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In a discussion on Douglas Bundy's essay thread, I said that I would post some comments here about Einstein's possible view on instants and instantaneous magnitudes. Although he must have been aware of them, as far as I know, Einstein is not on record as having ever talked about Zeno's paradoxes, so we are unable to look there for clues. His theories of special and general relativity are heavily...

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In a discussion on Douglas Bundy's essay thread, I said that I would post some comments here about Einstein's possible view on instants and instantaneous magnitudes. Although he must have been aware of them, as far as I know, Einstein is not on record as having ever talked about Zeno's paradoxes, so we are unable to look there for clues. His theories of special and general relativity are heavily reliant on continuous functions, and he regularly talked meaningfully, in a physical sense, of instants and exact positions and magnitudes, not taking issue with them. Three such examples, two from his 1905 paper on special relativity, and one from an essay written on the occasion of his 70th birthday in 1949, are included below:



"...that is to say that their [synchronized clocks] indications correspond at any instant to the 'time of the stationary system' at the places where they happen to be."

"If the electron is at rest at a given epoch, the motion of the electron ensues in the next instant of time according to the equations..."

-- On the Electrodynamics of Moving Bodies, 1905

"A problem arises only when we ascribe group-characteristics to a theory, i.e., if we assume or postulate that the same physical situation admits of several ways of description, each of which is to be viewed as equally justified. For in this case we obviously cannot ascribe complete objective meaning (for example the x [position] -component of the velocity of a particle or its x-coordinates) to the individual (not eliminable) magnitudes."



-- Albert Einstein: Philosopher-Scientist, 1949

In relation to not taking exception with instants, however, there is a rather notable exception. Somewhat at odds with the quote referenced just above, in the very same essay, while arguing the statistical interpretation of quantum theory was incomplete, we find this:

"To this the quantum theorist will reply: This consideration stands and falls with the assertion that there actually is such a thing as a definite time of disintegration of the individual atom (an instant of time existing independently of any observation). But this assertion is, from my point of view, not merely arbitrary but actually meaningless. The assertion of the existence of a definite time-instant for the disintegration makes sense only if I can in principle determine this time-instant empirically...

One may not merely ask: "Does a definite time instant for the transformation of a single atom exist?" but rather: "Is it, within the framework of our theoretical total construction, reasonable to posit the existence of a definite point of time for the transformation of a single atom?" One may not even ask what this assertion means. One can only ask whether such a proposition, within the framework of the chosen conceptual system - with a view to its ability to grasp theoretically what is empirically given - is reasonable or not. Insofar, then, as a quantum-theoretician takes the position that the description by means of a Psi-function refers only to an ideal systematic totality but in no wise to the individual system, he may calmly assume a definite point of time for the transformation. But, if he represents the assumption that his description by way of the Psi-function is to be taken as the complete description of the individual system, then he must reject the postulation of a specific decay-time."

In other words, here Einstein rejected instants of time because they were not empirically verifiable. On this basis, he also rejected the idea that an atom could decay at a definite exact time. His other comments, and lack of further related ones, however, suggest that the ramifications of this conclusion perhaps had not quite hit completely home.

During the discusson on Douglas' thread, one related to the reality of interval, I also mentioned about another quote, and I think it could be a good idea to include it here too.

"Time is what a clock measures"

The above quote is usually attributed to Einstein. That Einstein would have said this puzzled me, particularly as a key feature of the dilation of time in relativity was Einstein's treatment of time as being nothing more than that what a clock showed. On locating Einstein's original quote, it came as no surprise then to find that what he actually said, "Zeit ist das, was man an der Uhr abliest," translates to "Time is what one reads off the clock." Although the difference seems tiny, as "measure" says that time exists, while "reads" neither affirms or denies its existence, the diiference is actually quite big. My essay (and essay notes) should make the possible relevance of this point more clear.

Best wishes

Peter

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Hello Peter,

I enjoyed your essay--more than once! Thanks!

It seems you are searching for something--for a "deeper" mechanism--a more fundamental *physical* invariant powering time and all its arrows and assymetries.

