Discrete and Continuous

VLT image of the cometary globule CG4

A rational number, whether real, imaginary or complex is a discrete entity. A discrete entity has a boundary of value. A set of discrete entities is a collection of such values with boundaries.

A scale, dimension, or field is a continuous entity that is stretched out. A continuous entity has continuously varying values. There is no boundary that defines an exact value.

The moment you refer to a “number” you have placed a boundary around a value by identifying it.

A continuous scale, dimension, or field may be filled with a lot of precisely defined numbers, but these numbers remain separate from the scale.

When you say, “There is a continuous field of numbers,” all you are saying is that there is a continuous field that may be filled with numbers. That continuous field is still there when it is not filled with numbers. The numbers DO NOT MAKE the field.

There are no precisely identifiable irrational numbers. When one is looking at an irrational number one is looking at a piece of scale, dimension, or field.

A continuous function like SINE is by itself. It is not made up of numbers, though infinity of numbers may fit into any segment of it.

The bottom line is,

“INFINITY OF DISCRETE ENTITIES DO NOT TOGETHER MAKE A SINGLE CONTINUOUS ENTITY.

Space may be filled with points; but space is not the same thing as a set of points.

.

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Comments

  • vinaire  On February 7, 2015 at 10:12 PM

    Here is an interesting definition of “philosophers” by Isaac Asimov:

    “Those who attempted the rationalistic search for understanding, without calling in the aid of intuition, inspiration, revelation, or other nonrational sources of information, were the philosophers.”

    .

  • vinaire  On February 7, 2015 at 10:24 PM

    From Isaac Asimov:

    Mathematics: the study of the abstract relationships of form and number.

    Astronomy: the study of the position and the movements of the heavenly bodies.

    Geology: the study of the physical nature of the earth we live upon.

    Chemistry: the study of compostion and interaction of substances.

    Biology: the study of the structure, function and interrelationships of living organisms.

    Physics: to describe study of those portions of nature that remained after the above mentioned specialities were subtracted.

    A study of physics may be said to include, primarily, a consideration of the interrelationships of energy and matter.

    .

  • vinaire  On February 7, 2015 at 10:43 PM

    Ockham’s razor: the effort made to whittle away at unnecessary assumptions.
    .

  • vinaire  On February 8, 2015 at 9:58 AM

    Aristotle’s Theory of Motion:

    Forced motion must inevitably give way to natural motion, and that natural motion will eventually bring the object to its natural place. Once there, since it has no place else to go, it will stop moving. The state of
    rest, or lack of motion, is therefore the natural state.

    The laws governing the motion of heavenly bodies are different from those governing the motion of earthly bodies. Here the natural state is rest, but in the heavens the natural state is perpetual circular motion.

    .

  • vinaire  On February 8, 2015 at 10:01 AM

    Validity of a theory:

    One method of casting doubt upon any theory (however respected and long established) is to show that two contradictory conclusions can be drawn from it.

    • vinaire  On February 8, 2015 at 10:10 AM

      A second method of testing a theory, and one that has proved to be far more useful, is to draw a necessary conclusion from the theory and then check it against actual phenomenon as rigorously as possible.
      .

    • vinaire  On February 8, 2015 at 10:15 AM

      It is from Galileo that the birth of “experimental science” or “modern science” is usually dated.
      .

  • vinaire  On February 8, 2015 at 10:45 AM

    Newton’s first law of motion:

    A body remains at rest or, if already in motion, remains in uniform motion with constant speed in a straight line, unless it is acted on by an unbalanced external force.

    This law is also referred to as the principle of inertia.. It stresses upon the tendency of a state of motion to maintain itself.

    It would seem that the prnciple of inertia depends upon a circular argument. We begin by stating that a body will behave in a certain way unless a force is acting on it. Then, whenever it turns out that a body does not behave in that way, we invent a force to account for it.

    We may call inertia as “internalization or interiorization of force.”
    .

    • vinaire  On February 8, 2015 at 10:47 AM

      This force also spreads out as a field of gravity.

