This is definitely not a “big problem” in that we can use math regardless of what the outcome is.
It sounds like you’re arguing that semantic uniformity doesn’t matter, because we can change what “exists” means. But once you change what “exists” means, you will likely run into epistemological issues. If your mathematical objects aren’t physical entities capable of interacting with the world, how can you have knowledge that is causally related to those entities? That’s the dilemma of the argument above—it seems possible to get semantic uniformity at the expense of epistemological uniformity, or vice versa, but having both together is difficult.
Nobody has to believe that ordinary non mathematical langage contains a single well defined meaning of “exists”. And if “exists” is polysemous , then one of its meanings could be the meaning of mathematically-exists … it doesn’t have to have a unique meaning. Fictivism is an example: the theory that mathematically-exists means fictionally-exists, since ordinary language allows truth and existence to be used in reference to fictional worlds.
This is definitely not a “big problem” in that we can use math regardless of what the outcome is.
It sounds like you’re arguing that semantic uniformity doesn’t matter, because we can change what “exists” means. But once you change what “exists” means, you will likely run into epistemological issues. If your mathematical objects aren’t physical entities capable of interacting with the world, how can you have knowledge that is causally related to those entities? That’s the dilemma of the argument above—it seems possible to get semantic uniformity at the expense of epistemological uniformity, or vice versa, but having both together is difficult.
Nobody has to believe that ordinary non mathematical langage contains a single well defined meaning of “exists”. And if “exists” is polysemous , then one of its meanings could be the meaning of mathematically-exists … it doesn’t have to have a unique meaning. Fictivism is an example: the theory that mathematically-exists means fictionally-exists, since ordinary language allows truth and existence to be used in reference to fictional worlds.