Here are some thoughts from Tobias Dantzig, a favourite mathematician of Einstein's. I think they show why Zeno's paradoxes cannot be solved by quantising spacetime into Planck lengths or conceptual infinitessimals.
"Herein I see the genesis of the conflict between geometrical intuition, from which our physical concepts derive, and the logic of arithmetic. The harmony of the universe knows only one musical form - the legato
; while the symphony of numbers knows only its opposite, - the staccato
. All attempts to reconcile this discrepancy are based on the hope that an accelerated staccato
may appear to our senses as legato
. Yet our intellect will always brand such attempts as deceptions and reject such theories as an insult, as a metaphysics that purports to explain away a concept by resolving it into its opposite."
"The axiom of Dedekind - "if all points of a straight line fall into two classes, such that every point of the first class lies to the left of any point of the second class, then there exists one and only one point which produces this division of all points into two classes, this severing of the straight line into two portions" - this axiom is just a skillful paraphrase of the fundamental property we attribute to time. Our intuition permits us, by an act of the mind, to sever all time
into the two clasess, the past
and the future
, which are mutually exclusive and yet together comprise all of time, eternity
: The now
is the partition which separates all the past from all the future; any instant of the past was once a now
, any instant of the future will be a now
anon, and so any instant may itself act as such a partition. To be sure, of the past we know only disparate instants, yet, by an act of the mind we fill out the gaps; we conceive that between any two instants - no matter how closely these may be associated in our memory - there were other instants, and we postulate the same compactness for the future. This is what we mean by the flow of time.
Furthermore, paradoxical though this may seem, the present is truly irrational
in the Dedekind sense of the word, for while it acts as partition it is neither a part of the past nor a part of the future. Indeed, in an arithmetic based on pure time, if such an arithmetic was at all possible, it is the irrational which would be taken as a matter of course, while all the painstaking efforts of our logic would be directed toward establishing the existence of rational numbers.
Finally, when Dedekind says that "if we knew for certain that space was discontinuous, there would be nothing to prevent us, in case we so desired, from filling up its gaps in thought and thus making it continuous," he states post factum
. This filling-out process was accomplished ages ago, and we shall never dicover any gaps in space for the simple reason that we cannot conceive of any gaps in time."
Number - The Language of Science (182)
Pearson Education 2005 (1930)