Well, Moving Dimensions Theory provides such a mechanism! The fourth dimension is expanidng at the rate of c, or dx4/dt=ic.

In the attached paper, you...

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Hello Peter,

I enjoyed your essay--more than once! Thanks!

It seems you are searching for something--for a "deeper" mechanism--a more fundamental *physical* invariant powering time and all its arrows and assymetries.

Well, Moving Dimensions Theory provides such a mechanism! The fourth dimension is expanidng at the rate of c, or dx4/dt=ic.

In the attached paper, you will see that as the fourth dimension expands at the rate of c, as a probabilistic wavefornt of nonlocality that has a wavelength of Planck's length; an underlying mechanism is presented for time and all its arrows and assymetries. Also accounted for with MDT's fourth moving/exapnding dimension are entropy, all the dualities--time/space, matter/energy, wave/partice--and Heisenberg's and Huygens' principles.

Our whole lives they have been telling us that spacetime is a sea of quantum foam--that the so-called vacuum is buzzing with activity. Well, what is more natural then, to provide this activity, than a fourth dimension that is expanding relative to the three spatial dimensons, exhibiting wavelike properties. So it is that instead of using vibrating strings, why not use vibrating spacetime? It makes far more sense, and it unifies far more arenas of physics, than current approaches.

You would enjoy my paper, as it provides the mechanism you are seeking!

http://fqxi.org/community/forum/topic/238

"Time as an Emergent Phenomenon: Traveling Back to the Heroic Age of Physics by Elliot McGucken."

You write, "The realization that there was no such thing as an instant in time, and that an object in relative motion did not have a determined relative position, had some further implications."

Please see the attached paper. As time is an emergent phenomenon that arises because the fourth dimension is expanding relative to the three spatial dimensions (as the fourth expanding dimension carries the photons that power our watches/clocks), and because the expansion of the fourth dimension is wavelike, there is a degree of uncertainty in all measurements of space and time, as the fundamental character of the fourth expanding dimension is wavelike.

You write, "Perhaps the most obvious one related to the nature of time itself. If there was no such thing as an instant in time, I also realized that there could be no “flow” or passage of time, for without a continuous progression through indivisible instants over an extended duration, there could be no physical progression or flow."

MDT: dx4/dt = ic. x4 is the fourth dimension. t is the time measured on your watch or clock. i is the imaginary number. c is the velocity of light. For the first time in the history of relativity, *change* has benn woven into the fundamental fabric of spacetime.

You write, "In other words, there was nothing there, no temporal stepping-stone, for which time could possibly use to progress. Kind of similar to how an invisible ether was once assumed to permeate throughout the entire universe, Newton’s invisible river of time, assumed by many to enable motion and change as it proceeded, could not exist either. It had no water. Although we certainly feel as though we are swimming along with it, we are actually high and dry. Further, the same could be said for the existence of space, due to the lack of spatial points."

Please see the attached paper. MDT waves change, wavelike behavior, Planck's length, and the velocity of light into the fundamental fabric of spacetime, at long last providing a mechanism for time and all its arrows and assymetries acorss all realms.

"In his 1912 Manuscript on Relativity, Einstein never stated that time is the fourth dimension, but rather he wrote x4 = ict. The fourth dimension is not time, but ict. Despite this, prominent physicists have oft equated time and the fourth dimension, leading to un-resolvable paradoxes and confusion regarding time’s physical nature, as physicists mistakenly projected properties of the three spatial dimensions onto a time dimension, resulting in curious concepts including frozen time and block universes in which the past and future are omni-present, thusly denying free will, while implying the possibility of time travel into the past, which visitors from the future have yet to verify. Beginning with the postulate that time is an emergent phenomenon resulting from a fourth dimension expanding relative to the three spatial dimensions at the rate of c, diverse phenomena from relativity, quantum mechanics, and statistical mechanics are accounted for. Time dilation, the equivalence of mass and energy, nonlocality, wave-particle duality, and entropy are shown to arise from a common, deeper physical reality expressed with dx4/dt=ic. This postulate and equation, from which Einstein’s relativity is derived, presents a fundamental model accounting for the emergence of time, the constant velocity of light, the fact that the maximum velocity is c, and the fact that c is independent of the velocity of the source, as photons are but matter surfing a fourth expanding dimension. In general relativity, Einstein showed that the dimensions themselves could bend, curve, and move. The present theory extends this principle, postulating that the fourth dimension is moving independently of the three spatial dimensions, distributing locality and fathering time. This physical model underlies and accounts for time in quantum mechanics, relativity, and statistical mechanics, as well as entropy, the universe’s expansion, and time’s arrows."

