How Does Zeno's Paradox Influence Quantum Gravity and Modern Science?

[Chronos]

Here is an entertaining [albeit pretty tough in places] read. It's kind of wide ranging, flitting between logic, physics and the interleaved mathematical formalisms. I found it particularly interesting from a quantum gravity perspective. It certainly serves as a reminder to be careful before dismissing a counterintuitive result. Some paradoxes have teeth:

http://www.arxiv.org/abs/physics/0505042
Zeno meets modern science

 

[marcus]

it has a rhyming translation from the Russian poet Pushkin.

A bearded sage once said that there’s no motion.
His silent colleague simply strolled before him,--
How could he answer better?! -- all adored him!
And praised his wise reply with great devotion.
But men, this is enchanting! -- let me interject,
For me, another grand occurrence comes to play:
The sun rotates around us every single day,
And yet, the headstrong Galileo was correct.

http://spintongues.vladivostok.com/pooshkin3.htm

it has an historical essay on the "Stone-von-Neumann" theorem by Rosenberg

http://www.math.umd.edu/~jmr/StoneVNart.pdf

the last paragraph is inspiring (I omit the quote from Lopez):

"The main conclusion of this paper is that physics is beautiful. Questions aroused two and half millennium ago and scrutinized many times are still not exhausted. Zeno’s paradoxes deal with fundamental aspects of reality like localization, motion, space and time. New and unexpected facets of these notions come into sight from time to time and every century finds it worthwhile to return to Zeno over and over. The process of approaching to the ultimate resolution of Zeno’s paradoxes seems endless and our understanding of the surrounding world is still incomplete and fragmentary."

 

[mccrone]

A good paper indeed. But a pity that he dismisses Aristotle's approach so quickly without offering any real justification -

"In the case of Dichotomy it is even not clear how the movement can begin at all because there is no first step to be taken. Aristotle tried to resolve this situation by distinguishing potential and actual infinities: “To the question whether it is possible to pass through an infinite number of units either of time or of distance we must reply that in a sense it is and in a sense it is not. If the units are actual, it is not possible; if they are potential, it is possible”.

Zeno's paradoxes rely on limit states (such as discrete~continuous, or change~stasis) to be taken as actual states. Yet we know from modern physics that most physical limits can only be approach asymptotically. For instance, the Planck scale, the speed of light (if you are a mass), or absolute rest.

Cheers - John McCrone.

 

[Berislav]

Very interesting find. But, IMO, it's a bit of a stretch to equate Zeno's idea with an infinite number of measurments on a quantum system.

 

[setAI]

Zeno's mathematical concepts were made obsolete by the calculus- as it showed the computability of infinite terms in a finite time

 

[Chronos]

Agreed, but the paper raises some very interesting and relevant points. The experimental examples are fascinating. There is more to the universe than Newton's calculus.

 

[EnumaElish]

How would you refute Zeno's paradox physically (rather than mathematically)? I am thinking "as the (incremental) distance tends to zero, the time needed to cross that distance (with any nonzero speed) also tends to zero." Would this work as a purely physical argument?

 

[nightcleaner]

How would you refute Zeno's paradox physically (rather than mathematically)? I am thinking "as the (incremental) distance tends to zero, the time needed to cross that distance (with any nonzero speed) also tends to zero." Would this work as a purely physical argument?

Seems to me not a solution, since it would imply instantaneous transmissions, and ignore the speed of light.

However, the solution I would offer to the paradox is that the incremental distance cannot go to zero, but must be taken in some sort of fundamental chunk, so that the appearance of motion can resume. This solution, I think, is in line with Planck's action potential unit, h or h-bar.

nc

 

[EnumaElish]

Seems to me not a solution, since it would imply instantaneous transmissions, and ignore the speed of light.

Just the opposite. As long as the runner keeps a nonzero sublight average speed, the time will go to zero as distance does. Velocity at that point is 0/0 = indeterminate.

I think this works as a thought experiment in classical physics.

I am not saying I don't agree with your quantum approach.

 

[Chronos]

EE and nc, I think you're both right. What I saw in the paper is results that defy both classical and quantum expectations when we probe sub-planckian realms. That makes sense to me because we already know something is missing that is needed to connect the dots between both theories [GR and QT].

 

[NeutronStar]

Zeno's mathematical concepts were made obsolete by the calculus- as it showed the computability of infinite terms in a finite time

This is a huge misconception in the mathematical community. Calculus does absolutely nothing to address Zeno's concepts.

It is grossly incorrect to claim that calculus showed the computability of infinite terms in a finite time. If you actually study calculus and understand the rules all you are really doing is proving convergence, nothing more.

Calculus does not claim to complete an infinite number of operations in a finite way. To do so would be to do nothing other than claim that infinity is finite. Calculus does not do that and it does not address Zeno's concerns.

This is just a gross misunderstanding of the meaning and definitions of the calculus limit. This misunderstanding of the calculus limit is far too widespread and it's a real shame when any professional mathematicians support this idea, yet far too many of them do!

