Dear George,
Thanks for your thoughtful reply. I appreciate it. In relation to time and space being quantized, and as I mentioned on Carlo Rovelli's essay thread, by assuming that time and space are quantized, and come as minimum, indivisible intervals, one is also assuming the existence of instants in time and spatial points to bound and determine such intervals. If such instants and points existed, however, motion and change would be impossible. I naturally have no problem with Planck time and length - intervals beyond which clocks and rulers can no longer have meaning - but this does not mean that continuity ceases beyond this point, not does Planck time and length require the existence of instants and spatial points. Considering that Carlo is a founder of Loop Quantum Gravity, I was a little bit surprised (and delighted) by his response.
I wholeheartedly agree that infinities are mathematical entities that never occur in physical reality. That Hilbert quote is really nice one as well. However, the conclusion that instants and instantaneous magnitudes do not exist also gets ride of (physical) infinitesimals.
In relation to analysis (and non-standard analysis too), my arguments relating to the non-physicality of instants and instantaneous magnitudes apply to any formalized method where such things are assumed.
"I cannot understand how you conceptualise motion without the flow of time to underpin it. Motion is identified by something being in a different position at a later time than at an earlier time."
This is a faulty definition, as, by talking about later and earlier, it already assumes tense. A better definition might simply be, motion is defined by something having a changing relative position. Moreover, if one simply refers to the continuous readings of a clock, one needn't invoke or refer to tense, past or future etc. That is, one can still assign such readings an order without referring to time.
"But adopting a relative view of space or time will still leave one with change taking place as relative time flows."
I think the last four words of that sentence are unnecessary. Here is a related passage from my essay notes.
The question of interval's existence is a very subtle topic, however, especially so because an assertion that is physical continuity which is basic and fundamental rather than interval, could equably be reversed, and interval in time and space be said to have physical existence (something which could be "measured"), this making physical continuity, and as such, motion and change possible. This is because both are completely indistinguishable, in so far as they both represent exactly the same thing (i.e. the capability for events to be continuous). Yet, they are also mutually exclusive, in that only one can be fundamental and come first. This poses a brilliant paradox, as, and although the same cannot be said for a specific interval of time or space due to, by definition, their requiring the existence of instants and spatial points to bound and determine their respective values as intervals, this indistinguishability also makes it impossible, in purely logical sense anyway, to say whether it is physical continuity, and as such, motion and change, which is basic and fundamental (with interval having no physical existence), or if it is the existence of interval which makes continuity possible.
However, I feel that once this paradox has been brought to light, it becomes evident that the indistinguishability between the two is the major cause of people mistakenly attributing physical reality to interval in the first place. They have not realized that there is a more basic explanation. I also think it doubtful that Nature would go to the trouble of giving interval in time and space existence, when just the presence of motion would fulfil the same purpose. It is surplus to needs and Nature is not silly. Finally, unlike us, without a need to "measure" intervals of time and space, She also has no reason to bother attributing reality to them! In respect to Nature having no need to attribute reality to interval (as well as to time and space in general), a similar thing could be said in connection to the universe not being conscious.
Given that you seem to believe that instants in time exist, can I ask why you believe this? Where are they, these instants?
Best wishes
Peter