Platonism in Mathematics

The basic ontological question in the philosophy of mathematics deals with the issue of the existence of mathematical objects. According to the Platonist, mathematical objects exist independently of us.

For the Platonist, not only mathematical objects exist, they exist neces­sarily. That is, those objects that do exist necessarily do so. Mathematical objects, on this conception, could not have failed to exist. What is the source of this necessity? In other words, is there something that makes math­ematical objects necessarily existing entities? Clearly, there is no causal process involved here, given the causal inertness of mathematical objects. So, the necessity has some other origin. An answer that some Platonists, such as Gottlob Frege (1974), have offered emerges from the concepts involved. Concepts are here understood not as mental, psychological states, such as beliefs or mental images. Such states are subjective, and they depend on whoever has them. Your mental image of a watermelon may, but need not, have something in common with mine. Concepts, on the Platonist view, are not subjective images or experiences on our minds. Rather, they are objective, mind-independent objects whose existence does not depend on us.

There are different forms of Platonism. Here I will consider four of them: Fregean Platonism (Frege 1974, Hale and Wright 2001), Godelian Platonism (Godel 1964, Maddy 1990), Quinean Platonism (Quine 1960, Colyvan 2001), and structuralist Platonism (Resnik 1997, Shapiro 1997). Given that all of these views are particular forms of Platonism, they have in common the contention that mathematical objects (broadly understood here to include mathematical structures): exist; are mind-independent; and are abstract (that is, they are not physically extended). Despite these common features, there are significant differences among these views. I will explore some of them below.

2.1 Fregean Platonism

For the Fregean Platonist, the abstract character of mathematical objects emerges from the kind of thing they are: objects that fall under certain concepts. As we saw, Fregean concepts are abstract, mind-independent things; numbers and other mathematical objects inherit the same abstract character from the concepts under which they fall. The Fregean Platonist has no difficulty making sense of the objectivity of mathematics, given the mind­independence of the concepts involved in their characterization.

Part of the motivation for Frege's logicism was epistemological. The logicist approach could offer a suitable account of our knowledge of arithmetic. (I will return to this point below.) But probably Frege's main motivation was of an ontological nature. He wanted to provide the right answer about the nature of mathematical objects, and, in particular, provide the proper characterization of the concept of number. In fact, the bulk of Foundations of Arithmetic (Frege 1974) is dedicated to a thorough and extremely critical discussion of the accounts of the concept of number in Frege's time. Given what Frege (correctly) perceived to be a morass of confusion, unclearness, and incoherence that prevailed in the discussions of the foundations of arithmetic from the seventeenth through the nine­teenth centuries, he worked very hard to offer a clear, well grounded, and coherent logicist alternative. Due to the precise nature of the logical notions, Frege's reduction of arithmetic concepts to logical ones generated a clear-cut formulation of the former.

Central to Frege's strategy was the use of what is now called Hume's Principle: two concepts are equinumerous if and only if there is a one-to-one correspondence between them. Hume's Principle was used at various crucial points; for instance, to show that the number zero is different from the number one.

But how can one establish that Hume's Principle is true? Frege thought he could derive Hume's Principle from a basic logical law, which is called Basic Law V. According to this law, the extension of the concept F is the same as the extension of the concept G if and only if the same objects fall under the concepts F and G. Basic Law V seemed to be a fundamental logical law, dealing with concepts, their extensions, and their identity. It had the right sort of generality and analytic character that was needed for a logicist foundation of arithmetic.

There is only one problem: Basic Law V turns out to be inconsistent. It immediately raises Russell's paradox if we consider the concept is not a member of itself. To see why this is the case, suppose that there is such a thing as the set composed by all the sets that are not members of themselves. Let us call this set R (for Russell). Now let us consider whether R is a member of R. Suppose that it is. In this case, we conclude that R is not a member of R, given that, by definition of R, R’s members are those sets that are not members of themselves. Suppose, in turn, that R is not a member of R. In this case, we conclude that R is a member of R - since this is precisely what it takes for the set R to be a member of R. Thus, R is a member of R if, and only if, R is not a member of R. It then imme­diately follows that R is and is not a member of R - a contradiction.

Someone may say that this argument just shows that there is not such a thing as the Russell set R after all.^[40] So, what is the big deal? The problem is that, as Russell also found out, it follows from Frege's system using suitable definitions that there is a set of all sets that are not members of themselves.

