Realism and Naturalism, Mathematical
Realism and Naturalism, Mathematical
REALISM AND NATURALISM, MATHEMATICAL
Many versions of realism in mathematics are intimately related to versions of naturalism. The purpose of this article is to explore relationships between the various views, and, briefly, the main opposition to them. The focus here is exclusively on mathematics. So, for example, "Platonism" is to be read as "Platonism about mathematics." This entry does not claim to do justice to the subtle and detailed works of everyone who works in the philosophy of mathematics, or even everyone who defends versions of realism and/or naturalism. Instead, this entry seeks to provide a useful road map of an important part of the territory.
In broad terms, realism is the view that mathematics is objective: independent of the lives, customs, language, and form of life of mathematicians. This statement is deliberately indeterminate. What aspects of mathematics are being discussed? What, exactly, is it independent of? And what is it to be independent? What is it to be objective? In philosophy there is little that one can take for granted.
There are at least two forms of realism: realism in ontology, which concerns mathematical objects, and realism in truth value, which concerns mathematical truth. Realism in ontology is the view that mathematical objects, such as numbers, sets, functions, and geometric points exist independently of the mathematician. Prima facie, these mathematical objects do not occupy physical space; they exist eternally and are not created or destroyed; and they do not enter into causal relationships with either each other or with physical objects. Because Platonic forms share these features, realism in ontology is sometimes called "Platonism" or, as Geoffrey Hellman (1989) dubs it, "objects Platonism." This sort of Platonism is sometimes written with a lowercase "p," perhaps to mark some distance from Plato. For the realist in ontology, mathematical propositions are taken at face value, as statements about mathematical objects. The theorem that 101 is a prime number just is the statement that a given object, the number 101, enjoys a certain property, primeness. The sentence "101 is a prime number" has the same logical form as "Socrates is Greek." Most versions of realism in ontology have it that mathematical truth is necessary, in a deep metaphysical sense: If the subject matter of mathematics is as these realists say it is, then typical propositions about mathematical objects—the principles of pure mathematics, for example—do not suffer from the contingencies of science or ordinary statements about ordinary physical objects.
Probably the most difficult problems associated with realism in ontology are in epistemology (see Benacerraf 1973). The realist declares that mathematics is about a realm of prima facie abstract, causally inert, and eternally existing objects. How can human beings ever come to know anything about these objects? How can humans have reliable, justified beliefs about such objects? The way people come to know things about physical objects typically involves some sort of causal contact between people, the knowers, and the objects (e.g., seeing them). This is ruled out with mathematical objects. Presumably, most of the beliefs that mathematicians have about mathematical objects are true. Mathematicians are reliable indicators of how things are in the mathematical realm. How does one explain this reliability (see Field 1989, essay 7)?
One resolution to these problems is to postulate a special faculty that humans have, an intuition, that links humans to the mathematical realm. Such was Plato's own solution to the analogous problem concerning Forms. Some of the logician Kurt Gödel's (1944, 1964) remarks can be interpreted along these lines:
Despite their remoteness from sense experience, we do have something like a perception also of the objects of set theory, as is seen from the fact that axioms force themselves upon us as being true. I don't see any reason why we should have less confidence in this kind of perception, i.e., in mathematical intuition, than in sense perception … It should be noted that mathematical intuition need not be conceived of as a faculty giving an immediate knowledge of the objects concerned. Rather, it seems that, as in the case of physical experience, we form our ideas also of those objects on the basis of something else which is immediately given … It by no means follows … that the data of this … kind, because they cannot be associated with actions of certain things upon our sense organs, are something purely subjective … Rather they … may represent an aspect of objective reality, but, as opposed to the sensations, their presence in us may be due to another kind of relationship between ourselves and reality. (Gödel 1964, p. 484)
A philosopher who is inclined this way has the task of trying to square the presence of mathematical intuition with the current scientific view of a human being as a thoroughly physical organism in a physical universe.
Another strategy for epistemology comes from logicism, the view that mathematical truth is a species of logical truth. The epistemology for mathematics is thus the epistemology for logic. The most detailed developments are those of Gottlob Frege (1884, 1893) and Alfred North Whitehead and Bertrand Russell (1910). Of those, Frege was a realist in ontology, at least for arithmetic and analysis. So for Frege, logic has an ontology—there are "logical objects." Numbers are constructed out of logical objects.