I hope MDT aids in your further contemplations of spacetime!

Best,

Dr. E (The Real McCoy)

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My last reply to Elliot McGucken at 238 was mutilated when I begun to explain what essential details he was not aware of. Then he quoted Born in order to lecture me. I consider the following details important for all who are using complex physical quantities.

One safely arrives at the complex frequency domain and correctly returns only via several steps while it is common practice to skip...

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My last reply to Elliot McGucken at 238 was mutilated when I begun to explain what essential details he was not aware of. Then he quoted Born in order to lecture me. I consider the following details important for all who are using complex physical quantities.

One safely arrives at the complex frequency domain and correctly returns only via several steps while it is common practice to skip several steps for convenience.

Given, a physical quantity could be described with a function cos(x) with -oo < x < +oo, then Euler's identity reads 2 cos(x) = exp(ix) + exp(-ix). The two exp functions represent two phasors, which are rotating in complex plane anti-clockwise and clockwise, respectably. Transform into a complex domain means to arbitrarily omit either exp(x) or exp(-ix). It can also be considered like an arbitrary addition of either i sin(x) or -i sin(x). Electrical engineers decided roughly one hundred years ago to use a more reasonable sign. As a result, their transform is different from the originally chosen one which is still preferred in mechanics.

The transform into complex plane is tacitly performed with a complex integral transform, in particular with Fourier transform by multiplication with the complex kernel. Fourier transform of a real function of time yields a complex function of positive as well as negative frequency. Likewise Fourier transform of a real function of frequency would provide a complex function of negative as well as positive time. The same is true for the pair distance and wave number. Some people are calling wave numbers spatial frequencies. Momentum and position constitute a further pair.

In particular the negative frequencies puzzled engineers as well as physicists when quantum physics was introduced. Dirac explicitly wrote what all others thought: Frequencies cannot be negative. Indeed, negative frequencies are just an artifact of transform from physically correct original time domain into a complex frequency domain. They vanish with the return by performing inverse transform.

Interpretation of complex quantities like impedance and permeability in electrical engineering always relates to non-rotating phasors. Those who did not like demanding careful work were ready to forget that complex frequency domain, or also complex time domain, does strictly speaking not allow immediate physical interpretation. Instead of being aware of performing any transform they made an ansatz immediately in complex domain and did not pay attention to the non-identity of complex frequency domain and complex time domain.

They felt supported by mathematicians, who referred to complex plane, not an artificial tool of physicists and engineers when they correctly confirmed that the complex numbers enriched the ordinary ones, and the complex representation is the more general one.

Now I will point to a decisive peculiarity: Fourier's transform demands x either to range from -oo to +oo or to be included between perfect mirrors. Fourier himself dealt with a closed ring. Quantities like temporal or spatial distance do not make sense if one demands them to be actually infinite. Radius and elapsed time are reasonably limited to merely positive values. Oliver Heaviside, found a solution that boosted science and technology. He continued the function f(x>0) into the negative half-plane x

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Thanks Peter,

You write, "as I explained in my essay, as long as one recognizes that instants, instantaneous magnitudes, space-time points etc, do not exist. . ."

Yes--but why do points not exist in spacetime?

I propose that the anwser to this is "because the fourth dimension is expanding relative to the three spatial dimensions at the rate of c, or dx4/dt=ic." ...

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Thanks Peter,

You write, "as I explained in my essay, as long as one recognizes that instants, instantaneous magnitudes, space-time points etc, do not exist. . ."

Yes--but why do points not exist in spacetime?

I propose that the anwser to this is "because the fourth dimension is expanding relative to the three spatial dimensions at the rate of c, or dx4/dt=ic."

