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

There is an extremely popular notion that mathematical platonism (every mathematical object has a platonic existence) is well-defined. To bad that's false. Due to the incompleteness theorem, we know that any reasonable foundation for mathematics is either inconsistent (useless) or incomplete (does not prove every true sentence about natural numbers), and so cannot pin down a unique mathematical universe. Thus the popular definition of mathematical platonism is ill-defined, because "mathematical object" is ill-defined without reference to a specific foundational system. One can amend it to "there is a real-world interpretation of ZFC", but nobody can justify such a statement. So what then? So far all the empirical evidence that mathematics is relevant to the real world only justifies that there seems to be a real-world interpretation (maybe even merely approximately) of some very very tiny fragment of ZFC, maybe ACA or ATR (see Reverse Mathematics) or HOA (higher-order arithmetic). People who attempt to use the undisputed success of applied mathematics to justify all of modern mathematics are just wrong.

People who claim that every possible axiomatic system for mathematics have a platonic model, are even more wrong. Again by the incompleteness theorem, if PA is consistent then PA+¬Con(PA) is also consistent, but it proves itself inconsistent! So obviously some FOL theories are simply utterly false (with respect to the standard model of PA on which FOL syntax and deduction itself is based)! Worse still, there are people who claim that every mathematical structure exists. This claim is not even wrong because it is ill-defined in itself. There is no way anyone can define what mathematical structure means without reference to some formal system or some class of formal systems, and the mere definition of "formal system" needs to be based on a rudimentary amount of assumptions about finite strings.

The bottom line is that you need to define "mathematical entity" otherwise your question is ill-defined. But what I said already addresses some important issues that you need to understand if you want to undertake proper philosophy of mathematics.

To answer your question directly, the real numbers as constructed within a formal system of course do not exist in reality any more than a checkmate strategy from a given winning chess position. The question you should ask is whether there is an embedding of the mathematical structure of the reals into reality. Unfortunately, that is ill-defined, because different formal systems prove different things about real numbers!

For example, ACA0 suffices for all applied real analysis so far, because one can encode any real number as a subset of ℕ, and can also encode any real functions with countably many discontinuities as a subset of ℕ, and ACA0 is strong enough to facilitate manipulation of such encodings. But ACA0 has a model M comprising ℕ and arithmetical subsets of ℕ, equivalently subsets of ℕ whose membership can be computed using some finite Turing jump. Note that in this model M there are only countably many subsets of ℕ!!

Think about it. For applied real analysis we do not seem to need anything beyond ACA0, but there is a model M of ACA0 that has only countably many reals (per the encoding). Where did all the other 'reals' go? Observe that ACA0 does prove something like Cantor's theorem (encoded), namely "there is no surjection from ℕ to ℝ". There is no contradiction here; in the model M this just means there is no finite-jump-computable surjection from ℕ onto arithmetical sets, which is correct.

Now of course Z set theory proves that ℝ is uncountable, but that is just a statement encoded in the language of set theory. By Lowenheim-Skolem you know that if Z is consistent then Z has a countable model, so that should let you realize that "the reals" is not an absolute concept. Unless you come up with a privileged model of Z, you simply do not have any objective notion of "the reals", and hence asking whether "the reals" exist would not be a well-defined question.

Answer 2

"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 3

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.