 mathematics, philosophy of

Branch of philosophy concerned with the epistemology and ontology of mathematics.Early in the 20th century, three main schools of thoughtarose to account for and resolve the crisis in the foundations of mathematics. Logicism argues that all mathematical notions are reducible to laws of pure thought, or logical principles; a variant known as mathematical Platonism holds that mathematical notions are transcendent Ideals, or Forms, independent of human consciousness. Formalism holds that mathematics consists simply of the manipulation of finite configurations of symbols according to prescribed rules; a "game" independent of any physical interpretation of the symbols. Intuitionism is characterized by its rejection of any knowledgeor evidencetranscendent notion of truth. Hence, only objects that can be constructed (see constructivism) in a finite number of steps are admitted, while actual infinities and the law of the excluded middle (see laws of thought) are rejected. These three schools of thought were principally led, respectively, by Bertrand Russell, David Hilbert, and the Dutch mathematician Luitzen Egbertus Jan Brouwer (1881–1966).
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Introductionbranch of philosophy that is concerned with two major questions: one concerning the meanings of ordinary mathematical sentences and the other concerning the issue of whether abstract objects exist. The first is a straightforward question of interpretation: What is the best way to interpret standard mathematical sentences and theories? In other words, what is really meant by ordinary mathematical sentences such as “3 is prime,” “2 + 2 = 4,” and “There are infinitely many prime numbers.” Thus, a central task of the philosophy of mathematics is to construct a semantic theory for the language of mathematics. semantics is concerned with what certain expressions mean (or refer to) in ordinary discourse. So, for instance, the claim that in English the term Mars denotes the Mississippi River is a false semantic theory; and the claim that in English Mars denotes the fourth planet from the Sun is a true semantic theory. Thus, to say that philosophers of mathematics are interested in figuring out how to interpret mathematical sentences is just to say that they want to provide a semantic theory for the language of mathematics.Philosophers are interested in this question for two main reasons: 1) it is not at all obvious what the right answer is, and 2) the various answers seem to have deep philosophical implications. More specifically, different interpretations of mathematics seem to produce different metaphysical views about the nature of reality. These points can be brought out by looking at the sentences of arithmetic, which seem to make straightforward claims about certain objects. Consider, for instance, the sentence “4 is even.” This seems to be a simple subjectpredicate sentence of the form “S is P”—like, for instance, the sentence “The Moon is round.” This latter sentence makes a straightforward claim about the Moon, and likewise, “4 is even” seems to make a straightforward claim about the number 4. This, however, is where philosophers get puzzled. For it is not clear what the number 4 is supposed to be. What kind of thing is a number? Some philosophers (antirealists) have responded here with disbelief—according to them, there are simply no such things as numbers. Others (realists) think that there are such things as numbers (as well as other mathematical objects). Among the realists, however, there are several different views of what kind of thing a number is. Some realists think that numbers are mental objects (something like ideas in people's heads). Other realists claim that numbers exist outside of people's heads, as features of the physical world. There is, however, a third view of the nature of numbers, known as Platonism or mathematical Platonism, that has been more popular in the history of philosophy. This is the view that numbers are abstract objects, where an abstract object is both nonphysical and nonmental. According to Platonists, abstract objects exist but not anywhere in the physical world or in people's minds. In fact, they do not exist in space and time at all.In what follows, more will be said to clarify exactly what Platonists have in mind by an abstract object. However, it is important to note that many philosophers simply do not believe in abstract objects; they think that to believe in abstract objects—objects that are wholly nonspatiotemporal, nonphysical, and nonmental—is to believe in weird, occult entities. In fact, the question of whether abstract objects exist is one of the oldest and most controversial questions of philosophy. The view that there do exist such things goes back to Plato, and serious resistance to the view can be traced back at least to Aristotle. This ongoing controversy has survived for more than 2,000 years.The second major question with which the philosophy of mathematics is concerned is this: “Do abstract objects exist?” This question is deeply related to the semantic question about how the sentences and theories of mathematics should be interpreted. For if Platonism is right that the best interpretation of mathematics is that sentences such as “4 is even” are about abstract objects (and it will become clear below that there are some very good reasons for endorsing this interpretation), and if (what seems pretty obvious) sentences such as “4 is even” are true, then it would seem natural to endorse the view that abstract objects exist.The next section, Mathematical Platonism (mathematics, philosophy of), provides a sketch of the Platonist view of mathematics and how it has developed. The following section, Mathematical antiPlatonism (mathematics, philosophy of), provides a sketch of the alternatives to Platonism—that is, the various antiPlatonist views that are available to those who cannot bring themselves to believe in abstract