Mathematics of Space and Location

SL5

References:

What is Awareness, Scientifically?

A Proposed Measure of Motion

A New Explanation of Inertia

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When it comes to space and location it seems that the understanding of these concepts is based on idealized mathematical objects. The following is a description of Space from Wikipedia:

“Space is the boundless three-dimensional extent in which objects and events have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum known as spacetime. In mathematics, “spaces” are examined with different numbers of dimensions and with different underlying structures. The concept of space is considered to be of fundamental importance to an understanding of the physical universe. However, disagreement continues between philosophers over whether it is itself an entity, a relationship between entities, or part of a conceptual framework.”

The concept of space is not well understood. The location of a physical object in space is approximated by a mathematical point. The following is a description of Point from Wikipedia:

“In modern mathematics, a point refers usually to an element of some set called a space. More specifically, in Euclidean geometry, a point is a primitive notion upon which the geometry is built. Being a primitive notion means that a point cannot be defined in terms of previously defined objects. That is, a point is defined only by some properties, called axioms that it must satisfy. In particular, the geometric points do not have any length, area, volume, or any other dimensional attribute. A common interpretation is that the concept of a point is meant to capture the notion of a unique location in Euclidean space.”

The mathematical point is a primitive notion that is idealized per Euclidean space. The following is a description of Primitive Notion from Wikipedia:

“In mathematics, logic, and formal systems, a primitive notion is an undefined concept. In particular, a primitive notion is not defined in terms of previously defined concepts, but is only motivated informally, usually by an appeal to intuition and everyday experience. In an axiomatic theory or other formal system, the role of a primitive notion is analogous to that of axiom. In axiomatic theories, the primitive notions are sometimes said to be “defined” by one or more axioms, but this can be misleading. Formal theories cannot dispense with primitive notions, under pain of infinite regress.”

The mathematical point is a primitive notion, and its definition should be derived from the observed property of location in physical space.

The following is an analysis of physical space and the locations within it.

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The physical space is visualized as a “background,” which is populated by matter and energy. Einstein related these two elements with his famous equation: E = mc2. The key characteristic common to matter and energy is Inertia. The property of inertia is defined in Wikipedia as follows:

Inertia is the resistance of any physical object to any change in its state of motion, including changes to its speed and direction… Isaac Newton defined inertia as his first law in his Philosophiæ Naturalis Principia Mathematica, which states: The vis insita, or innate force of matter, is a power of resisting by which every body, as much as in it lies, endeavours to preserve its present state, whether it be of rest or of moving uniformly forward in a straight line.

According to Newton, inertia is an innate state of motion that resists change. This may be visualized by thinking of a spinning top, which maintains the speed and direction of its motion. For a matter particle, inertia is expressed by its mass.

Let’s look at how inertia may be expressed for an energy photon. From Einstein’s equations,  E = hf = mc2, the frequency is proportional to mass. A certain equivalence between frequency and mass exists in the region where wave-particle duality is observed. Since mass expresses inertia in matter particles, we may assume that frequency shall expresses inertia in energy photons.

Inertia is the innate state of motion that resists change. It is expressed through mass in matter particles, and frequency in energy photons.

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We visualize mass particles to be compact like golf balls. As the mass of a particle increases, the inertia gets increasingly centered at the “center of mass,” and the particle becomes harder to move. The ultimate in mass then provides a completely centered inertia.

Going in the opposite direction, as the mass of a particle decreases, the inertia becomes less centered, until inertia transitions into the frequency of a photon. In the region where particle-wave duality is observed, inertia is expressed through both mass and frequency.

We visualize photons as wave packets that do not have mass but they carry a frequency. The inertia of a photon cannot be expressed as being located at a “center.” That means that the innate state of motion of a photon oscillates over an appreciable range. It no longer resists change, except in its frequency of oscillation. As the frequency of a photon decreases, the oscillations spread out over a  larger range, and the photon becomes less discrete. The ultimate disappearance of frequency then provides a completely spread out non-discrete inertia.

Thus, we observe a scale of inertia (innate state of motion). At the upper end of this scale, inertia is totally discrete, unvarying and centered. This state of motion may be identified with a physical location. At the lower end of the scale, inertia is non-discrete, non-resisting and spread out. This state of motion may be identified with physical space.

