This article is inspired by the current debates about Platonism in the philosophy of mathematics and aims to clarify the terminology used in this field, primarily from the perspective of its ontological foundations. The difficulty lies in the fact that mathematics deals with special objects, if objects are recognized, and/or with special structures, if certain structures are recognized instead of objects. The purpose of this study is to demonstrate the incompatibility of Platonism and naturalism. The term «realism» is polysemantic. It encompasses not only Platonism (which is quite traditional), but also certain variants of naturalism, as we attempt to demonstrate in this article. Platonism is incompatible with nominalism, according to existing historical-philosophical classifications, but realism, in some versions, can likely be compatible with naturalism. The main «points of support» for our research will be the positions of Willard Quine and Hilary Putnam in the philosophy of mathematics, which, as we have attempted to demonstrate, consist of an attempt to combine certain forms of realism with: 1) naturalistic nominalism (W. Quine); 2) conceptualism (H. Putnam). Our solution to the classification problems is demonstrated using the example of W. Quine’s philosophy of mathematics, which has a direct connection both with the ideas of nominalistic metaphysics and with the ideas of a naturalistic-realistic interpretation of mathematical objects. By unraveling this knot, we will be able to clarify a number of metaphysical and ontological problems. A novel feature is our proof that W. Quine cannot be called a Platonist. Translating into Quine’s terminology, we can say that the operation of hypostatizing abstract objects, necessary for any Platonism, occurs at the level of the theory’s ideology, that is, exclusively at the level of language. The operation of endowing them with existence, however, is carried out not in language itself, but in a manner internal to theory – based on the quantification of free variables. This defines the boundaries between metaphysics and ontology sought in this article. In the example of W. Quine, we obtain an anti-Platonist metaphysics accompanied by a moderately realist ontology that allows for a plurality of naturalistic models.

have themselves a reality that exists outside space and time. As a result, the philosophical view that mathematical objects somehow exist on their own in Philosophy of mathematics is the branch of philosophy that deals with the nature of mathematics and its relationship to other areas of philosophy, particularly epistemology and metaphysics. Central questions posed include whether or not mathematical objects are purely abstract entities or are in some way concrete, and in what the relationship such objects have with physical reality consists. Major Aristotelian realism holds that mathematics studies properties such as symmetry, continuity and order that can be literally realized in the physical world (or in any other world there might be). It contrasts with Platonism in holding that the objects of mathematics, such as numbers, do not exist in an "abstract" world but can be physically realized. For example, the number 4 is realized in the relation between a heap of parrots and the universal "being a parrot" that divides the heap into so many parrots. Aristotelian realism is defended by James Franklin and the Sydney School in the philosophy of mathematics and is close to the view of Penelope Maddy that when an egg carton is opened, a set of three eggs is perceived (that is, a mathematical entity realized in the physical world). A problem for Aristotelian realism is what account to give of higher infinities, which may not be realizable in the physical world. The Euclidean arithmetic developed by John Penn Mayberry in his book The Foundations of Mathematics in the Theory of Sets also falls into the Aristotelian realist tradition. Mayberry, following Euclid, considers numbers to be simply "definite multitudes of units" realized in nature—such as "the members of the London Symphony Orchestra" or "the trees in Birnam wood". Whether or not there are definite multitudes of units for which Euclid's Common Notion 5 (the whole is greater than the part) fails and which would consequently be reckoned as infinite is for Mayberry essentially a question about Nature and does not entail any transcendental suppositions.

defends mathematical Platonism, asserting that numbers exist because the best scientific theories are ontologically committed to numbers. Possibility and necessity Ontology is the philosophical study of being. It is traditionally understood as the subdiscipline of metaphysics focused on the most general features of reality. As one of the most fundamental concepts, being encompasses all of reality and every entity within it. To articulate the basic structure of being, ontology examines the shared characteristics among all things and investigates their classif An ontological commitment of a person or a theory is an entity that exists according to them. For instance, a person who believes in God has an ontological commitment to God. Ontological commitments can be used to analyze which ontologies people explicitly defend or implicitly assume. They play a central role in contemporary metaphysics when trying to decide between competing theories. For example, the Quine–Putnam indispensability argument defends mathematical Platonism, asserting that numbers exist because the best scientific theories are ontologically committed to numbers. Possibility and necessity are… In… I…

Abstract In this chapter, I will discuss Benacerraf s epistemological argument against platonism, which holds, in a nutshell, that platonism cannot be right because it precludes the possibility of mathematical knowledge. I will begin by formulating the argument in what I think is the best possible way. I will then move on to a discussion of some possible solutions to the problem. This discussion will serve simultaneously as a historical survey of the answers that platonists have actually given to the epistemological problem and as a search of the logical space of possible solutions that might be given. I will concern myself with seven different proposed solutions to the problem: one developed by Godel; another by Maddy; a third by a host of contemporary platonists, most notably Parsons; a fourth by Wright and Hale; a fifth hinted at by Quine and developed by Steiner and Resnik; a sixth by Katz and Lewis; and finally, a seventh by Resnik and Shapiro. I will argue that the first five suggestions fail outright.

Abstract So far, I have tried to show that there is a version of mathematical platonism that avoids all of the important objections to that view. I will now try to do the same for mathematical anti-platonism. This will occupy me for all but the last few pages of part II. Now, I suppose that there are numerous arguments against mathematical antiplatonism (or, what comes to the same thing, in favor of mathematical platonism), but it seems to me that there is only one such argument with a serious claim to cogency. Thus, I will ignore all other arguments and consider only this single line of attack. The argument I have in mind is due to Frege. My presentation will be somewhat different from Frege’s, but the spirit of the argument is essentially the same.