Jonathan
I will pick up a few threads:
1 "For Physics to advance; we must set assumptions aside..."
Attempting to start from no assumptions, which is impossible, is about as bad as starting from flawed ones. The start point must be how is existence detected and hence, generically, how does that (ie physical existence) occur. If you went to woodwork classes, you would not take a toothpick and a lawnmower. Why not, these are tools. The answer being because they are useless given the fundamental nature of wood. Similarly, physics should first have established the fundamental nature of the physical existence it is investigating. And our physical existence is a spatial phenomenon which alters over time, existing in one state at a time.
2 "...we observe"
What sentient organisms sense, is different from what is. There are physically existent sensory realities (aka light, noise, vibration, etc) which, on the basis of their functional capability (which may well not be perfect as well) fulfil a role, caused by evolution, of providing representations of existent reality.
3 "If an object and observer share the same local frame of reference and neither is a supermassive object, we don't need a more complicated formula. Einstein's version reduces to the Newtonian form then, which is easily solvable. But his more general formulation gives meaningful insights where Newton's equations fail.."
Not so. As at 2 above, there is no sharing of frames. Existence and observation of existence are different. Furthermore, no two physically existent entities can be at the same spatial point at the same time, so there is always light and a delay whilst that light travels before being received, if it is. Einstein was wrong, he attributed this delay to a timing differential in physical reality.
4 "dimensionality is different at small and large scales"
Not so. As per above, reality functions in accord with simple all embracing rules, it is human philosophising of it which is the problem.
The first point is that: distance is solely determined by physically existent states, since it is the difference between them in respect of a spatial attribute, and differences do not exist physically. So distance can only involve physically existent states which exist at the same time. It is not possible to establish a distance, as opposed to some form of conceptual spatial relationship, between something which exists and something else which does not.
Therefore: any given distance is always unique, since it reflects a definitive physically existent circumstance at a given time. Notions which relate to the quantification of it in terms of space, or duration, and the comparison of one way with the other, are a fallacy, if they involve the presumption that there could be a difference. Whatever quantification methodology, there can only be one result.
And so: distance could be conceived as a single example of change, ie a difference. So it can be expressed, conceptually, in terms of duration incurred. The concept being that instead of expressing distance as the fixed spatial quantity which it is, it can alternatively be quantified as the duration which would have been incurred had any given entity been able to travel along it, either way. But it must be understood that there is no duration as such, this is just an alternative to, and the equivalent of, a spatial measure, ie a singular quantity. Failure to understand the absence of elapsed time in a physical reality results in the flawed application of the equation x = vt. Making this mistake reifies change, and hence duration.
In establishing what constitutes dimension, distance and space in our reality, we are using a reference which conceives of any given physical reality being divided into a grid of spatial positions. And the constituent physically existent states have definitive dimension/size/shape (ie spatial footprint), which can be defined as spatial positions 'occupied' at the given time of existence. Dimension is a specific aspect of spatial footprint, relating to the distance along any possible axis of that 'occupation'. So, three is the minimum number of spatial dimensions that is 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 is half the number of possible directions that the smallest substance in physical existence could travel from any single spatial point.
5 "Can empty spaces have a particular dimensionality at all?"
One has to be very careful in defining existent space, because space is really what is not something as defined. Or in the sense of interplanatory space, it is 'stuff' that is different from 'ordinary matter', but it is still 'stuff', and then the first point applies. In other words, there is 'stuff' of various types, that is what we know. Now, is there 'not stuff'? And that question about space goes back to the concept of the spatial grid above, ie as at any given time, are there spatial positions where there is absolutely nothing?
6 "The idea of emergent dimensionality has seen a resurgence.."
Any such form of explanation of physical reality should ring alarm bells, because these type of contrived concepts reflect a metaphysical stance. Sure, taking this particular point, in the very early stages of the physical existence known to us, it was fundamentally different, but that is different from asserting these metaphysical concepts. More importantly, whilst they have a veneer of science, they are no more valid than my assertion that our physical existence is actually a shoot em up game being run by green giants with six heads. My point being that, once one steps outside what is either directly or indirectly provable (that being based on sensory validity) then 'anything goes'. As per a recent exchange between Ben, Eckard & myself, when it comes to the question of a God, the scientific point is that such an entity is logically possible. If one states that God exists, then that is an assertion. We must presume, although we can do nothing to prove it, or indeed do anything about it, that we are within a closed system. What is, possibly, outside that is a matter for belief, not science.
7 "This escapes notice because people assume that higher-order dimensions are just like lower-order dimensions - only with more of them. It is untrue".
In this and what follows you have fallen into at least two traps. First, you have taken a non-existent entity, ie space and have reified it, by considering it as physically existent. Space is just an abstract concept, it is, in the minds of humans, the corollary of stuff. The question is, what is physically happening to stuff, which could substantiate these outcomes. The other trap is the representation of physical existence with devices and presumed associations which do not actually correspond with physical existence, ie spheres and particular mathematical constructs.
8 "The heated debate forced both sides to examine assumptions about what is most fundamental, and caused a lot of hidden assumptions to be made explicit, but it also raised awareness about little known areas of Math that have a lot to offer Physics"
Yes, sadly it did not get to the real bottom line, ie that the representational devices did not correspond with the fundamental characteristics of physical existence. It was just one metaphysical view vrs another. "Magical properties", yes quite, but I have yet to hear anyone explain how all this can relate to the 'structure' of physical existence.
9 "Entropy is often equated with disorder"
Such concepts are more than "misleading", they are wrong. By definition, in physical existence, there is no disorder, no indefiniteness, etc, etc. This is all in our minds. An example of that being what you say next: "entropy embodies a host of non-linear properties, or processes, because boundary conditions confining a system to linear regions change over". Nothing feeds back, or any other such concept. There is no future, and the past has ceased to exist. The "exciting and paradoxical things" you refer to are a function of people not understanding existential sequence and therefore getting various existent elements entangled, ie asserting relationships which did not actually exist.
10 "Finally; it is important to question the assumption that objects and phenomena are independent or discrete. The illusion of separateness is compelling, but it now appears that no two things are completely separated and no system is isolated"
This is one of the basic problems. But I would not have said "it now appears", I would have thought this was a statement of the obvious. The real differentiation there is between what was direct physical influence and everything else, otherwise we are in the 'if a butterfly flaps its wings' syndrome. The elementary substance of physical existence is discrete, but what exists is a particular physically existent state thereof at any given time. Above that level of elementary substance we are conceptualising abstract concepts, not things, apart from not properly differentiating substance and existent state of substance. For example: St Pauls Cathedral does not physically exist. It is an abstract concept. What we do is take certain superficial physical characteristics and reify them as an existent 'thing'. So as long as those identification features endure, we assert that the thing continues to exist. Indeed, even if some features alter, we just assert the thing still exists, but has changed. We only cease attributing it with existence when the defining features are no more, ie it is a pile of rubble!
Paul
PS: as with Ben, I would not advocate bothering with my essay, but look at the last post on my blog (18 November). This operationalises what I am saying, which is much more gritty, isn't it. I don't know how to do links, so I just dumped all 12 pages as a post. If you want a copy e-mail paulwhatsit@msn.com.
