I think this question is best understood as a scientific question, to be addressed by scientific theory and ultimately experiment. Others think it is a metaphysical issue, to be addressed by philosophy or intuition. A friend of mine feels that the question is a non-question, the very idea of time and/or space being discrete being a nonsense idea, and he does not accept my characterization of his approach as being metaphysical but "real." He insists I must explain how time or space could be quantized before accepting this as a valid question.
I don't know the answer, and the highest rated answer in one of many physics stack exchange threads on this topic (there are many!) being that the answer is not known. Central physical theories including General Relativity and Quantum Mechanics essentially assume that time and space are continuous by standard interpretations. But we know because these central theories are incompatible that they must not be the complete story. A number of leading contenders for a new Theory of Everything such as Quantum Gravity do propose the quantization of time and/or space. For some time now, the often proposed granularity of spacetime is derived from Plank's constant. However, critics of this point out that just because this is very very small (as we know must be required) and has units of spacetime, does not mean that IT is the ultimate quantization limit. Some recent experimental evidence suggests that if there is a quatization limit, it must be much smaller than even Plank's constant. Even that is contested. However FWIW physicists almost universally accept this as a "scientific" question to be settled by science rather than philosophy or intuition. And as a valid experimentally testable question, not nonsense.
Given my friend's hardened position, I'm finding myself to lean more and more in the opposite direction. And various thoughts and theories come to mind.
1) Measurement as commonly understood is essentially discrete. Generally we seek numbers to describe our measurments, and such numbers can only be finite.
1a) It would be possible to do something like measurement in an analog way. For example, I could set a mechanical spanner to a particular opening size, and as long as I do not interpret this numerically, you could argue it represents an infinitely small quantization size, though practically only a finite actual precision.
1b) Given that measurement is quantized, and will only ever have limited precision as well, you could argue that continuousness is only and ever a theory. Everthing we can actually measure is either explicity quantized or of limited precision.
1c) This suggests that while quantization CAN be proven (as we have done for masses and particles), and continuousness will only ever be a theory which can never be proven by experimental means, only disproven.
2) Following from that, "information" itself must be discrete. Perhaps this is limited to what you might call "representative" information such as numbers. Direct or physical information, such as the opening width of a mechanical spanner, or the elapsed time between two instantaneous events*, could potentially be continuous. However this does NOT apply to things where the physical "information" is materially based, only spatially or temporally based. For example vinyl records and analog magnetic tape recordings must ultimately be limited at least by the sizes of various particles, such as molecules of vinyl or magnetic domains.
(*Of course, no physical events can actually be instantaneous. However, we could define the "time" of the event as the time at which it starts, which sounds like it could be infinitely small. However any possible physical event involving matter, energy, or physical forces, even the "starting" time does not have infinite quantization. Everything that starts takes a finite time to start, such as the time it takes to generate a photon from an orbital shift.)
3) Ultimately, continuousness would suggest that there is infinite information in every spacetime difference. A universe of such infinities within even the smallest spacetime differences seems absurd to me.
I don't know the answer, and the highest rated answer in one of many physics stack exchange threads on this topic (there are many!) being that the answer is not known. Central physical theories including General Relativity and Quantum Mechanics essentially assume that time and space are continuous by standard interpretations. But we know because these central theories are incompatible that they must not be the complete story. A number of leading contenders for a new Theory of Everything such as Quantum Gravity do propose the quantization of time and/or space. For some time now, the often proposed granularity of spacetime is derived from Plank's constant. However, critics of this point out that just because this is very very small (as we know must be required) and has units of spacetime, does not mean that IT is the ultimate quantization limit. Some recent experimental evidence suggests that if there is a quatization limit, it must be much smaller than even Plank's constant. Even that is contested. However FWIW physicists almost universally accept this as a "scientific" question to be settled by science rather than philosophy or intuition. And as a valid experimentally testable question, not nonsense.
Given my friend's hardened position, I'm finding myself to lean more and more in the opposite direction. And various thoughts and theories come to mind.
1) Measurement as commonly understood is essentially discrete. Generally we seek numbers to describe our measurments, and such numbers can only be finite.
1a) It would be possible to do something like measurement in an analog way. For example, I could set a mechanical spanner to a particular opening size, and as long as I do not interpret this numerically, you could argue it represents an infinitely small quantization size, though practically only a finite actual precision.
1b) Given that measurement is quantized, and will only ever have limited precision as well, you could argue that continuousness is only and ever a theory. Everthing we can actually measure is either explicity quantized or of limited precision.
1c) This suggests that while quantization CAN be proven (as we have done for masses and particles), and continuousness will only ever be a theory which can never be proven by experimental means, only disproven.
2) Following from that, "information" itself must be discrete. Perhaps this is limited to what you might call "representative" information such as numbers. Direct or physical information, such as the opening width of a mechanical spanner, or the elapsed time between two instantaneous events*, could potentially be continuous. However this does NOT apply to things where the physical "information" is materially based, only spatially or temporally based. For example vinyl records and analog magnetic tape recordings must ultimately be limited at least by the sizes of various particles, such as molecules of vinyl or magnetic domains.
(*Of course, no physical events can actually be instantaneous. However, we could define the "time" of the event as the time at which it starts, which sounds like it could be infinitely small. However any possible physical event involving matter, energy, or physical forces, even the "starting" time does not have infinite quantization. Everything that starts takes a finite time to start, such as the time it takes to generate a photon from an orbital shift.)
3) Ultimately, continuousness would suggest that there is infinite information in every spacetime difference. A universe of such infinities within even the smallest spacetime differences seems absurd to me.
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