# is space infinitely divisible?

As a child I remember hearing the popular paradox presented by Zeno proposing that Achilles could never catch a tortoise in a race since he would have to traverse the infinite space between himself and the tortoise. After some pondering I concluded that the obvious answer to the dilemma is that the space is not infinitely divisible. Although in later years I learned that this paradox can be explained by a finite infinite series, I have always found it difficult to accept that just because a concept can be supported mathematically, it must therefor exist in reality. In the case of Zeno’s paradox, although finite infinite series do exist, I believe the simple explanation I constructed as a child to be a better answer since it is both a viable solution to the problem, while also being comprehendible at a practical and intuitive level that is often lost with mathematical constructs. A few years ago I learned that the current scientific position is that space is in fact infinitely divisible and I have been disturbed ever since.

Does Infinity exist? In the literature the overwhelming answer to this question by the scientific community is yes... on paper. It is interesting to consider that in spite of a prolonged search for infinity in nature, it remains elusive. Although the importance of the concept of infinity in the field of mathematics is indisputable, ironically, it represents perhaps the most compelling example of a mathematical constructs that is “true” yet non existent in reality. Although many will argue that infinity exists in the singularity at the heart of a black hole, or perhaps at in some fields, when these phenomena are examined they become logically incomprehensible, mathematically error ridden, and physically non existent. If one were to weigh the evidence with respect to the physical existence of infinity, the overwhelming weight points to its non existence. If infinity doesn’t exist than space must be finitely divisible. If so, of what is space divisible into, and what are the consequences of its finite divisibility?

In searching for a possible constituent of the fabric of space one could consider whether the oscillations of electro magnetic states observed by the motion of matter/energy through space are a reflection of the matter/energy “train” moving through space, or perhaps an illumination of the “tracks” on which it travels. As a photon of light, buckyball, or large celestial body travels through space, could it be passing through (occupying), a mesh like network of electro magnetic “cells”? These cells could consist of an electro core surrounded by a magnetic space that causes oscillations in state as the matter moves from one cell to another. Perhaps this process is similar to the duel particle process by which electrons jump between orbitals... i.e.) The energy is particulate in nature while in the middle of each cell but it never occupies the space between each cell as it moves from cell to cell... instead it is exhibits wave like properties between cells.

Dr Richard Feynman once stated that the double slit experiment contains the only mystery of quantum mechanics. If one considers the double slit experiment from within the framework of an electro magnetic mesh through which matter and energy travel the experiment may be able to be accounted for in a more conventional manner than current postulations. Imagine the borders of the electro magnetic cells constantly moving in relation to the material containing the double slits as the earth, laboratory, apparatus, and particles being fired at the screen constantly move through the mesh like network. Depending on the position of the cell abutting the screen relative to the slot, the particle could be in a state of solid, defined, particulate matter, or in a state of diffuse energy (electro or magnetic). If the particle hits the slot in the screen in a state of diffuse energy it will diffusely enter both slits and interfere with itself. If it hits the screen during the solid particle phase it will pass through one of the two slits or hit the barrier and travel through none at all.

Although I am not able to support these ideas mathematically due to a lack of training, the model I am proposing has some interesting intuitive possibiliteis. These include;

In such a model could movement from cell to cell produce the electromagnetic radiation seen as matter changes velocity/direction when acted on by a force, or by the oscillation of atoms? Perhaps there is an electro magnetic “lag” of the vector a particle was on before force was applied to change its direction.

Could the motion of matter through a network of electromagnetic cells produce gravitational forces? Interesting... the more matter massed together the more cells being “activated” the more gravity being produced. If this was true one would also expect an increase in velocity to result in an increase in gravitational forces since more cells would be activated per unit time.

Could time itself be a function of motion through electro magnetic cell with time being relative to motion through cells similar to the example of gravity mentioned above?

Perhaps some of these questions have been addressed with research done on electron orbitals and energy states?

Thanks for considering these ideas and providing feedback,

Josh.

