About the finite divisibility of physical objects: rather than the need of a distance interval to insert in a division, mentioning atoms as roughly indivisible parts of material objects, would be more directly clear, wouldn't it ?

"there is a limit to the number of extended points or fundamental lengths, but there is also no distance between those points, since that distance will also consist of points."

There is a limit to the number of atoms, but there is a distance between atoms. That distance does not consist of atoms ; it is subject to a quantum uncertainty, that is smaller than this inter-atom distance (in the case of a covalent bound), but still quite bigger than the Planck length, which is thus irrelevant here.

"where to cut is a problem" : I don't think butchers see it so.

"Then, as you turn the screw further, beyond the sub-atomic scale of quarks towards the fundamental scale of length, ~ 10-35 m, the unexpected happens. At the scale of the fundamental building block, the blue extension is ultimately made of a featureless length, same with the yellow, green and red strips. At that fundamental color-blind scale, how is the boundary to be demarcated, say between one blue strip and another blue strip or between a blue and a yellow strip?(...) reality is fundamentally One thing"

You are telling a lot about what happens at the Planck scale. How can you know that much on what it actually looks like at that scale ? Colored objects are made of atoms; the information of color is carried by these atoms. Thus the division defined by colors cannot go smaller than the inter-atom distance. There is no point to develop qualifications about what happens when trying to physically cut at smaller scales, since we have no way to even try physically cutting things there. All we can do is take our best mathematical theories of physics, which were verified by experiments, and discuss what they mathematically define at smaller scales (quantum fields, structure of nucleus...). But we still don't have verified mathematical theories of what happens at the Planck scale, so I see no sense of even developing a discussion about it, as you did, as if we did have things to say there. Personally, I think it is more interesting to discuss the things we know (quantum physics), which already have interesting consequences on the issue (position uncertainties of atoms at the atomic scales), than writing an essay about how things look like at the Planck scale (which we actually don't know) in ways that ignore the known quantum uncertainties at the atomic scale.

"possibly at a future time, [space] would collapse, reduce to the minimum length and disappear (Big crunch)" : this was considered among scientific hypotheses before the discovery of the acceleration of expansion by dark energy, but since this discovery, the Big crunch is not considered scientifically plausible anymore.

"Quantum mechanics, an object, e.g. an electron at a location in the atomic orbit can without seeming to traverse the 'space in between', arrive at a different location, the so called 'quantum jumping'"

The electron does not jump from any definite position to any other definite position, because it never has any definite position. "Orbits" are not states of definite position, but states of rather definite energy. To be more precise : quantum states of the atoms are in a Hilbert space where can be defined a Hamiltonian operator that distinguishes "distinct energy levels" by its eigenvectors. Depending on circumstances, we can consider a given atom to be in a specific energy level in the list of possible energy levels (eigenvectors of the Hamiltonian), but nothing generally obliges this. The energies of these levels are defined down to a quite thinner accuracy than the intervals between these energy levels themselves. However they are still not infinitely precise, since atoms are not isolated systems but interact with their environment, at least the electromagnetic field. In particular, the typical time it takes to emit a photon, defines a quantum uncertainty on the interval between energy levels themselves.

So, since orbitals are not states of definite position, any question of "space in between" is senseless.

"extended points may not be eternally existing physical objects"

Before wondering if they are eternally existing, a first question would be whether they are physical objects at all. Basically, "extended points" is just an English phrase. It is not clear if this phrase can make any sense in physics, and which one, depending on a theory of physics that we need to specify as a reference if we want to give our English phrases any precise meaning.

"but can appear and disappear spontaneously, or when induced to do so"

Do you allow for a quantum uncertainty on whether or not they did ?

"I crave the reader's indulgence here to temporarily set aside current doctrine on the eternal existence of points"

I never heard about any such doctrine, as concerns physics. (There is no such doctrine in maths either, since "eternal" should refer to a stability along time, but Euclidean geometry does not admit any concept of time in the first place, so that there is no quality of eternity either to be formulated and claimed in the mathematical language of geometry).

I might continue another time with the last pages, but to tell my general opinion: I see this essay as rather boring, with ideas more repeated than developed, and not well related enough to what is actually known about physics. On roughly the same topic, I found the essay by William T. Parsons, "Are Boltzmann Brains running Hilbert's Hotel?" much more interesting and relevant. See my comment there about the issue of the infinitely small.

18 days later

Akinbo,

I have been thinking more about your question. It seems to me that the problem reduces to whether or not integer numbers are a subset of real numbers. Here is what I mean by this ... integer numbers are treated as though they have an infinite amount of precision ... i.e., 1 is equal to 1.0000000000 ad infinitum. Most measurements are real numbers of some sort and they have a fixed amount of precision ... such as 1 inch = 2.54 centimeter or something similar. So, if it is possible for a real number to have an infinite amount of precision, then space can be infinitely divided. If integers are a subset of reals and integers have infinite precision, then reals should also have infinite precision. I was taught that integers are a subset of the reals ... but perhaps that is not actually true.

