Can "Infinity" ever be more than a mathematical abstraction?

Discussion in 'Physics & Math' started by Seattle, Jun 24, 2018.

  1. someguy1 Registered Senior Member

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    726
    You have a point, in that this is the argument of constructive mathematicians. If we remove the noncomputable points, then the intermediate value theorem is false. But a constructivist patches the problem by restricting continuous functions to computable functions and then the "constructive IVT" is true again. So you are in agreement with constructivism, the philosophy of math that says that every mathematical object must be computable (or constructible in some other sense. There are various flavors of constructivism as I understand it, but I'm not too familiar with the subject).

    https://en.wikipedia.org/wiki/Constructivism_(mathematics)

    A few mathematicians and physicists have attempted to frame modern physics in terms of constructive mathematics, but with very limited results to date. That's not to say we won't all feel differently about these matters in the future.

    I found an interesting looking paper on the subject, which I didn't read. https://arxiv.org/pdf/0805.2859.pdf. Pulling one bolded quote from the author's introduction: "algorithms must replace formulas." I trust this viewpoint would be satisfying to you. It's not mainstream but that doesn't make it wrong.

    The problem with mathematical constructivism is that it denies uncountable sets. So it throws out the last 140 years of mainstream mathematics; and with it, the conventional foundations of both classical and modern physics.
     
    Last edited: Jul 23, 2018
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  3. someguy1 Registered Senior Member

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    726
    Since Newtonian physics is not the actual physics of our world, then what about it? Even in classical physics, we can only measure things approximately. It's true that in Newtonian physics we have a philosophical belief that each object is in some particular position given by a 3-tuple of real numbers. But since that isn't the universe we live in, I fail to see the relevance to the discussion.

    Sure, you can claim it. But not in a manner consistent with known physics.

    Ok, measurement -- which you agree is necessarily approximate -- is all we have. I'm going to hold you to this.

    Well since you agreed that "measurement is all we have," and since in an earlier post you agreed that all measurement is necessarily approximate, I'd say that YOU proved yourself wrong. Here's your earlier quote:

    I would invite you to try to frame your argument in such a way that it's consistent with known physics and doesn't contradict itself. I'm sure you are trying to make a valid point, but your own statements contradict each other AND the known laws of physics.
     
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  5. arfa brane call me arf Valued Senior Member

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    What about the fact Newtonian physics says an object with mass has an exact centre of mass? What about the fact that the actual physics isn't something we perceive, or for that matter, measure? What about the fact that measurement is always classical (that is, Newtonian) despite what we know about quantum particles?

    Despite measurement being approximate (you need to "run" a quantum algorithm many times), I can still claim there are such things as exact distances, exact numbers of particles, etc.
    I just did, and I think it's perfectly consistent with known physics.

    I think there's a big difference between "actual" physics and what can be measured.
     
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  7. someguy1 Registered Senior Member

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    726
    Ah ... what about all that stuff? I can't connect any of it to whatever point you're trying to make. Can you try to put your ideas into some kind of coherent argument?

    If you agree that "the actual physics isn't something we perceive, or for that matter, measure?" then how on earth can you claim that a particle is in some particular place? You are honestly not making any sense.
     
  8. arfa brane call me arf Valued Senior Member

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    6,255
    Maybe I don't need to claim that. Unless of course the particle is Newtonian, has a centre of mass, a definite path etc.
     
  9. someguy1 Registered Senior Member

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    726
    You've claimed it several times. If you no longer claim it, then our conversation has reached resolution with you having conceded the point.

    What of it? The world's not Newtonian. And you just backed off your original claim. What do you want me to say?
     
  10. arfa brane call me arf Valued Senior Member

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    6,255
    So, you're saying there is no point in defining exact units, like the metre and second, because we can't measure anything with that kind of exactness? Since the position of a particle in motion isn't something we can talk about in this universe--the particle can only be said to be somewhere along any distance we want to measure?
    Defining the metre in terms of the speed of light and 1/299792458 seconds is a philosophical belief?
    Note that we choose the metre as a unit of distance. It's completely arbitrary but: we also decide that the speed of light has nothing to do with our choice, it's an independent "natural" thing, likewise we decide that the rate time "moves" is constant, so we can again choose arbitrary units.
    All of which suggests we need a really accurate clock to get a really accurate (up to uncertainty in wavelength) metre. I mention the uncertainty in wavelength because it's true for classical and for quantum wavelike things, therefore we also presume uncertainty is inescapable in any physical system.

