What does it mean when we say a mathematical object exists?

Question

I learned recently that there are mathematical objects that can be proven to exist, but also that can be proven to be impossible to "construct". For example see this answer on MSE:
Does the existence of a mathematical object imply that it is possible to construct the object?

Now my question is then, what does existence of an object really mean, if it is impossible to "find" it? What does it mean when we say that a mathematical object exists?

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Because the abstract nature of this question, and to make sure I understand myself what I'm asking, here is a more specific example.

Say that if have shown that there exists a real number $x$ that satisfies some property. I always assumed that this means that it possible to find this number $x$ in the set of real numbers. It may be hard to describe this number, but I would assume it is at least possible, to construct a number $x$ that is provable a real number that satisfies the property.

But if that does not have to be the case, if it is impossible to find the number $x$ that satisfies this property, what does this then mean? I just can't get my mind around this. Does that mean that there is some real number out there, but uncatchable somehow by the nature of its existence?

Answer 1

Arguably, your question cannot be answered in a satisfying way (unless you're a formalist).

Ultimately most mathematicians don't spend too much time thinking about ontology - a sort of "naive Platonism" may be adopted, although when pressed I think we generally retreat from that stance - but I think the "standard" meaning is simply, "The existence of such an object is provable from the axioms of mathematics," and "the axioms of mathematics" is generally understood as referring to ZFC. So, e.g., when we say "We can prove that an object with property $P$ exists," what we mean is "ZFC proves '$\exists xP(x)$.'" This is a completely formalist approach; in particular, it renders a question like

does that mean that there is some real number out there, but uncatchable somehow by the nature of its existence?

irrelevant, since there is no "out there" being referred to. It is also completely unambiguous (up to a choice of how we express the relevant mathematical statement in the language of set theory). Of course, if you give me a precise notion of "construct" (or "catch") then the question "Why does (or does?) ZFC prove the existence of a non-constructable (deliberately misspelled for clarity) object?" is something we can address, but now it's not really about the nature of mathematical existence but rather the nature of ZFC as a theory.

This response does ultimately just push the question back to why we privilege ZFC (and classical logic), and dodging this question (and any other formalist answer) grates against "realist" sensibilities. At the end of the day, the nature of existence is a philosophical, rather than mathematical, question; to my mind one of the main values of formalism is that it provides us with a language for doing mathematics which bridges philosophical differences. E.g. a large-cardinal-Platonist, an intuitionist, and an ultrafinitist will all agree with the statement "ZFC proves that there is an undetermined infinite game on $\omega$," regardless of their opinions on the statement "There exists an undetermined infinite game on $\omega$."

Answer 2

Concerning your question:

what does existence of an object really mean, if it is impossible to "find" it? What does it mean when we say that a mathematical object exists?

I would note that the underlying issue indeed centers on a concern with meaning as your wording suggests. How is it meaningful to talk about entities that are impossible to "find" as you put it? There are at least three approaches to this issue:

(a) Platonists believe that the meaning is given to such entities through their objective existence but in an abstract realm. Adherents of this belief would suspect those who don't share their faith as being "platonists on weekdays, formalists on the weekend".

(b) Formalists may question the coherence of the issue of meaning to begin with, and argue that one can only talk about mathematical meaning in a suitable formal framework (this is a bit of a dodge since one has yet to explain what "mathematical meaning" means).

(c) An increasingly popular approach that goes back at least to Benacerraf and Quine is to distinguish between procedure and ontology and argue that traditional questions like "what is a number?" focus on ontological issues that are less fruitful than procedural ones, whereas the real issues of meaning lie in the analysis of the procedures mathematicians employ and their usefulness in applications.

It seems to me that the other answer is a combination of (b) and (c) but it may be useful to separate them because the issues of meaning are more clearly addressed in (c) than in (b).

In (c), applications are understood in a broad sense that certainly includes physics but also other fields within mathematics itself. From this point of view, Felix Klein's unifying contributions to fields ranging from analysis to geometry to group theory would rate higher than Cantor/Weierstrass contributions to the foundations focusing on the nature of entities like "number" and "point". Some related discussions can be found in this 2017 publication in Foundations of Science and this 2017 publication in Mat. Stud.