Best Regards and Good Luck,

Gary Simpson

    Dear Gary,

    Thanks for your interest and comment. You are likely a better mathematician than myself but let me answer as best as I can.

    By the statement, "if it is possible for a real number to have an infinite amount of precision, then space can be infinitely divided", I take it that if a real number CANNOT have an infinite amount of precision, then space cannot be infinitely divided. By infinite amount of precision, I take this to mean as conventionally held that between any two real numbers there is always a third.

    Now, this definition of real numbers is an assumption, not a found and proved fact. That being the case, suppose this assumption is wrong? Supposing space is not infinitely divisible? What becomes of our physics and mathematics? Do we have a Plan B? It is this Plan B, that I have been working on and the focus of my essay this year and in 2013. The need for this Plan B also seems necessitated by the various paradoxes that plague Plan A.

    Here, let me again quote an eminent mathematician, Roger Penrose in his book, The Emperor's New Mind, p.113... "The system of real numbers has the property for example, that between any two of them, no matter how close, there lies a third. It is not at all clear that physical distances or times can realistically be said to have this property. If we continue to divide up the physical distance between two points, we should eventually reach scales so small that the very concept of distance, in the ordinary sense, could cease to have meaning. It is anticipated that at the 'quantum gravity' scale (...10-35m), this would indeed be the case", then further on,

    "We should at least be a little suspicious that (despite the logical elegance, consistency, and mathematical power of the real number system) there might be a difficulty of fundamental principle on the tiniest scales", and "This confidence - perhaps misplaced-..."

    In my current essay, I raised the issue how to divide a real number line, if there is always a third element between two elements and going by geometrical considerations these elements are uncuttable into parts, i.e. there is a point or number at each incidence of cutting and points (or the real number of infinite precision that you mention) cannot have parts (or an imprecise part thereof).

    I am not too sure I have made my point clear enough, but you are welcome to clarify any aspects. With Calculus - Revision 2.0, you are the boss! :)

    Thanks,

    Akinbo

    Akinbo,

    First, let me offer congratulations. It looks like you will make the finals. It is odd that no announcement has been made.

    Regarding my statement ... "If it is possible for a real number to have an infinite amount of precision, then space can be infinitely divided" ... The logical contrapositive is ... "If space cannot be infinitely divided then real numbers cannot have infinite precision" ... Your statement is the negation of mine rather than the contrapositive. A contrapositive is always true. A negation is not always true but often it is true. In this case, they feel about the same to me. The answer to your question is "Yes". In my opinion, there is a direct link between the divisibility of space and the precision of real numbers.

    Regarding the paper by Dr. Phips, I liked it. He raised many good issues. His explanation of GPS timing was very helpful to me. He also mentioned a lateral force associated with the flow of electricity that he believes is not presently considered. The essay by Dr. Neil Granaeau (spelling?) mentions the same force.

    One of the essays discusses the difference between mathematical infinity and physical infinity with physical infinity being impossible. He created a new symbol for physical infinity and called it "Bravo". The symbol is a "B" with a little tail on it. The inverse of his "Bravo" is similar to the smallest increment that you are seeking. Perhaps you should simply choose a symbol and define it as the smallest distance possible and then work through some of the math. Presumably, the Planck length will be related to the symbol that you choose. You can then define a set of alternate "Real" numbers that have a finite amount of precision. Quantization will be built into this number system.

    Here is something that might be of interest. In mathematics, there is a theorem named the Lagrange Four Squares Theorem. What it states is that any integer can be constructed from the sum of the squares of four integers. Basically, you start with the integer one and you can construct all the other integers. This is applicable to your question because all rational numbers can be produced as the ratio of two integers and all integers can be produced by the four squares method. Therefore, the only numbers that remain are the irrational reals such as e and pi and sqrt2, sqrt3 ... etc. It seems to me that the number system that you are looking for is very intimately related to irrational numbers.

    Best Regards and Good Luck,

    Gary Simpson

    a month later

    Akinbo,

    As requested, I've read the text of the SR essay that you provided. Firstly, let me state that I do not consider myself to be very knowledgeable regarding SR. I am aware of the paradoxes and I studied the subject very briefly in a Physics class in college. People who take a degree in Physics dedicate an entire semester or more to the subject. My background includes only a few lectures, some homework, and a test.

    I have a few comments and observations.

    You should put your name, a date, and a revision number in the title block on the front page.

    On page 3, you apply SR to photons. I don't think SR allows that. Your details are correct, but they contain an assumption that SR does not allow. You need a third reference frame to measure the motion of both photons. That reference frame is your point O. The problem here is that SR is limited to two reference frames at once. So, you can use O and one photon, or you can use both photons. But you cannot use O and both photons.

    So, the only thing that the observer at O can state is that the two photons returned at the same time after 2 years. My understanding of SR does not allow me to say how much time passed for the photons. It appears to me that they are timeless.

    Having said that, I agree with you. You SHOULD be able to use as many reference frames as you want. And you SHOULD get the same conclusion for every reference frame. That is one of the things that troubles me regarding SR.

