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."
13 comments:
I don't see why an infinite past precludes causality. You mention that two waves could never meet, but why? From every point in the past there is a finite amount of time to the present, so they can meet without a problem. Perhaps you are bothered that there is no "ultimate cause." That is, every chain of causation can be traced back indefinitely far into the past, but this is certainly logically possible. Indeed there are universes which are solutions to the Einstein general relativity equations where time extends infinitely far into the past and where causality operates. I agree that thermodynamics poses an issue for an infinitely old universe.
Time can extend infinitely far into the past if we use "now" as origin. But if the origin is infinitely far into the past, then the waves never arrive. I suppose we could surmise that the origin on the time axis is coupled with some ultimate cause.
I a universe where time extends indefinitely far into the past, there is no "origin" at the beginning. The universe simply exists. Nothing is postulated to have "caused" it. If you feel that's unsatisfying you can always say that a superior being created it, the whole infinitely long shebang, outside of the universe's own temporal framework.
Here is a model situation. Consider the universe to be the real number line (spatially). Time will also be paramterized by the real numbers. (So spacetime is just a plane.) Now we can have a wave be at position x=-t at time t and another wave at position x=t. They meet at time t=0. I don't see any paradox with this scenario.
The issue is the unidirectionalism of time. If time is indeed unidirectional -- and disregarding entropy concerns -- then "it" had to have proceeded year by year before our now. But the limit is never reached.
Think of someone with the potential to live forever born at some time t. He/she never traverses an infinite time, but only a finite time. So if he/she was born infinitely long ago, he/she never attains our "now."
In my example, where the waves are at position t and -t, time is unidirectional. There is no issue with them needing an infinite amount of time to reach t=0 because they did not exist infinitely long ago. This is a subtle distinction from "existing indefinitely far into the past." The waves exist indefinitely far into the past, but they did not exist infinitely long ago, just as there are real numbers which are indefinitely far away from the origin, but there is no real number which is infinitely far away.
Indefinitely means: "for any t, if t, then t-1 or t+1" (Assuming t is in years.)
But any t, with respect to our "now," is a finite time-distance away.
Now I guess you could use the ZF infinity axiom to assert that an eternity up to now could be said to exist. But, I think we're still in trouble when it comes to the alleged "arrow of time" (which of course doesn't exist in classical physics equations but which seems from that standpoint to be basically a function of entropy or probabilities).
"But, I think we're still in trouble when it comes to the alleged "arrow of time" (which of course doesn't exist in classical physics equations but which seems from that standpoint to be basically a function of entropy or probabilities)."
Okay, but that is a completely different issue than saying there is a contradiction because things take an infinite amount of time to do things. The issue of whether there could be constantly increasing entropy in a universe which has always existed may be a problem. I don't know enough physics to say definitively, but it's plausible.
I was acknowledging that the "arrow of time" is a tricky issue, and that entropy is part of that issue. But, I am not using entropy as part of my argument. I simply say that if time is unidirectional, then "it" must "flow" from the past to the present. But "it" is measured in terms of cycles around the sun. Now, if we postulate, for the moment, an eternally ancient sun, that sun shouldn't exist because only a finite number of cycles can have occurred -- if time only moves "forward."
Clearly the sun is not infinitely old. In a universe extending indefinitely far back in time, there's no reason the sun has to also be indefinitely old. In fact in our own universe the sun is younger than the age of the universe.
This is reminiscent of the Bohr-Einstein debate.
Of course the sun isn't infinitely old. I was trying to make a point.
Here's another way to think of the argument:
If we say that a spaceship in Euclidean flat space is sent out at the speed C on a straight line (ignoring for now gravitational effects) from Earth and assume the cosmos is infinitely roomy, then those of us on Earth know that the ship will never reach infinite mileage.
Now imagine an observer Al at the other end of the infinite line. (This can only be done if we fiddle with infinity via logic and set theory axioms, perhaps.)
Al's ship, launched along that line, will never reach Earth.
If we regard ourselves on Earth as infinitely far in Al's past, then we will never reach Al.
Similarly, a universe that is infinitely old presumably contains a set of finitely timed causes and effects. So the chain reactions from that set never reach us. Hence, we shouldn't even exist in an infinitely old cosmos where time flows unidirectionally.
If there were a point at t=-infinty, perhaps your argument would carry some force, but saying the universe is infinitely old is not the same as saying there is some time t=-infinity at which the universe existed.
I already gave an example of an infinitely old universe with a wave at x=t and x=-t. Clearly the waves intersect at time t=0. Yet there is no time at which the time of meeting is infinitely far in the future, so your argument does not apply.
So I suppose you mean that if you choose some finite point measured as -t, then from that point, time can flow forward. If so, then we can say the same for -(t+1). So there is always some finite point with respect to us from which time flows forward.
So, as you say, there is no point -t = infinity, and hence there is no point at which time t started flowing forward. Time never began to flow.
However, I am prepared to grant that it can be argued that time didn't need to begin to flow.
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