Instead of depicting matter as tiny, vibrating strings--something for which there is no proof for--why not admit the fact that the fourth dimension has nonlocal wavelike qualities, as attested to by both relativity and quantum mechanics?

Relativity relies on the photon being nonlocal, and so does quantum mechanics.

As contemplations on the photon lead Einstein to the theories of relativity and quantum mechanics that revolutionized our notions of space, time, and physical reality, this paper again turns towards the photon and Einstein’s original works to shed light on time. Various phenomena in Einstein's 1905 papers can be united with a simple postulate representing an underlying physical reality from where time itself emerges—the fourth dimension is expanding relative to the three spatial dimensions at the rate of c.

Consider the emission of a photon in free space. One second later, the photon has equal probability of being found anywhere upon a sphere with a radius of 186,000 miles, as the velocity of light is 186,000 miles/second. If we covered the surface of said sphere with detectors, one, and only one detector, would detect the photon. Although having traveled 186,000 miles through space, the photon will not have aged one iota, for time stops at the speed of light. It will not have moved one iota in the fourth dimension. And there lies a clue to the reality that the fourth dimension is expanding relative to the three spatial dimensions. For how can a photon propagate

186,000 miles in the three spatial dimensions, and yet not budge an inch in the fourth dimension, unless that fourth dimension is moving right along with it, just as a wave moves right along with a surfer?

Consider two interacting photons that propagate in opposite directions, as in experiments inspired by Bell’s Inequality and conducted by Aspect et al. One second later, each photon's polarization is measured at detectors separated by 372,000 miles. According to the laws of

quantum mechanics and numerous supporting experiments, the measurement at one detector instantaneously affects the measurement at the second detector. It is as if the photons are yet side-by-side during the measurement. This “spooky action-at-a-distance,” as Einstein called it, is not so spooky in the context of a fourth expanding dimension, for although separated by 372,000 miles, the photons yet inhabit a common locality in the fourth dimension, as the fourth dimension is expanding relative to the three spatial dimensions, distributing locality at the rate of c. So it is that both quantum and relativistic phenomena are accounted for with the simple elegance of the postulate: the fourth dimension is expanding relative to the three spatial dimensions.

Ergo the fourth dimension has both nonlocal and wavelike properties, as it expands at the rate of c, setting both c (the velocity of light) and h (Planck's constant).

The expansion of the fourth dimension manifests itself as a spherically-symmetric probabilistic wavefront expanding at c--exactly the mechanism underlying the motion of the photon, which happens to be described by a spherically-symmetric probabilistic wavefront expanding at c by quantum mechanics. And relativity tells us that the photon remains in the exact same place in the foruth dimensions, so the fourth dimension must be a spherically-symmetric probabilistic wavefront expanding at c. Ergo the fourth dimension is expanding at c relative to the three spatial dimensions, as a spherically-symmetric wavefront with wavelength of the Planck length. Thus it is impossible to measure anything with absolute certainty, as space-time itself has a wavelike character.

The three spatial dimension are only ever known by measurement, and all measurements entail the propagation of energy, it is but mass that exists purely in the fourth expanding dimension, which quantizes it via the fact that only discreet wavelengths are allowed, as the wavelength of the fourth dimension is Planck's Length.

Please see the attached paper, which suggests that while space is bent by matter, the expansion of the fourth dimension remains an invariant. Could it be that space is not bent by mass, and that mass instead expels the fourth dimension, warping its expansion? Perhaps. Curved space would imply that time is curved by the same amount as the space, even if the expansion of the fourth dimension remained an invariant, as we only know time by space. Perhpas mass is where the expansion of the fourth dimension cannot penetrate, and thus rest mass represents a physical entity that is foreign from the fourth dimension, as well as an entity which the fourth dimension never penetrates.

Please see the attached paer.

Thanks & best,

Dr. E

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Hello Peter,

Hope all is well down under in Australia!

“In questions of science, the authority of a thousand is not worth the humble reasoning of a single individual.” –Galileo Galilei

I think it's cool how MDT solves all the philosophical problems you perceive (misperceive in some cases) with a simple *physical* model, postulate, and equation: The fourth dimension is...