    On a related matter, what you present is part of the logic associated with "pair production". Photons are typically created in pairs. The reason is that the photon has momentum and momentum must be conserved.

    When AE defined the velocity of light, he used the round-trip time in a manner that is essentially what you present. To my thinking, this obscures something very important. For this thought experiment to happen in reality, the photons must be emitted at O. They must be absorbed at the mirrors. They must be emitted again at the mirrors. And they must be reabsorbed at their point of origin. Therefore, the absorption and emission character of light is built into SR. Essentially, I think that making sense or SR will require a solution to the mechanics of absorption and emission.

    I suppose it is possible that dark matter is a wave medium. But it is also possible that dark matter is not real but instead is the result of incorrect understanding of gravity.

    Best Regards and Good Luck,

    Gary Simpson

      Gary,

      I will incorporate some of your advice into any possible paper that may develop. Many thanks for your very useful comments.

      Generally speaking, and based on my understanding Galilean relativity predict that the two photons will return in 2 years, but Special relativity does not permit the addition of velocities to be above c, and so in that case the photons will return in 3 years. All in all I am satisfied with your comment.

      Regards,

      Akinbo

      Akinbo,

      The value of two years is based upon reference frame O. The three year value that you use mixes reference frame O on the trip to the mirrors and the reference frame of the photons on the trip back. You must use the same reference frame for both parts of the journey.

      Good Luck,

      g

      5 months later

      Greetings Akinbo -

      Are you close to Lagos? I passed through to catch a cheap flight to New York and then my home in Louisville KY on returning from Peace Corps service building fish ponds in Cameroon in 1992. Hope the maniacal Boko Haram menace poses no direct threat to your life and work.

      Pardon my long delay in replying to your comment (repeated below) on my Trick or Truth essay, but some personal struggles with divorce attorneys (and the US surveillance state's sniffing the cellphone-panties of its so-called allies) have led once again to a few months of incarceration. In full and lifelong but peaceful revolt against state-sponsored espionage and the global rule of politicians' manmade laws over the natural laws of science, I'm also battling the absurd inconsistencies in physics relating to big-bang creationism and to the zoo of alleged particles of zero dimension assumed to be simultaneously multidimensional waves??

      The predictive power and self-consistency of math and logic, driven by conflict in theory v experiment, prodded Einstein toward his novel assumption of light-speed constancy thereby refining Newton's magical action-at-a-distance into Riemannian spacetime curvature. I'm betting the same math and further novel assumptions - again driven by confusion such as dark matter, dark energy (and why not dark time?) - will do the same regarding priests Lemaitre, Guth and Hawking's magical inflation of reality from a mythical primeval-atom singularity.

      "Extraordinary claims require extraordinary evidence", and the night sky will provide it once technical sensitivity advances sufficiently (if it hasn't already?) to permit detection of fluctuations in the waves of CMB temperature anisotropies: these waves are not a static ancient postcard from the last scattering surface at cosmological reionization, but rather dance like the surface of a gentle sea in a cadence set by the collective stochastic swarming of galaxies through gravitation's 4 dimensionality. COBE, WMAP, PLANCK and WEBB aren't showing us baby-picture postcards from the remote coastal sea at eternity's birth, they're showing us we are floating dynamically on that very sea all around and within us in the form of unobservable spacetime curvature's galactic-scale geodesics and the quark-scale zero-point ether-energy of quantum jittering.

      Such novel ideas never take hold peacefully among the titanic professorial intellects battling ruthlessly for scientific funding supremacy in the zero-sum, winner-take-all, one-theory-wins-only-by-destroying-competitors game of taming reality into the cage of grand unification; an early adherent of continuous creation of our eternal universe named Giordano Bruno was burned alive by the Pope and his priests of fanaticism in the year 1600 while Galileo quieted his visionary courtship of Truth rather than be summarily assassinated by the monstrous power of religious superstition.

      The somnambulant spirits of Bruno and Hoyle fuel my revolt against the conventional witchcraft of limitlessly tunable accelerating inflation, a Rube-Goldbergian theory so malleable as to embrace any observational wrinkles whatsoever arising through observation to confront it. As an applied mathematician, the acceleratingly inflationary model of an expanding universe is beyond abominable, it is an outright rape of the elegant self-consistency and falsifiable predictability of relativity itself.

      Would like to respond to the issues you raise below if you're still active here at fqxi... do you have a gmail address? I'm at kevinbootes09ATgmailDOTcom

      (Hello Kevin,

      I would not know your motivation for writing in this poetic way. Perhaps, to hide the many hidden truths, which I find, from those who do not desire that truth return to our physics.

      I will not engage you on the Michelson-Morley experimental results which you pointed to.

      I will first like you to tell me where this wonderful tourist destination called FlatLand is?

      Second, concerning the photon, someone in this community, Armin Shirazi wrote a paper where he discussed a 'photon existence paradox', although he claims to still support SR. It goes somehow like this, if it is as it is claimed that time does not flow for a photon, then it follows that the time of emission of a photon is the same as the time of its absorption, how then can photon exist?

      All the best in the competition.

      Regards,

      Akinbo)

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