    • vinaire  On February 8, 2015 at 11:26 AM

      It is an inherent state of tension.

    • vinaire  On February 8, 2015 at 11:36 AM

      Asimov: “The term force comes from the Latin word for “strength,” and we know its common meaning when we speak of the “force of circumstance” or the “force of an argument” or “military force.” In physics, however, force is defined by Newton’s laws of motion. A force is that which can impose a change of velocity on a material body.”

      Force seems to be a “change producing element” that is also the change itself. The most fundamental force would be a change in the position in space.

      • Chris Thompson  On February 8, 2015 at 10:10 PM

        In electromagnetism and gravity, this force cannot be separated out from the space in which it acts.

  • vinaire  On February 8, 2015 at 11:42 AM

    Force introduces the concepts of “state,” “change,” and “direction.”

  • vinaire  On February 8, 2015 at 3:40 PM

    Newton’s second law of motion:

    The Acceleration produced by a particular force acting on a body is directly proportional to the magnitude of the force and inversely proportional to the mass of the body.

    mass: the quantity of inertia possessed by a body.

  • vinaire  On February 8, 2015 at 4:59 PM

    A force, to exist, must be exerted by something and upon something. It is obvious that something cannot be pushed unless something else is pushing.

    Anything exerting force must have force stored in it in some manner. So anything living, to be a source of force, must have force stored in it in some manner.

    • vinaire  On February 8, 2015 at 5:04 PM

      What is a potential of force? A compressed spring would be a potential of force.

    • vinaire  On February 8, 2015 at 6:04 PM

      The force is stored as energy. Mass is also a form of stored force, but of a different kind.

  • vinaire  On February 8, 2015 at 6:15 PM

    Newton’s third law of motion:

    Whenever one body exerts a force on a second body, the second body exerts a force on the first body. These forces are equal in magnitude end opposite in direction.

    Whenever there is a manifestation of change, there are equal and opposite forces.

    • vinaire  On February 8, 2015 at 7:48 PM

      A change resists itself by generating a structure. Inertia is inherent to change. The amount of inertia shows how much change has occurred.

    • vinaire  On February 9, 2015 at 8:11 AM

      The amount of matter, energy, dark matter, and dark energy shows how much change has occurred in this universe.

  • vinaire  On February 9, 2015 at 8:19 AM

    Creation and Asimov:

    But the word “created” is a difficult one for physicists (or for that matter philosophers) to swallow. Can anything really be created out of nothing? Or is one thing merely changed into a second, so the second comes into existence only at the expense of the passing into non-existence of the first? Or perhaps one object undergoes a change (from rest to motion, for instance) because, and only because, another object undergoes an opposing change (from rest to motion in the opposite direction, for instance). In this last case, what is created is not motion but motion plus “anti-motion.” and if the two together cancel out to zero, there is perhaps no true creation at all.

    .
    The idea of CREATION goes together with the ideas of GOD, the unmoved MOVER, the uncaused CAUSE. There is something very fishy about all these ideas. There is a resignation to I DON’T KNOW so I’ll make this thing up.

    • vinaire  On February 9, 2015 at 8:26 AM

      Not knowing converts into assumptions.

    • vinaire  On February 9, 2015 at 8:41 AM

      Impulse is a product of force and time
      Motion is a product of mass and velocity (measured as momentum).

      An impulse (ft) applied to a body at rest causes that body to gain a momentum (mv) equal to the impulse. More generally, if the body is already in movement, the application of an impulse brings about a change of momentum, equal to the impulse. In brief impulse equals change of momentum.

      Impulse converts into motion or momentum.

      • vinaire  On February 9, 2015 at 8:52 AM

        In elastic collisions, momentum converts to impulse, which converts back to momentum. The momentum is conserved.

  • vinaire  On February 9, 2015 at 9:55 AM

    Proof and Asimov:

    At this point you might feel that if the physicist proves the conservation of momentum from the third law of motion, and then proves the third law of motion from the conservation of momentum, he is actually arguing in a circle and not proving anything at all. He would be if that were what he is doing, but he is not.

