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New Theories / Re: How Many Numbers Exist?
« on: 16/10/2021 23:29:00 »
Hi.
There's nothing I know of that would prevent space from being continuous. Indeed, the standard formulation of General Relativity requires that space is continuous.
However, when you come to measure a rod there are practical limits in our ability to measure things that accurately. It also assumes the rod is some idealised body and not a quantum mechanical system. Treating the particles that make up the rod as quantum mechanical objects would put a theoretical limit on the ability to localise the ends of the rod.
If I recall correctly, it was this sort of limitation on the ability to localise a particle that lead to the first proposals that space (and time) may not be continuous since it almost becomes irrelevant: You can't localise a particle to one point in space unless it's momentum* → ∞, there is some uncertainty in it's position and therefore there isn't any great difference if you just split space up into discrete intervals rather than assuming it is continuous. However, I prefer to remain open to both possibilities. Space might be continuous or it might be discrete.
From a purely mathematical point of view. We often take derivatives or integrals with respect to some co-ordinate. This Calculus does assume the co-ordinate will be a continuous variable.
Best Wishes.
LATE EDITING: * See comment below by Hamdani Yusuf. "uncertainty in momentum".
What prevents us from expanding a 1 meter rod to 1 meter plus half Planck's length, eg. by heating it up?Are you asking me? BilboGrabbins was the one proposing that space cannot be divided into units smaller than the Planck length.
There's nothing I know of that would prevent space from being continuous. Indeed, the standard formulation of General Relativity requires that space is continuous.
However, when you come to measure a rod there are practical limits in our ability to measure things that accurately. It also assumes the rod is some idealised body and not a quantum mechanical system. Treating the particles that make up the rod as quantum mechanical objects would put a theoretical limit on the ability to localise the ends of the rod.
If I recall correctly, it was this sort of limitation on the ability to localise a particle that lead to the first proposals that space (and time) may not be continuous since it almost becomes irrelevant: You can't localise a particle to one point in space unless it's momentum* → ∞, there is some uncertainty in it's position and therefore there isn't any great difference if you just split space up into discrete intervals rather than assuming it is continuous. However, I prefer to remain open to both possibilities. Space might be continuous or it might be discrete.
From a purely mathematical point of view. We often take derivatives or integrals with respect to some co-ordinate. This Calculus does assume the co-ordinate will be a continuous variable.
Best Wishes.
LATE EDITING: * See comment below by Hamdani Yusuf. "uncertainty in momentum".