kant said:
Not so fast. Obviously, if we start with a mathematical model of nothing. The a priori postulate in such model is that it is void of space-time, matter, energy, real numbers, and the laws of nature. The mathematical model would be in some sense a blank piece of paper.
Can someone help me out a little here? I'm having a hard time following this.
Say that to build, or define, a mathematical model, we start with a set. This is our domain and contains all of the individual things that we want to consider. For example, our individuals might be numbers, sets, dogs, colored objects, members of a family, or the stops on a certain subway line.
On top of the domain, we can define some relations and operations. For example: if our domain contains dogs, we can partition the domain into all of the different dog breeds; for numbers, we can define zero, one, and multiplication; for a family, we can define a family tree; for subway stops, we can put them in order by the order in which the train visits them.
This basic setup, a domain with some relations and operations defined on it, is how I understand "mathematical model". So can someone tell me what a "mathematical model of nothing" would be? Does that simply mean that the domain is empty?
Also, I'm having a hard time with how an empty model could be "void of space-time" too. Is your domain there a vector space? If the domain is empty, how could you tell it apart from another model with an empty domain? In all formulations of set theory that I have seen (I am borrowing some modest set-theoretic ideas here), there can be only one empty set. This follows from the assumption (axiom of extensionality) that if two sets have exactly the same members, they are considered to be equal, or the same set.
1 I think this is a useful idea. Your domain is set, and the relations and operations that you define on your domain are also sets and are built using the members of the domain. So how can you tell two empty models apart? Don't they have exactly the same members: none?
Or maybe someone can explain in what other way the terms are meant. To me, a mathematical model is an abstract object and a blank piece of paper is not; it's a concrete object. I realize, or presume, that this abstract vs. concrete distinction is just a tool to help us divide up our perceptions. But I still find it to be a useful tool, so could someone help straighten that out? How can a mathematical model be like a piece of paper?
1. There is some difference between equality and identity, but I don't suspect we are to that subtlety yet.