    .
    The bottom line is that there is observation. All relationships are also observed. All observation is basically tautological. One can either observe or not observe.

    An offort to prove is simply an effort to point to certain relationships. For the other person, it is simply a matter of observation.

    The bottom line is the ability to observe what is there. Any proof is secondary.

  • vinaire  On February 9, 2015 at 8:29 PM

    More basic to linear velocity is the angular velocity of rotation. The velocity of a location very far from the axis of rotation appears to be linear.

    A common object has dimensions. Whenever the force applied to the object is away from its center of mass, rotational motion is generated.

    • vinaire  On February 9, 2015 at 8:45 PM

      In rotational motion, inertia depends not only on mass, but also on the distribution of mass relative to the axis of rotation.

    • vinaire  On February 10, 2015 at 8:59 AM

      When motion appears to be linear, the axis of rotation is infinitely distant.

      The moment of inertia for linear motion is, therefore, infinite.

      Light travels in straight lines. It has no mass, but it has inertia because it maintains a certain fixed frequency. The inertia of light may be small but its moment of inertia may be seen to be nearly infinite.

  • vinaire  On February 10, 2015 at 9:28 AM

    When a person does work, such as, lifting weight, then that work comes from somewhere. That stored thing that converts to work is called ENERGY.

    • vinaire  On February 10, 2015 at 4:02 PM

      Work comes from an application of force that results in a displacement of mass.

  • vinaire  On February 10, 2015 at 4:58 PM

    Conservation of Energy is the ultimate Law in Physics.

    • vinaire  On February 12, 2015 at 1:45 PM

      Energy may be stored as tension in a wire, or as compression in a spring.

      Energy may also be stored as gravitational, sound or heat potential.

      Energy may be stored as electromagnetic, chemical or nuclear potential.
      .

  • vinaire  On February 12, 2015 at 9:12 PM

    Asimov” “As it turned out, sound and heat are forms of kinetic energy after all. In the case of sound, the atoms and molecules making up the air, or any other medium through which sound travels, move back and forth in an orderly manner. In this way, waves of compression and rarefaction spread out at a fixed velocity. Heat, on the other hand, is associated with the random movement of the atoms and molecules making up any substance. The greater the average velocity of such movement, the greater the intensity of heat.”
    .

  • vinaire  On February 13, 2015 at 8:12 AM

    Asimov: “As was pointed out at the start of the book, then are two ways, in general, of avoiding the problem of action at a distance. One is to suppose particles streaming across a space that might then be considered as empty; the other is to suppose waves being propagated through a space that is not really empty. Both types of explanation were advanced for light in the latter half of the seventeenth century.”
    .

  • vinaire  On February 13, 2015 at 3:03 PM

    Asimov: “When the wave theory of light sprang back into prominence with Young’s experiment, it soon enough became clear that it light were only considered transverse waves, rather than longitudinal waves, polarized light could easily be explained. By 1817, Young had come to that conclusion, and it was further taken up by a French physicist, Augustine Jean Fresnel (1788 -1827). In 1814, Fresnel had independently discovered interference patterns, and -he went on to deal with transverse waves in a detailed mathematical analysis.”
    .

  • vinaire  On February 13, 2015 at 3:36 PM

    Asimov: “In connection with the ether, the difference in properties between transverse and longitudinal waves becomes important. Longitudinal waves can be conducted by material in any state: solid, liquid or gaseous. Transverse waves, however, can only be conducted through solids, or, in a gravitational field, along liquid surfaces. Transverse waves cannot be conducted through the body of a liquid or gas. It was for this reason that early proponents of the wave theory of light, assuming the ether to be a gas, also assumed light to consist of longitudinal waves that could pass through a gas rather than transverse waves that could not.

    When the question of polarization, however, seemed to establish the fact that light consisted of transverse waves, the concept of the ether had to be drastically revised. The ether had to be a solid to carry transverse light waves; it had to be a substance in which all parts were fixed firmly in place.