In attempting to define the natural numbers and the general notion of natural number, Frege (1884, §63) proposed the following principle, which has become known as "Hume's principle":
For any concepts F, G, the number of F 's is identical to the number of G 's if and only if F and G are equinumerous.
Two concepts are equinumerous if they can be put in one-to-one correspondence. Frege showed how to define equinumerosity without invoking natural numbers. In the end, he balked at taking Hume's principle as the ultimate foundation for arithmetic, and went on to provide an explicit definition of the natural numbers in terms of concepts and their extensions. The number two, for example, is the extension (or collection) of all concepts that hold of exactly two elements. Unfortunately, the inconsistency in Frege's theory of extensions, as shown by Russell's paradox, marked a tragic end to Frege's logicist program.
Variations of Frege's approach are vigorously pursued in the early twenty-first century, in the work of Crispin Wright, beginning with (1983), and others such as Bob Hale (1987) and Neil Tennant (1997). The idea is to bypass the treatment of extensions and to work with Hume's principle, or something like it, directly. On this neo-Fregean approach, Hume's principle is taken to be an explanation of the concept of "number." It is an implicit definition, true by stipulation. Frege's own technical development shows that the Peano postulates can be derived from Hume's principle in a standard, higher-order logic. Indeed, the only essential use that Frege made of extensions was to derive Hume's principle—everything else concerning numbers follows from that.
Another popular strategy for epistemology comes from an overarching hypothetical-deductive approach. The argument begins with the observation that virtually all of science is formulated in mathematical terms. One cannot believe in the truth of physics, say, without also accepting the mathematics that occurs in it. Thus, mathematics is confirmed to the extent that science is. In short, because mathematics is indispensable for science, and because science is well-confirmed and (approximately) true, one can conclude that mathematics is well-confirmed and true as well. This "indispensability argument" is attributed to W. W. O. Quine; a clear articulation is found in Hilary Putnam's Philosophy of Logic (1971, ch. 5) (see also Colyvan 2001).
According to structuralism, the subject matter of arithmetic, for example, is the pattern common to any infinite system of objects that has a distinguished initial object, which plays the role of zero, and a successor relation or operation that satisfies the induction principle. The arabic numerals exemplify this natural number structure, as does an infinite sequence of distinct moments of time, an infinite sequence of discrete points in space, and so on. Similarly, real analysis is about the real number structure, set theory is about the set-theoretic-hierarchy structure, and topology is about topological structures. According to the ante rem version of this view, the natural number structure, for example, exists independently of whether it has instances in the physical world, or any other world for that matter (see Shapiro 1997, Resnik 1997, also Parsons 1990). This is an ontological realism. The number six, for example, is a place in the natural number structure, the seventh place (if one begins with zero). Because, on the view in question, the structure exists objectively, then so do its places. Structuralists have proposed various epistemological strategies, ranging from pattern recognition, linguistic abstraction, implicit definition (much like neo-logicism), and postulation via indispensability (with the Quinean). One line, shared with the full-blooded platonism articulated by Mark Balaguer (1998), holds that the realm of structures is so robust that every coherent axiomatization is true of at least one structure. So the sticky problem concerning knowledge of mathematical objects reduces to knowledge of the coherence of an axiomatization.
the opposition: antirealism in ontology
Speaking logically, the opponents of realism in ontology fall into two camps. One group holds that numbers, functions, sets, points, and the like exist, but not objectively. Mathematical objects are not independent of the mind, language, conventions, or the form of life of the mathematician or the mathematical/scientific community. According to traditional intuitionism, for example, mathematical objects are mental constructions (e.g., Brouwer 1912, 1948; Heyting 1956). This is an idealism of sorts. Some intuitionists have explicitly Kantian roots, tying mathematical construction to the forms of pure intuition (typically of time). Another ontological antirealist view sees mathematical objects as social constructions.
