Does Zeno's paradox proof the continuity of spacetime?
Zeno's paradox seems to umply that motion is not possible. The paradox is easily resolved though by pointing to time intervals that get smaller if smaller space intervals are chosen in the formulation of the paradox. Like that there is no ground to make motion impossible.
If space or time are discrete though then there will be intervals from which a particle in motion can't travel to the next interval. If the basis unit of time is a Planck-time (10exp-43(s)) then how can time be measured between these intervals? In every interval everything is motionless so what determines the transition to the next interval in which things are slightly different? How can a particle move from space interval (a Planck interval is about 10exp-33(m)) to a from this interval spatially disconconnected next interval?
Now there are physical theories that include exactly discrete spacetime structures to account for a quantized version of gravity. One of these theories is loop quantum gravity. Such a discrete spacetime shouldn't be considered though as ordinary spacetime litterally built up from almost tangent chuncks of spacetime. But discreteness is involved. This doesn't automatically mean that continuity is gone though. Maybe there is a spacetime process going on between the intervals that we can't perceive. It looks like discreteness but behind the scenes there is continuity. But spacetime is no longer diffeomorph, which is a prerequisite in general relativity.
Does Zeno's paradox prove that this has to be the case? I mean, does the fact that things can move trough spacetime prove that there is continuity on every level? Can there be processes outside 4D spacetime that determine how each new interval must look like?
If space or time are discrete though then there will be intervals from which a particle in motion can't travel to the next interval. If the basis unit of time is a Planck-time (10exp-43(s)) then how can time be measured between these intervals? In every interval everything is motionless so what determines the transition to the next interval in which things are slightly different? How can a particle move from space interval (a Planck interval is about 10exp-33(m)) to a from this interval spatially disconconnected next interval?
Now there are physical theories that include exactly discrete spacetime structures to account for a quantized version of gravity. One of these theories is loop quantum gravity. Such a discrete spacetime shouldn't be considered though as ordinary spacetime litterally built up from almost tangent chuncks of spacetime. But discreteness is involved. This doesn't automatically mean that continuity is gone though. Maybe there is a spacetime process going on between the intervals that we can't perceive. It looks like discreteness but behind the scenes there is continuity. But spacetime is no longer diffeomorph, which is a prerequisite in general relativity.
Does Zeno's paradox prove that this has to be the case? I mean, does the fact that things can move trough spacetime prove that there is continuity on every level? Can there be processes outside 4D spacetime that determine how each new interval must look like?
Comments (53)
Should have implications for Calculus, infinitesimals to be precise. What say you?
Me say: Interesting, nodding head! Me think a bit. Prishon think a lot these days. Prishon's head aches litlle. Head doesnt wanna think so much! But Prishon dont care what head says. I let him think a bit bout you comment!
Nope. Limits of measurements are physical problems.
Not if the physical space has a Natural limit of continuity.
Quoting TheMadFool
I suppose physical space might. I'm not saying it does.
But infinitesimals are rarely used as such in math that is not non-standard analysis. However, just recently I employed a step to prove a theorem of sorts in which a second order term was ignored, similar to NSA.
Zeno showed that space and objects in them appear to be both finite and infinite at the same time. This is why Kant called this an Antimony (an impasse)
Which theorem? Whats NSA?
Im not sure I understand. Space is infinitely small and of finite size at the same time? You can impose a row of small imaginary measuring poles on space (like on time), with decreasing mutual distances, and thus cut it up in ever smaller parts but its the question how far you can go. Can we infer from the fact that things can move at all that space is continuous? Or time? Loop quantum gravity (quantum loop gravity) assumes there is a minimum time interval in which time is absent. What determines that a Planck time has passed? An external clock that ticks a Planck time and then gives a sign to change?
Nonstandard analysis
If space is not infinitely divisible than there is a space that can't be divided. But that wouldn't be space. So space is infinitely dividable. So it has infinite parts. Yet it has finite length. Therefore the infinite is finite and the finite is infinite. Antimony 2 of Kant
It could be that on the micro micro micro micro micro micro scale space is not smooth anymaore. Well, smooth maybe, but foamy, so to speak. The virtual graviton can take care of this. Or the quantum Nature of spacetime itself. Pointlike gravitons cause trouble though. Thats why strings were invented. Or quantum loops of space. There is research going on to unify these two.
If you are not familiar with a certain area of mathematics it would make little sense. Has to do with the convergence of infinite compositions of parabolic linear fractional transformations (having indifferent fixed points) that converge to the identity.
Non-standard analysis.
Who says Im unfamiliar with them?
Prove you are not! :cool:
What you want me to prove? It are just transformations of the unit disc in the complx plane. A bit old already though. ?
What have tbey to do with the discreteness of space? Excuse eventual spelling. Im on phone.
Sorry if I did a bit not so nice to you! I like math but in relation to physics Im a bit fed up with it. For example, the whole Higgs mechanism is based on it while the mechanism is non-existent. ?
