> if object X models object Y, then I’m going to say that X is Y
If you haven't read to the end of the post, you might be interested in the philosophical discussion it builds to. The idea there, which I ascribe to, is not quite the same as what you are saying, but related in a way, namely, that in the case that X models Y, the mathematician is only concerned with the structure that is isomorphic between them. But on the other hand, I think following "therefore X is Y" to its logical conclusion will lead you to commit to things you don't really believe.
> But on the other hand, I think following "therefore X is Y" to its logical conclusion will lead you to commit to things you don't really believe.
I would love to hear an example… but before you do, I’m going to clarify that my statement was expressing a notion of what “is” sometimes means to a mathematician, and caution that
1. This notion is contextual, that sometimes we use the word “is” differently, and
2. It requires an understanding of “forgetfulness”.
So if I say that “Cauchy sequences in Q is R” and “Dedekind cuts is R”, you have to forget the structure not implied by R. In a set-theoretic sense, the two constructions are unequal, because you use constructed different sets.
I think this weird notion of “is” is the only sane way to talk about math. YMMV.
I think the problem with insisting on using "is" that way is that you then can't distinguish between two things you might reasonably want to express, i.e. "is isomorphic to"/"has the same structure as" and "refers to the same object". I totally agree that math is all about forgetting about the features of your objects that are not relevant to your problem (and in particular as the post argues things like R and C do not refer to any concrete construction but rather to their common structure), but if you want to describe that position you have to be able to distinguish between equality and isomorphism.
(Of course using "is" that way in informal discussion among mathematicians is fine -- in that case everyone is on the same page about what you mean by it usually)
> I think the problem with insisting on using "is" that way is that you then can't distinguish between two things you might reasonably want to express, i.e. "is isomorphic to"/"has the same structure as" and "refers to the same object".
It’s reasonable to want to express that difference in specific circumstances, but it would be completely unreasonable to make this the default.
For example, I can say that Z is a subset of Q, and Q is a subset of R. I can do this, but maybe you cannot—you’ve expressed a preference for a more rigid and inflexible terminology, and I don’t think you’re prepared to deal with the consequences.
If you haven't read to the end of the post, you might be interested in the philosophical discussion it builds to. The idea there, which I ascribe to, is not quite the same as what you are saying, but related in a way, namely, that in the case that X models Y, the mathematician is only concerned with the structure that is isomorphic between them. But on the other hand, I think following "therefore X is Y" to its logical conclusion will lead you to commit to things you don't really believe.