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Zeno of Elea's Four Paradice

Doug Renselle's Quantum Comments and Solutions

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"We have here a very striking and general example of the breakdown of classical mechanics —
not merely an inaccuracy in its laws of motion, but an inadequacy of its concepts to supply us
with a description of atomic events
." Page 3 of 314 total, including index.

P. A. M. Dirac
in his
The Principles of Quantum Mechanics
OUP - 1958

Plutarch on Zeno of Elea


Zeno of Elea's Paradice

Quantum Comments1

Arrow Paradox

There is no motion since what is moving must arrive at a midpoint before arriving at an endpoint.

Henri Louis Bergson describes this as a self~absurd dialectical, formal, canonical, mathematical, mechanical "movement by immobilities." See Bergson's Time and Free Will, Creative Evolution, and An Introduction to Metaphysics. Doug - 19Mar2010.

This paradox makes a deluded classical presumption that reality is both an infinitely divisible spatial extensity and that infinitesimal reductions of that infinitely divisible extensity are demarcable by ideal 'dimensionless' classical 'points' and that a moving object can 'stop' at classically demarcated points. (Classically 'stop' means zero momentum.)

Another way to make our statement is that this, Zeno's first, paradox assumes that classical motion and time are subject to analyticity. Via that assumption, any classical motion is, due its spatial extensity, thus stoppable for purposes of analysis, and not continuous. Similarly for time.

Stoppability of continuous motion is source of this proposition's paradox.

We see here two separate contexts treated as one:

  1. stopped reality, and
    (infinitesimal, stopped time and motion)
  2. moving reality
    (Bergsonian durational time and motion).
1. contd.


In Quantonics, we choose to view Zeno's first paradox as his own derision of classicists' assumptions of both analytic motion stoppability and analytic temporal stoppability in a single context. Our chosen views derive from our studies of works of Henri Louis Bergson, William James, et al. (See links below.)

Readers should know that our choices/views run wholly counter to classical interpretations of Zeno's intent. An implication of our quantum-enhanced version of Zeno's view, assuming we are correct, is that he intuited macroscopic quantum uncertainty of any moving entity's position and momentum. To a classical mind, absent quantum thinking modes, noncommutativity of Poisson's bracket of position and momentum is a paradox. We infer that Zeno intuited this issue. Most classicists today are incapable of this level of intuition.

1 contd.


Most physicists, today, view quantum uncertainty as only a microscopic phenomenon and describe it mathematically/analytically like this:

pq - qp = [p, q] = ih

where p, q, i, and h are classical objects (see dichons) and [p, q] is Poisson's bracket AKA a classical analytic commutation relation. Most physicists presume this expression is only valid at physical atomic and subatomic 'levels' of classical reality. Zeno, in our view, and agreeing with Quantonics, is saying that Poisson's bracket, AKA quantum umcærtainty applies at all levels of quantum reality.

unQELRed text

QELRed text

Aside - 21Mar2005:

We need to clarify our use above, tentatively highlighted bold, of "Most physicists...only valid..."

Physicists show this relation many 'di' fferent ways. Often that equals sign is shown as a greater-than-equals. Latter, in our view, is more appropriate, in general. Indeed, seldom does one 'see' any experimental result where only 'one' Planck quantum appears (we do not possess 'scientific' accoutrements capable of such classical 'measurement.').

What one usually sees is a result which divided by 'N' gives a fairly high precision (not 'accuracy' due quantum uncertainty itself, of which our 'equation' above is a 'measure') Planck quantum result.

Pure classicists even disagree with (deny, decry, abhor) Planck's result. Classical-quantumists believe that our 'equation' only holds at meso-atomic, atomic, and subatomic levels of reality. They believe that quantum uncertainty "irons itself out" in reality's macrocosm. See our scaling of uncertainty at our critique of Margenau's treatise on probability and likelihood: What is Wrong with Probability as Value.

In Quantonics we believe, and believe that we have demonstrated well, that quantum uncertainty scales all reality and is quintessentially not restricted just to atomic (inorganic) scales of reality.

We shall QELR this red text in 30-60 days.


End aside.

 Aside - 21Mar2005:

Wæ nææd t¤ clarihfy ¤ur usæ ab¤ve, tæntatihvæly highlighted b¤ld, ¤f "Most physicists...only valid..."

Physicists show this relation many ¤mnihfferænt ways. Oftæn that equals sign issi sh¤wn as a græhter-than-equals. Lattær, ihn ¤ur vihew, issi m¤re appr¤priatæ, ihn gænæral. Ihndææd, seld¤m d¤æs ¤næ 'sææ' any e[pærimæntal ræsult where ¤nly '¤næ' Planck quantum appæars (wæ d¤ n¤t p¤ssæss 'scihæntihfihc' acc¤utræmænts capablæ ¤f such classical 'measurement.').

