The real number system is chosen in physics for its mathematical utility, simplicity, and elegance, together with the fact that it accords, over a very wide range, with the physical concepts of distance and time. It is not chosen because it is known to agree with these physical concepts over all ranges. One might well anticipate that there is indeed no such accord at very tiny scales of distance or time. It is commonplace to use rulers for the measurement of simple distances, but such rulers will themselves take on a granular nature when we get down to the scale of their own atoms. This does
The real number system is chosen in physics for its mathematical utility, simplicity, and elegance, together with the fact that it accords, over a very wide range, with the physical concepts of distance and time. It is not chosen because it is known to agree with these physical concepts over all ranges. One might well anticipate that there is indeed no such accord at very tiny scales of distance or time. It is commonplace to use rulers for the measurement of simple distances, but such rulers will themselves take on a granular nature when we get down to the scale of their own atoms. This does not, in itself, prevent us from continuing to use real numbers in an accurate way, but a good deal more sophistication is needed for the measurement of yet smaller distances. We should at least be a little suspicious that there might eventually be a difficulty of fundamental principle for distances on the tiniest scale. As it turns out, Nature is remarkably kind to us, and it appears that the same real numbers that we have grown used to for the description of things at an everyday scale or larger retain their usefulness on scales much smaller than atoms – certainly down to less than one-hundredth of the ‘classical’ diameter of a sub-atomic particle, say an electron or proton – and seemingly down to the ‘quantum gravity scale’, twenty orders of magnitude smaller than such a particle! This is a quite extraordinary extrapolation from experience. The familiar concept of real-number distanc...
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This reminds me of Gauge theory in mathematics where means of measurement are not universal that discrepancies can occur from two different observations.
Common examples would be lbs and kilograms.
Measuring quantum behaviour.
Measuring the lightyear distance between you and an object from one position in space, with another measuring the same object from another position in space, not accounting for discrepancies in light bending due to the curvature of space time, gravitational waves, so on so forth.