objects. Finally, the last section, Mathematical Platonism: for and against (mathematics, philosophy of), presents the best arguments for and against Platonism.Mathematical PlatonismFormal definitionMathematical Platonism, formally defined, is the view that (a) there exist abstract objects—objects that are wholly nonspatiotemporal, nonphysical, and nonmental—and (b) there are true mathematical sentences that provide true descriptions of such objects. The discussion of Platonism that follows will address both (a) and (b).It is best to start with what is meant by an abstract object. Among contemporary Platonists, the most common view is that the really defining trait of an abstract object is nonspatiotemporality. That is, abstract objects are not located anywhere in the physical universe, and they are also entirely nonmental, yet they have always existed and they always will exist. This does not preclude having mental ideas of abstract objects; according to Platonists, one can—e.g., one might have a mental idea of the number 4. It does not follow from this, though, that the number 4 is just a mental idea. After all, people have ideas of the Moon in their heads too, but it does not follow from this that the Moon is just an idea, because the Moon and people's ideas of the Moon are distinct things. Thus, when Platonists say that the number 4 is an abstract object, they mean to say that it is a real and objective thing that, like the Moon, exists independently of people and their thinking but, unlike the Moon, is nonphysical.Abstract objects are also, according to Platonists, unchanging and entirely noncausal. Because abstract objects are not extended in space and not made of physical matter, it follows that they cannot enter into causeandeffect relationships with other objects.Platonists also claim that mathematical theorems provide true descriptions of such objects. What does this claim amount to? Consider the positive integers (1, 2, 3,…). According to Platonists, the theory of arithmetic says what this sequence of abstract objects is like. Over the years, mathematicians have discovered all sorts of interesting facts about this sequence. For instance, Euclid proved more than 2,000 years ago that there are infinitely many prime numbers among the positive integers. Thus, according to Platonists, the sequence of positive integers is an object of study, just like the solar system is an object of study for astronomers.Now, so far, only one kind of mathematical object has been discussed, namely, numbers. But there are many different kinds of mathematical objects—functions, sets, vectors, circles, and so on—and for Platonists these are all abstract objects. Moreover, Platonists also believe that there are such things as settheoretic hierarchies and that set theory describes these structures. And so on for all the various branches of mathematics. In general, according to Platonists, mathematics is the study of the nature of various mathematical structures, which are abstract in nature.Platonism has been around for over two millennia, and over the years it has been one of the most popular views among philosophers of mathematics. Yet, for most of the history of philosophy, mathematical Platonism was stagnant. In the late 19th century Gottlob Frege (Frege, Gottlob) of Germany, who founded modern mathematical logic, developed what is widely thought to be the most powerful argument in favour of Platonism; but he did not alter the formulation of the view. Likewise, in the 20th century Kurt Gödel (Gödel, Kurt) of Austria and Willard Van Orman Quine (Quine, Willard Van Orman) of the United States introduced hypotheses in an attempt to explain how human beings could acquire knowledge of abstract objects—but again, neither of these thinkers altered the Platonist view itself. (Gödel's hypothesis was about the nature of human beings, and Quine's hypothesis was about the nature of empirical evidence.)Nontraditional versionsDuring the 1980s and '90s, various Americans developed three nontraditional versions of mathematical Platonism: one by Penelope Maddy, a second by Mark Balaguer (the author of this article) and Edward Zalta, and a third by Michael Resnik and Stewart Shapiro. All three versions were inspired by concerns over how humans could acquire knowledge of abstract objects.According to Maddy, mathematics is about abstract objects, and abstract objects are, in some important sense, nonphysical and nonmental, though they are located in space and time. Maddy developed this idea most fully in connection with sets. For her, a set of physical objects is located right where the physical objects themselves are located. For instance, if there are three eggs in a refrigerator, then the set containing those eggs is also in the refrigerator. This might seem eminently sensible, and one might wonder why Maddy counts as a Platonist at all; that is, one might wonder why a set of eggs counts as a nonphysical object in Maddy's view. In order to appreciate why Maddy is a Platonist (in some nontraditional sense), it is necessary to know something about set theory—most notably, that for every physical object, or pile of physical objects, there are infinitely many sets. For instance, if there are three eggs in a refrigerator, then corresponding to these eggs there exists the set containing the eggs, the set containing that set, the set containing that set, and so on. Moreover, there is also a set containing two different sets—namely, the set containing the eggs and the set containing the set containing the eggs—and so on without end. Thus, combining the principles of set theory (which Maddy wants to preserve) with Maddy's thesis that sets are spatiotemporally located implies that if there are three eggs in a given refrigerator, then there are also infinitely many sets in the refrigerator. Of course there is only a finite amount of physical stuff in the refrigerator. More specifically, it contains a rather small aggregate of eggstuff. Thus, for