Reflections, Reflections

I have been fascinated with the Time question for a few days ever since I stumbled on it. I haven't had hardly any time to read about this subject, but I did do a little search on what other authors have contemplated about time. The most interesting I found in other writings so far, which I had not thought about, was about our own perception of time. Without memory, we would have no sense that time that has passed. Is that completely true? Haven't given too much thought of how it could be a wrong statement, but in important ways, it's true. But even before finding a text about how humans perceive time, and how memory plays a fundamental part in this perception ability, I did think the following, still related to my Zeno paradox reflections or Alessandra's Grand Time Paradox: time does not move (explained in my previous post linked above). :-)

My thinking path first passed by this wonderfully ingenious extension of the arrow paradox:

Even if the arrow moves it cannot hit any point!

The probability of the arrow landing in a certain region of the target is equal to the ratio of that region's area to the total target area. As the region gets smaller the probability gets less, and there is zero probability of the arrow's point landing on any particular point of the target. Then since this is true for all points the arrow cannot hit the target.

That is lovely! Not only an arrow in flight does not move, it can never hit a target!
Sweet.

Then I came upon Lynds, who recently said the answer to the arrow paradox is that the position of the arrow can never be determined, since we are talking about infinite divisions of time and the corresponding location of the arrow in flight.

In thinking about Lynds' solution, I figured he was wrong.

Compare two arrows, one blue that covers a certain distance over 10 seconds, one red stays at the same place for the same 10 seconds. The blue arrow covers a distance of exactly 10 arrow lengths. What is different between the two arrows? The position of the blue arrow was constantly changing over the 10 seconds, while the red arrow’s position is the same. For motion to happen, there needs to be time (the passing of time). There is no motion without time. What Zeno did was to eliminate the passing of time, and he simply took a snapshot of an instant of those 10 seconds, not the entire 10 seconds. If you freeze time, there is no motion. It is exactly what we do with a photographic camera. Imagine the arrow flying in front of you and you snap a picture. Then you show the picture to someone and say, "“See, the arrow wasn'’t flying, it was static all the time.”" The problem is with the "“all the time"” that was added.

When did the blue arrow move? Over the entire 10 seconds. It doesn’'t matter how much you subdivide the 10 seconds, the arrow is always moving constantly, to the infinitesimal subdivision, and its position is therefore proportional and we can calculate it (considering an ideal context, where it’s just a mathematical problem, and you aren'’t dealing with air, mass, wind, resistance, etc.). So where was the arrow at 5 seconds? At the middle of the distance. Subdividing the distance doesn’'t make the arrow stop, it's just a finer measurement. Like the Achilles race paradox.

Therefore my conclusion is diametrically opposite to Lynds'. You can precisely determine where the arrow is at every instant, to infinity. I don't know enough math to be able to say for certain if our mathematics allows us calculate the position when the time division hits infinity (I don't know infinity math), but I would think that is a mathematical shortcoming problem, not a logic of how to solve the paradox problem.

Mr. Efthimios Harokopos, a Greek scientist and researcher of Zeno’s paradoxes, claims there is a hidden contradiction in the argument made by Lynds, which leads, in the best case, in a fallacious argument.

Mr. Harokopos said: "Lynds states that without a continuous and chronological progression through definite indivisible instants in time over an extended interval of time, there can be no time progression (physical or flow of time). But just earlier in the paper, Lynds argues that by nature, a precise instant in time does not have a duration so neither a progression of instants in time and motion in such a progression would be a perpetuation of static frozen motion and as such continuity is impossible. Obviously these two statements are contradictory and Lynds attempts to answer Zeno's paradoxes by introducing another paradox (or better to say contradiction). The absence of a precise static instant in time underlying a physical process could simply mean that time is continuous and all physical magnitudes are continuous as postulated in classical mechanics. In an attempt to deny such a straightforward approach and introduce the concept of indeterminacy for continuity or, as stated in the title of the paper, indeterminacy versus discontinuity, Lynds commits a contradiction and possibly a straw man fallacy. On one hand Lynds claims precise static instants in time do not exist and on the other he asserts that a chronological progression through definite instants of time is required for the flow of time. One can easily see that Lynds' thesis leads to a contradiction in the case of a particle of constant mass M moving with a constant velocity V, which has a precise momentum vector equal to P= MV, at every instant of time or interval. According to Lynds, certainty is traded off for continuity and this applies to all physical values. But his conclusion does not apply in the example I just gave, as no trade off is required for a body with a constant velocity to be moving continuously. Therefore Lynds has not discovered a law of physics and if such trade-off is present under some conditions it does not relate to motion and to Zeno's paradoxes. More importantly, Lynds' conclusions are not falsifiable by observation and do not provide a predictive capacity. In this light, his claims are of a metaphysical nature and in the best case possible such claims relate to some hypothetical mechanism of physical reality, which, even if present, is not required in making calculations and predictions."

In summary, Harokopos argues that the concept of indeterminacy versus discontinuity is of no real value in resolving Zeno’s paradoxes. According to Harokopos, the paradoxes are not about the non-existence of precise static instants in time and precise physical values but about the impossibility of motion in general in a continuous or discrete space-time.

Curiously, this is exactly what I joked about in my Grand Time Paradox:
"A precise instant in time does not have a duration so neither a progression of instants in time and motion in such a progression would be a perpetuation of static frozen motion and as such continuity is impossible."

The difference is I consider the above a paradox and not reality.

Additionally, Eric Engle wrote:

Lynds fails to consider other possibilities than that motion be continuous and time discontinuous. What if motion were also discontinuous? Then Zeno's arrow could occupy locus l1 at time t1 and locus l3 at t2 without ever transiting locus l2. If motion were a series of very tiny (even infinitely tiny?) "jumps" (teleportations if you will) then Lynds' reductio fails. This "teleportation" model does in fact appear to reflect sub-atomic physics where, as I understand, particles mysteriously appear and disappear as if teleported. If we presume motion is in fact discontinuous then we are in no way compelled to admit Lynds' argument by reductio, that time cannot be divided into discrete elements.

I can't comment on this potential teleportation phenomenon. I don't think neither motion nor time are discontinuous. In addition to being continuous (which is how time seems to me), nothing could happen if there was no time.

Time is what allows for different moments, so, before even matter can exist, there is time.

That leads to: time is primordially necessary for life to exist. I had never thought about that before.

Time, fascinating.



related links:
Michael Taber's similar reflections. My own next post on this subject. On the math side: Infinity is for Children---and Mathematicians!

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