Suppose space and time are non-discrete. Do we have good reason to think that they form an uncountable continuum of the real-number sort? One might first speculate: Perhaps points in space have coordinates that are triples of rational numbers (in some coordinate system)? That would, however, make it impossible to rotate an object by 45 degrees: the coordinates after such a rotation would no longer be rational numbers. And that's implausible. But there are bigger countable sets than that of rational numbers that one might invoke that would get out of problems like that. So why suppose our space and time have the structure of the real numbers?
I guess I think that if you're going to have a non-discrete continuum, it's a little weird to have gaps in the number line. The thought would be, a non-real-valued continuum would call out for an explanation.
ReplyDeleteMaybe that's not much of a reason?
Well, it might beg the question to say that there would be "gaps". Let's say we have the rational numbers. Then, we have gaps in the senses:
ReplyDelete1. One can extend the field to include more stuff.
2. There are "naturally definable" numbers that are missing, like the square root of two.
3. There are dedekind cuts that don't correspond to a number.
Well, 1 is always true: the real numbers, the hyperreals, they all can be extended by inserting more "numbers". So that's not an issue.
Now, 2 doesn't seem that strong. After all, nobody should complain that the real line doesn't contain a square root of negative one, or that space doesn't contain a distance greater than D (for some D), if space is finite. Besides, I think the right version of the proposal will include all the naturally definable numbers in the continuum like the square root of two.
I think 3 may be the best way of getting at the intuition, but it may beg the question.
Regarding 1, isn't there a difference between a field's having 'gaps' and a field's being extensible in some way? (Perhaps X has 'gaps' if it's a subset of Y and for any x1 and x2 in X there's a y in Y where x1 < y < x2.)
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