Maddy the various sets built up out of this eggstuff are all distinct from the aggregate itself. In order to avoid contradicting the principles of set theory, Maddy has to say that the sets are distinct from the eggaggregate, and so even though she wants to maintain that all these sets are located in the refrigerator, she has to say that they are nonphysical in some sense. (Again, the reason that Maddy altered the Platonist view by giving sets spatiotemporal existence is that she thought it was necessary in order to explain how anyone could acquire knowledge of abstract objects. See below Mathematical Platonism: for and against (mathematics, philosophy of).)According to Balaguer and Zalta, on the other hand, the only versions of Platonism that are tenable are those that maintain not just the existence of abstract objects but the existence of as many abstract objects as there can possibly be. If this is right, then any system of mathematical objects that can consistently be conceived of must actually exist. Balaguer called this view “fullblooded Platonism,” and he argued that it is only by endorsing this view that Platonists can explain how humans could acquire knowledge of abstract objects.Finally, the nontraditional version of Platonism developed by Resnik and Shapiro is known as structuralism. The essential ideas here are that the real objects of study in mathematics are structures, or patterns—things such as infinite series, geometric spaces, and settheoretic hierarchies—and that individual mathematical objects (such as the number 4) are not really objects at all in the ordinary sense of the term. Rather, they are simply positions in structures, or patterns. This idea can be clarified by thinking first about nonmathematical patterns.Consider a baseball defense, which can be thought of as a certain kind of pattern. There is a left fielder, a right fielder, a shortstop, a pitcher, and so on. These are all positions in the overall pattern, or structure, and they are all associated with certain regions on a baseball field. Now, when a specific team takes the field, real players occupy these positions. For instance, during the early 1900s Honus Wagner (Wagner, Honus) usually occupied the shortstop position for the Pittsburgh Pirates. He was a specific object, with spatiotemporal location. However, one can also think about the shortstop position itself. It is not an object in the ordinary sense of the term; rather, it is a role that can be filled by different people. According to Resnik and Shapiro, similar things can be said about mathematical structures. They are something like patterns, made up of positions that can be filled by objects. The number 4, for instance, is just the fourth position in the positive integer pattern. Different objects can be put into this position, but the number itself is not an object at all; it is merely a position. Structuralists sometimes express this idea by saying that numbers have no internal properties or that their only properties are those they have because of the relations they bear to other numbers in the structure; e.g., 4 has the property of being between 3 and 5. This is analogous to saying that the shortstop position does not have internal properties in the way that actual shortstops do; for instance, it does not have a height or a weight or a nationality. The only properties that it has are structural, such as the property of being located in or near the infield between the third baseman and the second baseman.Mathematical antiPlatonismMany philosophers cannot bring themselves to believe in abstract objects. However, there are not many tenable alternatives to mathematical Platonism. One option is to maintain that there do exist such things as numbers and sets (and that mathematical theorems provide true descriptions of these things) while denying that these things are abstract objects. Views of this kind can be called realistic versions of antiPlatonism. Like Platonism, they are still versions of mathematical realism because they maintain that mathematical theorems provide true descriptions of some part of the world.In contrast to realistic versions of antiPlatonism, there is also an antirealist view known as mathematical nominalism. This view rejects the belief in the existence of numbers, sets, and so on and also rejects the belief that mathematical theorems provide true descriptions of some part of the world.The two main alternatives to Platonism, then, are realistic antiPlatonism and nominalism. These alternatives are described more fully in the following two sections.Realistic antiPlatonismThere are two different versions of realistic antiPlatonism, namely, psychologism and physicalism. Psychologism is the view that mathematical theorems are about concrete mental objects of some sort. In this view, numbers and circles and so on do exist, but they do not exist independently of people; instead, they are concrete mental objects—in particular, ideas in people's heads. As will become clearer below (in the section Mathematical Platonism: for and against (mathematics, philosophy of)), psychologism has serious problems and is no longer endorsed by many philosophers; nonetheless, it was popular during the late 19th and early 20th centuries, the most notable proponents being the German philosopher Edmund Husserl (Husserl, Edmund) and the Dutch mathematicians L.E.J. Brouwer (Brouwer, Luitzen Egbertus Jan) and Arend Heyting.Physicalism, on the other hand, is the view that mathematics is about concrete physical objects of some sort. Advocates of this view agree with Platonists that there exist such things as numbers and sets, and, unlike adherents of psychologism, they also agree that these things exist independently of people and their thoughts. Physicalists differ from Platonists, however, in holding that mathematics is about ordinary physical objects. There are a few different versions of this view. For example, one might hold that geometric objects, such