The other way to reject ontological realism is to hold that there are no distinctive mathematical objects at all. There simply are no numbers, sets, functions, points, and so on. This is called nominalism. Again, it comes in two varieties. On one of them, mathematical assertions keep a straightforward, face-value reading. So the statement that every natural number is prime is vacuously true, because there are no natural numbers. "Seven is prime" is either false or lacks truth-value, depending on how nondenoting singular terms are handled. On this view, mathematical objects are likened to characters and objects in fiction. The sentence that seven is prime is of a piece with "Miss Marple is nosy." Of course, fictionalists do not recommend that mathematicians settle their questions via the literal, face-value reading of their assertions. Either they advert to a "truth in mathematics" akin to "truth in the story" for fiction, or else they provide some other purpose for mathematics beyond seeking mathematical truth (see Field 1989). Fictionalism is an error-theory about mathematics.
The other variety of nominalism provides alternate, non-face-value readings of mathematics. So the statements of mathematics come out true or false, without presupposing a mathematical ontology. The modal structuralist, for example, reads a statement such as "there are infinitely many prime numbers" as "any exemplification of the natural number structure has infinitely many places, each of which satisfied the property of being prime in that structure." Charles Chihara (1990, 2004) provides versions of various mathematical theories in terms of possible linguistic constructions. One interesting issue concerns the relationship between the "nominalized" assertions and their original counterparts (see Burgess and Rosen 1997).
realism in truth-value
These nominalistic programs lead to another major type of realism concerning mathematics. Georg Kreisel is often cited as suggesting that the important questions in the philosophy of mathematics do not concern the existence of mathematical objects, but rather the objectivity of mathematical assertions. Let us define realism in truth-value to be the view that mathematical statements have objective and nonvacuous truth-values independent of the minds, languages, and conventions of mathematicians.
Once again, the opponents to this view logically fall into two categories, depending on what is being denied. The radical opposition holds that mathematical statements have no nonvacuous truth-values at all. The fictionalist, noted above, is the primary and perhaps only occupant of this category. It is difficult to conceive of a projectivism or expressivism concerning mathematics.
The less radical versions of truth-value irrealism allow that mathematical statements have truth-values, but these are not independent of the minds, languages, and conventions of mathematicians. The traditional intuitionists, as described above, fit this bill. Because, for them, mathematical objects are mental constructions, mathematical assertions relate to the activity of construction. Contemporary intuitionists, following Michael Dummett (1977, 1978) also fit this bill, holding that all truths are knowable, on broadly semantic grounds.
Realism in ontology is naturally allied with realism in truth-value. To get from the former to the latter, one just insists that the sentences of mathematics be read literally, at face value. If, for example, "seven" is a genuine singular term, and the sentence "seven is prime" is objectively true, then, it seems, "seven" denotes something, namely, the number seven. And it exists objectively. Conversely, a realist in ontology gets to realism in truth-value by insisting that the typical propositions concerning the interrelations of the mind-independent mathematical objects are themselves objective.
Nevertheless, the connections between these realisms are not forced by logical connections that are obvious to all. As noted, many nominalists are realists in truth-value. They reject the face-value reading of mathematical assertions. At least one prominent philosopher of mathematics goes in the opposite direction. Neil Tennant (1987, 1997) holds that mathematical objects exist objectively, of necessity, and yet he adopts a Dummettian antirealism concerning truth-value.
Unfortunately, the word "naturalism" has become something of a term of art, and it is hard to find a common theme that underlies every view that goes by that name. Perhaps most of them share a certain deference to the natural sciences. Quine characterizes naturalism as "the abandonment of first philosophy" and "the recognition that it is within science itself … that reality is to be identified and described" (Quine 1981, p. 72; see also 1969). The idea is to see philosophy as continuous with the sciences, not prior to them in any epistemological or foundational sense.
The naturalist accepts the existence of the theoretical entities, such as forces and electrons, that occur in the most up-to-date scientific theories. Current science describes the world in such terms, and it runs against the theme of naturalism to reject them on philosophical grounds, adopting some sort of instrumentalism or constructive empiricism. When it comes to mathematics, however, naturalists differ. As seen with the aforementioned indispensability argument, Quine himself accepts mathematics to the extent—but only to the extent—that it is needed in science. It is impossible to do physics, or just about any other science for that matter, without invoking real analysis. So the theorems of real analysis are confirmed to the extent that the various scientific theories are confirmed, and these theories are the best ones available. So Quine accepts the truth of real analysis. Moreover, some of the traditional, Platonic themes have naturalistic counterparts. For example, the eternity of mathematical objects corresponds to the fact that mathematical assertions are not inflected with tense.