But not everything was lost. Although Frege acknowledged the problem, and tried to fix it by introducing a new, consistent principle, his solution ultimately did not work.^[41] However, there was a solution available to Frege. He could have jettisoned the inconsistent Basic Law V and adopted Hume's Principle as his basic principle instead. Given that the only use that Frege made of Basic Law V was to derive Hume's Principle, if the latter were assumed as basic, one could then run, in a perfectly consistent manner, Frege's reconstruction of arithmetic. In fact, we could then credit Frege with the theorem to the effect that arithmetic can be derived in a system like Frege's from Hume's Principle alone. Frege's approach could then be extended to other branches of mathematics.^[42]

2.2 Godelian Platonism

If, for the Fregean, arithmetic is ultimately derivable in second-order logic plus Hume's Principle, for someone like Kurt Godel, the truth of basic mathematical axioms can be obtained directly by intuition (see Godel 1964, Maddy 1990). Frege thought that intuition played a role in how we come to know the truth of geometrical principles (which, for him, following Kant, were synthetic a priori);^[43] but arithmetic, being derivable from logic, was analytic. For Godel, however, not only the principles of arithmetic, but also the axioms of set theory can be apprehended directly by intuition. We have, Godel claims, “something like a perception of the objects of set theory” (1964, 485). That is, we are able to “perceive” these objects as having certain properties and lacking others, in a similar way to that in which we perceive physical objects around us. That we have such a perception of set-theoretic objects is supposed to be “seen from the fact that the axioms [of set theory] force themselves upon us as being true” (Godel 1964, 485).

But how exactly does the fact that the axioms of set theory “force them­selves upon us as being true” support the claim that we “perceive” the objects of set theory? Godel seemed to have a broad conception of perception, and when he referred to the objects of set theory, he thought that we “perceived” the concepts involved in the characterization of these objects as well. The point may seem to be strange at first. With further reflection, however, it is not unreasonable. In fact, an analogous move can be made in the case of the perception of physical objects. For example, in order for me to perceive a tennis ball, and recognize it as a tennis ball, I need to have the concept of tennis ball. Without the latter concept, at best I will perceive a round yellow thing - assuming I have these concepts. Similarly, I would not be able to recognize certain mathematical objects as objects of set theory unless I had the relevant concepts. The objects could not be “per­ceived” to be set-theoretic except if the relevant concepts were in place.

Now, in order to justify the “perception” of set-theoretic objects from the fact that the axioms of set theory are forced upon us as being true, Godel needs to articulate a certain conception of rational evidence (see Parsons 2008, 146-8). On Godel's view, in order for us to have rational evidence for a proposition - such as an axiom of set theory - we need to make sense of the concepts that occur in that proposition. In making sense of these concepts, we are “perceiving” them. Mathematical concepts are robust in their characterization, in the sense that what they stand for is not of our own making. Our perception of physical objects is similarly robust. If there is no pink elephant in front of me right now, I cannot perceive one.^[44] And you cannot fail to perceive the letters of this sentence as you read it, even though you might not be thinking about the letters as you read the sentence, but what the latter stands for. The analogy between sense percep­tion and the “perception” of concepts is grounded on the robustness of both. The robustness requires that I perceive what is the case, although I can, of course, be mistaken in my perception. For instance, as I walk along the street, I see a bird by a tree. I find it initially strange that the bird does not move as I get closer to the tree; only to find out, when I get close enough, that there was no bird there, but a colorful piece of paper. I thought I had perceived a bird, when in fact I perceived something else. The perception, although robust - something was perceived, after all - is fallible. But I still perceived what was the case: a piece of paper in the shape of a bird. I just mistook that for a bird, something I corrected later. Similarly, the robustness of our “perception” of the concepts involved in an axiom of set theory is part of the account of how that axiom can force itself upon us as being true. By making sense of the relevant set-theoretic concepts, we “perceive” the latter and the connections among them. In this way, we “perceive” what is the case among the sets involved. Of course, similarly to the case of sense perception, we may be mistaken about what we think we perceive - that is part of the fallibility of the proposal. But the “perception” is, nevertheless, robust. We “perceive” something that is true.

This account of “perception” of mathematical concepts and objects is, in fact, an account of mathematical intuition. Following Charles Parsons (2008, 138-43), we should note that we have intuition of two sorts of things. We have intuition of objects (e.g., the intuition of the objects of arithmetic), and we have intuition that some proposition is true (e.g., the intuition that “the successor of a natural number is also a natural number” is true). The former can be called “intuition of,” and the latter “intuition that.” In the passage quoted in the first paragraph of this section, Godel seems to be using the intuition that the axioms of set theory force themselves upon us to support the corresponding intuition of the objects of set theory. The robustness of both intuitions involved here is a central feature of the account.

2.3 Quinean Platonism

The Godelian Platonist explores some connections between the “perception” of mathematical objects and perception of physical entities. The Quinean Platonist draws still closer connections between mathematics and the empirical sciences. If you have a strong nominalistic tendency, but find out that, in the end, you cannot avoid being committed to the existence of mathematical objects when you try to make sense of the best theories of the world, you probably are a Quinean Platonist. W.V. Quine himself was such a Platonist (see Quine 1960). Even when he acknowledges the indispensable role that reference to mathematical objects plays in the formulation of our best theories of the world, Quine insists that he is committed to only one kind of mathematical object: classes. All the other mathematical objects that he needs, such as numbers, functions, and geometrical spaces, can be obtained from them.