as circles, are regions of actual physical space. Similarly, sets might be claimed to be piles of actual physical objects—thus, a set of eggs would be nothing more than the aggregate of physical matter that makes up the eggs. Moving on to numbers, one strategy is to take them to be physical properties of some sort—for example, properties of piles of physical objects, so that, for instance, the number 3 might be a property of a pile of three eggs. It should be noted here that many people have endorsed a Platonistic view of properties. In particular, Plato thought that, in addition to all the red things he observed in the world, there exists an independent property of redness and that this property was an abstract object. Aristotle, on the other hand, thought that properties exist in the physical world; thus, in his view, redness exists in particular objects, such as red houses and red apples, rather than as an abstract object outside of space and time. So in order to motivate a physicalistic view of mathematics by claiming that numbers are properties, one would also have to argue for an Aristotelian, or physicalistic, view of properties. One person who has developed a view of this sort since Aristotle is the Australian philosopher David Armstrong.Another strategy for interpreting talk of numbers to be about the physical world is to interpret it as talk about actual piles of physical objects rather than properties of such piles. For instance, one might maintain that the sentence “2 + 3 = 5” is not really about specific entities (the numbers 2, 3, and 5); rather, it says that whenever a pile of two objects is pushed together with a pile of three objects, the result is a pile of five objects. A view of this sort was developed by the English philosopher John Stuart Mill (Mill, John Stuart) in the 19th century.NominalismNominalism is the view that mathematical objects such as numbers and sets and circles do not really exist. Nominalists do admit that there are such things as piles of three eggs and ideas of the number 3 in people's heads, but they do not think that any of these things is the number 3. Of course, when nominalists deny that the number 3 is a physical or mental object, they are in agreement with Platonists. They admit that if there were any such thing as the number 3, then it would be an abstract object; but, unlike mathematical Platonists, they do not believe in abstract objects, and so they do not believe in numbers. There are three different versions of mathematical nominalism: paraphrase nominalism, fictionalism, and what can be called neoMeinongianism.The paraphrase nominalist view can be elucidated by returning to the sentence “4 is even.” Paraphrase nominalists agree with Platonists that if this sentence is interpreted at face value—i.e., as saying that the object 4 has the property of being even—then it makes a straightforward claim about an abstract object. However, paraphrase nominalists do not think that ordinary mathematical sentences such as “4 is even” should be interpreted at face value; they think that what these sentences really say is different from what they seem to say on the surface. More specifically, paraphrase nominalists think that these sentences do not make straightforward claims about objects. There are several different versions of paraphrase nominalism, of which the best known is “ifthenism,” or deductivism. According to this view, the sentence “4 is even” can be paraphrased by the sentence “If there were such things as numbers, then 4 would be even.” In this view, even if there are no such things as numbers, the sentence “4 is even” is still true. For, of course, even if there is no such thing as the number 4, it is still true that, if there were such a thing, it would be even. Deductivism has roots in the thought of David Hilbert (Hilbert, David), a brilliant German mathematician from the late 19th and early 20th centuries, but it was developed more fully by the American philosophers Hilary Putnam (Putnam, Hilary) and Geoffrey Hellman. Other versions of paraphrase nominalism have been developed by the American philosophers Haskell Curry and Charles Chihara.Mathematical fictionalists agree with paraphrase nominalists that there are no such things as abstract objects and, hence, no such things as numbers. They think that paraphrase nominalists are mistaken, however, in their claims about what mathematical sentences such as “4 is even” really mean. Fictionalists think that Platonists are right that these sentences should be read at face value; they think that “4 is even” should be taken as saying just what it seems to say—namely, that the number 4 has the property of being even. Moreover, fictionalists also agree with Platonists that if there really were such a thing as the number 4, then it would be an abstract object. But, again, fictionalists do not believe that there is such a thing as the number 4, and so they maintain that sentences like “4 is even” are not literally true. Fictionalists think that sentences such as “4 is even” are analogous in a certain way to sentences like “Santa Claus lives at the North Pole.” They are not literally true descriptions of the world, but they are true in a certain wellknown story. Thus, according to fictionalism, arithmetic is something like a story, and it involves a sort of fiction, or pretense, to the effect that there are such things as numbers. Given this pretense, the theory says what numbers are like, or what they would be like if they existed. Fictionalists then argue that it is not a bad thing that mathematical sentences are not literally true. Mathematics is not supposed to be literally true, say the fictionalists, and they have a long explanation of why mathematics is pragmatically useful and intellectually interesting despite the fact that it is not literally true. Fictionalism was first proposed by the American philosopher Hartry Field. It was then developed in a somewhat different way by Balaguer, the