References:
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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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