We may define this universe as gradients of inertia (innate states of motion), at one end of which is physical space, and at the other end of which is a physical location.

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The primitive notion of a mathematical point should then approximate the properties of a physical location as a discrete, unvarying and centered state of motion.

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Comments

  • Chris Thompson  On January 14, 2015 at 7:59 AM

    The center of the galaxy is the most located and the least space?

    • vinaire  On January 14, 2015 at 8:46 AM

      Yes, that conclusion follows this analysis. One finds a black hole located at the center of the galaxy.

  • Chris Thompson  On January 14, 2015 at 7:59 AM

    I wonder if black holes were formed by accretion.

    • vinaire  On January 14, 2015 at 8:50 AM

      Black holes seemed to be formed with an accretion of inertia.

  • Chris Thompson  On January 14, 2015 at 8:02 AM

    Then space and mass are inversely proportional?

    • vinaire  On January 14, 2015 at 8:52 AM

      It is more precise to look at it in terms of the scale of inertia. Mass is just one form of inertia.

  • Chris Thompson  On January 14, 2015 at 8:35 AM

    I liked the idea of primitive notion. I was familiar with this notion but had no word for it.

  • Chris Thompson  On January 14, 2015 at 8:51 AM

    Accretion is also a good word. Gravity as a word is laden with primitive notions. I hope live to see it properly understood, not just properly measured. Or possibly David Mermin’s admonishment to “shut up and calculate” achieves the actual understanding of gravity? That is a good game, to calculate and to “nail it.” And yet, somehow unsatisfying, incomplete.

    I liked your idea of idealized geometry. That is a good way to describe the analogies of space. Space, like gravity is not yet philosophically understood. And yet it is well enough understood to fling an earth-made space craft out into space past two planets gathering energy and speed on a 10 year trip through the solar system to catch up to, pace, and land on Comet 67-P.

  • vinaire  On January 15, 2015 at 6:28 AM

    From “Topological Space” in Wikipedia:

    “In topology and related branches of mathematics, a topological space is a set of points, along with a set of neighbourhoods for each point, that satisfy a set of axioms relating points and neighbourhoods. The definition of a topological space relies only upon set theory and is the most general notion of a mathematical space that allows for the definition of concepts such as continuity, connectedness, and convergence. Other spaces, such as manifolds and metric spaces, are specializations of topological spaces with extra structures or constraints. Being so general, topological spaces are a central unifying notion and appear in virtually every branch of modern mathematics. The branch of mathematics that studies topological spaces in their own right is called point-set topology or general topology.”

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    Comment:
    Topological space is defined in terms of points. This brings up the question about the size and shape of points and how they are connected. Points are discrete. So how can they lead to the continuity of space?
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  • vinaire  On January 15, 2015 at 6:38 AM

    From Wikipedia:

    The Law of Continuity is a heuristic principle introduced by Gottfried Leibniz based on earlier work by Nicholas of Cusa and Johannes Kepler. It is the principle that “whatever succeeds for the finite, also succeeds for the infinite”. Kepler used it to calculate the area of the circle by representing the latter as an infinite-sided polygon with infinitesimal sides, and adding the areas of infinitely many triangles with infinitesimal bases. Leibniz used the principle to extend concepts such as arithmetic operations, from ordinary numbers to infinitesimals, laying the groundwork for infinitesimal calculus.

    • vinaire  On January 15, 2015 at 6:44 AM

      The infinite is not made up of finite, just like space is not made up of points. This makes the above principle quite interesting.

      This principle is true because both space and point are made up of motion. This makes “continuity” the property of motion regardless of its inertia.

  • vinaire  On January 15, 2015 at 6:52 AM

    From Wikipedia:

    “The transcendental law of homogeneity (TLH) is a heuristic principle enunciated by Gottfried Wilhelm Leibniz… Henk J. M. Bos describes it as the principle to the effect that in a sum involving infinitesimals of different orders, only the lowest-order term must be retained, and the remainder discarded. Thus, if a is finite and dx is infinitesimal, then one sets

    a + dx = a.