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Hello Peter,

Hope all is well down under in Australia!

“In questions of science, the authority of a thousand is not worth the humble reasoning of a single individual.” –Galileo Galilei

I think it's cool how MDT solves all the philosophical problems you perceive (misperceive in some cases) with a simple *physical* model, postulate, and equation: The fourth dimension is expanding relative to the three spatial dimensions at the rate of c, or dx4/dt=ic.

I think you will agree that *physical* theory ought contain *physical* models representing *physical* reality which leads to naturally-emergent *physical* phenomena that we can *physically* observe and measure. I think you will also agree that the purpose of physics has ever been to provide novel physical theories representing previously unheralded phsyical aspects of our universe, captured in simple, immutable postulates and equations, which oft unify disparate physical phenomena in a common, underlying, deeper *physical* model. MDT does all this with dx4/dt=ic.

MOVING DIMENSIONS THEORY: EXALTING EINSTEIN’S ELEMENTARY FOUNDATIONS OF RELATIVITY & SCHRODENGER’S CHARACTERISTIC TRAIT OF QUANTUM MECHANICS

by Dr. Elliot McGucken

http://fqxi.org/community/forum/topic/238

"A physical theory can be satisfactory only if its structures are composed of elementary foundations. The theory of relativity is ultimately as little satisfactory as, for example, classical thermodynamics was before Boltzmann had interpreted the entropy as probability. –Einstein in a letter to Arnold Sommerfield on January 14th, 1908. CPAE, Vol. 5, Doc. 73:"

"When two systems, of which we know the states by their respective representatives, enter into temporary physical interaction due to known forces between them, and when after a time of mutual influence the systems separate again, then they can no longer be described in the same way as before, viz. by endowing each of them with a representative of its own. I would not call that one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought. By the interaction the two representatives [the quantum states] have become entangled." –Schrödinger

Moving Dimensions Theory’s simple postulate, physical model, and equation account for both “relativity’s elementary foundations,” which Einstein stated we yet needed, and Schrödinger’s “characteristic trait” of quantum mechanics—entanglement. Both relativity (it's two postulates and Minkowski/Einstein spacetime metric) and quantum entanglement/nonlocality naturally emerge from MDT's simple postualte and equation, which weaves change into the fundamental fabric of spacetime for the first time in the history of relativity.

Peter--you write, "In relation to space-time having no physical existence, this is far from a revelation. Einstein himself held this view, and often appears to have been careful to make a note of it."

This is completely untrue. Never, never did Einstein state that space-time has no *physical* existence. His formulation of relativity is based on the very opposite of this. Have you ever read his papers or The Meaning of Relativity?

Indeed--Einstein's entire framework of relativity was built upon the fact that dimensions are *physically* real and that they have a *physical* reality, and that they could *physically* bend, warp, and *move*. It was this bold insight that now helps power our invaluable GPS systems. Whenever you fly on an airplane, you are *physically* leveraging and making use of the *physical* reality Einstein pursued and exalted. You are acknowledging the physical reality of space, time, and dimensions, whether you admit to it in your philosophy or not.

"Vocatus atque non vocatus, Deus aderit."

"Called or not called , GOD will be present.” – Inscription on Gravestone of Professor Dr. Carl G. Jung



"There is more in this heaven and earth than is dreamt of in your philosophy." --Hamlet

"CHAPTER XXXII: THE STRUCTURE OF SPACE ACCORDING TO THE GENERAL THEORY OF RELATIVITY: According to the general theory of relativity, the geometrical properties of space are not independent, but they are determined by matter. Thus we can draw conclusions about the geometrical structure of the universe only if we base our considerations on the state of matter as being something that is known." --Einstein's Meaning of Relativity

Space has geometrical properties. And it has a *physical* reality which can bend starlight, attract apples to the earth, hold satellites in orbit, and inform their GPS systems. Space and time are as real as starlight, apples, and satellites. Now I know that one can get a Ph.D. in philosophy by saying that nothing is real but for tenure, ST/LQG, sabbaticals, and summers off in summer homes, but this is physics--the physics which assumes the physical realtiy that your computer and internet connection depend on.

