Does Zeno's paradox proof the continuity of spacetime?

Prishon

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?

TheMadFool

Planck-time (10exp-43(s)) — Prishon

Should have implications for Calculus, infinitesimals to be precise. What say you?

Prishon

What say you? — TheMadFool

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!

jgill

Should have implications for Calculus, infinitesimals to be precise. What say you? — TheMadFool

Nope. Limits of measurements are physical problems.

Prishon

Nope. Limits of measurements are physical problems. — jgill

Not if the physical space has a Natural limit of continuity.

jgill

Not if the physical space has a Natural limit of continuity. — Prishon

infinitesimals to be precise — 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.

Gregory

↪Prishon

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)

Prishon

However, just recently I employed a step to prove a theorem of sorts in which a second order term was ignored, similar to NSA. — jgill

Which theorem? Whats NSA?

Prishon

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) — Gregory

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?

Gregory

↪Prishon

Nonstandard analysis

Gregory

↪Prishon

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

Prishon

If space is not infinitely divisible than there is a space that can't be divided — Gregory

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.

jgill

Which theorem? Whats NSA? — Prishon

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.

Prishon

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. — jgill

Who says Im unfamiliar with them?

jgill

Who says Im unfamiliar with them? — Prishon

Prove you are not! :cool:

Prishon

↪jgill

What you want me to prove? It are just transformations of the unit disc in the complx plane. A bit old already though. ☺

Prishon

↪jgill

What have tbey to do with the discreteness of space? Excuse eventual spelling. Im on phone.

Prishon

↪jgill

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. ☺

Gregory

↪Prishon

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

Gregory

↪jgill

What are parabolic linear fractional transformation? Is it something I should fear lol, jk

jgill

It are just transformations of the unit disc in the complx plane. A bit old already though. ☺ — Prishon

Yes, they've been around the block a few times. Sounds like you know what you are talking about!
:lol:

jgill

What are parabolic linear fractional transformation? Is it something I should fear lol, jk — Gregory

Infinite Compositions of Möbius Transformations

These are the same things as LFTs. LFTs have geometric, matrix, and analytic theories. Olde Goodies.

jgill

What have they to do with the discreteness of space? — Prishon

About as much as anything on The Philosophy Forum.

I like math but in relation to physics Im a bit fed up with it — Prishon

I would think this would put you in an impossible position if you are serious about physics. :roll:

Prishon

I would think this would put you in an impossible position if you are serious about physics. :roll: — jgill

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.

Prishon

About as much as anything on The Philosophy Forum — jgill

And that is how much?

TheMadFool

infinitesimals to be precise
— 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. — jgill

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.

Prishon

↪TheMadFool

Infinitesimals are funny things. What about velocity, dx/dt (is there mathJax here?)? You think its a real physical quantity?

jgill

Infinitesimals are the subject of modern non-standard analysis, and were one of Leibniz's playthings. Occasionally a calculus course is taught from that perspective, but most calculus is taught from the limit point of view developed by Cauchy and Weierstrass.

About as much as anything on The Philosophy Forum — jgill

And that is how much? — Prishon

:rofl:

Prishon

Infinitesimals are the subject of modern non-standard analysis, and were one of Leibniz's playthings. — jgill

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.

Metaphysician Undercover

Math is the cause for getting the physics wrong... — Prishon

Finally, someone on tpf who speaks my language.

I mean, does the fact that things can move trough spacetime prove that there is continuity on every level? — Prishon

I don't see how you derive this conclusion.

Can there be processes outside 4D spacetime that determine how each new interval must look like? — 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.

Prishon

I don't see how you derive this conclusion — Metaphysician Undercover

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.

Does Zeno's paradox proof the continuity of spacetime?

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