    Similarly,

    u dv + v du + du dv = u dv + v du,

    where the higher-order term du dv is discarded in accordance with the TLH. “

    • vinaire  On January 15, 2015 at 7:02 AM

      Euclidean Geometry bases its study of space on a homogenous distribution of points of the same order. Thus, there is a consistency in the measurements of length, area and volume.

    • vinaire  On January 15, 2015 at 7:08 AM

      The order of a point depends on the degree of centered-ness of motion. The more centered a point is the more pointed it would be.

      An infinitesimal would be extremely centered or pointed.

    • vinaire  On January 15, 2015 at 7:15 AM

      A point is just the opposite of space. However, calculus tries to approximate the continuity of space through the use of infinitesimal points that are infinitesimally close together.

      We may say that calculus approximates continuity through “connectedness” of discrete objects.

      • vinaire  On January 15, 2015 at 7:27 AM

        Here “connectedness” may be determined by closeness.

        This brings into question the concepts of “straight line” and “angles” at atomic scales.

  • vinaire  On January 15, 2015 at 7:32 AM

    From Wikipedia:

    Set theory is the branch of mathematical logic that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used in the definitions of nearly all mathematical objects.

    The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s. After the discovery of paradoxes in naive set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known.

    Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, with an active research community. Contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals.

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    • vinaire  On January 15, 2015 at 7:36 AM

      Basically, we are using a set of points to approximate space.

      The interesting new observation is that “space” and “point” are diametrically opposite manifestations of motion.

    • vinaire  On January 15, 2015 at 8:17 AM

      It is like an attempt to understand the infinite through lots of finites.

    • vinaire  On January 15, 2015 at 8:30 AM

      A lot of discretes put together can never be the same as continuous, because the two are of totally opposite character.

      The more discrete something is, the less continuous it is.

      We are trying to make a point more and more pointed (discrete), and then collect more and more of these points together to create a continuous space.

      This is a built in contradiction that generates an appearance of space, which is not space.

      Space is generated from a single point by making it less and less centered, and more and more dispersed.

  • vinaire  On January 15, 2015 at 8:31 AM

    Quantum Mechanics is basically puzzling over the transition of inertia from mass (centered-ness) to frequency (spreading-ness).

  • vinaire  On January 15, 2015 at 8:44 AM

    If mass is like a spinning motion, and frequency is like an oscillatory motion, then how does spinning motion transition to oscillatory motion?

    • vinaire  On January 15, 2015 at 8:54 AM

      An oscillating charge generates electromagnetic waves. The motion of electromagnetic waves combines translation and oscillation to generate a corkscrew type motion.

      The spirals of corkscrew motion (like the threads on a screw) come closer and closer as the frequency of oscillations increases.

      At some point the spirals of corkscrew motion come so close that they get jammed into each other and collapse into a spinning motion.

    • vinaire  On January 15, 2015 at 9:00 AM

      The translation seems to reduce as oscillations increase. At some threshold, the oscillations are so high that translation is almost negligible and spinning comes about.

      Translation = constant / oscillation.

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      • vinaire  On January 15, 2015 at 9:32 AM

        A parallel of this relationship lies in mental space where time takes much longer to flow the moment attention gets fixated.

        Flow of time = constant / Fixation of attention

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      • vinaire  On January 15, 2015 at 9:56 AM

        Flow of time depends on how a change is perceived

    • vinaire  On January 15, 2015 at 9:15 AM

      As spinning motion starts to slow down a threshold is crossed, and spinning transitions into a corkscrew motion.

      It is the nature of this threshold that seems to be of immense interest.

  • vinaire  On January 15, 2015 at 9:06 AM

    A point may be visualized as a totally focused point of a laser.

    A space my be visualized as totally unfocused and spread out light.

    Now think of somebody using lots of these focused points of light to approximate totally unfocused and spread out light

  • vinaire  On January 18, 2015 at 8:18 AM

    At the interface of Physics and Metaphysics lie the subjects of Mathematics and Logic. These subjects structure our awareness.

    We see discrete objects and a collection of those objects. This brings about the sense of numbers, sets and cardinality.

