A Universe Infinite in Scale with No Action-at-a-Distance

Tom Van Flandern Proposes a Five-Dimensional Infinite Universe

This post follows on from these two posts:

1. Christiaan Huygens Proposes an Infinite Series of Levels to the Universe

2. Michael Faraday and James Clerk Maxwell on Action-at-a-Distance

In the first post, I explained that Christiaan Huygens (1629-1695) had proposed that the luminiferous aether might be composed of an infinite series of levels, each level made up of particles finer than those of the previous level.

In the second post, I explained that the medium used by Michael Faraday (1791-1867) to explain electric induction supposed action-at-distance between the particles of said medium. I concluded as follows:

My conclusion is that there may well be, at every level of the universe, a medium composed of finer particles, making up a lower level. Should there be a lowest level, then the interactions between particles at that lowest level would have to be action-at-a-distance.

In this post, I will consider the possibility that there be no lowest level. I will present the arguments Tom Van Flandern (1940-2009) gives in the first two chapters of his book, Dark Matter, Missing Planets & New Comets¹, recommended by one of my readers after I published the first post. In this book, Van Flandern argues that in fact there are an infinite number of levels to the universe, and that action-at-a-distance is a logical impossibility.

Van Flandern’s reasoning is purely a priori and deductive. He begins his discussion by presenting Zeno’s Paradox, named in honour of the ancient Greek philosopher Zeno (c.490 BCE - c.430 BCE).

Zeno’s Paradox deals with the ultra-small structure of space and time. In its essence, the paradox notes that if a moving body is in a specific place at every instant, then there is no instant when it is in transition from one place to another; and therefore motion is impossible. Since this contradicts everyday experience, it is called a paradox.

The same paradox can be expressed in a different form: to move from point A to point B one must first complete the trip to the midpoint. Having reached that far, one must next reach the new midpoint of the remaining distance. But however far one has traveled, one must first travel half the remaining distance before one can travel all of it. Hence one can never reach point B, because an infinite number of “half-the-distance” steps are required. [p.7]

His answer to the Zeno Paradox is that both space and time are infinitely divisible:

The only apparent way is to assume that space and time are infinitely divisible, because the assumption of a minimum possible space or time interval would require discontinuities of motion or existence. [p.2]

He then considers the possibility that this solution might itself contradict Zeno’s Paradox and explains that this is not a problem:

But is this not also ruled out by Zeno’s argument? The problem is with our intuitions: while it is easy for us to imagine a finite whole as composed of an infinite number of parts, it is difficult to imagine an infinite number of components being assembled into a finite whole. As is well known in mathematics, an infinite series can have a finite sum. [pp.9-10]

He gives the well known example of the infinite sum 1/2 + 1/4 + 1/8 + 1/16 + ···, which equals the value 1. He concludes with this argument:

Viewed in this way, it may be seen that the body is at every instant at some specific point on a space-time line. And once again the points in the interval can be put into a one-to-one correspondence with numbers between zero and one. So even though the distance traveled by the body in zero time is zero, it is nonetheless possible to traverse a finite distance in a finite time, each interval consisting of an infinite number of time instants and space points. To clarify this conclusion, it is possible for substances to be unchanging at every instant, yet changed after a finite interval, only if there are an infinite number of steps in the interval! [p.11]

Having dealt with space and time, Van Flandern moves on to matter, presenting a new form of the paradox:

There is another form of Zeno’s Paradox that applies to masses: “If bodies are infinitely divisible, then contact should be impossible.” For example, when macroscopic bodies seem to touch, they actually consist of mostly empty space at the atomic level; so it must be their atoms which actually touch. But atoms are themselves composed of smaller particles and mostly empty space, so it must be these smaller constituents which actually touch. But if matter is infinitely divisible, this argument can be prolonged indefinitely, and nothing can ever actually touch. [p.12]

Van Flandern resolves this paradox in the same manner as for space and time:

It should be apparent from these considerations that postulating a “minimum possible unit of substance” is no more logically palatable than a “minimum possible unit of space or time.” Substance must be infinitely divisible, as must space and time; or else the paradoxes quickly lead to unresolvable logical dilemmas. But how then can matter ever experience “contact,” if everything which might experience contact is itself composed of smaller substances? The resolution of this paradox would seem to be analogous to that for space-time. If the substance of bodies always gets denser (more substance per unit volume) at smaller and smaller scales, then in the limit as dimensions approach zero, density approaches infinity and substances approaching each other must make “contact.” (That is, at infinite density, they cannot be “transparent” to other substance). In the real universe, the density of matter greatly increases as scale decreases. Hence the ratio of mass to volume in electrons is enormously greater (about 10^10 g/cc) than the same ratio for matter in ordinary human experience (of order 1 g/cc), which in turn is enormously greater than the ratio for the entire visible universe (10^-31 g/cc). “Contact” is therefore possible for infinitely divisible matter, as long as the smaller and smaller particles continue to increase in density with sufficient rapidity, without limit. [pp.12-13]

So Van Flandern is proposing, as does Huygens, that there be an infinite series of levels of particles, the particles at each level smaller than those at the previous level. What is new with Van Flandern is that he also proposes that the density of particles increases in each lower level. The limit of this series is infinite density:

By analogy with the proposed resolution of Zeno’s paradoxes for space and time, the paradox for mass is resolved, apparently necessarily, by the conclusion that substance must be infinitely divisible and that it must approach infinite density as size decreases toward zero dimensions. This conclusion is reached by reasoning alone; it is reinforced by the observation that matter does in fact increase rapidly in density as scale becomes smaller over a range of more than 40 orders of magnitude in the observable universe. [p.14]

Van Flandern then moves on to considering this series of levels in the opposite direction:

From the preceding considerations it seems altogether reasonable, and in a way compelling, to deduce that space, time, and substance are all infinitely divisible, because the consequences of the alternative are logically absurd. But if they are infinitely divisible on the smaller scale, what about the larger scale?

Recall our earlier argument that the entire visible universe would have undefined scale in space, time, and mass, unless such scale is provided by the presence of other substance in the greater universe beyond. That argument must remain true without limit. So no matter how large a universe we consider, its scale for space, time, and mass must eventually become undefined unless there is a greater universe around it to provide meaning to its dimensions. We conclude that all five dimensions [three space dimensions, plus time and scale dimensions] are surely as infinite on the large scale as they must be on the small scale. [p.18, my emphasis]

The emphasized argument is reminiscent of the one used by Giordano Bruno (1548-1600). See my post Giordano Bruno on the One and the Infinite.

So Van Flandern’s summary is as follows:

• The physical universe has five, and only five, dimensions. 

• The universe is infinite in extent in all five dimensions. [p.xxxii]

Now we can come back to the question of action-at-a-distance. In my post on Faraday and Clerk Maxwell on action-at-a-distance, I stated that should there be a lowest level of particles, then the latter would have to interact at a distance, leaving open the question of what would happen should there be no lowest level.

Reading Van Flandern’s arguments in favour of the infinite series of levels of particles ultimately having infinite density, I think it can be argued that in this framework, then action at a distance would not be necessary. And this is, in fact, his position:

Forces such as gravity or magnetism, which appear to act at a distance, must actually consist of something which passes between the source body and the affected body. In other words, “action at a distance” in its purest form, where something local affects something remote without any causative agent passing between the two, must logically be impossible. In the cases of gravity and magnetism, this postulate is reinforced by the observation that the force exists at all distances from the source, dropping in strength with the square of distance, suggesting an outward propagation from that source. [p.31, my emphasis]

I think that Tom Van Flandern’s reasoning provides a framework for a universe with no action-at-a-distance. It is only with Huygens’s infinite series of levels that such action could possibly be avoided.

The debate continues, and it is intriguing….

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1. Tom Van Flandern. Dark Matter, Missing Planets & New Comets: Paradoxes Resolved, Origins Illuminated. Berkeley, California: North Atlantic Books, 1993.