# 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 comprehensible 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 possibilities. 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?

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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 Z 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
Possible duplicates: physics.stackexchange.com/q/9720/2451 , physics.stackexchange.com/q/35674/2451 , and links therein. Also related: physics.stackexchange.com/q/64197/2451 –  Qmechanic Jul 25 '13 at 7:36

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
@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
@JoshMacKay Indulge me a little on this topic to put forward an idea very dear to my physics thinking. We REALLY DO need the complex maths, because we have evolved mainly to see and understand the patterns that we came across in our evolutionary home - the wet savannas of late neogene and early quaternary Africa. This fact inevitably hinders our pattern recognition: we are prejudiced by what we learnt in that environment. Logic and mathematics are probably the only way we can overcome those prejudices, and build intuition for other patterns ... –  WetSavannaAnimal aka Rod Vance Jul 25 '13 at 7:59

We have developed an atomist model of the universe with the quantum being the smallest packet of energy. If matter and energy are quantized, one might make the qualitative leap that space is quantized as well. Quantized space does not violate any physical principles.

To assume that space is infinitely divisible when energy and matter are not seems to be a luxury.

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the one thing to remember, though, is that space, unlike energy or matter, is a dimension (or rather, multiple dimensions). A dimension, to speak slightly artistically, is at its very core, a physical realization of mathematics. There is nothing in nature closer to being a purely mathematical concept than a dimension. It's not a big leap to realize that as much as there must be countably infinite numbers between 0 and 1, there must also be countably infinite positions between two spatial coordinates. –  ACuriousJim Jul 25 '13 at 14:40
though there may be infinitely many countable positions between two quantum states, those positions always remain unoccupied. Just as an electron may exist in an excited orbital and a ground state, it cannot occupy intermediate states. For all intents, those intermediate states cannot be counted and so do not exist in reality. –  aquagremlin Jul 26 '13 at 4:44
This is just getting silly –  ACuriousJim Jul 26 '13 at 15:07
jim, there are not a countable infinite numbers between 0 and 1 in a dynamic universe.. only an understanding in our highly evolved brains of the mathematical construct of infinity and how it applies to numbers. There are not an infinite number of anything, including numbers. Until we have "invented" the infinite numbers they don't exist.. only the concept. This is deeper than just not having named the numbers.. the very notion of naming all the numbers generates a pretty big logical error.. ie, there will always be one more unnamed number to name. –  Josh MacKay Oct 9 '13 at 3:05
Different flavours of this same error always come up with when considering infinity in reality. –  Josh MacKay Oct 9 '13 at 3:06

The question kind of jumps around a bit, but I'll try to answer what I can in an intuitive and not complicatedly mathematical way.

Starting with the main issue at hand, is space infinitely divisible. I am going to say yes, absolutely. To prove it, allow me to present a relatively simple logical argument. Let us suppose space were not infinitely divisible. That would mean that in a given inertial reference frame, there must be a smallest possible distance. Furthermore, it would mean that all motions within said reference frame must be integer multiples of this smallest distance, otherwise I could travel a non-integer multiple forward and an integer multiple backward and wind up at a distance less than the smallest one from my starting point. Because space is 3-dimensional, I can naturally assume that moving at right angles to one's current trajectory would be allowed, since it merely represents moving in a different Cartesian direction. So let us picture a fundamental massive particle (I don't care which one) moving very slowly in this frame. Since any displacements over time or space are limited to integer multiples of our smallest distance, the smallest amount this particle can move is one "square" ahead; let's picture it doing that now. Then, due to thermal motions, our particle moves one square to the right. Now, how would I describe the new location of the particle relative to the old location? One block forward and one block right? I can only say it must be a non-integer multiple of our smallest distance; something between 1 and 2. But hold on, you say, perhaps that's allowable as long as no single motions are non-integer multiples. To that I say this, I never did specify what direction "forward" or "right" relates to. The truth is, there is no privileged directions, so at any given time, that particle should have been able to travel in any direction. That means this particle, to get back to its initial position (a necessarily allowable motion), could move in a straight line for that non-integer multiple distance. Even if we restrict motion to integer multiples, it would travel towards the starting point one block and the net displacement would be less than the smallest distance possible. This means that there must, therefore, be a smaller distance than what we defined as the smallest. But, since we defined no number, there must always be a smaller distance than whatever we choose to be the smallest. Thus, space must be infinitely divisible. Perhaps not infinitely measurable, but definitely infinitely divisible.

I would attempt answering the rest, but that's more of a statement of personal theories. All well and good, but not really addressable in the operational context of this website. Besides, I think my last point renders any answer I give as moot.

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"yes, absolutely"? If Infinity exists please give me a real life example of it. It's not possible because it does't exist and can't exist in a dynamic reality. We as intelligent beings can conceptualize what infinity is and even apply the concept to real world problems, but that is not proof of its existence. Your "proof" above that space is infinitely divisible is flawed. Why could matter not exist between two cells? I see no reason why it could not, in my ramblings above I considered that matter/energies motion through spaces' smallest constituents might be the electromagnetic effect. –  Josh MacKay Oct 9 '13 at 2:31
... that is to say that as matter passes from space cell to space between space cell to space cell it would exist as matter, energy, and matter respectfully. If the borders between cells were more of a gradient than a hard border it would very much resemble the electromagnetic effect... and possible explain the double slit experiment in an more einsteiniun fashion. –  Josh MacKay Oct 9 '13 at 2:42
... I am somewhat saddened that my creative thinking is not within the operational context of this website. –  Josh MacKay Oct 9 '13 at 2:46
@JoshMacKay ultimately which unprovable ideology you believe is up to you. I stipulated that matter could not exist between the two cells because that would constitute a distance smaller than what we had defined as the smallest distance. If this were allowed then the proof that we could not have a smallest distance would be trivial; it would be the same as the proof for no smallest distance between two real numbers. You posit that between space cells it could exist as energy. That would imply that a "between" exists and thus there are distances smaller than the smallest... –  ACuriousJim Oct 9 '13 at 13:38
the picture you have painted is not of quantized space made up of unit blocks the size of the smallest distance possible, it is instead continuous, infinitely divisible space through which matter moves as energy on distances below a certain threshold. For space to truly not infinitely divisible, there must be a smallest unit such that smaller units of distance do not exist. Furthermore, I am not trying to tell you that infinity exists, I am showing that an infinitesimal exists –  ACuriousJim Oct 9 '13 at 13:44

## protected by Qmechanic♦Jul 25 '13 at 7:29

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