Yes classical space is what is illogical. Zeno was the first to prove it. Space in order to remain itself has to be divisible. Yet infinite sections means a distance has infinite parts, a finite length, and takes infinite slices of time to traverse. Which makes no sense. So a loop or something is needed to explain it and I'm sure many physicists have good ideas on this
What are parabolic linear fractional transformation? Is it something I should fear lol, jk
Yes, they've been around the block a few times. Sounds like you know what you are talking about!
:lol:
Infinite Compositions of Möbius Transformations
These are the same things as LFTs. LFTs have geometric, matrix, and analytic theories. Olde Goodies.
About as much as anything on The Philosophy Forum.
Quoting Prishon
I would think this would put you in an impossible position if you are serious about physics. :roll:
Why? Dont you think ideas come first? Math is the cause for getting the physics wrong as I explained in the context of the Higgs dield an the Higgs mechanism.
And that is how much?
I thought calculus was about infinitesimals - a controversial concept no doubt but if memory serves, two mathematicians defined it so that it ceased to be an issue.
Infinitesimals are funny things. What about velocity, dx/dt (is there mathJax here?)? You think its a real physical quantity?
Quoting Prishon
:rofl:
Was his penis that small?...Just joking
? Hi there! Iiiiim back! How are they treated in NSA? I had some rest and have no more headache.
Finally, someone on tpf who speaks my language.
Quoting Prishon
I don't see how you derive this conclusion.
Quoting Prishon
I believe that this is the proper conclusion, and what it indicates is that the conception of 4D spacetime is inadequate. What is required is a proper analysis which separates space from time, allowing one to be discrete, and the other continuous. So for example, "processes outside 4D spacetime" implies time outside of spactime, because processes require time. Such processes would be non-spatial, because the concept of "spacetime" is space based, tying time to space. Therefore we need to release time from space, making it the 0 dimension instead of dimension 4, properly non-spatial, allowing for a continuous time, complete with non-spatial processes, along with a discrete space.
Well, if space is not continuous, arent there gaps to stop the motio? Of course discrete space is not constructed by gluing together planck sized chuncks. Its more complicated.
That doesn't resolve Zeno's paradox. There is more to Zeno's paradox than the oft stated claim that it would take an infinite amount of time to traverse an infinite number of points.
Consider the notion of counting every [math]{1}\over{2^n}[/math] between 0 and 1 in ascending order. Simply saying that if it takes [math]n[/math] seconds to count from [math]0[/math] to [math]0.5[/math], [math]{n}\over{2}[/math] seconds to count from [math]0[/math] to [math]0.25[/math], etc. and using the convergent series to show that the sum is finite doesn't show that it's possible to perform such a count.
There's the far more practical problem of where such a count starts. There is no first [math]{1}\over{2^n}[/math] to count after 0, and so you can't even start counting. There is no first [math]{1\over{2^n}}m[/math] point to move to, and so you can't even start moving.
A solution is that motion isn't continuous; it's discrete. There is some smallest unit of movement, e.g. the Planck length, and that such movement doesn't involve passing through some halfway point.
Zeno had four paradoxes and he needed all of them for the reason you suggest. He assumed that time is either discrete or continuous and time likewise. He considered all combinations - four possibilities in all. Those pre-Socratics didn't have calculus or anything but speculative theory of matter: but they were no slouches with logic.
I have no idea; all I know is infinitesimals are like near death experiences: deadish but not quite dead, if you know what I mean.
Yes. Newton and consortes did indeed introduce near-death or even death experiences, introducing them! Those unlucky kids at high school having to absorb them. Good if they wanna commit suicide. Thats the sunny side maybe.
An oxymoron for a classicist
Here's a good intro to the subject: Nonstandard Analysis
Quoting Metaphysician Undercover
I shudder when I say this, but there might be something to this idea. Just a feeling, since the two are so different.
Can you word your feeling? You feel it is right? Are not space and time separated but living together?
Why? This leads nowhere, nor does it prove anything. It does, however, resemble something I looked into a few months ago concerning convergence of infinite compositions in the complex plane.
But I don't think Zeno's paradox should go beyond the problems it may cause to some of our intuitions about space, i.e. real life affairs.
Because it's comparable to passing every [math]{1\over{2^n}}m[/math] point between [math]0m[/math] and [math]1m[/math] in order. For the same reason that the count is impossible, so too is the movement. The impossibility has nothing to do with the length of time it would take and so isn't solved by referencing a convergent series of time intervals.
Gaps do not necessarily stop motion. that would only be the case if motion is continuous. Doesn't quantum mechanics demonstrate that it is probably the case that the motion of fundamental particles in not continuous. And if the motion of fundamental particles is not continuous, why not consider that the motion of any body is not continuous. That motion is continuous was simply an assumption of convenience. Then the required mathematics was produced to support that assumption.