In a section reverently titled "No Time, No Space," in your essay, you write, "The realization that there was no such thing as an instant in time, and that an object in relative motion did not have a determined relative position, had some further implications. Perhaps the most obvious one related to the nature of time itself. If there is was no such thing as an instant in time, I also realized that there could be no “flow” or passage of time, for without a continuous progression through indivisible instants over an extended duration, there could be no physical progression or flow."

I do not grasp this logic on several levels. De Broglie showed that all electrons and matter have wave-like properties, and Heisenberg's Uncertainty Principle showed that due to the wave-like nature, it was impossible to measure the absolute postition at an absolute time with absolute precision. But how does this imply in any way the impossibility of the flow of time?

You write, "for without a continuous progression through indivisible instants over an extended duration, there could be no physical progression or flow." Why not? Moving Dimensions Theory postulates that the fourth dimension is expanding relative to the three spatial dimensions at c, weaving wave-like behavior and motion into spacetime itself, as shown in the attached paper.

You write, "In other words, there was nothing there, no temporal stepping-stone, for which time could possibly use to progress. Kind of similar to how an invisible ether was once assumed to permeate throughout the entire universe, Newton’s invisible river of time, assumed by many to enable motion and change as it proceeded, could not exist either. It had no water. Although we certainly feel as though we are swimming along with it, we are actually high and dry. Further, the same could be said for the existence of space, due to the lack of spatial points." The sweeping analogies and extrapolations here lose me.

Indeed, we need a new model of spacetime which accomodates change and the *physical* reality we perceive, and MDT provides the new *physical* reality, accounting for space, time, and motion--providing a *physical* foundation for all of relativity, quantum entanglement, entropy, and time and all its arrows and assymetries.

Too, too many physicits and philosophers would prefer to simply receive tenure and titles by throwing out time and space--by disposing of physics--in our fallen, postmodern world whence we have abandoned the heroic quest of undertsanding *physical* reality; as well as the gold standard and god standard; leading to our current financial crises. Thus, it is easier for them to dismiss Einstein's, Bohr's, and Feynman's spirits, along with Einstein's relativity amd Schrodenger's characteristic trait of quantum mechanics, than it is to unite them all in Moving Dimensions Theory. Now many have used the fact that Moving Dimensions Theory resembles neither LQG nor String Theory to catsigate and impugn it, but the fact that it resembles neither ST nor LQG is actually a grand compliment in its favor. For while ST & LQG lack postulates and equations despite their hundreds of millions in funding, MDT proposes a simple postulate and equation: "The fourth dimension is expanding relative to the three spatial dimensions at c, or dx4/dt=ic." From this time, all of relativity, entanglement, nonlocality, entropy, and time and all its arrows and assymetries naturally emerge.

Please see the attached for further development/commentary/philosophy.

Hope all is well!

Best,

Dr. E (The Real McCoy)

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Dear Peter,

I think I agree almost entirely with what you wrote in your paper “Time for a change”. I came to the same conclusions regarding the inexistence of time and space in observer-independent reality – but by a different path.

Just some comments about your paper.

“The paradoxes of Achilles and the Tortoise and the Dichotomy, on the other hand, have generally been...

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Dear Peter,

I think I agree almost entirely with what you wrote in your paper “Time for a change”. I came to the same conclusions regarding the inexistence of time and space in observer-independent reality – but by a different path.

Just some comments about your paper.

“The paradoxes of Achilles and the Tortoise and the Dichotomy, on the other hand, have generally been thought to be solved by the summing of an infinite series, a mathematical technique developed by Cauchy, Weierstrass, and the German mathematician, Richard Dedekind. In relation to the paradoxes, this means the summing of an infinite series of progressively small time intervals and distances, so that the time taken for Achilles to reach his goal and overtake the Tortoise, or to traverse the said distance in the Dichotomy, is, in fact, finite. The faulty logic in Zeno's argument is seen to be the assumption that the sum of an infinite number of terms is always infinite, when in fact, an infinite sum, for instance, 1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ..., can be mathematically manipulated to be shown equal to a finite number, or in this case, equal to 2. Therefore, Zeno's infinitely many subdivisions of any distance to be traversed, can be mathematically reassembled to give the desired finite answer, and the body in apparent motion in the paradoxes said to have reached its said impossible goal.”

