# Must mathematical entities necessarily exist? [duplicate]

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?

• What concept of necessity/impossibility are you working with? Possible worlds? Something else? Commented Apr 26, 2021 at 23:06
• According to Quine, anything can be made necessarily existed, a more interesting question is if mathematical entities necessarily exist then what other existence(s) have to be eliminated or adjusted in order for the whole scientific field to hold... Commented Apr 27, 2021 at 1:14
• The SEP mathematical platonism article says here that there's a potential difference between "truth-value realism"--the idea that there are necessary truths about mathematical propositions even if we never discover them--and math. platonism in the sense of mathematical entities "existing" as abstract objects. Not sure if truth-value realists who reject math. platonism are rejecting Quine's criterion, using a logic other than 1st-order, or what. Commented Apr 27, 2021 at 18:09

"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.

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.

• "Almost no one would say that physical objects or mental states exist necessarily" Apart from the widespread mathematical-platonists, like Max Tegmark, who consider mathematics 'more real' than any given reality which is contingent on universal math, which are then 'necessary' to existence in principle? Commented Apr 27, 2021 at 18:51
• I am confused by your comment. Platonism today pretty much just means the position that abstract objects exist and that mathematical objects are abstract objects. Platonism has nothing to do with the metaphysics of the physical or the mental. And why would believing that abstract objects are "more real" than contingent objects imply that they don't think those contingent objects are contingent? Commented Apr 27, 2021 at 20:51
• The question is "Are mathematical entities necessarily existing objects". By en.wikipedia.org/wiki/Necessity_and_sufficiency#Necessity Tegmark would say mathematics like geometry are necessary. My point is you imply this is an almost unheard-of view, but it is in fact common. Mathematical Platonism is the position that abstract objects exist independently of minds, ie prexist minds, and are necessary precursors. Commented Apr 27, 2021 at 21:06
• I think you have dramatically misunderstood what I wrote. I ended by saying that this position is probably the most common after materialism/physicalism. What is almost unheard of is the position that physical or mental objects are necessary, but that is not part of Platonism. Maybe you could explain where the misunderstanding comes from so I can fix the answer. Commented Apr 27, 2021 at 21:14
• I've modified the answer. Please let me know if it clears up your misunderstanding. Commented Apr 27, 2021 at 21:38