The post below generated a lively controversy (see comments).
So now we serve up Zeno deluxe:
Consider a measuring stick. We have the segment from points 0 to 1 -- [0,1] -- which is a finite distance. Nevertheless, we can map the entire set of positive integers plus 0 onto this interval.
Now suppose we have an infinitesimal spaceship able to traverse each infinitesimal distance between n and n+1 at some infinitesimal speed. The ship would move at a finite speed from an infinitesimal perspective but that speed would be infinitely slow from our perspective. That is, an infinitesimal observer at point 1 would have to wait forever for the infinitesimal ship to cover the distance of 1 unit.
Now suppose we argue that Earth now is an infinitesimal object located at point 1. In that case, there could be a 0 point that is infinitely far in the past from our perspective but only finitely far in the past from the perspective of a "higher space."
Kryptograff contains Paul Conant's thoughts on scientific and mathematical matters. Conant is a journalist who holds no scientific degrees. This blog was set up after problems at the previous address: http://kryptograff.blogspot.com. Please check there for previous posts.
Thursday, March 6, 2008
Monday, March 3, 2008
Zeno was right
It's only in the last few decades that the Big Bang theory has been accepted and the cosmos dated as being on the order of 1010 years old.
Prior to the acceptance of that theory, many scientists held that the cosmos had been around forever, though perhaps the problem of entropy might raise a question about that idea.
But now if we measure time in terms of earth years, and consider that each year can be represented as a member of the set N of whole numbers, and assume that time flows in one direction, then we are left with the paradox that we can't be here. That is, infinity never occurs as a compilation of finite steps.
The only way that we can posit an eternal past is to use our present time as origin and extrapolate backward. But, that scenario doesn't account for the usual idea of our current situation as dictated by the convergence of sequences of mechanical causes. If, say, we think of the earth as it is at 1:55 p.m. EST, March 3, 2008, as represented by the intersection of two (or any number) of waves at that time, then those waves, if we assume they move forward in time, can never have arrived in order to cross. [I'm using the wave analogy as shorthand for sets of causes and effects.]
Similarly, quantum theory vindicates Zeno.
We measure the gravitational potential energy of a swing raised to height y as mgy. But, energy is quantized. This means that we cannot raise the swing to any height between 0 and y, but only to heights which incorporate Planck's constant h. That is, there is not a continuous and infinite number of points to which the swing can be raised but a finite number of points (in the trillions, of course).
So what happens to the swing when it passes from point yn to yn+1? I suppose you would say it makes a quantum jump. It exists momentarily at height yn and would not be defined as moving during this brief time interval. Yet it doesn't move in a classical sense when it rises to yn+1 because it can't exist at a height yn < ym < yn+1.
The puzzle is compounded by the fact that the swing is composed of trillions of quantum particles each subject to quantum uncertainty. I suppose we could take the center of mass of the swing to be what is measured. That would rise only to specified heights and that would be construed as "quantum jumping."
Prior to the acceptance of that theory, many scientists held that the cosmos had been around forever, though perhaps the problem of entropy might raise a question about that idea.
But now if we measure time in terms of earth years, and consider that each year can be represented as a member of the set N of whole numbers, and assume that time flows in one direction, then we are left with the paradox that we can't be here. That is, infinity never occurs as a compilation of finite steps.
The only way that we can posit an eternal past is to use our present time as origin and extrapolate backward. But, that scenario doesn't account for the usual idea of our current situation as dictated by the convergence of sequences of mechanical causes. If, say, we think of the earth as it is at 1:55 p.m. EST, March 3, 2008, as represented by the intersection of two (or any number) of waves at that time, then those waves, if we assume they move forward in time, can never have arrived in order to cross. [I'm using the wave analogy as shorthand for sets of causes and effects.]
Similarly, quantum theory vindicates Zeno.
We measure the gravitational potential energy of a swing raised to height y as mgy. But, energy is quantized. This means that we cannot raise the swing to any height between 0 and y, but only to heights which incorporate Planck's constant h. That is, there is not a continuous and infinite number of points to which the swing can be raised but a finite number of points (in the trillions, of course).
So what happens to the swing when it passes from point yn to yn+1? I suppose you would say it makes a quantum jump. It exists momentarily at height yn and would not be defined as moving during this brief time interval. Yet it doesn't move in a classical sense when it rises to yn+1 because it can't exist at a height yn < ym < yn+1.
The puzzle is compounded by the fact that the swing is composed of trillions of quantum particles each subject to quantum uncertainty. I suppose we could take the center of mass of the swing to be what is measured. That would rise only to specified heights and that would be construed as "quantum jumping."