangle do not sum to 180°, parallel straight lines do cross, etc.

Let's take that last one: how can parallel straight lines cross? Well, what exactly is the definition of "straight"? We can all recognize a straight line when we see it (in a flat space), but what property defines it uniquely as "straight"? The definition of straight is this: pick a point on the line and take a tiny step in the direction of the line at that point. If you are still on the line, and you can do this at any point on the line, then it's straight. Parallel straight lines are thus two lines that fit this criterion, and the "tiny step" is in the same direction for both lines.

Note that in a flat space, straight parallel lines are the same distance apart everywhere, but this is not true for curved space. Back to the curved surface of the earth, we apply our definition of "straight", and we get a curved line! Specifically, we get a great circle. Any other type of line on the surface of the earth will not fit this definition. Longitude lines are straight, latitude lines are not. Parallel lines do cross each other on the surface of the earth (you might think latitude lines are parallel, but that is not so).

My point in all this long-windedness is that we are taught to think in terms of euclidean (flat) geometry, but there is absolutely no reason to think that the universe has to conform to our conceptions about it. Indeed, as I have illustrated with the surface of the Earth example, there are simple cases well within our experience where it does not conform. When moving out into realms where our common sense simply does not apply, we need to be careful that we have properly generalized our conceptions (like the straight line), and then we need to take those tiny steps forward, and test to see if we are still on the line. If we pass those tests, then we need to open our experience up to new realms and widen our definition of common sense. Once you train yourself with math and experiment, the weirdest behavior of a quantum particle will also make sense to you, and then (and only then) can you start to trust your intuition about what "seems" right. Until then you are just trying to draw straight lines on the curved surface of the earth.

Yours,

Don

<font size=-1>[ This Message was edited by: DoctorDon on 2002-08-22 13:49 ]</font>