The existence of mathematical entities is one of the main topic of philosophy of math in the twentieth century.
Carlo Cellucci explicitly questions the topic in his "Indiscrete Variations on Gian-Carlo Rota’s Themes" (in: E. Damiani, O. D’Antona, V. Marra, F. Palombi (eds.), From Combinatorics to Philosophy. The Legacy of G.-C. Rota, Springer, New York, 2009.). Cellucci notes that there are four possible answers to this question:
(i) Mathematical objects exist;
(ii) they do not;
(iii) We don’t know;
(iv) The question is irrelevant to mathematical practice or meaningless.
Cellucci states that (iv) is a very sensible answer (it was given in XVII and XVIII century by many thinkers - e.g. Descartes, Locke, Hume).
Also G.C. Rota reply to this question according to (iv). As cellucci puts it, "he states that it does not matter whether mathematical items exist, and probably it makes little sense to ask the question. If someone proved beyond any reasonable doubt that mathematical items did not exist, that would not affect the truth of any mathematical statement. Discussions concerning the existence of mathematical objects are motivated by deep-seated emotional cravings for permanence which are of psychiatric rather than philosophical interest".
For Cellucci moves beyond (iv) and argues for the thesis that mathematical objects are 'hypotheses'. In particular, Cellucci states that "a more satisfactory account of the nature of mathematical objects can be given stating that mathematical objects are hypotheses tentatively introduced to solve mathematical problems. A mathematical object is the hypothesis that a certain condition is satisfable" and "if, in the course of reasoning, the condition turns out to be satisfable, we say that the object `exists', if it turns out to be unsatisfable, we say that it `does not exist'. Thus speaking of `existence' is just a metaphor". Therefore, "there is no more to mathematical existence than the fact that mathematical objects are hypotheses tentatively introduced to solve mathematical problems. Such hypotheses are in turn a problem to be solved, it will be solved by introducing other hypotheses, and so on. Thus solving a mathematical problem is a potentially infinite task".
Cellucci's thesis is explicitely related to Plato's view (Plato, Republic, VI 510 c 2-5), according to which mathematicains hypothesize mathematical objects and they also hypothesize properties of such objects. "Therefore mathematical hypotheses concern either mathematical objects or their properties".
For, hypotheses characterize identity of mathematical entities (and "identity is characterized by the hypothesis, and characterized differently by di fferent hypotheses") but say nothing about their existence. In turn, as Cellucci stresses, hypotheses don't characterize the identity of mathematical items completely and conclusively but only partially and provisionally: their identity is always open to new determinations. It's so because mathematical objects are open to interactions with other objects, from which new properties may arise.
Moreover Cellucci notes that 'hypotheses' in this sense are di fferent from fictions:
1) fictions are explicitly not real, hypotheses are aimed at reality. "They are meant to provide an adequate approach to a still unknown or not perfectly known reality, although, as in the case of proofs by reductio ad absurdum, they eventually may fail to provide an adequate approach. The determination of reality given by a hypothesis is provisional, but one can try to make it stable showing that the hypothesis is plausible, that is, compatible with the facts of experience".
2) fictions are merely thinkable, hypotheses are supposed to be possible: "they must agree with the facts of experience. Only then hypotheses can be said to provide an adequate approach to reality. On the contrary, fictions are not supposed to be possible, and hence are not rejected if they do not agree with the facts of experience".
Cellucci's thesis is explicitely related to Plato's view (Plato, Republic, VI 510 c 2-5), according to which mathematicains hypothesize mathematical objects and they also hypothesize properties of such objects. "Therefore mathematical hypotheses concern either mathematical objects or their properties".
For, hypotheses characterize identity of mathematical entities (and "identity is characterized by the hypothesis, and characterized differently by di fferent hypotheses") but say nothing about their existence. In turn, as Cellucci stresses, hypotheses don't characterize the identity of mathematical items completely and conclusively but only partially and provisionally: their identity is always open to new determinations. It's so because mathematical objects are open to interactions with other objects, from which new properties may arise.
Moreover Cellucci notes that 'hypotheses' in this sense are di fferent from fictions:
1) fictions are explicitly not real, hypotheses are aimed at reality. "They are meant to provide an adequate approach to a still unknown or not perfectly known reality, although, as in the case of proofs by reductio ad absurdum, they eventually may fail to provide an adequate approach. The determination of reality given by a hypothesis is provisional, but one can try to make it stable showing that the hypothesis is plausible, that is, compatible with the facts of experience".
2) fictions are merely thinkable, hypotheses are supposed to be possible: "they must agree with the facts of experience. Only then hypotheses can be said to provide an adequate approach to reality. On the contrary, fictions are not supposed to be possible, and hence are not rejected if they do not agree with the facts of experience".
So, according to this view, hypotheses provide an only partial and provisional characterization of mathematical objects and new determinations are possible through interactions between hypotheses and experience: "from the interactions of the given mathematical objects with other objects, new facts may emerge which may suggest to modify or completely replace the hypothesis by which the identity of the given mathematical objects had been characterized. This is a potentially endless process, for mathematical objects are always open to interactions with other objects".
One of the most stimulating points of Cellucci's view is the fact that if mathematical items are hypotheses, is there a difference between mathematical and physical entities? Or between mathematical and certain chemical items?