 

[Locrian]

Why do people discuss zeno's paradox so much? I could propose any number of mathematical systems that, because they are not based on mathematics that represent real physical systems, generate erroneous predictions. Would people discuss those for years to come?

 

[setAI]

Calculus does not claim to complete an infinite number of operations in a finite way. To do so would be to do nothing other than claim that infinity is finite. Calculus does not do that and it does not address Zeno's concerns.

I would suggest that you study Calculus a bit more- and then study Cantor-

 

[NeutronStar]

I would suggest that you study Calculus a bit more- and then study Cantor-

I have studied calculus in great detail. I have also studied the work of Georg Cantor for over 30 years. I might also mention that I've found Zeno's paradoxes quite interesting since my early youth.

I am quite confident of my position. Thank you for your suggestion though.

 

[EnumaElish]

My immediate reaction is to agree with NeutronStar. Suppose one were to write down (or look at, or visualize) every element of the series \left\{\frac1x \right\}; it'd still take an eternity. Even skipping a finite number of elements in every step wouldn't help.

 

[setAI]

the 'zeno error' is that it ignores proportionality of TIME: any non-zero distance will be covered in a directly proportional non-zero time- and the supposedly “never” reached zero-distance would be traversed in zero time- thus instantaneously- and because the calculus has shown us that a sum of infinitely many terms can yield a finite result- an object always arrives when and where it should even with infinite resolution because the time adds up the same no matter how you slice it

 

[EnumaElish]

But that's not math, that's physics. How do you know it'll take zero time? Math doesn't really tell you that. "Proportionality of time" doesn't sound mathematical; it doesn't even sound theoretical; it sounds empirical or experiential. Doesn't proportionality depend on the process being contemplated in the thought experiment? I can define a discrete process which takes a constant length of time for each and every element of an infinite (yet convergent) series, and we would never get "there."

 

[NeutronStar]

the 'zeno error' is that it ignores proportionality of TIME: any non-zero distance will be covered in a directly proportional non-zero time- and the supposedly “never” reached zero-distance would be traversed in zero time- thus instantaneously- and because the calculus has shown us that a sum of infinitely many terms can yield a finite result- an object always arrives when and where it should even with infinite resolution because the time adds up the same no matter how you slice it

This argument has been given many times over the ages. This is not a valid argument for calculus. Calculus does NOT claim that you will ever reach a "zeroth" time or space interval. On the contrary all that calculus shows it that the process continues to converge to a particular value. It really doesn't claim that it will ever get there. In fact, in come cases calculus actually states that a process can never get to the limit! Yet we still say that the limit itself exists because that's where it is converging.

Personally I think that Max Planck has gone a lot further to proving that Zeno was correct. The universe really is quantized. Max Planck showed that energy is quantized. Emmy Noether and Werner Heisenberg have both shown that energy and time have a conservation relationship. Therefore if energy is quantized, then time must also be quantized.

Albert Einstein showed time and space are a single fabric. Therefore if time is quantized then space must be quantized also.

There you have it. Zeno was right all along. So why do people continue to try to prove him wrong? He's my hero! He was right! Leave him alone.

 

[setAI]

if we are talking about the physicial universe- YES it is most likely discrete- with Planck time/area defining the 'pixels'- but the metric of space-time is emergent from relationships and is not a fundamental continuity [coming form an acceptance of background independence] so zeno was right but for the wrong reasons- it is still possible to perform infinite computations in a hypothetical finite continuous space- the fibonacci golden section shows that- any process can be fractally condensed to maintain proportionality in principle without limit-
the notion of impossible motion due to infinite time to move the distance is absurd- the time is ALWAYS proportianal

 

[EnumaElish]

Of course it is absurd for a "continuous process" (e.g. defined as "constant average speed per unit distance," I guess); but not all processes are continuous, and the definition of a continuous process is physical, not mathematical.

 

[mccrone]

The whole point about Zeno is that you get stuck in an eternal flip-flopping between discrete and continuous if you try to prove either one to be fundamental.

As with any broken symmetry, you have to go to a higher dimensionality to restore the hidden commonality.

In metaphysics, the dichotomy of discrete~continuous becomes unified at the "higher temperature" of vagueness. Or in physics, questions about the continuous and the discrete disappear into the quantum uncertainty of the Planck scale - the higher temperature that melts spacetime. In mathematics...well, mathematicians are still very confused over how to handle this. You will find recent attempts to employ fuzzy logic (Zadeh), paraconsistent logic, etc, as ways around discrete~continuous.

Zeno's paradox cannot be solved from within as its axioms are faulty (the claim that either one or other, dicrete or continuous, change or stasis, location or motion, is monadically fundamental). So you have to step back from this assertion in a way that allows the contradictory ideas of the discrete and the continuous to be reunited in a deeper symmetry.

 

[Chronos]

I think Zeno had a valid point. But it is not a point that can be solved by traditional math. To me, it suggests our concept of mathematics is fundamentally flawed. I sometimes wonder - do we need a quantum theory of mathematics?

 

[selfAdjoint]

I sometimes wonder - do we need a quantum theory of mathematics?

Combinatorics?