Essay Abstract

In this essay, I attempt to raise and address some questions concerning the procedure of dividing extension mathematically, which is mental and the actual act of doing so, which is physical. By extension is implied a length, whether of matter or of a distance. And by dividing is implied the procedure of creating parts. Inevitably, such a discussion would touch on the continuity or otherwise of extended parts. I end the essay by moving the motion that we exorcise the lingering millennia old Parmenidean spell cast on our mathematics and physics and allow that whatever exists can perish.

Author Bio

I am a practising physician with keen interest in foundational physics topics. I have authored a published paper and other unpublishable ones. I also enjoys 'dialectic' with physicists over the internet.

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A little grammatical correction and a small question arising from what discuss in my essay...

CORRECTION: The last sentence in my Bio should have read: I also enjoy 'dialectic' with physicists over the internet.

A LITTLE QUESTION: I had posed a question to Pentcho Valev on Feb. 4, 2015 @ 19:51 GMT on FQXi forum topic: The limits of mathematics, which I am also interested in discussing here.

When we write 2 3 = 5,

is it a very, very, very high probability that when we add 2 and 3 we get 5 or is it a certainty?

Thanks for reading and commenting.

    Dear Akinbo,

    You have certainly taken an elemental notion and worried it to death. But as you point out "Till this day the issues raised here have not been fully settled in physics and philosophy..." A few observations. On page 7 you mention that cutting of matter results in "the creation of space between the divided parts". But was the space already there, simply filled by the matter? Is what you create "empty space"?

    Your point about curvature implying composite is interesting. I don't quite understand your discussion of extension as starting to exist and ceasing to exist and the discrete nature of otherwise syrupy space becoming manifest. I tend to view the field or continuum as the fundamental reality, which I guess is your "syrupy space". I found your essay on comparison of the 'math versus physics' meaning of division well-written and fascinating. You discuss Zeno's Dichotomy Argument. My essay concerns a dichotomy, that of whether spin only exists as the dichotomy of 'up' and 'down' (or a superposition thereof) or whether this is simply a characteristic of the state of the "measured" spin based on alignment or anti-alignment with the measuring field.

    You say "current mathematical doctrine does not appear to have allowance for what is to perish." Similarly current physics does not like the idea of information perishing. But if a "non-aligned spin" enters the magnetic field and becomes aligned, either the information associated with its original orientation vanishes or it is transformed into a positional deflection, which as I note implies QM is incomplete.

    Judging by my downward trending score this not a popular argument, but as it should be experimentally testable, low scores will not change the fact that quantum mechanics may be proved to be incomplete. I invite you to read my essay and comment.

    Good to see you back!

    Edwin Eugene Klingman

      Dear Sir,

      Your definition of division is too restricted. You can universalize it by following the definition of number in our essay. There is a limit (call it the Planck scale if you want) up to which a number can be divided. Physically, it is the quarks. Zero is not nothingness (no magnitude), but something that does not exist at here-now. One is without similars, where the dimensions are fully perceptible. Infinity is not a very big number, but something without similars, where the dimensions are not fully perceptible. Thus, no mathematics is possible using infinity, as everything becomes indistinguishable. Similarly, we have shown that mathematics except multiplication by zero leaves the number unchanged. Regarding e = mc^2, please refer to our essay.

      Mathematics is the quantitative description of Nature, whereas physics is its physical aspects. Hence both are related and cannot be separated. Multiplication and division are non-linear accumulation and reduction of partly similar related objects. A knife does not divide bread; it is only an instrument for the operation of division, which is taking away a part from the whole. This leads to discontinuity of the whole bread. You also admit it when you say: "The parts become discrete entities and no longer continuous, with "something" separating them". The process of division by a knife requires energy to be applied to it to displace a piece from the whole. Since energy is applied by the hand, an equal force acts on it, though it may not be evident always.