American philosopher Gideon Rosen, and the Canadian philosopher Stephen Yablo.The last version of nominalism is neoMeinongianism, which derives from Alexius Meinong (Meinong, Alexius), a late19th century Austrian philosopher. Meinong endorsed a view that was supposed to be distinct from Platonism, but most philosophers now agree that it is in fact equivalent to Platonism. In particular, Meinong held that there are such things as abstract objects but that these things do not have fullblown existence. Philosophers have responded to Meinong's claims by making a pair of related points. First, since Meinong thought there are such things as numbers, and since he thought that these things are nonspatiotemporal, it follows that he was a Platonist. Second, Meinong simply used the word exist in a nonstandard way; according to ordinary English, anything that is exists, and so it is contradictory to say that numbers are but do not exist.Advocates of neoMeinongianism agree with Platonists and fictionalists that the sentence “4 is even” should be interpreted at face value, as making (or purporting to make) a straightforward claim about a certain object—namely, the number 4. Moreover, they also agree that if there were any such thing as the number 4, then it would be an abstract object. Finally, they agree with fictionalists that there are no such things as abstract objects. In spite of this, neoMeinongians claim that “4 is even” is literally true, for they maintain that a sentence of the form “The object O has the property P” can be literally true, even if there is no such thing as the object O. Thus, neoMeinongianism consists in the following (seemingly awkward) trio of claims: (1) mathematical sentences should be read at face value, as purporting to make claims about mathematical objects such as numbers; (2) there are no such things as mathematical objects; and yet (3) mathematical sentences are still literally true. NeoMeinongianism, in the form described here, was first introduced by the New Zealand philosopher Richard Sylvan, but related views were held much earlier by the German philosophers Rudolf Carnap (Carnap, Rudolf) and Carl Gustav Hempel (Hempel, Carl Gustav) and the British philosopher Sir Alfred Ayer (Ayer, Sir A.J.). Views along these lines have been endorsed by Graham Priest of England, Jody Azzouni of the United States, and Otavio Bueno of Brazil.In sum, then, there are essentially five alternatives to Platonism. If one does not want to claim that mathematics is about nonphysical, nonmental, nonspatiotemporal objects, then one must to claim either (1) that mathematics is about concrete mental objects in people's heads (psychologism); or (2) that it is about concrete physical objects (physicalism); or (3) that, contrary to first appearances, mathematical sentences do not make claims about objects at all (paraphrase nominalism); or (4) that, while mathematics does purport to be about abstract objects, there are in fact no such things, and so mathematics is not literally true (fictionalism); or (5) that mathematical sentences purport to be about abstract objects, and there are no such things as abstract objects, and yet these sentences are still literally true (neoMeinongianism).logicism, intuitionism, and formalismDuring the first half of the 20th century, the philosophy of mathematics was dominated by three views: logicism, intuitionism, and formalism. Given this, it might seem odd that none of these views has been mentioned yet. The reason is that (with the exception of certain varieties of formalism) these views are not views of the kind discussed above. The views discussed above concern what the sentences of mathematics are really saying and what they are really about. But logicism and intuitionism are not views of this kind at all, and insofar as certain versions of formalism are views of this kind, they are versions of the views described above. How then should logicism, intuitionism, and formalism be characterized? In order to understand these views, it is important to understand the intellectual climate in which they were developed. During the late 19th and early 20th centuries, mathematicians and philosophers of mathematics became preoccupied with the idea of securing a firm foundation of mathematics (mathematics, foundations of). That is, they wanted to show that mathematics, as ordinarily practiced, was reliable or trustworthy or certain. It was in connection with this project that logicism, intuitionism, and formalism were developed.The desire to secure a foundation for mathematics was brought on in large part by the British philosopher Bertrand Russell (Russell, Bertrand)'s discovery in 1901 that naive set theory contained a contradiction. It had been naively thought that for every concept, there exists a set of things that fall under that concept; for instance, corresponding to the concept “egg” is the set of all the eggs in the world. Even concepts such as “mermaid” are associated with a set—namely, the empty set. Russell noticed, however, that there is no set corresponding to the concept “not a member of itself.” For suppose that there were such a set—i.e., a set of all the sets that are not members of themselves. Call this set S. Is S a member of itself? If it is, then it is not (because all the sets in S are not members of themselves); and if S is not a member of itself, then it is (because all the sets not in S are members of themselves). Either way, a contradiction follows. Thus, there is no such set as S.Logicism is the view that mathematical truths are ultimately logical truths. This idea was introduced by Frege (Frege, Gottlob). He endorsed logicism in conjunction with Platonism, but logicism is consistent with various antiPlatonist views as well. Logicism was also endorsed at about the same time by Russell and his associate, British philosopher Alfred North Whitehead (Whitehead, Alfred North). Few people still endorse this view, although there is a neologicist school, the