Naturalized epistemology is the application of Quinean naturalism to the study of knowledge. The philosopher sees the human knower as a thoroughly natural being within the physical universe. Any faculty that the philosopher invokes to explain knowledge must involve only natural processes amenable to ordinary scientific scrutiny.
This theme exacerbates the epistemic problems with realism. Platonic apprehension of a detached mathematical universe is ruled out from the start, as a nonnatural process. The challenge to the ontological realist is to show how a physical being in a physical universe can come to know about abstracta such as mathematical objects. There may be no refutation of realism in ontology, but there is a deep challenge to it. The advocate of indispensability cites the role of mathematics in science. The idea is that mathematics is known the same way that science is. However, it is not enough to leave it at that. The advocate of realism in ontology should delimit the exact role that mathematics plays in science. How, for example, is it possible for a casually isolated realm of abstracta to shed light on the interactions of physical matter? An answer to this would go a long way toward solving the epistemological puzzles.
Notice that, at best, the indispensability argument delivers the truth of the principles of real analysis. If one assumes that science is objective, then there is realism in truth-value. It is not clear that the Quinean naturalist is also committed to realism in ontology, despite Quine's own tendencies in that direction. This depends on whether naturalism requires the philosopher to accept the pronouncements of mathematical science at face value. Quine famously calls for regimentation of ordinary and scientific discourse, to clean up the ontological commitments. One can see some of the aforementioned nominalistic programs in this spirit. Some of them show (or try to show) how mathematics can be true without presupposing the existence of distinctively mathematical objects (Hellman 1989). And this truth is all that is needed in science, or so the argument goes.
Other nominalists take issue with the indispensability argument itself. They show how science could proceed without mathematics, or at least without mathematics as it is standardly understood (Field 1980, Chihara 2004). This is also perhaps in the spirit of naturalism.
Quine's own realism extends to real analysis, functional analysis, and perhaps a bit more. But it stops there. Quine does not accept the truth of the higher reaches of set theory unless and until it finds application in science. In fact, Quine goes so far as to recommend the adoption of a restrictive axiom in set theory (V=L), because it simplifies higher-set theory, noting that simplicity is a criterion of theory acceptance in science. This is despite most set-theorists' rejection of this axiom. It is ironic that Quine, the naturalist, feels comfortable dictating something to mathematicians on philosophical grounds.
other versions of naturalism
Penelope Maddy's (1997) and John Burgess's and Gideon Rosen's (1997) versions of naturalism defend a deferential attitude towards mathematics much like the one Quine shows toward science. They note, first, that mathematics has its own methodology, distinct from so-called scientific method, and that this methodology has proven successful over the centuries. The success of mathematics is measured in mathematical, not scientific terms. Moreover, if mathematicians gave serious pursuit only to those branches known to have applications in natural science, much of the mathematics known in the twenty-first century would not exist, nor would the science. The history of science is full of cases where branches of pure mathematics eventually found application in science (see Steiner 1997). That is to say, the overall goals of the scientific enterprise have been well-served by mathematicians pursuing their own disciplines with their own methodology, ignoring science if necessary. Thus, one does not need a direct inferential link between a piece of mathematics and sensory experience before accepting the mathematics as a legitimate part of the web.
On general naturalistic grounds, Burgess and Rosen adopt a realism in ontology for mathematics. For them, the convenience of the face value reading of mathematical propositions counts in its favor. Someone who proposes a nominalistic reconstruction must defend their account on accepted scientific, or mathematical grounds. That is, they must show that the ontology-free versions of mathematics are better mathematics and/or better science. Foregoing philosophical puzzles concerning epistemology do not count. Maddy is more circumspect, arguing that naturalism does not demand a realist interpretation of mathematics.
The varieties of naturalism treated here might be dubbed methodological because they focus on the methods of science, adopting those to traditional philosophical questions. Nominalism, as construed here, is an expression of another, ontological variety of naturalism. The thesis is that the only things that exist are the material objects of science, and the only properties people need to consider are the material properties of those objects. Alternately, the only objects in which people are licensed to believe are those with which they causally interact. Mark Colyvan (2001, ch. 3) calls this the eleatic principle. Another issue that separates naturalists—or at least philosophies that go by that name—is whether all legitimate knowledge is empirical. In the spirit of radical empiricism, in the manner of John Stuart Mill, Quine has launched a sustained attack on a priori knowledge. Not every contemporary naturalist follows suit. Bernard Linksy and Edward Zalta (1995) argue that the proper interpretation of science requires a more traditional Platonism, according to which mathematical propositions are synthetic a priori. Clearly, an article such as this can do no more than scratch the surface of these rich and wonderful topics.