      Division ad infinitum is only imagined and not possible even mathematically. Since "one" is without similars, it becomes "many" in division. In a fraction, the new one becomes the unit, out of which the denominator indicates the total number of the new 'one' out of which the numerator indicates the number of 'one's taken out. Thus, it can continue only till the minimum describable quantity (fundamental scale of length, ~ 10^-35m, as you put it). For example, you cannot divide a quark further. Mathematics also stops here.

      When you say "the extremities of a line are points", you admit its existence as with or without similars (one or many points separated by space). Introduction of space is cutting a line. Points are discrete. If line is the locus of a point, it implies continuity and not discreteness. Thus, the principle applied for a fraction mentioned above applies.

      When you talk about dividing line and divided line of an object, you imply interaction of two axes. They are parts of dimension of the same object (occupying three dimensional space or two dimensions of the graph) representing two different dimensions. Thus, they are not separate objects. If you talk about intersecting lines, then also they would divide the space on the graph or a paper. Hence they can co-exist. The knife creates space for itself by dividing the bread, which no longer remains one object. If you cut a rock with the blade, it will not give space and your operation becomes futile.

      Distinguishing one part from the other is fundamental to the number theory; hence mathematics and quantitative physics. The simplest answer to Zeno's paradox is that velocity is related to the mass of the body that is moving, the energy used (force applied) to move it and the total density of and the totality of the energy operating on the field. These are all mobile units against the back drop of the field that is static with reference to these. Middle of the distance is related to the frame of reference, which is relatively static with reference to the other mobile aspects. Thus, it is like comparing position and momentum. They do not commute. Hence there is no paradox, which is borne out of experience. While the middle of the distance is gradually reduced, the velocity is not reduced by the same proportion. Hence the runner will reach the end point.

      Regards,

      basudeba

        Dear Sir,

        Should we be concerned about scoring points or finding the truth? We go for the later. Scoring points is harming progress of science, as often papers are presented by incrementally building on 'accepted theories', even when such theories have been known to be untrue or non-existent like dark energy and extra dimensions, as pointed out in our essay. There is a need for reviewing and rewriting science.

        Regards,

        basudeba

        Thanks very much Edwin Klingman for stopping by and more importantly for pointing to areas that need clarification.

        On your question, "But was the space already there, simply filled by the matter? Is what you create "empty space"?"

        Thanks for raising this, as it is something that may similarly agitate others. So let me explain by first asking: Going by Euclid's definition and what historically led to it, can there be more than one point at a point? Can two points, one relating to matter and the other relating to "empty space" occupy the same point? If that were so, will it be correct to still say there are no parts at a point and that a point is indivisible into parts, one of matter, the other empty space?

        On the question of 'starting to exist and ceasing to exist', this is a reasonable position you might take, if you believe that what exists cannot perish. Do you believe the cosmology that the universe can start to exist and cease to exist? Do you accept the quantum description of virtual particles popping into existence and subsequently perishing? If you do not, I won't blame you. But if you do, then my essay uses this to challenge the idea of the continuum as the fundamental reality. The universe, which is apparently a continuous extension, and which if it is still expanding, all of its extension are not of the same "age". Possibly also at a Big crunch, some of the extension will perish before others. If parts of the universe's extension, are of different ages of existence, then the universal seemingly continuous extension will not taste the same (using wine for analogy, old wine will taste better than new). "Time" will bring discreteness to the otherwise continuous and syrupy universal wine.

        I will be commenting shortly on your essay, which I have read along with the long 'dialectic' with Tim.

        If you still have time, you may wish to tell me how to cut a 'continuous' line, which consists of an infinite number of uncuttable points. The usual description is that between any two points, there is a third, but in continuum view I am yet to hear that at any point there are others.

        Best regards,

        Akinbo

        Dear Basudeba,

        Thanks for reading my essay and the lengthy reply. It makes one happy when fellow seekers after truth take from their precious time to do this. In replying, I will be using parts of your response to interrogate the issues in contention, while not claiming to understand all the statements in your post.

