Y's opinion: ***If Reality Is Just A Set Of Numbers then the physical universe does not exist.***
My opinion: If reality is just a set of numbers, then the physical universe would be the only thing that existed. There would be no consciousness. The physical universe, complete with its people and their works, would look and behave exactly as they do in our universe except that all the people would be zombies and would experience no subjectivity.
Y's opinion: ***mathematics is a set of numbers. ***
My opinion: Mathematics is not a set of numbers but a set of statements. Numbers are concepts defined by some of those mathematical statements.
Y's opinion: ***Personally I do not believe that mathematics is reality.***
My opinion: Me neither. I believe that physical (read "describable") reality is mathematics.
Y's opinion: ***math can describe much of reality, plus probably alot more that is not real.***
Me: Math can describe all of physical reality and nothing but possible realities except for consciousness, and none of consciousness. (Please parse carefully.)
Y: ***Math includes all possible assumptions and results in many different systems, or realities if you wish.***
Me: I disagree. One system of math can only include consistent assumptions. There are many contradictory assumptions which cannot be included within a single math system. There can be many different math systems. Any math systems which are consistent could result in separate physical realities, each of which must obey the laws of physics, as Dick has proved.
Y: ***Whereas our universe presumably is described by a mathematics that is based on a particular set of assumptions,***
Me: I agree, those assumptions being at least the Peano Axioms which deliver numbers.
Y: ***...like the ones that Stafford used.***
Me: I disagree. You mistake the tools of mathematics for the objects of mathematics. Dick used the tool kit provided by the English language, the rules of logic, the mathematical system of arithmetic with its numbers, and the theorems of analysis including calculus and matrix algebra. Using these tools, he proved that any universe which is communicable must adhere to his constraint. He made no assumption on the universe except that it be communicable.
Y: ***In short we live in one particular physical reality described by one system of mathematics. ***
Me: I agree.
Y: ***For example, Stafford believes, so he says, that dark matter does not exist and that the observations suggesting the existence of dark matter are really do to electromagnetic effects.***
Me: I think you are correct about Dick's belief on this point, but I will let him speak for himself.
Y: ***At the present level of studies in cosmology, these alternatives are different assumptions as to what exists and the math describing the alternatives is quite different from each other.***
Me: I agree. However, the differing assumptions as to what exists, and the different math describing these alternatives has absolutely nothing to do with the development of Dick's result.
Y: ***But presumably only one corresponds to reality. ***
Me: Only one corresponds to our physical reality, i.e. the one physically accessible to us.
Y: ***A more basic example would be universes where different sets of dimensions compactify so that the resulting space-times differ. E.G., the 26-D string theory has two time dimensions.***
Me: The notion of "compactifying" is nonsense (remember, these are my opinions!) and I don't think these examples qualify as "basic", but they are good examples of possibilities for our universe.
Y: ***That would be an interesting universe to live in.***
Me: I agree. It is very interesting living in this, our universe, which I believe contains at least two time dimensions and several (5 to 25) spatial dimensions.
Y: ***We can do the mathematics***
Me: I agree. I fervently hope that someone will do it someday. On many occasions I have asked Dick to do it, but he hasn't yet seen the need.
Y: ***we cannot imagine what it would be like to live in 2 time dimensions.***
Me: Baloney. It is very easy to imagine. Imagine a world in which movies were made sort of the way we make them, but in addition, the characters in the movie itself were conscious of the movie making process (with its time dimension) in addition to the world of the movie (with its completely separate time dimension).
Shame on you for believing that our imaginations are so limited.