main proponents of which are the British philosophers Crispin Wright and Robert Hale.Intuitionism is the view that certain kinds of mathematical proofs (namely, nonconstructive arguments (mathematics, foundations of)) are unacceptable. More fundamentally, intuitionism is best seen as a theory about mathematical assertion and denial. Intuitionists embrace the nonstandard view that mathematical sentences of the form “The object O has the property P” really mean that there is a proof that the object O has the property P, and they also embrace the view that mathematical sentences of the form “notP” mean that a contradiction can be proven from P. Because intuitionists accept both of these views, they reject the traditionally accepted claim that for any mathematical sentence P, either P or notP is true; and because of this, they reject nonconstructive proofs. Intuitionism was introduced by L.E.J. Brouwer (Brouwer, Luitzen Egbertus Jan), and it was developed by Brouwer's student Arend Heyting and somewhat later by the British philosopher Michael Dummett. Brouwer and Heyting endorsed intuitionism in conjunction with psychologism, but Dummett did not, and the view is consistent with various nonpsychologistic views—e.g., Platonism and nominalism.There are a few different versions of formalism. Perhaps the simplest and most straightforward is metamathematical formalism, which holds that ordinary mathematical sentences that seem to be about things such as numbers are really about mathematical sentences and theories. In this view, “4 is even” should not be literally taken to mean that the number 4 is even but that the sentence “4 is even” follows from arithmetic axioms. Formalism can be held simultaneously with Platonism or various versions of antiPlatonism, but it is usually conjoined with nominalism. Metamathematical formalism was developed by Haskell Curry (Curry, Haskell Brooks), who endorsed it in conjunction with a sort of nominalism.Mathematical Platonism: for and againstPhilosophers have come up with numerous arguments for and against Platonism, but one of the arguments for Platonism stands out above the rest, and one of the arguments against Platonism also stands out as the best. These arguments have roots in the writings of Plato, but the proPlatonist argument was first clearly formulated by Frege, and the locus classicus of the antiPlatonist argument is a 1973 paper by the American philosopher Paul Benacerraf.The Fregean (Frege, Gottlob) argument for PlatonismFrege's argument for mathematical Platonism boils down to the assertion that it is the only tenable view of mathematics. (The version of the argument presented here includes numerous points that Frege himself never made; nonetheless, the argument is still Fregean in spirit.)From the Platonist point of view, the weakest antiPlatonist views are psychologism, physicalism, and paraphrase nominalism. These three views make controversial claims about how the language of mathematics should be interpreted, and Platonists rebut their claims by carefully examining what people actually mean when they make mathematical utterances. The following brings out some of the arguments against these three views.Psychologism can be thought of as involving two central claims: (1) numberideas exist inside people's heads and (2) ordinary mathematical sentences and theories are best interpreted as being about these ideas. Very few people would reject the first of these theses, but there are several wellknown arguments against accepting the second view. Three are presented here. First is the argument that psychologism makes mathematical truth contingent upon psychological truth. Thus, if every human being died, the sentence “2 + 2 = 4” would suddenly become untrue. This seems blatantly wrong. The second argument is that psychologism seems incompatible with standard arithmetical theory, which insists that infinitely many numbers actually exist, because clearly there are only a finite number of ideas in human heads. This is not to say that humans cannot conceive of an infinite set; the point is, rather, that infinitely many actual objects (i.e., distinct numberideas) cannot reside in human heads. Therefore, numbers cannot be ideas in human heads. (See also infinity for Aristotle's distinction between actual infinities and potential infinities.) Third, psychologism suggests that the proper methodology for mathematics is that of empirical psychology. If psychologism were true, then the proper way to discover whether, say, there is a prime number between 10,000,000 and 10,000,020 would be to do an empirical study of humans to ascertain whether such a number existed in someone's head. This, however, is obviously not the proper methodology for mathematics; the proper methodology involves mathematical proof, not empirical psychology.Physicalism does not fare much better in the eyes of Platonists. The easiest way to bring out the arguments against physicalistic interpretations of mathematics is to focus on set theory. According to physicalism, sets are just piles of physical objects. But, as has been previously shown, sets cannot be piles of physical stuff—or at any rate, when mathematicians talk about sets, they are not talking about physical piles—because it follows from the principles of set theory that for every physical pile, there corresponds infinitely many sets. A second problem with physicalistic views is that they seem incapable of accounting for the sheer size of the infinities involved in set theory. Standard set theory holds not just that there are infinitely large sets but also that there are infinitely many sizes of infinity, that these sizes get larger and larger with no end, and that there actually exist sets of all of these different sizes of infinity. There is simply no plausible way to take this sort of mathematical theorizing about the infinite to be about the physical world. Finally, a third problem with physicalism in Platonists' eyes is that it also seems