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Speaking of the discrete orbits of electrons, Bertrand Russell asks the following:

"Do we know that, between one orbit and the next, other orbits are geometrically possible? Einstein has led us to think that the neighbourhood of matter makes space non-Euclidean; might it not also make it discontinuous?" (Bertrand Russell, The Analysis of Matter)

It seems to me that there might be two possibilities here:

  1. The property of discontinuity might be attributed to space itself, determining the path of electrons presumably by not providing possible positions where it does not exist. The curvature of space could also be seen a property determining the path of light.
  2. The discreteness and curvature are determined by some (possibly unknown) factor independent from space.

Consider the following: Russell speaks of a region between one orbit and another. The idea of discontinuous space also assumes a region between where space exists and where it doesn't. Usually, we would also designate such an empty area as being "space" as well, so this idea suggests space within space; i.e. one space that has the property of being discontinuous within another that lacks such a property. In the same way, we could think of curved space as having a recognizable curvature in virtue of it existing within space which lacks any curvature. We recognize such things by means of contrast.

(Please note that these observations are only some things to consider. I'm not offering any opinion.)

My question: Is there any reason to prefer the first possibility to the second? Or, is there any reason not to identify the two possibilities and say that the "space within space" is the unknown factor independent from the space within which it's contained?

  • 1
    If String Theory is right, there seems to be a minimum length anything can have and still exist. Folks are looking for models of that, that don't make for all the intricacy of string theory. One alternative is Wheeler's notion of 'quantum foam' -- that space-time is 'grainy' at that resolution, made up of isolated points between which things must jump, and that two pieces of space that close together don't exist. Of course we can only picture that as points in space, but that is just us. – jobermark Dec 10 '16 at 17:49
  • Space/time discreteness would occur (in current theory) at the Planck scale, which is some 10^20-10^25 times smaller than atomic dimensions. So they're likely not related. Bohr's 1913 Old Quantum Theory explained discrete energy levels as corresponding to orbits with whole number of wavelengths. Otherwise, the wave-like electrons would destructively interfere with themselves and just vanish analogously to interference fringes. (Bohr's Nobel Prize formalism is a bit different and more complicated, but it can be boiled down to that for descriptive purposes.) – John Forkosh Dec 11 '16 at 4:34
  • The problem can be more subtle; for example, in one geometric picture of Yang-Mills Theories one can simply see spacetime as being made of infinitesimal points, so continuous in the usual sense, but also the points have structure - so they're also discrete. – Mozibur Ullah Dec 11 '16 at 10:25
  • In string-theory space is discretised into tubes around which strings can be wound - but there is no void between tubes; interesting question. – Mozibur Ullah Dec 11 '16 at 10:28
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    It's a fabulous question but impossible to answer briefly. I follow your logic and agree with it, but a solution would take us straight into Kant, Weyl, Buddhist philosophy and the idea that space is not metaphysically real, and this would mean a lot of work. But stick with it. It's always the simple and obvious questions that best reveal the flaws in our world-view. – PeterJ Sep 4 '17 at 13:45
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Aristotle talks about the place of a place which has interesting and suggestive parallels of your space in a space; he also says that it is a difficult problem to which he wasn't offering a solution.

The status of space as a continuum or not has a long pedigree, going back to Zeno, which in one reading is suggesting that space cannot be a continuum - ie infinitely divisible; I think its interesting that a mathematician of Weyls stature was felt it incumbent to re-think what the mathematical contiuum should be in the light of new discoveries (and questions) about the space and motion that QM brought along in its wake.

Its also worth noting that Aristotle pointed out that discontinuous space in which the discontinuity refers to voids there arises the problem of effect at a distance; which to him, discounted this notion of discontinuity.

Much later, Ibn Rushd (Avveroes), in one of his commentaries on Aristotles Physics, proposed a notion of contiguity to replace that of continuity (at least in the phenomenal realm, he still held that continuity held in the celestial), to account for change that differed in degree.

