 Independence says that mathematical objects, if there are any, are independent of intelligent agents and their language, thought, and practices. We will discuss what this thesis might amount to, and how it might be defended, in Section  4.   3. Objections to Mathematical Platonism   A variety of objections to mathematical platonism have been developed. Here are the most important ones. 3.1 Epistemological access   The most influential objection is probably the one inspired by Benacerraf (1973). What follows is an improved version of Benacerraf’s objection due to Field  (1989).[12]  This version relies on the following three premises.   Premise 1. Mathematicians are reliable, in the sense that for almost every mathematical sentence S, if mathematicians accept S, then S is true.   Premise 2. For belief in mathematics to be justified, it must at least in principle be possible to explain the reliability described in Premise 1.   Premise 3. If mathematical platonism is true, then this reliability cannot be explained even in principle.     If these three premises are correct, it will follow that mathematical platonism undercuts our justification for believing in mathematics.   But are the premises correct? The first two premises are relatively uncontroversial. Most platonists are already committed to Premise 1. And Premise 2 seems fairly secure. If the reliability of some belief formation procedure could not even in principle be explained, then the procedure would seem to work purely by chance, thus undercutting any justification we have for the beliefs produced in this way.   Premise 3 is far more controversial. Field defends this premise by observing that “the truth-values of our mathematical assertions depend on facts involving platonic entities that reside in a realm outside of space-time” (Field 1989, p. 68) and thus are causally isolated from us even in principle. However, this defense assumes that any adequate explanation of the reliability in question must involve some causal correlation. This has been challenged by a variety of philosophers who have proposed more minimal explanations of the reliability claim. (See Burgess &amp; Rosen 1997, pp. 41–49 and Lewis 1991, pp. 111–112; cf. also Clarke-Doane 2016. See Linnebo 2006 for a  critique.)[13] 3.2 A metaphysical objection   Another famous article by Benacerraf develops a metaphysical objection to mathematical platonism (Benacerraf 1965, cf. also Kitcher 1978). Although Benacerraf focuses on arithmetic, the objection naturally generalizes to most pure mathematical objects.   Benacerraf opens by defending what is now known as a structuralist view of the natural numbers, according to which the natural numbers have no properties other than those they have in virtue of being positions in an ω-sequence. For instance, there is nothing more to being the number 3 than having certain intrastructurally defined relational properties, such as succeeding 2, being half of 6, and being prime. No matter how hard we study arithmetic and set theory, we will never know whether 3 is identical with the fourth von Neumann ordinal, or with the corresponding Zermelo ordinal, or perhaps, as Frege suggested, with the class of all three-membered classes (in some system that allows such classes to exist).   Benacerraf now draws the following conclusion: Therefore, numbers are not objects at all, because in giving the properties …of numbers you merely characterize an abstract structure—and the distinction lies in the fact that the “elements” of the structure have no properties other than those relating them to other “elements” of the same structure. (Benacerraf 1965, p. 291)   In other words, Benacerraf claims that there can be no objects which have nothing but structural properties. All objects must have some non-structural properties as well. (See Benacerraf 1996 for some later reflections on this argument.)   Both of the steps of Benacerraf’s argument are controversial. The first step—that natural numbers have only structural properties—has recently been defended by a variety of mathematical structuralists (Parsons 1990, Resnik 1997, and Shapiro 1997). But this step is denied by logicists and neo-logicists, who claim that the natural numbers are intrinsically tied to the cardinalities of the collections that they number. And the second step—that there can be no objects with only structural properties—is explicitly rejected by all of the structuralists who defend the first step. (For some voices sympathetic to the second step, see Hellman 2001 and MacBride 2005. See also Linnebo 2008 for discussion.) 3.3 Other metaphysical objections   In addition to Benacerraf’s, a variety of metaphysical objections to mathematical platonism have been developed. One of the more famous examples is an argument of Nelson Goodman’s against set theory. Goodman (1956) defends the Principle of Nominalism, which states that whenever two entities have the same basic constituents, they are identical. This principle can be regarded as a strengthening of the familiar set theoretic axiom of extensionality. The axiom of extensionality states that if two sets x and y have the same elements—that is, if ∀u(u ∈ x ↔ u ∈ y)—then they are identical. The Principle of Nominalism is obtained by replacing the membership relation with its transitive  closure.[14]  The principle thus states that if x and y are borne ∈* by the same individuals—that is, if ∀u(u ∈* x ↔ u ∈* y)—then x and y are identical. By endorsing this principle, Goodman disallows the formation of sets and classes, allowing only the formation of mereological sums and the application to the standard mereological operations (as described by his “calculus of individuals”).   