What ¤næ usuahlly sææs issi a ræsult which dihvihdæd by 'N' gihves a faihrly high precisi¤n (n¤t 'accuracy' duæ quantum umcærtainty ihtsælf, ¤f which ¤ur 'equati¤n' ab¤ve issi a '¤mniht¤r') Planck quantum ræsult.

Pure classicists even disagree with (deny, decry, abhor) Planck's result. Classical-quantumists believe that our 'equation' only holds at meso-atomic, atomic, and subatomic levels of reality. They believe that quantum uncertainty "irons itself out" in reality's macrocosm. Sææ ¤ur scaling ¤f umcærtainty at ¤ur crihtiquæ ¤f Margenau's treatihsæ ¤n pr¤babilihty amd lihkælih¤¤d: What is Wrong with Probability as Value.

Ihn Quantonics wæ bæliævæ, amd bæliævæ that wæ have dæm¤nstratæd wæll, that quantum umcærtainty scalæs ahll ræhlihty amd issi quintæssæntiahlly n¤t ræstrihcted just t¤ at¤mihc (ihn¤rganihc) scalæs ¤f ræhlihty.


End aside.

In Quantonics we quantumly/quantonically apply our hermeneutics to achieve a more quantum real:


where p, q, i, and h are quantons and our quantum BAWAM, EIMA hermeneutic for i is a recursive quantum square root. (We assume all 'mathematics' in our Quantonics notation are ~quantum. E.g., classical concept 'minus' and quantum meme 'minus' are entirely omnifferent one another.)

1 contd.


Zeno, in our view, is telling us that:

Arrow_Position: A_P and
Arrow_Momentum: A_M
form a Poisson bracket:


where, in anihmatæ quantum macroscopic reality, which really has absolute motion, we may may n¤t k-n¤w A_P and A_M classical analytic, stoppable 'states.' Also see our Boolean Logic is Distributive. Readers can infer, for macroscopic sensibilities, scaled multiples of h above. Insertion of, say N, may suffice.

1 contd.


A very easy solution to this paradox is similar to our TRUE/FALSE 'paradox' and our CHIMERA 'paradox:' create a motion context and a stoppability context. Read detail of these solutions at our SOM Connection. A graphic of these bi-contextual solutions appears here: Many Truths to You.

This approach permits solution of paradice 2 and 3.

See Bergson in his Creative Evolution topic 40 (large page with animations) and his Time and Free Will topics 15, 19, 22, 23, and 34 on Zeno's paradice.

One renowned quantum physicist has been shown by James Gleick to understand thoroughly what Doug means by quantum~ "many truthings:"

"Feynman forever remembered the confrontation between the two men, their faces eerily illuminated by the glow from the light table used to view the photographic plates. Bethe looked at one plate and said that the gas of the cloud chamber seemed to be swirling, distorting the curvatures. In the next plate, and the next, and the next, he saw different sources of potential error. Finally they came to a clean-looking photograph, and Bethe mentioned the statistical likelihood of errors. Schein said that Bethe's own formula predicted only a one-in five chance of error. Yes, Bethe replied, and we've already looked at five plates. To Feynman, looking on, it seemed like classic self-deception: a researcher believes in the result he is seeking, and he starts to overweight the favorable evidence and underweight the possible counterexamples. Schein finally said in frustration: You have a different theory for every case, while I have a single hypothesis that explains all at once. Bethe replied, Yes, and the difference is that each of my many explanations is right and your one explanation is wrong." Quoted from page 305 of Gleick's Genius, Vintage, 1993 paperback. Doug's bold and color.

Bethe makes it clear that every perspective of any portion of reality is always omniffering from observer to observer, location to location, time to time, etc. Essentially, yes quintessentially, a quantum epiphany of "many truthings" directly experienced in quantum~reality. Also see Doug's now ancient Sophism Connections.

Doug has used "Many truths" you, as his Lila Squad adieux, since late 1997. Often he has been asked why he uses that adieux. Simply it's quantum. He should have used "Many Truthings," however, since classical truths tend toward ideal Platonic social state and social common sense grammatical 'statement' Ism. Quantumly "truthings" are waves and thus must be expressed phasementally using phasementings.