Now, it's worth recalling that Einsteins theory of gravity modelled the spacetime continuum as 4-dimensional in an intrinsic way, and that more recently there has been the discovery of non-smoothable 4-manifolds, these are known as exotic 4-manifolds, and very intriguingly the phenomenon only occurs in this dimension, not for lower ones, and nor for higher:

The existence of exotic 4-dimensional real space R4* is proven by a remarkable combination of topology, geometry and analysis ... we describe a striking property of R4* ... there exists a compact set C so that no smoothly embedded 3-sphere in R4* surrounds C ... so in the differentiable structure of R4*, spheres near infinity are very jagged.

D. Freed & K.Uhlenbeck - Instantons and Four-Manifolds (1984)

To picture this, put a penny on a table, we can quite easily draw a circle around it without touching the coin or drawing on top of it; now what Freed & Uhlenbeck say is if we give space this exotic geometry then we would be unable to do this, the line drawn will enter the coin and exit it; and it doesn't matter how far away we begin, for example a mile away, or as far as the moon; so this unsmoothable exotic geometry on four-dimensional space is 'very jagged'.

So Russell is in very good company to suggest this possibility.

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I'm hoping that someone might have something to contribute, but since nobody has yet responded, I'll offer a tentative answer. At least in Bertrand Russell's day, it seems that there was no way to answer this question either way. Rather than speaking of paths he spoke of geodesics, and rather than speaking of an independent factor, he spoke of forces. Given that forces are thought to be an independent factor from the space containing the paths which they determine, his comparison of ideas is essentially the same as mine:

"It is not quite clear why the man who uses forces with a conventional geometry should be regarded as making a 'mistake,' while the man who says that free particles travel in geodesics, and to justify himself has a queer geometry, is thought to be saying some substantially more accurate. It is true that we must not conceive 'force' as an actual agency, as the older mechanics did; it is merely part of the method of describing how bodies move. But as soon as this is recognized, it is a mere question of convenience whether we speak of forces or not." (Bertrand Russell, The Analysis of Matter, pg. 76,77)

"From this point of view, I prefer the variable space in which bodies move in geodesics to a Euclidean space with a field of force. But I cannot regard the question is one concerning the facts." (Bertrand Russell, The Analysis of Matter, pg. 80)

It's significant that he treats Euclidean geometry as a question of preference or convenience, because if it were a property of space, i.e. as if space were some sort of a substance whose shape is determined by this property, it wouldn't simply be a matter of preference. However, treating this question in the way he does also makes no ontological claims about forces, because he treats them not as having any substantive reality, but merely as a dispensable explanatory device with respect to how bodies move. This idea is further elaborated elsewhere:

"But gradually it was increasingly realized that 'force' is merely a connecting link between configurations and accelerations; that, in fact, causal laws of the sort leading to differential equations are what we need, and that 'force' is by no means necessary for the enunciation of such laws." (Bertrand Russell, The Analysis of Matter, pg. 19)

My reading of Russell may give rise to some objections because he does make some significant claims about the necessary connection between time and space as well as geometry being an empirical fact. However, I don't believe that it could be asserted that these claims arise because of some ontological status of space itself. On the contrary, to the extent that they are empirical and grounded in reality, they arise because of the physical presence of bodies within it:

"[T]he character of space-time in any region depends upon circumstances which can only be ascertained empirically—namely, the distribution of matter in the neighborhood." (Bertrand Russell, The Analysis of Matter, pg. 78,79)

Russell's question about the continuity of space might be considered in the same way, namely, that it is not to attribute any ontological fact to space but merely serves as a description of how bodies move, and the determination of this movement might be attributed to the presence other bodies in the neighborhood.

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Maybe we will never know from an ontological point of view. Physics might construct models either way, that are more and more accurate, but will that mean that "space IS continuous/discontinuous"? That being said, I would ascribe no epistemological to philosophy or metaphysics here. I would say it's squarely a question for physics, but that it might be more subtle - physicists are extremely cautious about the statements they make about nature.