However, Goodman’s defense of the Principle of Nominalism is now widely held to be unconvincing, as witnessed by the widespread acceptance by philosophers and mathematicians of set theory as a legitimate and valuable branch of mathematics. 4. Between object realism and mathematical platonism   Object realism says there exist abstract mathematical objects, whereas platonism adds Independence, which says that mathematical objects are independent of intelligent agents and their language, thought, and practices. This final section surveys some lightweight forms of object realism that stop short of full-fledged platonism. 4.1 How To Understand Independence   A natural gloss on Independence is the counterfactual conditional that, had there not been any intelligent agents, or had their language, thought, or practices been suitably different, there would still have been mathematical objects.    This counterfactual independence (as we may call it) is accepted by most analytic philosophers. To see why, consider the role that mathematics plays in our reasoning. We often reason about scenarios that aren’t actual. Were we to build a bridge across this canyon, say, how strong would it have to be to withstand the powerful gusts of wind? Sadly, the previous bridge collapsed. Would it have done so had the steel girders been twice as thick? This form of reasoning about counterfactual scenarios is indispensable both to our everyday deliberations and to science. The permissibility of such reasoning has an important consequence. Since the truths of pure mathematics can freely be appealed to throughout our counterfactual reasoning, it follows that these truths are counterfactually independent of us humans, and all other intelligent life for that matter. That is, had been there been no intelligent life, these truths would still have remained the same.    Pure mathematics is in this respect very different from ordinary empirical truths. Had intelligent life never existed, this article would not have been written. More interestingly, pure mathematics also contrasts with various social conventions and constructions, with which it is sometimes compared (Cole 2009, Feferman 2009, Hersh 1997). Had intelligent life never existed, there would have been no laws, contracts, or marriages—yet the mathematical truths would have remained the same.    Thus, if Independence is understood merely as counterfactual independence, then anyone who accepts object realism should also accept platonism.    It is doubtful that this understanding of Independence is sufficient, however. For Independence is meant to substantiate an analogy between mathematical objects and ordinary physical objects. Just as electrons and planets exist independently of us, so do numbers and sets. And just as statements about electrons and planets are made true or false by the objects with which they are concerned and these objects’ perfectly objective properties, so are statements about numbers and sets. In short, mathematical objects are just as “real” as ordinary physical objects (if not even more so, as Plato thought).    Let us now consider some views that reject this stronger understanding of Independence in terms of the mentioned analogy. These views are thus lightweight forms of object realism, which stop short of full-blown platonism.  4.2 Plenitudious platonism   One lightweight form of object realism is the “full-blooded platonism” of Balaguer 1998. This view is characterized by a plenitude principle to the effect that any mathematical objects that could exist actually do exist. For instance, since the Continuum Hypothesis is independent of the standard axiomatization of set theory, there is a universe of sets in which the hypothesis is true and another in which it is false. And neither universe is metaphysically privileged. By contrast, traditional platonism asserts that there is a unique universe of sets in which the Continuum Hypothesis is either determinately true or determinately false.[15]   One alleged benefit of this plenitudinous view is in the epistemology of mathematics. If every consistent mathematical theory is true of some universe of mathematical objects, then mathematical knowledge will, in some sense, be easy to obtain: provided that our mathematical theories are consistent, they are guaranteed to be true of some universe of mathematical objects.   However, “full-blooded platonism” has received much criticism. Colyvan and Zalta 1999 criticize it for undermining the possibility of reference to mathematical objects, and Restall 2003, for lacking a precise and coherent formulation of the plenitude principle on which the view is based. Martin (2001) proposes that different universes of sets be amalgamated to yield a single maximal universe, which will be privileged by fitting our conception of set better than any other universe of sets.   A different version of plenitudinous platonism is developed in Linsky &amp; Zalta 1995 and a series of further articles. (See, for instance, Linsky &amp; Zalta 2006 and other articles cited therein.) Traditional platonism goes wrong by “conceiv[ing] of abstract objects on the model of physical objects” (Linsky &amp; Zalta 1995, p. 533), including in particular the idea that such objects are “sparse” rather than plenitudinous. Linsky &amp; Zalta develop an alternative approach on the basis of the second author’s “object theory”. The main feature of object theory is a very general comprehension principle which asserts the existence of a plenitude of abstract objects: for any collection of properties, there is an abstract object which “encodes” precisely these properties. In object theory, moreover, two abstract objects are identical just in case they encode precisely the same properties. Object theory’s comprehension principle and identity criterion are said to “provide the link between our cognitive faculty of understanding and abstract objects” (ibid., p. 547). (See Ebert &amp; Rossberg 2007 for critical discussion.) 