ruveyn wrote:
Not so. How did we get from Euclidean Geometry to non Euclidean Geometry. How did we get from commutative algebras to non commutative algebras (e.g. the algebra of matrices and linear transformations. Non commutative).
Remember, we are talking about "laws" of mathematics. So they should be statements, like "2 + 2 = 4". Can you give me an example of a mathematical statement which is true in some possible worlds but false in others?
I don't see how any of those topics you raised allow you to come up with such a statement.
Take Euclidean geometry, for example. Any statement about Euclidean geometry would be something like
Quote:
In Euclidean geometry, Pythagoras' theorem is true.
Well, that statement is true in any possible world, regardless of whether or not Euclidean geometry is a good model for some aspect of the
physical laws of that world.
So maybe you would try to come up with a statement like
Quote:
Pythagoras' theorem is true.
But I claim that this statement is actually meaningless, since it does not contain a reference to what type of geometry we are talking about.
Well, maybe you would try to use a statement like
Quote:
Euclidean geometry is a good model for some aspect of the physical laws of the world which I am in.
Well, this is a statement which is true in some possible worlds but not in others. However, it can't really be called a "law of mathematics", since it is actually about the
application of mathematics, not about mathematics itself.