    A matter-centric frame of reference is a particle-based system. It is based on the ideas of objects as being massive and discrete. Each object has mass that is centered. As mass increases, it gets more centered. The more centered is the mass; the harder it is to change its position.

    The property of being discrete may be represented by numbers. The mass may be represented by a set of particles. The center of mass may be represented by a point. The centeredness of the mass may be represented by the cardinality of the set.

    On the other hand, an ether-centric frame of reference is a space-based system. It is based on the idea there is no mass and no center of mass either. There is only uniform continuity spread all around. Inertia appears only as a disturbance. However, as this disturbance increases to a certain limit, it turns into mass.

    The disturbance has a frequency. But the “discreteness” of frequency is not the same as the discreteness of particles. The use of numbers hides this fact.

    Both frequency and mass represent inertia. But if the cardinality of mass is considered positive, the cardinality of frequency seems to be negative.

    The Cardinality of Uncountable Sets may be given negative values.

  • vinaire  On January 18, 2015 at 8:28 AM

    A summary and a question to mathematicians:

    I am not a mathematician. To me mathematics simply provides tools to understand the real world. I am checking if topological space is the right tool to understand the physical space.

    As an engineer, I do not feel comfortable with the notion that the “continuum” is a collection of points. Points are discrete. Space is a continuum. In my view, discrete elements cannot logically represent a continuum. This notion of approximating space was used by Newton through the use of integral calculus. This may have been adequate for classical mechanics but I do not think that it can serve quantum and wave mechanics.

    I compared the primitive notion of point to a physical location. The physical location is associated with the location of a particle in physical space. A particle has mass that can be represented by a center of mass. This center of mass represents physical location quite well in mathematical calculations made in classical mechanics. Mass represents inertia, which is “a state of motion that maintains itself.”

    Heavier particles are harder to move. One may say that as inertia increases, the particles get more centered. Thus, with increasing inertia, the location in physical space gets more sharply defined. I don’t think that the mathematical point reflects this property of physical location.

    If we go in the direction of decreasing inertia, the center of mass gets less centered. When we reach the region where wave-particle duality is observed, the center of mass loses its meaning. The mathematical point can no longer be used to represent physical location in space.

    Here we have inertia transitioning from mass of a particle to the frequency of a wave-packet (photon). In the region of electromagnetic radiation, a photon has no mass but it still has some inertia. A light wave bends as it passes by a heavy heavenly body. That inertia now gets expressed in terms of frequency. The inertia of a photon cannot be expressed as being located at a “center” because of an appreciable wavelength involved. That means that the innate state of motion of a photon oscillates over an appreciable range. The concept of a mathematical point to build up the notion of space is no longer applicable. The classical mathematics of “point-space” no longer works.

    If we continue to move in the direction of decresing inertia, we find ourselves moving in the direction of decreasing frequency on the electromagnetic spectrum. As frequency decreases the wavelength increases. That means that inertia is spread over a larger range. This is opposite to the “centered-ness” of inertia as mass. We now have “spreading-ness” of inertia in terms of decreasing frequency and increasing wavelength, which cannot be modeled using a mathematical point.

    We can see that as frequency approaches zero, the “spreading-ness” of inertia approaches infinity, and we have a condition, which we may call “space” that forms the background of all energy and matter, This makes me believe that space is a primitive notion in its own right, and that it is not a collection of points.

    Is topological space treated as a primitive notion in mathematics?
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    http://www.quora.com/Is-topological-space-capable-of-bringing-about-the-understanding-of-physical-space-at-atomic-dimensions
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    • vinaire  On January 18, 2015 at 8:34 AM

      The mathematical point may be assigned the property of “cardinality”. When the cardinality is an integer it represents “centered-ness”. When the cardinality is the inverse of integer (a unit fraction) it represents “spreading-ness”.
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  • vinaire  On January 18, 2015 at 11:56 AM

    A location in space, like the center of mass of an object, is best referenced by itself, There are no absolute reference point in space.

    The closest one may come to an absolute reference point is the center of mass of the universe.

    The reference point for any sector of space may be determined as the center of mass of that sector.

    The idea of location applies only to objects with mass. It does not apply to energy waves whose locations are spread over a wide region. Therefore they cannot be pin-pointed.

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