I do not evaluate “there exists a city larger than Paris” and “there exists a number greater than 17″ in the same way.
The first statement is a proposition about the world.
The second statement is a proposition about my map of the world.
Given a set of axioms, we say a mathematical statement is true if it’s true in some model of the axioms.
If the statement is derivable from the axioms, it’s true in all models of the axioms.
However, given a set of axioms A, we can add another set of axioms B (such that B is consistent with A) to produce A’.
A model is an implementation of the axioms.
Let M be a model of A.
Let M’ be a model of A’. M’ is also a model of A. It is possible for a given statement X to be true in M’, but not in M. This means that X is not derivable from A.
When we say X is true in M, we can reason about “truth” in a model the same way we reason about truth in reality.
Presumably you answered the question about Paris without needing to go look at Paris, or even its wikipedia page. If this is the case, then I would argue that the question about Paris and the question about 17 were both resolved, in your head, as propositions about your map of the world.
I agree that this is more or less common usage of the word “true,” when applied to mathematical statements and given some axioms. But we could just as well call this “the fleem property”—a mathematical statement has the fleem property, given some axioms, if it appears in a model of the axioms. After all, the word “true” is already used to talk about correspondences between our map of the world and the world—why would humans mix up the fleem property with truth?
I do not evaluate “there exists a city larger than Paris” and “there exists a number greater than 17″ in the same way.
The first statement is a proposition about the world.
The second statement is a proposition about my map of the world.
Given a set of axioms, we say a mathematical statement is true if it’s true in some model of the axioms.
If the statement is derivable from the axioms, it’s true in all models of the axioms.
However, given a set of axioms A, we can add another set of axioms B (such that B is consistent with A) to produce A’.
A model is an implementation of the axioms.
Let M be a model of A.
Let M’ be a model of A’. M’ is also a model of A. It is possible for a given statement X to be true in M’, but not in M. This means that X is not derivable from A.
When we say X is true in M, we can reason about “truth” in a model the same way we reason about truth in reality.
Presumably you answered the question about Paris without needing to go look at Paris, or even its wikipedia page. If this is the case, then I would argue that the question about Paris and the question about 17 were both resolved, in your head, as propositions about your map of the world.
I agree that this is more or less common usage of the word “true,” when applied to mathematical statements and given some axioms. But we could just as well call this “the fleem property”—a mathematical statement has the fleem property, given some axioms, if it appears in a model of the axioms. After all, the word “true” is already used to talk about correspondences between our map of the world and the world—why would humans mix up the fleem property with truth?
I’ll make a series of stubs that present my answer soon.
The second stub is this.