        You say my definition of division is too restricted. That may be so, but could you expand the meaning? Your suggestion that this can be universalized by following the definition of number and at the same time saying there is a limit (call it the Planck scale if you want) up to which a number can be divided does not seem to mathematically apply. For instance, to a mathematician there is nothing unusual dividing an extension say of 10-35m into a million places, each 10-41m in length. So, there is no such limit in mathematics, contrary to your opinion that "mathematics also stops here".

        In your 4th paragraph, Yes I do support the existence of points in physical reality, not just in the mathematical realm. The geometric point being the fundamental unit of extension, i.e. space. When you therefore say "one or many points separated by space", would that separating space itself not consist of points?

        In your 5th paragraph, you said points of the dividing line and the divided line can co-exist. Do you mean that there can be a multitude of discrete points at the same point? Using your term, can there be many 'ones' in a 'one'? If so, would they not add up to something other than 'one'?

        On Zeno's paradox, I agree tentatively to your use of velocity to find a solution. But what is really at stake is the number of positions between Atalanta and his destination, and not his speed. No matter how high his velocity, if there are are an infinite number of positions to be traversed before reaching his goal, Atalanta will not be able to reach the end point since as you point out in your words, 'infinity is not a very big number, but something without similars, where the dimensions are not fully perceptible', BUT in Atalanta's case, distance to destination is fully perceptible and measurable before the race starts, and it has similars with other racetracks in my opinion. Atalanta will therefore get nearer and nearer his end point but never taking that LAST, fractional step that crosses the finish line. As I said in my essay, mathematical solutions have been proffered, although that last fractional step is remains mysterious still even in those solutions.

        I will check more about your fascinating ideas when I read your essay over the weekend. And by the way, do you think your 'One' can perish?

        Best regards,

        Akinbo

        Dear Sir,

        This is in addition to our earlier post. We will reply to the points raised by you separately.

        When you look at the various colors in the strip, you could distinguish the different wave lengths of radiation emanating from out of it. When you reach "fundamental extended lengths which being indivisible" is not distinguishable from others, you are looking at the radiation of minimum wave-length. But what about the mass that emits radiation? It is like looking at various macro objects. When you reach quark level, all look same. Only at this scale, mass and energy become equal. But then everything requires a ground or base or medium to exist or propagate. That is the so-called dark energy (which you call 'syrupy' space). Just like the smooth and persistent background structure does not interact with the object, the so-called dark energy also does not interact with mass/energy. It did not show in the M&M experiment because light is a transverse wave, which is background invariant.

        What you call 'dividing line' cannot be of zero breadth, because, physically, it must be distance between two points in three dimensional space. Mathematically, it has to be drawn on a paper, which also is not one dimensional. The problem arises because you are ignoring the background. This mistake appears as the problem of dark energy. Just like emphasizing red-shift to infer expanding universe (which is now questioned after discovery of galactic blue-shift and merger) has led to the problem of dark matter. The galaxy rotation curve can be easily explained with Keplerian mathematics if we accept that the universe is static but rotating on its axis. Starting with 'big-bounce' on the background structure, it can explain 'inflation' differently. This harmonizes with your view that "space appears not to be an eternally existing entity" and "time as the separator of minimum lengths", though it has to be explained properly.

        Quantum jumping is not that mysterious. Science knows all about electrons except what it is? The same is true for photon also. This creates all confusion. Photon is the point of intersection of the electric and magnetic planes with their direction of motion. When a ship moves in sea, it expels water in front sideways, which reunites at the back. A similar phenomenon is seen in magnetospheric reconnection. Something similar happens with quantum jumps. It becomes visible only at shifting intersections with the right orientation.

