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Thread: Calculability

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    Calculability

    So, a recent bit of discussion here lit a spark in my mind to investigate and ponder on the nature of calculability. Of course, I have read next to nothing on the topic, just scratched perhaps at a wiki article? I'm not even sure I went that far.

    I'm a lazy sod, after all.

    But on that subject, it stuck me that for something to be a number, to be real perhaps not in the "math" jargon but actually *real*,, it must be able to be expressed in some way. It's declaration must be possible by some means even if not through "calculation".

    So, off the top of my head, I can only come up with one type of way a number may be actually real, and not be calculated; it must be measured. But measured numbers have a limit to their meaningfulness, insofar as quantum numbers go: there is a maximum specific meaningful precision to the speed of light. And even of G.

    So, is there a set that can be described, but not calculated, that is not merely "measured" and thus finite in complexity given a finite reference frame?

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    When dealing with measured values there is no significance beyond the accuracy of the values.

    You need enough buffer digits isoyou do not loose accuracy in a chain of calculations, if that is what yo are getting at. Before computers when doing hand calculations you had to figure out how far to carry it out. Or how many digits of a constant like PI to use.
    It almost sounds like you are talking about Turing Machines and cmputability.

    From a Theory Of Computation class I took it is said if a problem can be solved it must be Turing commutable, meaning no limits on memory and time. In modern computer terms, if you can not code it as an algorithm on a computer it is not solvable.

    An ifinite number of operations on an infinite set would not be Turing computable as I see it.

    Measurements can not be infinite. A ruler is limted by the number of atoms that can be chaind together. Electronic measurements are ultimatly limited by the quantized electron.

    Numbers are the measurement. Philosophically I do not see numbers as real othr than something in our brains. You do not need numbers for a ruler. Just an arbitrary length.
    Last edited by steve_bank; 05-21-2021 at 08:49 PM.

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    Quote Originally Posted by Jarhyn View Post
    ... there is a maximum specific meaningful precision to the speed of light. And even of G.

    So, is there a set that can be described, but not calculated, that is not merely "measured" and thus finite in complexity given a finite reference frame?
    I'm not clear on the question. I think you're distinguishing between "mathematical numbers" and (the ratios of) physical measurements. Is that correct?

    Nitpick: The speed of light in SI meters per SI second is known exactly!
    Quote Originally Posted by Système international (d'unités)
    The metre (meter) is currently defined as the length of the path traveled by light in a vacuum in 1/299,792,458 of a second.

    The second is defined as being equal to the time duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the fundamental unperturbed ground-state of the caesium-133 atom.
    Until 2019 the SI kilogram was a hold-out, being defined as the mass of a certain platinum-iridium cylinder in a certain climate-controlled French vault (which was slowly evaporating!). But just as the meter is now defined by fixing the speed of light to be a specific finite constant, so the kilogram is now defined as whatever it needs to be to make Planck's Constant = 6.62607015×10-34 kg m2s-1 exactly.

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    Fair dinkum thinkum bilby's Avatar
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    Quote Originally Posted by Swammerdami View Post
    Quote Originally Posted by Jarhyn View Post
    ... there is a maximum specific meaningful precision to the speed of light. And even of G.

    So, is there a set that can be described, but not calculated, that is not merely "measured" and thus finite in complexity given a finite reference frame?
    I'm not clear on the question. I think you're distinguishing between "mathematical numbers" and (the ratios of) physical measurements. Is that correct?

    Nitpick: The speed of light in SI meters per SI second is known exactly!
    Quote Originally Posted by Système international (d'unités)
    The metre (meter) is currently defined as the length of the path traveled by light in a vacuum in 1/299,792,458 of a second.

    The second is defined as being equal to the time duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the fundamental unperturbed ground-state of the caesium-133 atom.
    Until 2019 the SI kilogram was a hold-out, being defined as the mass of a certain platinum-iridium cylinder in a certain climate-controlled French vault (which was slowly evaporating!). But just as the meter is now defined by fixing the speed of light to be a specific finite constant, so the kilogram is now defined as whatever it needs to be to make Planck's Constant = 6.62607015×10-34 kg m2s-1 exactly.
    Sure. But how long is a metre?

    It's impossible to measure the speed of light, because it's a defined quantity. And the same applies to the duration of a second. But as a result, it's very impressive to determine a more accurate distance for the metre.

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    C is measurable as is anything via an arbitrary set of reference points. WE have SI and fundamental definitions.

    All quantifiable measurements are relative to a standard. SI is of value because anyone enywhere can derive the stanrd refernce standards independently. Mass was the last item.

    No measurements are absolute and all refences have a repeatability-error band. An inherent uncertainty.

    There are no absolutes. If I measure 1kg of potatoes in a store it just means it equals the kg standard.