    If that were so, then if a portion of the ether were distorted at-right angles to the motion of a light beam (as seemed to be required if light were a transverse wave phenomena), the forces tending to hold that portion in place would snap it back. That portion would overshoot the mark, snap back from the other direction, overshoot the mark again, and so on. (This is what happens in the case of water waves, where gravity supplies the force necessary for snapback, and in sound waves, where intermolecular forces do the job.)

    The up and down movement of the ether thus forms the light wave. Moreover, the rate at which a transverse wave travels through a medium depends on the size of the force that snaps back the distorted region. The greater the force, the faster the snap back, the more rapid the progression of the wave. With light traveling at over 186,000 miles per second, the snap-back must be rapid indeed, and the force holding each portion of the ether in place was calculated to be considerably stronger than steel.

    The luminiferous ether, therefore, must at one and the same time be an extremely tenuous gas, and possess rigidity greater, than that of steel. Such a combination of properties is hard to visualize, but during the mid-nineteenth century, physicists worked hard to work out the consequences of such a rigid-gas and to establish its existence. They did this for two reasons. First, there seemed no alternative, if light consisted of transverse waves. Secondly, the ether was needed as a reference point against which to measure motion. This second reason is extremely important, for without such a reference point, the very idea of motion becomes vague, and all of the nineteenth century development of physics becomes shaky.”
    .

    • vinaire  On February 13, 2015 at 3:42 PM

      Here the thinking seems to be conditioned to reflect that a “fixed point” ought to be moving up and down. What if there is nothing fixed? There doesn’t have to be a defined entity that is moving up and down. There simply could be a play of tension in a field.
      .

  • vinaire  On February 13, 2015 at 4:00 PM

    Asimov: “By mid-nineteenth century, there was strong reason to suppose that no material object anywhere in the universe represented a state of absolute rest, and that absolute motion could not therefore be measured relative to any material object. This might have raised a serious heart chilling doubt as to the universal validity of Newton’s laws of motion, on which all of nineteenth century physics was based. However, a material object was not needed to establish absolute motion.”
    .

    • vinaire  On February 13, 2015 at 6:14 PM

      (1) Closest to absolute motion would be motion relative to itself, with no external point of reference needed.

      (2) Rotational motion shall be one such condition, where the axis of rotation refers only to itself.

      (3) In rotational motion, only a force may be felt that radiates outward. It is balanced by a force that radiates inwards.

      (4) The two ends of a rotating axis may act as the two opposite poles. Lines of force may originate from one pole and end at the other pole.

    • vinaire  On February 13, 2015 at 6:32 PM

      A corkscrew type of motion may only reference to itself. This may be the motion applicable to electromagnetic wave.

      The corkscrew motion rotates on its own axis. It advances in the direction of its axis. It advance is determined by the speed of rotation and the thread length (wavelength).

      In case of light that is moving forward in a corkscrew type fashion, its translational speed shall depend on the product of its angular speed and wavelength.

      c = ωλ, where, as λ decreases, ω increases as well as inertia.

      Increase in inertia depends on the increase in rotation speed ω. The translational speed c decreases as inertia increases.

      Therefire, c is not constant. λ decreases faster than ω increases. Therefore, c decreases as ω increases and λ decreases.

    • vinaire  On February 13, 2015 at 6:35 PM

      See Conjecture about Energy and Space
      .

      • vinaire  On February 13, 2015 at 6:38 PM

        Magnetic field seems to provide inertia to the motion of electric charge.
        .

      • vinaire  On February 13, 2015 at 6:56 PM

        In a simple harmonic motion, the kinetic energy (KE) changes into potential energy (PE), which changes back to KE and so on. Oscillation is the result of KE and PE transforming into each other in a periodic fashion. Here inertia in the form of mass is interacting with the gravitational field.

        In case of an electromagnetiuc wave, the oscillation comes about with electrical and magnetic energies transforming into each other in a periodic fashion. Here the inertia is due to the rotational part of the corkscrew motion, that is interacting with space.

        An electromagnetic wave seems to be like a screw that is threading itself through space as it rotates.

        Thread

        Please see http://en.wikipedia.org/wiki/Electromagnetism#Fundamental_forces

        .

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