Quoting jgill
It wouldn't be the first time we agreed on something, even though the two of us are so different.
Quoting Gregory
Space is a concept, developed from studying the properties of bodies. It is not a container, but has been deemed as a necessary condition for motion, as a body needs a place, space, to move to. Time is not the effect of motion, but it is also a required condition for motion. Traditionally, space was conceived of as static, as an object and its properties were something static. But Aristotle demonstrated the need to allow for change, and motion if our conceptions are to be real representations. This produced the need to integrate the two distinct conceptions, space, and time, as the two necessary conditions for motion.
Only if they're infinite. Math does need a better explanation of this imo. Saying infinite steps has a finite sumation doesn't answer the paradox
Quoting Metaphysician Undercover
Which mathematics demonstrate space can be discrete? Isn't this contrary to the very definition of space? As I said a loop of some kind is a better idea
Quoting Metaphysician Undercover
Aristotle didn't believe in space or time, just forms. Space is a physical container and humans use the concept of time to understand how relativity works within space. Aristotle was right actually in that space and time are both phantoms but modern physics doesn't work with these absolute ideas anymore
My intuition wants to say there is no real passage of time, and that this all occurs in the same space (or lack thereof, as it were) at once.
Ultimately, I think we can take bites out of truth and be led where we may, but ultimate knowledge is just out of our league.
That's my point as well. Time is a mystical concept that is helpful in physics but there is really no stuff called time. Physics deals with stuff. Time and space, understood in an absolute sense, are a kind of Platonic heaven, designed to help people see this world as a Platonic place. That's philosophy though, not empirical thought
Here's something I've looked into on numerous occasions that bears some resemblance to this topic. Instead of dealing with 0 to 1 or 1 to 0, this is a sequence that goes from n to 1 where n is unbounded.
[math]{{F}_{n}}(z)={{F}_{n-1}}({{f}_{n}}(z)),\text{ }{{F}_{1}}(z)={{f}_{1}}(z)[/math], [math]{{f}_{n}}(z)\to f(z)[/math]
The question is does [math]{{F}_{n}}({{z}_{0}})\to L[/math] as [math]n\to \infty [/math] ?
So it might seem that n being unbounded raises a similar issue of where to start since backward recursion is involved? But an example of where this appears in math literature is in the analytic theory of continued fractions, and it is quite solvable.
Agree completely. Probabilistic notions maybe especially...contrived formulas for Platonic, as you put it, "realities" that are neither here nor there.
It's all rigidly fixed academia, though.
I think humans can't understand the world at all unless something remains a mystery. Once I think I understand everything suddenly nothing makes sense. It's assumed we know what material existence is so we posit other realities. But Heidegger asked, "do we really know what 'to be' means?" At such a point one forgets about other realities and does science, but Platonic ideas always creep in nonetheless
I said the mathematics supports the assumption of continuity. "That motion is continuous was simply an assumption of convenience. Then the required mathematics was produced to support that assumption.
Quoting Gregory
I conclude that you haven't read Aristotle's "Physics".
I have read the Physics. There is no middle ground between absolute time and space on one hand and relational theory. Aristotle rejected the former, calling it a void, and so falls in the other camp
And youre not being clear about continuity and discreteness. Space can't be discrete. Space necessarily has parts. You say mathematics backed up motion being continuous and yet this was exactly Zeno's point.
You said "Aristotle didn't believe in space or time", though Bk.4 of his "Physics" indicates that he believed in both "place" and "time". Though he rejected the prevailing conception of "void", this does not mean that he did not believe in "space", because he replaced "void" with the more comprehensive and practical "place". And, he stated that "time" has two distinct senses, primarily it is a measurement, and secondarily it is the thing measured. In modern usage this separation is not maintained and equivocation is the result. When pressed for an explanation, most people simply deny the second, 'there is no such thing as time', as something which is being measured. You can see this in Einstein's famous quote where he states that time is a persistent illusion.
Quoting Gregory
I don't see what you're objecting to. If space necessarily has parts, then we must conclude that it is discrete, as each part is a distinct and therefore discrete entity. If space were continuous, then it would have no parts, as being partitioned means that it is divided, therefore necessarily not continuous.
Einstein did not deny that place and time exist in the Aristotelian sense. People who believe in the universe believe in this, but it is a relational theory and in Aristotle it was the quintessence instead of spacetime that provided the means for parts to talk to each other in the language of space and time. Your attempt to find a middle ground between absolute plenum and relational theory doesnt work as I already pointed out. Aristotle believed in relationship theory but God(s) held the relations together through the 5th element
Quoting Metaphysician Undercover
No because each part of space has parts which have parts which have parts which have parts... to infinity.
Quoting Metaphysician Undercover
You have discrete and continuous mixed up. Discrete is pointsize. Continuous is infinitely divisible, which even Aristotle said was the case. Discrete space doesn't exist. The question is how to understand infinite divisibility because it leads to problems as Zeno showed