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Good question but very difficult to answer. Before we find any evidence that supports continuity or discontinuity, we need to clarify what continuity means.

"Infinitely divisible" does not necessarily imply continuity. A series of rationals is infinitely divisible and was thought to be sufficient to represent all the points on a continuous line, but the discovery of incommensurables suggests that there are gaps between rationals.

Cantor's definition of continuous series are series that are Dedekindian and contain ℵ_0 as a median class, in virtue of which the series of reals are continuous. But still, definitions like this do not preclude further discoveries of "gaps."

I think we need someone who is well-versed with Principia Mathematica to revise "the Principles," revisiting the same problems in the Principles by employing the tools and insights provided in Principia, using Principia not as a new dogma but as a stepping stone. He who can do this will have his name remembered for generations to come.

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In order to have a really coherent discussion about these things, it is difficult to avoid some usage of mathematics -- the thoughts involved are very precise and esoteric, and mathematics is good at expressing such things without ambiguity.

We typically model "spacetime" as a manifold; a collection of points together with a bunch of rules (coordinate functions) for assigning each point to some point in (usually) Euclidean space. In string theory, we can use 'compactified manifolds' which themselves have other manifolds packed into each point on their surface to model a 'higher dimensional' reality in which many of the dimensions (all but three) have been packed into these extra manifolds.

The question of whether spacetime is continuous or discontinuous, in this context, amounts to a question of what coordinate function domain we want to use on the manifolds. If we use coordinate functions which map into the real numbers (or Euclidean space as above) we can determine 'what the manifold' looks like by checking its Gaussian curvature, looking at the local metric for spacetime represented as a tensor, so on and so forth -- this is largely the domain of general relativity.

We can have the coordinate functions map into some other space, however, and this will produce fundamentally different types of manifold. These structures have points with nothing 'in between' them if we choose the correct coordinate domain besides the reals -- consider that for any two real numbers x and y such that x < y, there is always some third real number z such that x < z < y. If we move to the integers instead of the reals, however, for any integer n there is clearly no other integer between n and n+1. This is how we model the notion of nothing being 'in-between' two points in space time -- we pull the coordinates for our space time from some place with an ordering structure that permits two objects to have nothing ordered in-between them, then look at the corresponding objects on our space time manifold.

The question of whether or not there 'is space' between two orbits can be asked clearly in this context -- we ask whether the coordinate functions for our space time are pulled from Euclidean space (so there is space between the orbits which is just inaccessible to the particles traveling on the manifold), or whether they are pulled form some discretely ordered space (in which case there is actually 'no space' to occupy in there). Hopefully this can help answer your question of how to think about the problem, as this is a an open problem to date in physics and as such has no widely accepted answer.

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Russell's idea "might it [matter] not also make it [space] discontinuous?" is wrong (like many of his statements).

While we do not know whether space is grained or continuous, we cannot determine the nonexistence of space between two orbits from quantum mechanics. For unbounded states, i.e., for free moving particles with positive energy, there is no quantization. They can, according to Schrödinger or Heisenberg and even Bohr, cross the space between two orbits and exist at any point of their trajectory.

Further Bohr's theory has turned out insufficient. Electrons are not orbiting on fixed circles or ellipses but are existing as "clouds" with positive probability in every distance from the nucleus.

The grained structure of space however is suggested by the following: According to modern physics there exists nothing unobservable or unmeasurable (an electron has no position or momentum unless we measure it - and it does not exist with higher precision than we can measure it, according to the uncertainty relations). In order to define distance, we have to measure it. This can be done most precisely with interference of light. The shortest lightwaves are best suited. But even if we could collect all energy of the universe into one single photon, its wavelenght would be larger than 10^(-100) m. Therefore it appears nonsensical, from this standpoint of modern physics, to believe in shorter distances.

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