Balaguer, M. Platonism and Anti-Platonism in Mathematics. Oxford: Oxford University Press, 1998. Account of realism in ontology and its rivals.
Benacerraf, P. "What Numbers Could Not Be." Philosophical Review 74 (1965): 47–73. Reprinted in Benacerraf and Putnam, Philosophy of Mathematics, 272–294. One of the most widely cited works in the field; argues that numbers are not objects, and introduces an eliminative structuralism.
Benacerraf, P. "Mathematical Truth." Journal of Philosophy 70 (1973): 661–679. Reprinted in Benacerraf and Putnam, Philosophy of Mathematics, 403–420. Another widely cited work; argues that realism in ontology has formidable epistemological problems.
Benacerraf, P., and H. Putnam, eds. Philosophy of Mathematics. 2nd ed. Cambridge, U.K.: Cambridge University Press, 1983. A far-reaching collection containing many of the central articles.
Brouwer, L. E. J. "Consciousness, Philosophy and Mathematics." 1948. In Philosophy of Mathematics: Selected Readings. 2nd ed., edited by P. Benacerraf and H. Putnam, 90–96. New York: Cambridge University Press, 1983.
Brouwer, L. E. J. Intuitionisme en Formalisme. Groningen, Germany: Noordhof, 1912. Translated as "Intuitionism and Formalism." In Philosophy of Mathematics: Selected Readings. 2nd ed., edited by P. Benacerraf and H. Putnam, 77–79. New York: Cambridge University Press, 1983.
Burgess, J. "Why I Am Not a Nominalist." Notre Dame Journal of Formal Logic 24 (1983): 93–105. Early critique of nominalism.
Burgess, J., and G. Rosen. A Subject With No Object: Strategies for Nominalistic Interpretation of Mathematics. Oxford: Oxford University Press, 1997. Extensive articulation and criticism of nominalism.
Chihara, C. Constructibility and Mathematical Existence. Oxford: Oxford University Press, 1990. Defense of a modal view of mathematics, and sharp criticisms of several competing views.
Chihara, C. A Structural Account of Mathematics. Oxford: Oxford University Press, 2004. Sequel to Chihara, Constructibility and Mathematical Existence, with particular focus on the application of mathematics.
Colyvan, M. The Indispensability of Mathematics. Oxford: Oxford University Press, 2001. Elaboration and defense of the indispensability argument for ontological realism.
Dummett, M. Elements of Intuitionism. Oxford: Oxford University Press, 1977.
Dummett, M. "The Philosophical Basis of Intuitionistic Logic." In Truth and Other Enigmas. Cambridge, MA: Harvard University Press, 1978. Reprinted in Benacerraf and Putnam, Philosophy of Mathematics, 97–129; in Hart, The Philosophy of Mathematics, 63–94. Influential defense of intuitionism.
Dummett, M. Elements of Intuitionism. Oxford: Oxford University Press, 1977. Detailed introduction and defense of intuitionistic mathematics.
Field, H. Science Without Numbers. Princeton, NJ: Princeton University Press, 1980. A widely cited defense of fictionalism, by attempting to refute the indispensability argument.
Field, H. Realism, Mathematics and Modality. Oxford: Blackwell, 1989. Reprints of Field's articles on fictionalism.
Frege, G. Die Grundlagen der Arithmetik. Breslau, Germany: Koebner, 1884. Translated by J. Austin as The Foundations of Arithmetic. 2nd ed. New York: Harper, 1960. Classic articulation and defense of logicism.
Frege, G. Grundgesetze der Arithmetik. Vol. 1. Hildescheim, Germany: Olms, 1893. More technical development of Frege's logicism.
Gödel, K. "Russell's Mathematical Logic." 1944. In Philosophy of Mathematics, edited by P. Benacerraf and H. Putnam, 447–469. New York: Cambridge University Press, 1983. Much-cited defense of realism in ontology and realism in truth value.
