Really Steve, So Heisenberg's uncertainty principle is incorrect.I enjoy the topic of the thread.
Even two electrons are ideally not equal to each other except (potentially) in the context of mass and electrical charge, but they're not equal in terms of positions in space or time.
In a very real sense, there can't exist two things that are in all respects identical because they would be perceived the same as one and this should even be true of thought - if there's a manner two perceive them as two separate things then there should be at least one property that separates them.
This X and this X are not identical in that they occur at different locations within this sentence.
Similar to observations in quantum mechanics, I think assumptions of equality arise from the replacement of some distinct property by an unknown which makes those objects appear superimposed as indistinguishable from each other.
But an interesting thought is that, similar to Plato's cave, there may exist physical quantities larger than 1 and if something could truly only be measured as some none 1 to 1 ratio but there was no way to separate the individual elements constructing this ratio, then it may be that there exist objects that are effectively equal to each other in a mathematical sense, yet not identical or distinguishable.
A physical number 2 would ideally be constructed from a pair of inseparable and indistinguishable things - they could not be separated into 1 + 1 except in some abstract manner. So if we had a ratio of 2 to 1 between some trees, for example 6 trees on one block and 3 trees on another block, these trees are not "equal" in the sense of an identity - each tree is unique and distinct and we could count them individually, but if there existed a physical property that was in a 2 to 1 ratio with some other property, then one of these properties could be composed of 2 (or more) things and if these were not constructed from selecting to group separate observations as a single property, then it should imply the existence of two inseparable things ... but I have a feeling that ultimately it may not be possible to measure things in this manner because we could always ask what measurements constructed the 2 to 1 ratio and it would appear that it would have to arise from measuring at least 3 separate things, even if it was somehow ultimately only a detection of a single event, it would have to had the potential to at least been detected by 3 previously observed (or even imagined) detectors and each of these should be unique and distinct in some fashion.
Likely the closest we can get to an equality is the relationship that exists between (imagined) unknowns, though as soon as any of them become known, they're no longer identical (I think that's where the infinite to finite conversion happens - infinities are made by recurring processes that don't need to change and could be interpreted as having remained identical over time, though within different input/output contexts and so each application of such a process creates a relationship between inputs/outputs that is qualitatively identical in every such grouping, even if it is not working with identical elements - in this case, it's still not a demonstrable identity at any moment as the information flowing through it remains unique over time, but after extracting the commonalities between these events, then we can deduce with some confidence that an identical process has recurred across these elements, even if it's reflected in unique manners at each moment)
How do you measure an electron's speed and position at the same time?
Oh and: is an electron a wave or a particle, or both?
Is nature truly measurable, or was Heisenbergs uncertainty only a tip of the real berg?
What do you think?
Thanks,
=
MJA


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