4.3 Lightweight semantic values   Assume that object realism is true. For convenience, assume also Classical Semantics. These assumptions ensure that the singular terms and quantifiers of mathematical language refer to and range over abstract objects. Given these assumptions, should one also be a mathematical platonist? In other words, do the objects that mathematical sentences refer to and quantify over satisfy Independence or some similar condition?   It will be useful to restate our assumptions in more neutral terms. We can do this by invoking the notion of a semantic value, which plays an important role in semantics and the philosophy of language. In these fields it is widely assumed that each expression makes some definite contribution to the truth-value of sentences in which the expression occurs. This contribution is known as the semantic value of the expression. It is widely assumed that (at least in extensional contexts) the semantic value of a singular term is just its referent.   Our assumptions can now be stated neutrally as the claim that mathematical singular terms have abstract semantic values and that its quantifiers range over the kinds of item that serve as semantic values. Let’s focus on the claim about singular terms. What is the philosophical significance of this claim? In particular, does it support some version of Independence? The answer will depend on what is required for a mathematical singular term to have a semantic value.   Some philosophers argue that not very much is required (Frege 1953, Dummett 1981, Dummett 1991a, Wright 1983, Hale &amp; Wright 2000, Rayo 2013, and Linnebo 2012 and 2018). It suffices for the term t to make some definite contribution to the truth-values of sentences in which it occurs. The whole purpose of the notion of a semantic value was to represent such contributions. It therefore suffices for a singular term to possess a semantic value that it makes some such suitable contribution.   This may even open the way for a form of non-eliminative reductionism about mathematical objects (Dummett 1991a, Linnebo 2018). Although it is perfectly true that the mathematical singular term t has an abstract object as its semantic value, this truth may obtain in virtue of more basic facts which do not mention or involve the relevant abstract object. Compare for instance the relation of ownership that obtains between a person and her bank account. Although it is perfectly true that the person owns the bank account, this truth may obtain in virtue of more basic sociological or psychological facts which do not mention or involve the bank account.   If some lightweight account of semantic values is defensible, we can accept the assumptions of object realism and Classical Semantics without committing ourselves to any traditional or robust form of platonism. 4.4 Two further lightweight forms of object realism    We conclude by describing two further examples of lightweight forms of object realism that reject the platonistic analogy between mathematical objects and ordinary physical objects.   First, perhaps mathematical objects exist only in a potential manner, which contrasts with the actual mode of existence of ordinary physical objects. This idea is at the heart of the ancient notion of potential infinity (Lear 1980, Linnebo &amp; Shapiro 2017). According to Aristotle, the natural numbers are potentially infinite in the sense that, however large a number we have produced (by instantiating it in the physical world), it is possible to produce an even larger number. But Aristotle denies that the natural numbers are actually infinite: this would require the physical world to be infinite, which he argues is impossible.    Following Cantor, most mathematicians and philosophers now defend the actual infinity of the natural numbers. This is made possible in part by denying the Aristotelian requirement that every number needs to be instantiated in the physical world. When this is denied, the actual infinity of the natural numbers no longer entails the actual infinity of the physical world.    However, a form of potentialism about the hierarchy of sets continues to enjoy considerable support, especially in connection with the iterative conception of sets (Parsons 1977, Jané 2010, Linnebo 2013, Studd 2013). No matter how many sets have been formed, it is possible to form even more. If true, this would mean that sets have a potential form of existence which distinguishes them sharply from ordinary physical objects.    Second, perhaps mathematical objects are ontologically dependent or derivative in a way that distinguishes them from independently existing physical objects (Rosen 2011, Donaldson 2017). For example, on the Aristotelian view just mentioned, a natural number depends for its existence on some instantiation or other in the physical world. There are other versions of the view as well. For example, Kit Fine (1995) and others argue that a set is ontologically dependent on its elements. (This view is also closely related to the set-theoretic potentialism mentioned above.)  