to imply that mathematics is an empirical science, contingent on physical facts and susceptible to empirical falsification. This seems to contradict mathematical methodology; mathematics is not empirical (at least not usually), and most mathematical truths (e.g., “2 + 3 = 5”) cannot be empirically falsified by discoveries about the nature of the physical world.Platonists argue against the various versions of paraphrase nominalism by pointing out that they are also out of step with actual mathematical discourse. These views are all committed to implausible hypotheses about the intentions of mathematicians and ordinary folk. For instance, deductivism is committed to the thesis that when people utter sentences such as “4 is even,” what they really mean to say is that, if there were numbers, then 4 would be even. However, there simply is no evidence for this thesis, and, what is more, it seems obviously false. Similar remarks can be made about the other versions of paraphrase nominalism; all of these views involve the same idea that mathematical statements are not used literally. There is no evidence, however, that people use mathematical sentences nonliterally. It seems that the best interpretation of mathematical discourse takes it to be about (or at any rate, to purport to be about) certain kinds of objects. Furthermore, as has already been shown, there are good reasons to think that the objects in question could not be physical or mental objects. Thus, the arguments outlined here seem to lead to the Platonistic conclusion that mathematical discourse is about abstract objects.It does not follow from this that Platonism is true, however, because antiPlatonists can concede all these arguments and still endorse fictionalism or neoMeinongianism. Advocates of the neoMeinongian view accept the eminently plausible Platonistic interpretation of mathematical sentences while also denying that there are any such things as numbers and functions and sets; but then neoMeinongians want to claim that mathematics is true anyway. Platonists argue that this reasoning is absurd. For instance, if mermaids do not exist, then the sentence “There are some mermaids with red hair” cannot be literally true. Likewise, if there are no such things as numbers, then the sentence “There are some prime numbers larger than 20” cannot be literally true either. Perhaps the best thing to say here is that neoMeinongianism warps the meaning of the word true.The one remaining group of antiPlatonists, the fictionalists, agree with Platonists on how to interpret mathematical sentences. In fact, the only point on which fictionalists disagree with Platonists is the bare question of whether there exist any such things as abstract objects (and, as a result, the question about whether mathematical sentences are literally true). However, since abstract objects must be nonphysical and nonmental if they exist at all, it is not obvious how one could ever determine whether they exist. This is the beauty of the fictionalists' view: they endorse all of the Platonists' arguments that mathematics is best interpreted as being about abstract objects, and then they simply assert that they do not believe in abstract objects. It might seem very easy to dispense with fictionalism, because it might seem utterly obvious that sentences such as “2 + 2 = 4” are true. On closer inspection, however, this is not at all obvious. If the arguments discussed above are correct—and Platonists and fictionalists both accept them—then in order for “2 + 2 = 4” to be true, abstract objects must exist. But one might very well doubt that there really do exist such things; after all, they seem more than a bit strange, and what is more, there does not seem to be any evidence that they really exist.Or maybe some evidence does exist. This, at any rate, is what Platonists want to claim. Platonists have offered a few different arguments as refutations of fictionalism, but only one of them, known as the indispensability argument, has gained any real currency. According to the indispensability argument, wellestablished mathematical theorems must be true because they are inextricably woven into the empirical theories that have been developed and accepted in the natural sciences, and there are good reasons to think that these empirical theories are true. (This argument has roots in the work of Frege and has been developed by Quine and Putnam.) Fictionalists have offered two responses to this argument. Field has argued that mathematics is not inextricably woven into the empirical theories that scientists have developed; if scientists wanted, he has argued, they could extract mathematics from their theories. Furthermore, Balaguer, Rosen, and Yablo have argued that it does not matter whether mathematics is indispensable to empirical science because even if it is, and even if mathematical theorems are not literally true (because there are no such things as abstract objects), the empirical theories that use these mathematical theorems could still provide essentially accurate pictures of the physical world.The epistemological argument against PlatonismThe epistemological argument is very simple. It is based on the idea that, according to Platonism, mathematical knowledge is knowledge of abstract objects, but there does not seem to be any way for humans to acquire knowledge of abstract objects. The argument for the claim that humans could not acquire knowledge of abstract objects proceeds as follows:● (1) Humans exist entirely within spacetime.● (2) If there exist any abstract objects, then they exist entirely outside of spacetime.