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...All other quantities can be treated as continua (ums?) ...
there must be a discrete step, regardless of how small that step is, which implies quantisation
Quotethere must be a discrete step, regardless of how small that step is, which implies quantisationThat Mathematical argument doesn't really hold. If an interval can be arbitrarily small, then the quantity is infinitely variable - or continuous. We assume that we can differentiate safely, for instance, and you can't do that with a non continuous function.Although the planck time is "the smallest measurable unit" I don't think that forces the conclusion that time, itself is not continuous - just a limitation of measurement. But this is an 'angels on a pinhead' type argument, in any case. Calculations involving time use ordinary Maths and that does not involve quantisation (?) but I could be convinced otherwise. Certainly the quanta of time would be a lot smaller (in terms of the actual values involved) than those of Energy, which are pretty plain to see (literally, on a very dark night).
How would you actually define a continuous range, then?I did an 'Analysis' course in my first year Maths (1963!!!!) and such matters a open intervals, closed intervals, continuous functions etc etc came into it. There are some very rigorous arguments involved.Of course, Maths is Maths and Time is Time but, if you use Mathematical arguments to discuss the nature of Time, you need them to be watertight.
Just how much mathematics is there in a comparison between being exactly equal to zero and not being exactly equal to zero?
A TV picture is not a good example to use because it is fundamentally Sampled, in the first place.
... Better still, the progress of an object through space.
aamof, information theory tells us that you can measure to any degree of precision you want as long as the signal to noise ratio is good enough. So a yactometer of change would still represent a (albeit very small) change in phase of a Terraherz signal, which could, in principle, still be measured.
Displaying an image which was sampled at one rate and then displayed at another rate involves the assumption that it is, in fact, continuous. The 'best' standards conversion involves some fancy filtering. The image is never, in fact continuous but the assumption is made that it is. Vector graphics and text characters (Truetype, Postscript) , are drawn at the time and the 'original' is assumed to be continuous.I do agree as to the nonsense of measuring to that ridiculously fine accuracy. However, the lower limit is, essentially, a practical one and not a fundamental one - even Heisenberg doesn't put a limit as long as you are only measuring one dimension.But Maths is merely a construct / model and isn't 'real'.
In the end, I think the issue is precisely all about Maths being a construct/model and not being real.
Which makes me wonder - somewhere else in the Universe, could someone be using an entirely different kind of Maths?