    Also note that QM is not a physical theory, it's a kind of logic.
     
    Last edited: Jul 24, 2018
  11. someguy1 Registered Senior Member

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    726
    Ah, the "so you're saying" game followed by something I didn't say. I don't think you're making a coherent point and now you're just playing games. I've responded to the substantive points you made and can't add any more.
     
  12. arfa brane call me arf Valued Senior Member

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    6,255
    You said Newtonian physics isn't the universe we live in. You asked how on earth we can say a particle is in a particular place.

    I'm saying the description of a quantum particle, unlike the description of a Newtonian particle (which has a centre of mass), doesn't even describe something physical. I'm trying to call you on your apparent claim that QM describes the real universe, when it doesn't describe the measurable universe.

    We see 'Newtonian' dots on a screen, we think of momentum and energy, a path through space, because we know where the particle was when it hit the screen at a certain accurately measurable time. QM doesn't describe this dot thing. At all.
     
    Last edited: Jul 24, 2018
  13. someguy1 Registered Senior Member

    Messages:
    726
    To be clear, I don't really say those things. They are my understanding of what modern physicists say, bearing in mind that I am not a physicist nor do I know much about the subject other than what I read online and in SciAm over the years. But yes, that's my understanding. That the world's not Newtonian, and under some interpretations of QM, a point has no position at all till we measure it.

    This is a point of philosophy you need to take up with the physicists. I did link the Wiki page earlier on interpretations of QM.


    I make no such claim nor have I ever made such a claim. On the contrary I've strenuously argued the opposite many times in this thread already.

    QM is known standard accepted physics. That's all I say it is. Some think it describes a real world "out there," others think it only approximates a real world "out there" that we will never exactly know, and still others deny there's anything "out there" at all. These are philosophical issues. I take no position in this thread, although personally I tend toward's #2. There's a world "out there" but it's stranger than we'll ever know. At best we can get better theories that predict experiments to more and more decimal places. The current record is 12, in an experiment of QED.

    Ok by me. Nothing to do with my argument. My argument doesn't depend on QM being true. Don't you realize that nobody knows what QM means? This is a key issue in the modern philosophy of physics. We have this theory that gives 12 decimal places but none of it makes any sense. "Nobody understands quantum mechanics." -- Richard Feynman said that!

    I hope you are coming to appreciate this point. We have a theory that predicts but that makes no sense. For a century the smartest people in the world are trying to figure out what it means.



    Of course for our local conditions, Newtonian physics does quite well. Well enough to send men to the moon, to explain those little bouncing balls on thread things. Here it is, appropriately named Newton's Cradle. https://www.amazon.com/ScienceGeek-...ocphy=9031303&hvtargid=pla-442521121181&psc=1

    And since all physical science is a game of approximation, then any approximation that's good enough for local conditions is fine. But you know, the copies of the master kilogram that are kept around the world, tend to drift over time. Atoms evaporate from the surface even in a high quality vacuum. No physical apparatus is perfect.

    Do you feel that I'm addressing or at least understanding your points a little? Of course the world is Newtonian "for all intents and purposes." [Not, "for all intensive purposes," a common misunderstanding and misspelling of the phrase].

    But saying that the National Bureau of Standards has a one kilogram bar of pure platinum that defines what it means to be a kilogram, I'm not disagreeing with that. I just don't see what it has to do with anything. The official kilogram in Switzerland or wherever they keep it, is not evidence that a particle, whatever that is, can be in a certain location, whatever that is, measured by an arbitrary real number. You are very very far from making that kind of connection. And honestly, I do wonder why you bring up an example so trivial and off-the mark. Like "Oh are YOU SAYING that because measurement is approximate, I shouldn't teach my kid how a ruler works?" Jeez man, raise the level a bit. Try to hit my brain and not my buttons.
     
    Last edited: Jul 24, 2018
  14. arfa brane call me arf Valued Senior Member

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    6,255
    I think you should think about the difference between a physical description (for an exact distance for example), and actual measurement. They're quite different.