        Time is not a dimension in the same sense as the other three space dimensions, which are invariant under mutual transformation. We can exchange any of the space dimensions with any other without disturbing the structure, i.e., the interface between the external relational space with the internal structural space. But such transformation is not possible with time. Thus, we describe these as a set of six components: being (situation leading to its creation; or as you put it, motion - the substance of existence), becoming (its creation itself; or as you put it, time), growth (due to addition of other molecules, which, along with the two other factors following, can be in the three spatial dimensions), transformation (as a result), transmutation (due to the same effect - incompatible addition), destruction (change of form as a consequence; or as you put it, duration - the instance of existence). The motion transformations are perpetual (due to inertia) and deterministic processes. The duration transformations are action induced by the freewill of a conscious agent. This answers your question "what if what is can perish?" Evolution is cyclic.

        Regards,

        basudeba

        Dear Sir,

        Mathematics is the science of numbers that describes quantitative aspects of Nature. Nature is what exists, is intelligible (knowable) and communicable (describable). Numbers are properties of all substances by which we differentiate between similars. If there are no similars, it is one; otherwise it is many, which is successive perceptions of 'one's at high speed. Below Planck scale, you cannot perceive anything. Thus, number stops there.

        Separation of points by space would not be points, as space is the interval between objects that exist. Point has existence, though no dimensions. Thus, space is the interval between points also.

        Since everything is three dimensional only, there cannot be a multitude of discrete points at the same point, as point has no extended dimension, but the intersection of three of its components.

        The distance between Atlanta and his destination is neither infinite nor changing, whereas his own position has been replaced by momentum, which is ever approaching destination covering fixed positions. The paradox would have been valid, had the space also expanded or his velocity reduced proportionately.

        Regarding whether one can perish, please read in our other comment.

        Regards,

        basudeba

        Hi Akinbo,

        I still don't quite get the "two points, one relating to matter and the other to 'empty space', occupying the same point." I see points as conceptual overlays on reality. The coordinate system is defined to consist of these conceptual points, which identify location. What is actually at that location, whether matter or 'empty space' is irrelevant. I don't see the "two points" as colliding or in any way interfering with each other. They aren't "real" in my view.

        The question of whether the universe could start to exist is tougher. It's hard for me to understand this, but it's impossible for me to understand the universe that "always" existed, and the 'known' facts don't seem to support this. But now that it's here, I don't see it going away entirely. Nor do I have much faith in a 'cyclical' universe and none at all in the multiverse. One unitary universe does it for me.

        And no, I don't believe that virtual particles pop in and out of existence. If there is highly concentrated energy, as at the LHC, it can 'spawn' particles, but they don't pop in and out of empty space. It's more like the energy 'condenses' into relatively stable local configurations. Feynman's pictures represent perturbation terms in an expansion, and are not necessarily to be taken literally.

        I experience and am aware of continuity, and I do not believe that, other than through limiting processes of calculus, math has much to do with the physical continuity of reality, as it is a symbol system of truths, so I don't try too hard to formulate an accurate conceptual narrative to tell you how to "cut a 'continuous' line, ... consist[ing] of an infinite number of uncuttable points". But I very much enjoy reading your attempts to do so.

        And I'm happy with your wine analogy.

        I thank you for reading my essay and for your comments. John Cox left me a little note after my response to you to "lighten up". I'll try to do so. Going against the flow of current beliefs is tiring and frustrating, but as John points out, selling my ideas is probably not best accomplished by attacking others ideas disrespectfully.

        Edwin Eugene Klingman

        Akinbo,

        A pretty good read. Thank you.

        I will say that there is another way to think about division. That is as the inverse operation of multiplication. This effectively sidesteps many of your concerns since the act of division simply returns the thing that is divided to its pre-multiplication state.

        Obviously, that ignores situations such as those that are the subject of your essay ... namely things that cannot clearly divided, or whose division is ambiguous.