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why was this up voted? – user2963 May 13 '12 at 16:37
The best counterargument is that there is no evidence of a smallest possible distance. There are some theories that postulate fundamental length scales (LQG, for instance, quantizes areas at a fundamental length scale), but, as yet, there is no reason to believe any of this. On the other hand, the mathematics of differential differentials and contiuous functions is much, much cleaner than finitary matheamatics of systems with a large N. It's simiply easier to talk about the continuous interval (0,1) than it is to talk about $10^{34}$ discrete subintervals. – Jerry Schirmer May 13 '12 at 16:41
Hi Josh, and welcome to Physics Stack Exchange! Since you've written so much, it's hard to see the essence of what you're really asking without looking at the title. Could you try to edit the question down to be more easily readable by eliminating some of the less important details? – David Zaslavsky May 13 '12 at 19:52
Thanks for your feedback. I apologize for my rambling question. What I am really asking is if infinity were to not exist in reality could we then conclude that space must be comprised of finite bits and what would be consequences if this were the case. – Josh MacKay May 13 '12 at 21:35

One cannot do physics without mathematical abstractions, and in particular without the notion of infinity. But infinity cannot be observed - it must be extrapolated from what we observe. Thus all numbers measured are in fact rational, though we know that the diagonal of a square of rational side will be irrational, and hence have an infinite number of digits that is not periodic.

The physical theories that summarize our current theoretical understading of physics (namely quantum field theory and general relativity) assume that space is parameterized by three continuous variables, and hence is in prnciple infinitely divisible. But due to the Heisenberg uncertainty relation, arbitrarily small features cannot be observed except by expending arbitrarily much energy. (This is why the colliders used to probe the detials of the structure of enlementary particles must work with very high energies, and rising costs limit even more what can could be done in principle.)

However there are widespread speculations that in quantum gravity the small-scale structure of space and time should have to change. But due to lack of experiments and the difficuties of theoretical work, no consensus can be expected in the near future, and for a long time to come.

Regarding your ideas: Already to predict the effect of motion through electromagnetic cells requires that you adhere to the standard way of describing electromagnetism, which requires that you accept that space is continuously parameterized. Physics without continuity in space and time is bound to be severely crippled....

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Mathematical abstractions are clearly needed in physics however mathematics should be kept in perspective... Because something can be rationalized by mathematics doesn't mean it describes reality... It only means it could be describing reality. Although mathematics can prove a logical assumption incorrect, its absence from logical reasoning doesn't preclude the possibility of it being correct. General relativity was presented with very little mathematics supporting Einstein's ideas. Todays physics leans too heavily on uncomprehendable math rather than comprehendible reason. – Josh MacKay May 13 '12 at 15:49
I am not proposing adherence to the standard way of describing electromagnetism, quite the opposite. I am wondering if it is possible that electromagnetic waves are the unmasking of the constituients of space time as matter/energy travels through it rather than the self propagating electromagnetic wave described by standard electromagnetic theory. – Josh MacKay May 13 '12 at 15:59
The mathematics of current physics correctly describes reality with very few exceptions in cosmpo;logy, which are at present ill understood. If you propose a change in the description of electromagnetic waves you have the huge task before you to make sure that the vast amopunt of knowledge that matches the current successful description will also satisfy your alternative. - General relativity was only hardly compehended math before Eddington measured the deflection of light at a solar eclipse. Your judgment on today's physics is similarly unfounded. – Arnold Neumaier May 13 '12 at 18:16
Dr Neumaier, I am not meaning to pass any judgment on todays physics or the elegant mathematics behind it. Many people, my self included, with an interest in theoretical physics who are not career physicists find the divergence of physics from a tangible and comprehendible science to an abstract, mathematically driven science disheartening. I guess I am still holding on to the hope of one day being able to truly understand the nature of things... perhaps things aren't that simple and this will never be the case. Thanks for entertaining my ideas. – Josh MacKay May 13 '12 at 22:01
@JoshMacKay: But you do pass judgement when you say ''Todays physics leans too heavily...''. It has to! To truly understand the nature of things is to understand the mathematics involved in their explanations. Mathematics is just the language of precise concepts, and complex concepts such as ''the nature of things'' in general simply demand complex concepts, hence complex mathematics. – Arnold Neumaier May 14 '12 at 13:00
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