Doug offers the Hans Bethe exemplar as just that: a quantum exemplar of many truthings. It is one of few Doug has found during his last 15-20 years of quantum philosophical research. So far it is a best exemplar, excepting a similar memeo of quantum~macroscopic~uncertainty, Doug has found. All truthings, due their quantum~partial, quantum~wave, and quantum~stochastic natures, are genuinely quantum~uncertain...macroscopically and otherwise.

Hope this helps you grasp quantum~gn¤stic essence of "many truthings."

Thank you for reading,

Doug - 23Jul2008.

1 contd.


Zeno's 1st paradox offers some vague yet unsubtle sensations of Bohrian complementarity. Allow us to make a similar nexus via a quotation from Max Jammer's The Philosophy of Quantum Mechanics, pp. 92-93:

"[Jammer quotes Bohr:] 'While energy and momentum

[presumed commutative parameters; quantum energy and momenta emerq n¤ncommutative Poisson brackets]

are associated with the [classical] concept of particles, and, hence, may be characterized according to the classical point of view by definite space-time coordinates, the period of vibration and wave-length refer to a plane harmonic wave train of unlimited extent

[I.e., Bohr ~orbit quantum probability distribution; 'particle-momentum' becomes arbitrarily macroscopic!]

in space and time.' [Jammer:] A connection with the ordinary mode of description, Bohr continued, can be established with the aid of the superposition principle

[to us Bohr's notion of quantum superposition appears appropriately non-classical/sophist here, i.e., EIMA]

1 contd.


since it enables us to identify

[poor word; re-introduces classical Aristotelian excluded-middle, e.g., a tautologous identity A=A]

wave packets with particles, in view of de Broglie's well-known result:


according to which the group velocity of the wave field is equal to the translational velocity of the particle associated with the field. The [Poisson bracket] association of a particle with a wave packet, Bohr pointed out, demonstrates the complementarity character of the description almost ad oculos;

[on sight, by direct observation, on inspection, eidetic, etc.]

for [Bohr:] 'the use of wave groups is necessarily accomplished by a lack of sharpness in the definition of period and wave-length, and hence also in the definition of the corresponding energy and momentum....'" Bohr's quotes are from his September, 1927 Como lecture sequel.

Our bracketed comments, italicization of thelogos, and impure ASCII in place of Greek symbols.

Having read that quote, you may be asking as we have, "How could Bohr have ever viewed quantum complementarity as 'exclusive?'" We believe that he did n¤t. His use of "exclusive" was merely a palliative for his then overwhelmingly classical peers, like Einstein. See our Absoluteness as Uncertainty. Subsequent remarks by Jammer bring back Bohr's own special brand of exclusivity. Classical languages are traps which impose exclusive dialectical predilections on their users. Beware all analytic languages! (Question: Was William James Sidis' Vendergood an analytic, dialectic language? Does anyone know? Contact us if you do know. We currently do not have access to Vendergood.)

Also, you should be able to make a very strong nexus of your own with Zeno's 1st paradox.


Achilles Tortoise Paradox

That which is running slower shall never be passed by that which is running quicker. Why? Quicker must arrive at a point where slower already departed and thus slower must always maintain a distance advantage over quicker.

See our comments above regarding paradox 1.

See Bergson's Time and Free Will, Topic 23.

See James' Some Problems of Philosophy, Chapter X., and Chapter XI. Surrounding pages in these two chapters are worthy of your further study.


Racetrack Paradox

That which is at a place cannot move at a place which it is not. That which is at a place cannot move at that place where it is. But a moving object is always at the place at which it is. Specifically, said object is, at any instant, at rest. But if said object is not moving at any instant then it is at rest.

See our comments above regarding paradox 1.

See Bergson's Creative Evolution, Topic 40 (large page with animations) on Zeno, "...movement is made of immobilities."

Ponder how this paradox's assumptions outright deny quantum reality's probability distributions of 'place' and 'motion.' You may be able to QTM relate this putative as akin Aristotle's excluded-middle. In quantum reality, 'place' and 'motion' analogues have quantum anihmatæ EIMAs.


Stadium Paradox

Allow three objects of equal dimensions called A, B, and C. A at velocity x and C at velocity -x are passing B in opposing directions. A requires t to pass B's dimension and 1/2T to pass C's dimension. Implication: A requires both T and 1/2T to traverse an identical size.

Even to classicists this is no paradox (unless we assume A, B, and C are classically dimensionless point objects). Classical velocity is dx/dt. Opposing velocities traversing an equal 'stopped' distance A require 'identical' time. Opposing velocities traversing their mutual equal 'moving' distances A each require half of A's stopped traversal time.

Here we have at least three separate velocity contexts, and even classicists treat them as separate subcontexts (stopped, +dx/dt, and -dx/dt) within a larger framework context (which classically assumes common, homogeneous, independent classical 'time'.)

