Must mathematical entities necessarily exist? [duplicate]

Question

Are mathematical entities necessarily existing objects? That is to say, it is impossible for e.g. the real numbers not to exist. Have any philosophers talked about this topic?

Answer 1

"The most that can be expected from any model is that it can supply a useful approximation to reality: All models are wrong; some models are useful".

~ George E. P. Box

By this argument it doesn't matter, whether the constructs exist or not, since what's meaningful is, whether they're useful.

One can convey that all mathematics is models. Natural numbers are a model. A function is a model. The + operator is a model.

Does it then matter, whether they exist or whether they're useful?

A question about, whether e.g. "one" or "two" exist is unanswerable I find. Again, because the main interest is not, whether "one" or "two" exist, but whether they give useful information. One could perhaps also argue that functions are meaningless, unless they're applied to objects are somehow meaningful. E.g. in statistical trial one would also discover relationships, which are not accurate. And thus the function's existence in itself does not make "observations of function" existing, since there are also function mappings that do not exist empirically. I.e. the mere existence of the objects is not enough.

Answer 2

Some people don't believe that mathematical objects exist at all. We might call these people mathematical fictionalists. Those who believe they exist, generally believe they are one of three things: physical objects (such as symbols or brain states), mental states, or abstract objects. The position that mathematical objects are physical things has been called formalism or nominalism, and is associated with empiricism and naturalism/physicalism. I don't know if anyone today claims that mathematical objects are mental objects without also claiming that mental objects are physical objects, but this was once a position called conceptualism.

The position that mathematical objects are abstract objects is called Platonism or mathematical realism (I prefer the later, because modern Platonism is a lot different from classical Platonism). Abstract objects are objects that exist at no particular time or place and that are causally inert except through the mind. They include things like numbers, sets, and (abstract) propositions. Only mathematical realists would be likely to say that mathematical objects exist necessarily because almost no one would say that physical objects or mental states (or fictional objects) exist necessarily.

Not everyone agrees that there is such a thing as modal existence; that is, not everyone agrees that there is a difference between existing necessarily vs. existing contingently. Mathematical realists who believe in modal existence, would generally say that abstract objects--and therefore mathematical objects--exist necessarily.

My (somewhat vague and untutored) sense is that most philosophical thinkers today who are not materialists/physicalists would fall into this category.