Eckard
As expected, I now have a headache as well as a sore knee. I can specify the relevant fundamental nature of what is being investigated (ie our reality), and therefore what needs to be modelled with representational devices (eg mathematics). But having spent the day reading, I still certainly cannot make any judgement about the validity of specific constructs, nor indeed could I discern their underlying view as to how our reality is constituted.
The ontological characteristics of dimension, distance, and space
As previously stated, our reality is a sequence of physically existent states of the substances which comprise it. Only one such state can exist at a time, because for the successor to exist, the predecessor must cease. That is, at any given time within any given sequence, there is only ever one state (aka the present). Since the previous state (aka the past) has ceased to exist, and the next state in the sequence (aka the future) does not exist. Furthermore, that existent state must be physically definitive, and involve no change, otherwise existence, and alteration thereto, which is known to occur, could not do so.
In establishing what constitutes dimension, distance and space in our reality, it must be recognised that we are conceptualising any given physical reality (ie physically existent state) as if it was being divided into a grid of spatial positions. The 'mesh' size of this grid must be equivalent to the smallest elementary physical substance in our reality, so that the spatial position of any physical substance is identifiable.
Only physically existent states exist. That is, concepts either reflect that physicality, or are an artefact of it. By definition, any given physically existent state must have definitive dimension/size/shape (ie spatial footprint), this being a function of its constituent physical substance. That, with reference to the conceptual grid, can be defined as spatial positions 'occupied'.
It could be argued that a direct comparison between states is possible, and therefore there is no need for the concept of a grid. This is a fallacy, because logically the two circumstances are the same. The physically existent state used as a reference is just a surrogate grid. Indeed, in order to ensure compatibility with other comparisons, that state would have to be maintained as the reference (ie a de facto grid).
'Mapping' other states that were existent at the same given time, would reveal not only, obviously, both the spatial footprint of those states and their comparability, but also, distance. That is an artefact, a function of the physicality, and the particular selection of, the existent states involved. It is a difference, defined by comparison. There can be no distance between existent states which existed at different times. Their relative spatial position at their respective time of occurrence could be compared. Distance is usually measured between the two nearest dimensions of the existent states, but could include any combination of dimensions. And depending on the spatial relationship of the states involved, distance could involve spatial separation, or, if one state is within another, their spatial relationship, again in both cases with respect to specified dimensions.
Dimension is a specific aspect of spatial footprint. It relates to the distance along any possible axis. So three is the absolute minimum number of spatial dimensions that is still ontologically correct at the highest level of conceptualisation of any given physical reality. But is not what is physically existent. At that existential level, the number of possible dimensions that any given physical reality has is half the number of possible directions that the smallest substance in our reality could travel from any given spatial point.
Finally, space, in the sense of 'nothing', since otherwise it is distance. Again the start point is that only physically existent states are known to exist. Logically, there is the possibility that a not-physically existent state exists, ie at any given time there is a spatial position(s) that is(are) not occupied by physical substance. But this has not been proven yet. It is critical to differentiate this concept of 'nothing' from physical substance which just has different properties, because irrespective of how different the substances are, they are still substance.
The epistemological representation of dimension, distance, and space
As expected, I still cannot comment on any given mathematical construct in terms of its validity as a representational device of the above. Neither did I feel I made any real progress in discerning what form of reality Hilbert and Dedekind were presuming, ie following up on your comment that Hilbert 'shared the view that the distinction past and future is merely an illusion'. Without then understanding the content I could at least have made some comment, on the assumption that the content of the mathematical construct reflected the form of reality presumed. That particular notion of our reality being incorrect, indeed, it is probably its most critical feature.
So, at the risk of stating the blindingly obvious, the task is to develop models which properly correspond with the ontological characteristics of our reality (as above), which includes assessing the ones already being used. It has to be noted that in much of the material I read, there was a predominant concentration on intrinsic validity, ie whether the construct was internally valid. Obviously any given model cannot be internally inconsistent, but the search for 'internal perfection' is tantamount to assuming that our reality is an abstract concept and therefore what is required is a model that has no pre-conditions and 'covers all the options'. But, our reality does have definite, and discernable, characteristics, which need to be properly reflected in any device purporting to represent it.
Paul