On the Quinean conception, Platonism is a matter of ontological honesty. Suppose you are a scientific realist about science; that is, you take scientific theories to be true (or, at least, approximately true), and you think that the terms in these theories refer. So, for example, in quantum mechanics, you acknowledge that it is crucial to refer to things like electrons, protons, photons, and quarks. These posits are an integral part of the theory, and positing their existence is central in the explanation of the behavior of the observable phenomena. These explanations include making successful predictions and applying quantum theory in various contexts.

Now, in the formulation of quantum mechanics, it is indispensable to refer not only to electrons and other quantum particles, but also to math­ematical objects. After all, it is in terms of the latter that we can characterize the former. For instance, there is no way to characterize an electron but in terms of a certain group of invariants. These invariants are particular mathematical functions (particular mappings). So, to acknowledge com­mitment to the existence of electrons but deny the existence of the corres­ponding mathematical functions is to take back what needs to be assumed in order to express what such a physical object is. It is to indulge in double thinking, in ontological dishonesty. Quine's indispensability argument is an attempt to get us straight - particularly the scientific realists among us. According to this argument:

(P1) We ought to be ontologically committed to all and only those enti­ties that are indispensable to our best theories of the world.

(P2) Mathematical entities are indispensable to our best theories of the world.

(C) Therefore, we ought to be ontologically committed to mathematical entities.^[45]

The point of the indispensability argument is to ensure that mathematical and physical posits be treated in the same way. In fact, on the Quinean picture, there is no sharp divide between mathematics and empirical science. Both are part of the same continuum and ultimately depend on experience. There is only a difference in degree between them. Certain mathematical theories are presupposed and used in various branches of science, but, typically, the results of scientific theories are not presupposed in the construction of mathematical theories. In this sense, the theories in mathematics are taken to be more general than the theories in science. Mathematics also plays a fundamentally instrumental role in science, helping the formulation of scientific theories and the expression of suit­able relations among physical entities.

Of course, not every mathematical theory has indispensable applications in science. For example, theories about inaccessible cardinals in set theory - that is, roughly, sets that cannot be reached, even in principle, from other sets below in the hierarchy of sets - do not seem to have found indispensable applications. For Quine, such theories constitute mathematical recreations, and do not demand ontological commitment, that is, commitment to the existence of the entities they posit. The situation is entirely different, however, when we consider those mathematical theories that are used in science. Given that we cannot even formulate the corresponding scientific theories without using (and quantifying over) mathematical objects - such as functions, numbers, and geometrical spaces - the existence of the latter is required as much as the existence of physical posits - such as neutrons, protons, and positrons. Platonism is the outcome of an honest understanding of the ontology of science.

2.4 Structuralist Platonism

The crucial feature of mathematical structuralism is to conceptualize math­ematics as the study of structures, rather than objects. Different forms of structuralism provide different accounts of structure (see, e.g., Resnik 1997, Shapiro 1997). But, crucially for the structuralist, it does not matter which mathematical objects one considers; as long as they satisfy the relevant struc­ture it will be sufficient to explain the possibility of mathematical knowledge.

We find this move in Michael Resnik's defense of structuralism. To explain the possibility of mathematical knowledge, Resnik introduces the notion of a template, which is a concrete entity - including things such as draw­ings, models, blueprints - and is meant to link the concrete aspects of our experience with abstract patterns (Resnik's term for structure). The crucial idea is that there are structural relations (such as an isomorphism^[46]) between templates and patterns that allow us to represent the latter via the former. In particular, it is because there are such structural relations between patterns and templates that mathematicians can use proofs - the process of creating and manipulating concrete templates via certain operations - to generate information about abstract patterns (Resnik 1997, 229-35). And given that mathematicians only have access to templates, no direct access to positions in patterns - that is, no direct access to mathematical objects

- is presupposed in Resnik's picture.

A significant feature of patterns, on Resnik's view, is the fact that the positions in such patterns are incomplete. This means that there is no fact of the matter as to whether these positions have certain properties or not. Consider, for example, the second position in the natural number pattern (for simplicity, call that position “the number two”). It is not clear that there is a fact of the matter as to whether this position - the number two

- is the same as the corresponding position in the real number pattern. In other words, it is not clear that there is a fact of the matter as to whether the number two in the natural number pattern is the same as the number two in the real number structure. After all, the properties that a position in a pattern has depend on the pattern to which it belongs. In the natural number pattern, the number two has the third position in the pattern - that is, the number three - as its immediate successor. But this is not the case in the context of the real number pattern. Of course, in the real number pattern, the immediate successor of the number two that is also a natural number is the number three. But to say that this is the same property as the one in the natural number pattern is already to assume that the cor­responding numbers are the same, which is the point in question. As a result, it is not clear how one could decide issues such as these.^[47]

For Resnik, rather than a problem, this incompleteness of the positions in a pattern (or structure) is a significant feature of mathematics. Ultimately, we will not be able to decide several issues about the identity of the posi­tions in structure. We can decide, however, issues about the mathematical structures themselves, where such incompleteness typically does not emerge. That is expected on a structuralist account. After all, what we get to know when we know mathematics does not have to do with the nature of math­ematical objects, but rather the structures (or patterns) they are part of.

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