The mathematical “refutation” of Zeno’s argument rests entirely on the alledged assumption that “the sum of an infinite series is always infinite”

1. I notice that no reference is ever provided as where to find this assumption in Zeno’s works.

2. Zeno’s argument is based on the implicit assumption that a finite distance is infinitely divisible. Then what is this finite distance if not the sum of the infinite series of parts into which it can be divided ? In other words, the formula (1 = 1/2+1/4+1/8…) is implicitly but necessarily present in Zeno’s argument. Then, the “calculus solution” just demonstrates that

(1=1/2+1/4+1/8…) = ( 1/2+1/4+1/8…=1).

Well, for my part, I am too shy to call this a tautology. But some more callous people might.

I think the calculus solution is just an elegant way of sweeping the problem under the carpet. It is an unsettled problem and science does not like to admit that it is at least partly based on unsettled problems. That is why the dominant view in the scientific community is to ignore the challenge by pretending it has been settled.

But the problem is still here : it concerns our impossibility to conceive continuity and motion in their own terms – because we can think only in discrete terms, i.e. in the terms of their negation. If we could describe continuous motion in another way than as sequences of discrete events, there would be no Zeno’s paradoxes.

“The same fault applied to the Arrow paradox’s proposed solution by calculus, as the “limit” of the arrow’s velocity at the instant is never actually reached; strictly speaking, the arrow does not have a velocity at the instant. It cannot, as by definition, an instant has no duration, so the arrow cannot have a velocity at an instant if there is no interval of time during which it could cover a distance. Indeed, to say that it could have one, was like saying than a stationary body could at the same time be moving, or that a multi-sided polygon could also be a circle. Furthermore, as a continuous function is a static and indivisible mathematical entity, by invoking this model, people were essentially agreeing that motion did not exist and was some sort of strange subjective illusion.”

Saying that a moving body is at any given instant in a determined position amounts to say that at this instant it covers a zero distance (it is in a space point). If the distance travelled by the body is infinitely divisible, it contains no zero distance. There is no such thing as a point in space (a determined exact position) – and for the same reason, an instant in time.

I may add that if the said body is at every instant in a determined position, where it covers a zero distance, then the sum of all determined positions is also a zero distance : if there is such thing as an instant in reality, movement is impossible.

“The realization that there was no such thing as an instant in time, and that an object in relative motion did not have a determined relative position, had some further implications. Perhaps the most obvious one related to the nature of time itself. If there was no such thing as an instant in time, I also realized that there could be no “flow” or passage of time, for without a continuous progression through indivisible instants over an extended duration, there could be no physical progression or flow.”

I am not sure I understand you well here. Why should a continuous progression necessarily proceed “through indivisible instants”? Either it proceeds from an instant to the next (and it is not continuous), or it proceeds continuously. If there is no time in reality, there can be no flow of time.

An instant is a time point, i.e. a zero time duration. If there are no instants, there are no zero time durations. If there are no zero time durations, there are no zero space lengths, where the body can be at rest. In other words, in reality everything is changing continuously, however small the duration considered. But we cannot perceive continuous changing as such : when a body moves from A to B, we perceive it at rest in A, let us say for 1/25 of a second, then at rest in B, etc, never in between, however close A and B are. With the help of a fast-filming camera we can catch it in between, we can perceive it in C, in D, etc – but once more at rest for say 1/100 of a second, and so on. We always perceive continuous motion as a succession of fixed spatial configurations. But such fixed configurations do not exist in reality, they exist only in our representation of reality.

Space and time do exist - but only in our perception of reality (in phenomenal reality, as Kant said). They are not observer-independent (that is why times are different for two observers moving relatively to each other). The main problem is that we are trapped in an inadequate philosophical frame, that of classical realism, which assumes that reality and its perception are one and the same thing.

Best regards

Pierre Sabatier

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