If time were indeed heterogeneous and different (see omnifference) for each 'velocity' context, but treated as homogeneous then we, from any classical conspective, would have a temporal paradox. An example might be time as stable/stopped for B, future-istic for A, and past-istic for C. Results in this case would be paradoxical when 'analyzed' in one global temporal context.

Again, we see ughly old SOM rearing its arrogant head. As we have said over and over and over, "SOM is
incorrect and use of SOM in our thing-king makes most of our work products questionable as to their general utility."

Recall that we have said SOM assumes: One Global Context (OGC), One Independent Time, and Ideal Classical State.

All of those assumptions deny quantum reality!

We think Zeno was trying to show us SOM's absurdities about 2450 years ago, but no one, including Aristotle could
understand him. Why couldn't Aristotle understand? Aristotle is a SOMite. He wears SOM's CTM blinders!

Once we understand Quantonics' n¤nclassical perspectives (QTMs) of quantum reality, we understand that:
reality is n¤t analytic, reality is n¤t classically state-ic (i.e., analytically stoppable), reality is abs¤lute quantum flux.

Essentially, we can use those three statements to answer Zeno's paradice:

Paradox 1 - Reality is n¤t classically state-ic.
Paradox 2 - Reality is n¤t classically analytic.
Paradox 3 - Reality is abs¤lute quantum flux.
Paradox 4 - Reality is n¤t classically analytic.

In addition, our Quantonics heuristic and hermeneutic conjecture is that all
classical paradice are soluble via application of multiple con(m)texts.

Thank you for reading,

(Subsequent rev's. 19Nov2002 through 19Mar2010 - Doug.) (1 of 2)


Note 1 - For alternate, more classical perspectives see: Ancient Greek Philosophy - From Thales To Aristotle edited by S. Marc Cohen et al., The Cambridge Dictionary of Philosophy edited by Robert Audi, The Oxford Dictionary of Philosophy by Simon Blackburn, etc. Also, use Google to search for: "Zeno of Elea".


To contact Quantonics write to or call:

Doug Renselle
Quantonics, Inc.
Suite 18 # 368 1950 East Greyhound Pass
Carmel, INdiana 46033-7730

©Quantonics, Inc., 2002-2014 Rev. 19Mar2010  PDR Created 16Nov2002  PDR (2 of 2)
(19Nov2002 rev - Minor grammar alteration in red text in second comment para. of paradox 1.)
(10Jan2003 rev - Update and extend paradox 1 comments.)
(21Jan2003 rev - Repair commutation notation for actual representation of commutation of p and q.)
(21Jan2003 rev - Link Poisson Bracket to Quanton as Probability.)
(4Feb2003 rev - Add Paradox 1 comments link from 'noncommutativity' to our quantum-remediated 'commutative.')
(14Mar2003 rev - Add page top Dirac quote and paradox 1 red text comments.)
(21Jun2003 rev - Add Max Jammer quotes and his Bohr quotes under paradox 1.)
(14Jul2003 rev - Spelling correction.)
(27Sep2003 rev - Repair quantum square root link under paradox 1.)
(10Dec2003 rev - Substitute higher quality gifs for h-bar. Reformat 1-4 solutions just above for alignment.)
(13Apr2004 rev - Add missing spaces around an h-bar gif.)
(14Jul2004 rev - Add dox 1 link to discussion of macroscopic uncertainty; revert h-bar GIFs to MT Extra font.)
(21Mar2005 rev - Extend our comments on Poisson's brackets, under 1st paradox.)
(2Jun2005 rev - QELR 21Mar2005 aside.)
(16-17May2006 rev - Add an absolute motion link under our paragraph, "Zeno, in our view, is telling us..." Reset legacy red text.)
(2Jul2006 rev - Add link to Plutarch on Zeno of Elea.)
(6,9Sep2006 rev - Superpose 'ad oculos' link. Minor repairs to 21Mar2005 QELRed aside text.)
(21Dec2006 rev - Add 'Zenos Macroscopic Quantum Uncertainty' anchor. Reset legacy red markup text.)
(13Sep2007 rev - Repair contact link. Reformat.)
(4Apr2008 rev - Minor reformating.)
(23,27Jul2008 rev - Add Bethe quote from Gleick's Genius re: "many truthings." Add "many truthings" anchor. Fix missing text in latest update.)
(27May2009 rev - Repair some formats. Make page current. Reset legacy markups.)
(19Mar2010 rev - Add page top ref's to Bergson's texts on "movement by immobilities.")