Hi Ian,
In the toy 2 particle universe you describe, only the following information can exist (from the point of view of each object):
1. Categories of information (about "itself" and the other object) e.g. mass or a more fundamental category of information;
2. Information interrelationships;
3. Quantity information associated with the information categories.
A physicist could represent this information and these information relationships with law-of-nature mathematical-type equations; and the quantities would be represented by numbers.
It's clear that reality utilizes "categories of information" like mass, momentum, charge etc. And some of these categories are not fundamental, but are in effect built out of relationships between existing, more fundamental, information categories. Seemingly, at least one information category must be considered to be a "first principle" of reality.
Similarly, it is clear that reality utilizes information interrelationships. We attempt to represent such interrelationships with symbols like "= + x", and at least some of these interrelationships must also be considered to be "first principles" of reality.
When it comes to quantity, which we represent with numbers, we must also assume that there is an underlying simplicity. "Quantity" describes something that really exists in fundamental reality.
But the objects in fundamental reality can make no distinctions relating to number of objects: there can potentially only be distinctions made relating to quantity. The reason for this is made clear by mathematician Louis H. Kauffman's article "What is a Number?":
"Two classes are said to be similar if there is a one-to-one correspondence between the members of the one class and the members of the other class. . . [But] How do you know to take a given form as foreground against which to make the comparisons?. . . None of these questions can be addressed by a formal system alone. The ability [to] perform the relations indicated by these questions is a prerequisite to being able to make any formal system at all."
Correspondence and comparison are complex many-step operations: clearly there can be no such ur-counting behind-the-scenes at the level of fundamental reality. Classes and sets have a hidden underlying complexity: sets have a complicated underlying infrastructure of axioms, defined symbols, defined equivalence relations, ordering etc. If one assumes that some sort of simplicity underlies physical reality, one must conclude that the quantities found to exist in fundamental physical reality cannot be modelled by the sets that represent the properties of number systems.
On the other hand, a number seen as a ratio seems to be possible for fundamental reality. To give another toy example: assuming a toy law-of-nature relationship could be represented as "a+bc =d", then a (rational) number relationship could be represented as "(a+a+a)/a" . I'm saying that the substance of the reality underlying quantity could be as simple as the reality underlying laws-of-nature.
(I attempt to explain in my essay how the complex numbers, algebraic irrational numbers, and non-algebraic numbers found in nature might be seen in the light of the above way of looking at numbers.)
Cheers,
Lorraine