        Something that I have never understood regarding the concept of a Planck length ... I understand that it is intended to introduce a certain graininess to the universe, but the electron is considered to be a point particle in the truest sense. Therefore, why does the universe need to be grainy? Surely the electron is part of the universe.

        Oddly enough, I have thought of the possibility of motion being a creator and destroyer of space. I have also thought of the possibility that a particle is created in the direction in which it is moving and destroyed in the direction from which it moves. The two general ideas seem to be pretty complimentary.

        Best Regards,

        Gary Simpson

          Gary,

          Thanks for offering me this fresh insight and new way to think about division. From this perspective, it would appear clear that that which is to be divided or multiplied must therefore have some finite NON-ZERO value. If that is the case, are you then of the opinion that there is no limit to division?

          The Planck length is suggested to be in some sense that limit to division, a fundamental sort of length.

          What do you understand by point particle? Is it a particle that has mass and is of zero dimension and thus infinite density? Or does your point have a dimension? Many mathematicians postulate that the point is of zero dimension. In my 2013 essay I discussed the history behind this idea.

          I am happy you have yourself contemplated these ideas about motion. The two ideas are complimentary as you said. Indeed, the second is the way motion is depicted on a computer and TV screen. I similarly thought about the possibility before opting for what I discuss in my essay, which also resolves the other Arrow paradox by Zeno, which you can view here and here.

          Regards,

          Akinbo

          Akinbo,

          To me, it is possible in mathematical abstraction to divide something as many times as desired. In practice, this is not true. So, it seems that you have identified a difference between mathematics and physics. Namely the concept of divisibility (or non-divisibility).

          So perhaps the Planck Length has meaning for empty space but not for particles such as protons or neutrons.

          It is easier to multiply numbers than it is to divide them. I was taught multiplication before I was taught division. So, division is thought of as the inverse operation of multiplication. That makes it easy to divide something if that something is the result of a multiplication. This is perhaps not useful if something can not easily be divided ... for example half of a proton. Of course, a distance could be half of the diameter of a proton.

          Best Regards,

          Gary Simpson

          Regarding the electron, my understanding is that it has no physical dimension and that it has mass. Therefore, it has infinite density. My inference from this is that we do not have a correct understanding of the electron. I have presented some speculation regarding the electron in two papers that are posted to viXra.org. In the first post in my forum, I list the web address of those papers.

          Thank you for reading my essay.

          I am thinking - now - that many physical laws, describing a mathematical world, does not have access to the real world (if describing material objects); there is ever a minimum dimension where the divisibility ad infinitum is not applicable. So that a mathematical object (the physical law) have a limit of applicability in the real world.

          Depends on how you have defined the operation and what you mean with the question. Provided that you mean the normal addition law in the natural numbers, it's a certainty, if you ask for mathematical proof. If you ask for the physical/observational truth, you will ever only get a high probability.

          Sophia,

          Thanks for your comment. In other words, I get from your comment that Mathematical truth is not necessarily the same as Physical/Experimental truth, even though they may be close.

          As I mention in y essay, even for Mathematical truth there is an unstated, underlying assumption that things that are being added cannot perish during the addition process. Is this a certainty even for mathematical truth?

          As I commented elsewhere, I speculate that it is not a certainty that 2 3 = 5 but a very, very, very high probability, with the probability increasing with the size of the object being counted and reducing with the size of the object. In other words, 2 house 3 houses = 5 houses is more likely to be correct than 2 electron 3 electrons = 5 electrons. Not necessarily because of the experimental difficulty in identifying an electron but as I discuss in my essay, it is more likely for an electron to perish than for a whole house during the process of counting to determine the sum total.

          Regards,

          Akinbo

          Dear Akinbo,

          Thank you for the comment you made about my essay WHY THE REAL UNIVERSE IS NOT MATHEMATICAL My essay explains how the real Universe is occurring. Reality does not have an abstract uncommon, but abstract interesting perspective. I did not mistake an abstract image of an abstract object for the abstract object itself. You did that. Abstract images may be abstractly conveyed by abstract traveling abstract light, however, real light can only appear provided it is seen as adhering to a real surface that is traveling at the constant speed of surface.