    It's why I can claim I have an exact distance I can measure, although any measurement will be approximate if I use a standard metre as the basis. Especially today where the metre is defined in terms of a rational fraction of the second, and the second is defined in terms of an exact number of transitions (oscillations) of an atomic clock. So is it possible to define an exact unit of distance and claim it isn't a rational value? That isn't measuring a distance, but defining it.

    And physicists are trying to redefine all the physical units in terms of other natural constants, i.e. trying to define an exact value for them.
     
  15. arfa brane call me arf Valued Senior Member

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    6,255
    I guess what I'm trying to say there is that I think physics is two things, there are the equations of motion (or stasis), and there is measurement.

    But confusingly there is also quantum measurement, actually I think this is just another way to say entanglement.
     
  16. someguy1 Registered Senior Member

    Messages:
    726
    Perhaps you can give a specific example so I can get what you mean.

    Are you perhaps talking about the "real physics out there" versus the historically contingent and approximate human physics? Yes? No?

    Can you give an example of how you would do that? With an arbitrary real number? In other words sketch the experimental apparatus and how it would work.

    Also, do you claim you could measure any real number distance? Or just a computable distance? What is the set of real numbers that you could measure exactly.



    Ok. Here is a specific point on which we disagree. Your measurement will be approximate even if I let you choose the unit. You can even choose it after the measurement. So you can call that "one measurement" if you like and claim it's exact. But it isn't. You have measured anything exactly. Else please explain how you would go about doing this.

    Like I say: You tell me how the measurement is taken, and I'll let you define your units any way you like. Your measurement will still be approximate unless you show an example of your measurement idea.

    Values of constants have nothing to do with this. Just tell me how you would measure an exact value.
     
  17. Speakpigeon Registered Senior Member

    Messages:
    978
    How do we deduce for any Real number that if it is a non-computable number then it is not a Rational number? What would be preventing a non-computable number from having a rational value?
    EB
     
  18. arfa brane call me arf Valued Senior Member

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    6,255
    Sure.

    Newton's constant is exact, but measuring it is bound by accuracy.
    Well, you don't measure real numbers in physics, you measure distances, areas and volumes, and intervals of time.
    Yes, and choosing an exact unit means I can claim the distance is exact, which is not about measuring it.
    I wouldn't claim a measurement is exact, and I haven't. I've claimed a distance is exact--the metre. Hence any unit I like to choose is also exact.
     
    Last edited: Jul 24, 2018
  19. arfa brane call me arf Valued Senior Member

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    6,255
    The speed of light has everything to do with measuring a distance, and everything to do with defining (choosing) an exact distance. Because the second is also exactly defined.

    You don't measure an exact distance unless you have a clock as accurate as the standard--the one that defines or describes the second exactly. But the measurement will have a value which is approximate, you don't get an exact value because you need to build physical devices to measure how long it takes for light to get from A to B. These will always introduce some error in measurement.
     
    Last edited: Jul 24, 2018
  20. someguy1 Registered Senior Member

    Messages:
    726
    The grade school long division algorithm is the program that shows that n/m is computable. Any rational is computable.
     
  21. someguy1 Registered Senior Member

    Messages:
    726
    You haven't troubled yourself to come up with a specific experiment that would measure any physical value with infinite precision. Just more handwaving.

    By the way, regarding Newton's gravitational constant G:

    "The measured value of the constant is known with some certainty to four significant digits. "

    "https://en.wikipedia.org/wiki/Gravitational_constant"

    So tell me how you would measure something -- ANYTHING -- with infinite precision.
     
    Last edited: Jul 24, 2018
  22. arfa brane call me arf Valued Senior Member

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    6,255
    As I said above, physics is two things, equations and measurement. You don't seem to have taken this on board or tried to convince me I'm wrong.

    Equations mean the constants (such as the speed of light in vacuo) must have exact values; measurement means dealing with precision and accuracy.
    An electron has an exact amount of mass, etc.
     
  23. iceaura Valued Senior Member

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    28,952
    The intermediate value theorem would still hold for every function even potentially useful for perceiving or modeling physical cause, effect, probability, dimension, or field. (Because it would have to accept input values, for starters).
    So any infinities that show up would be, when abstracted, computable. So no holes - physically.
     

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