    C is measured within SI. Any reference system will do.

    Velocity is distance/time. A csr can be measured going 20km/hour. Photons in light are traveling at C in m/s.

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    Veteran Member Treedbear's Avatar
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    Quote Originally Posted by Swammerdami View Post
    ...
    Nitpick: The speed of light in SI meters per SI second is known exactly!
    Quote Originally Posted by Système international (d'unités)
    The metre (meter) is currently defined as the length of the path traveled by light in a vacuum in 1/299,792,458 of a second.

    The second is defined as being equal to the time duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the fundamental unperturbed ground-state of the caesium-133 atom.
    ...
    The meter is known exactly in terms of the speed of light and the frequency of the cesium atom, which are both assumed to be constants. But the speed of light, like the cesium atom, simply is what it is.

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    Fair dinkum thinkum bilby's Avatar
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    Quote Originally Posted by steve_bank View Post
    C is measurable as is anything via an arbitrary set of reference points. WE have SI and fundamental definitions.

    All quantifiable measurements are relative to a standard. SI is of value because anyone enywhere can derive the stanrd refernce standards independently. Mass was the last item.

    No measurements are absolute and all refences have a repeatability-error band. An inherent uncertainty.

    There are no absolutes. If I measure 1kg of potatoes in a store it just means it equals the kg standard.

    C is measured within SI. Any reference system will do.

    Velocity is distance/time. A csr can be measured going 20km/hour. Photons in light are traveling at C in m/s.
    c is NOT measurable. c is exactly 299792458ms-1, and no measurement can ever change that.

    The metre is measurable. If you determine how quickly photons cross a given space, then you are measuring the distance travelled, not the speed of the photons.

    Your intuition here is leading you astray. It sounds nonsensical; But it is true nonetheless. That's why we have metrologists, rather than just asking an engineer for his opinion.

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    Fair dinkum thinkum bilby's Avatar
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    Quote Originally Posted by Treedbear View Post
    Quote Originally Posted by Swammerdami View Post
    ...
    Nitpick: The speed of light in SI meters per SI second is known exactly!
    Quote Originally Posted by Système international (d'unités)
    The metre (meter) is currently defined as the length of the path traveled by light in a vacuum in 1/299,792,458 of a second.

    The second is defined as being equal to the time duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the fundamental unperturbed ground-state of the caesium-133 atom.
    ...
    The meter is known exactly in terms of the speed of light and the frequency of the cesium atom, which are both assumed to be constants. But the speed of light, like the cesium atom, simply is what it is.
    The speed of light isn't assumed to be constant; It is defined as constant. Something that Einstein demonstrated to be a sensible thing to do.

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    From relativity C will b appear constant regardless. of how you measure it.

    Obviously not practical. I start a sand clock and flash a light. Someone at a distance sees the light and flashes a light. When you see the light you stop the sand clock. The distance is 100 arm lengths away, it being the length io the arm if the king. The second is a measure of change, but so is a sand clock.

    Count the grains of sand and you have velocity in grains/arm lengths. Seconds based on an atomic clock is arbitrary, but it is convenient and reproducible.

    The key to SI is being reproducible anywhere. Even Mars.

    SI is harmonized or normalized such that all fundamental units and derived units are consistent. That includes mass, time, and distance. It was not always so.

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    Fair dinkum thinkum bilby's Avatar
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    Quote Originally Posted by steve_bank View Post
    From relativity C will b appear constant regardless. of how you measure it.

    Obviously not practical.
    Just because you think that what you believe is obvious, is no reason to accept that it is true. Indeed, it should be a warning flag to you that you're about to make a gross blunder into Dunning-Kruger territory.
    I start a sand clock and flash a light. Someone at a distance sees the light and flashes a light. When you see the light you stop the sand clock. The distance is 100 arm lengths away, it being the length io the arm if the king. The second is a measure of change, but so is a sand clock.

    Count the grains of sand and you have velocity in grains/arm lengths. Seconds based on an atomic clock is arbitrary, but it is convenient and reproducible.
    Wrong. The velocity is 299792458ms-1. What you have measured is the length of your arms.

    The key to SI is being reproducible anywhere. Even Mars.
    That's certainly one of its strengths.

    SI is harmonized or normalized such that all fundamental units and derived units are consistent. That includes mass, time, and distance. It was not always so.
    True. But completely irrelevant to your understandable rookie error in assuming that it's possible to measure the speed of light. It's not possible. The speed of light is 299792458ms-1 regardless of any measurement you make. Your experiment measures the length of your unit of distance (using a low accuracy measurement of time, and so probably giving a very crude result).

    Metrology is neither a democracy, nor a philosophical free-for-all. Opinion is irrelevant with regard to matters of fact.

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