          Only an abstract universe could come from an abstract nothing.

          Regards,

          Joe Fisher

            Dear Akinbo Ojo,

            Wow. You obviously put a lot of thought into this essay and I congratulate you on your effort. I found it very challenging to read partly because I couldn't see the "point" (no disrespect, but pun intended). I realize that it is typical of philosophers to pick a detail such as you did and obsess over it until they have pulled every possible string to its limit, but as an applied physicist, it is hard for me to follow without a clear end in sight.

            However, in regards to your hypothesis about time, you said, "Of course, not everybody accepts the cosmological theory, so we are assuming its correctness." Well, in fact, I disagree with the big-bang theory as well as interpretations of singularities as being black holes in spacetime. As I explained in my essay (Doctors of the Ring...) these are mis-perceptions of unified concepts (metaphorically represented by the "golden ring"). If you look at a circle from its side (or as a shadow on Plato's cave) you will see a line. And if there is a particle moving continuously on the ring, the shadow would make it look like an oscillation from one end to the other and back. If you believe that perspective, you will be trapped (under the spell) and won't rise above the plane of opposites to see their unity.

            Because the whole cosmological theory is based upon the notion that space is 3D and time is 1D, which is a lop-sided expansion of s = ct, correct mathematics has led to a lop-sided interpretation. It's a foregone conclusion.

            I have been working on a model of space-time-motion (which I am about to submit for publication) that represents space and time as conformal projections of motion onto a two-dimensional S-T plane.

            *Motion is what is real; it gives matter form. The word motion represents a complementary (unified) concept, i.e. "motion" is a single word used to express complementary antonyms (moving and not moving or at rest); yet the moving state can be expressed in terms of gradable parameters (displacement (s) and clock-time (t)),

            *The gradable parameters, s and t numerate (i.e. quantize) and denominate (i.e. reference to standard time scale) the moving state to provide a gradable spectrum by the ratio, v=s/t;

            *The speed of light in natural units is simply the state at which the ratio is 1:1

            Although I don't follow your logic, I think I agree with your statement that "time is the separator and conferor of discreteness on space". My reason has more to do with the fact that time is the scale that denominates motion, thus giving it a unit magnitude of change. The "curse of Zeno" may be resolved simply by shifting the reference from zero to unity. Any unit of measurement is defined by the "unit", i.e. one unit, not zero. "Zero" represents nothingness so any physically measurable model that includes zero length or displacement, is subject to Zeno's paradox. The concept of time is also subject to the "curse". A unit of time is "measured", although it is not a physical observable, by the motion of a clock (or grains of sand etc). So setting t=0 for analysis of a moving particle means zero relative motion, which is the rest state, so the moving model does not apply.

            I hope I made some sense, but it is very difficult to explain. If you are interested, you should read my first draft of the Space-time-motion diagram at http://vixra.org/abs/1402.0045. I have made some changes, mainly to cut down on the philosophy and background material, so check back in a few days to get the latest version.

            Oh yes, as for Parmenides philosophy: clearly I disagree with his view that change is impossible, but I agree that existence is timeless because time is nothing more than a scale for eternal (timeless - no beginning and no end) change.

            Best regards,

            Ted

              Your work convincingly relates that mathematics and physics are more artful.

              Sincerely,

              Miss. Sujatha Jagannathan

                Hello Akinbo,

                I was thinking about your idea of minimum physical length and whether probabilities of 2+3=5 exist depending of the length of the object. It could be useful to put objects into perspective to see if there is consistency.

                Maybe you can create a single log scale going from the maximum size object we know (universe) to the minimum size object we known. On this scale, you can highlight from what size we are starting to have issues with standard mathematics. Maybe it has already been done?

                Regards,

                Christophe