Gödel, K. "What is Cantor's Continuum Problem." 1964. In Philosophy of Mathematics, edited by P. Benacerraf and H. Putnam, 470–485. New York: Cambridge University Press, 1983. Much-cited defense of realism in ontology and realism in truth value.
Hale, Bob. Abstract Objects. Oxford: Basil Blackwell, 1987. Detailed development of neo-logicism, to support Wright (1983).
Hart, W. D., ed. The Philosophy of Mathematics. Oxford: Oxford University Press, 1996. Collection of articles published elsewhere.
Hellman, G. Mathematics Without Numbers. Oxford: Oxford University Press, 1989. Articulation and defense of modal structuralism.
Heyting, A. Intuitionism: An Introduction. Amsterdam: North Holland, 1956. Readable account of intuitionism.
Hodes, H. "Logicism and the Ontological Commitments of Arithmetic." Journal of Philosophy 81 (1984): 123–149. Another roughly Fregean logicism.
Linksy, B., and E. Zalta. "Naturalized Platonism versus Platonized Naturalism." Journal of Philosophy 92 (1995): 525–555.
Maddy, P. Naturalism in Mathematics. Oxford: Oxford University Press, 1997. Lucid account of naturalism concerning mathematics, and its relation to traditional philosophical issues.
Maddy, P. Realism in Mathematics. Oxford: Oxford University Press, 1990. Articulation and defense of realism about sets.
Maddy, P. "Three Forms of Naturalism." In Shapiro, Oxford Handbook of Philosophy of Mathematics and Logic, 437–459. Contrast with Quinean naturalism.
Parsons, C. "The Structuralist View of Mathematical Objects." Synthese 84 (1990): 303–346. Reprinted in The Philosophy of Mathematics, edited by W. D. Hart, 272–309. Oxford: Oxford University Press.
Putnam, H. Philosophy of Logic. New York: Harper Torchbooks, 1971. Source for the indispensability argument for ontological realism.
Quine, W. V. O. Ontological Relativity and Other Essays. New York: Columbia University Press, 1969.
Quine, W. V. O. Theories and Things. Cambridge, MA: Harvard University Press, 1981.
Resnik, M. Mathematics as a Science of Patterns. Oxford: Oxford University Press, 1997. Full articulation of a realist-style structuralism.
Schirn, M., ed. Philosophy of Mathematics Today. Oxford: Oxford University Press, 1998. Proceedings of a conference in the philosophy of mathematics, held in Munich in 1993; coverage of most of the topical issues.
Shapiro, S., ed. Oxford Handbook of Philosophy of Mathematics and Logic. Oxford: Oxford University Press, 2005. Up-to-date articles covering the discipline.
Shapiro, S. "Philosophy of Mathematics." In Philosophy of Science Today, edited by Peter Clark and Katherine Hawley, 181–200. Oxford: Oxford University Press, 2003.
Shapiro, S. Philosophy of Mathematics: Structure and Ontology. Oxford: Oxford University Press, 1997. Extensive articulation and defense of structuralism.
Shapiro, S. Thinking about Mathematics: The Philosophy of Mathematics. Oxford: Oxford University Press, 2000. Popularization and textbook in the philosophy of mathematics.
Steiner, M. The Applicability of Mathematics as a Philosophical Problem. Cambridge, MA: Harvard University Press, 1997.
Tennant, N. Anti-Realism and Logic. Oxford: Oxford University Press, 1987. Articulation of antirealism in truth value, realism in ontology; defends intuitionistic relevance logic against classical logic.
Tennant, N. The Taming of the True. Oxford: Oxford University Press, 1997. Detailed defense of global semantic antirealism.
Weir, A. "Naturalism reconsidered." In Oxford Handbook of Philosophy of Mathematics and Logic, edited by S. Shapiro, 460–482. Oxford: Oxford University Press.
Whitehead, A. N., and B. Russell. Principia Mathematica. Vol. 1. Cambridge, U.K.: Cambridge University Press, 1910.
Wright, C. Frege's Conception of Numbers as Objects. Aberdeen, Scotland: Aberdeen University Press, 1983. Revival of Fregean logicism.
Stewart Shapiro (2005)