● (3) Therefore, it seems that humans could never acquire knowledge of abstract objects.There are three ways for Platonists to respond to this argument. They can reject (1), they can reject (2), or they can accept (1) and (2) and explain why the very plausible sounding (3) is nonetheless false.Platonists who reject (1) maintain that the human mind is not entirely physical and that it is capable of somehow forging contact with abstract objects and thereby acquiring information about such objects. This strategy was pursued by Plato and Gödel. According to Plato, people have immaterial souls, and before birth their souls acquire knowledge of abstract objects, so that mathematical learning is really just a process of recollection. For Gödel, humans acquire information about abstract objects by means of a faculty of mathematical intuition—in much the same way that information about physical objects is acquired through sense perception.Platonists who reject (2) alter the traditional Platonic view and maintain that, although abstract objects are nonphysical and nonmental, they are still located in spacetime; hence, according to this view, knowledge of abstract objects can be acquired through ordinary sense perceptions. Maddy developed this idea in connection with sets. She claimed that sets of physical objects are spatiotemporally located and that, because of this, people can perceive them—that is, see them and taste them and so on. For example, suppose that Maddy is looking at three eggs. According to her view, she can see not only the three eggs but also the set containing them. Thus, she knows that this set has three members simply by looking at it—analogous to the way that she knows that one of the eggs is white just by looking at it.Platonists who accept both (1) and (2) deny that humans have some sort of informationgathering contact with abstract objects in the way proposed by Plato, Gödel, and Maddy; but these Platonists still think that humans can acquire knowledge of abstract objects. One strategy that Platonists have used here is to argue that people acquire knowledge of abstract mathematical objects by acquiring evidence for the truth of their empirical scientific theories; the idea is that this evidence provides reason to believe all of empirical science, and science includes claims about mathematical objects. Another approach, developed by Resnik and Shapiro, is to claim that humans can acquire knowledge of mathematical structures by means of the faculty of pattern recognition. They claim that mathematical structures are nothing more than patterns, and humans clearly have the ability to recognize patterns.Another strategy, that of fullblooded Platonism, is based on the claim that Platonists ought to endorse the thesis that all the mathematical objects that possibly could exist actually do exist. According to Balaguer, if fullblooded Platonism is true, then knowledge of abstract objects can be obtained without the aid of any informationtransferring contact with such objects. In particular, knowledge of abstract objects could be obtained via the following twostep method (which corresponds to the actual methodology of mathematicians): first, stipulate which mathematical structures are to be theorized about by formulating some axioms that characterize the structures of interest; and second, deduce facts about these structures by proving theorems from the given axioms.For example, if mathematicians want to study the sequence of nonnegative integers, they can begin with axioms that elaborate its structure. Thus, the axioms might say that there is a unique first number (namely, 0), that every number has a unique successor, that every nonzero number has a unique predecessor, and so on. Then, from these axioms, theorems can be proven—for instance, that there are infinitely many prime numbers. This is, in fact, how mathematicians actually proceed. The point here is that fullblooded Platonists can maintain that by proceeding in this way, mathematicians acquire knowledge of abstract objects without the aid of any informationtransferring contact with such objects. Put differently, they maintain that what mathematicians have discovered is that, in the sequence of nonnegative integers (by which is just meant the part or parts of the mathematical realm that mathematicians have in mind when they select the standard axioms of arithmetic), there are infinitely many prime numbers. Without fullblooded Platonism this cannot be said, because traditional Platonists have no answer to the question “How do mathematicians know which axiom systems describe the mathematical realm?” In contrast, this view entails that all internally consistent axiom systems accurately describe parts of the mathematical realm. Therefore, fullblooded Platonists can say that when mathematicians lay down axiom systems, all they are doing is stipulating which parts of the mathematical realm they want to talk about. Then they can acquire knowledge of those parts simply by proving theorems from the given axioms.Ongoing impasseJust as there is no widespread agreement that fictionalists can succeed in responding to the indispensability argument, there is no widespread agreement that Platonists can adequately respond to the epistemological argument. It seems to this writer, though, that both fullblooded Platonism and fictionalism can be successfully defended against all of the traditional arguments brought against them. Recall that Platonism and fictionalism agree on how mathematical sentences should be interpreted—that is, both views agree that mathematical sentences should be interpreted as being statements about abstract objects. On the other hand, Platonism and fictionalism disagree on the metaphysical question of whether abstract objects exist, and an examination of the foregoing debate does not provide any compelling reason to endorse or reject either view (though some reasons have proved plausible and attractive enough to persuade people to take sides on this question). In fact, humanity seems to be cut off in principle from ever knowing whether there are such things as abstract objects. Indeed, it seems to this writer that it is doubtful that a correct answer even exists. For it can be argued that the concept of an abstract object is so unclear that there is no objective, agreedupon condition that would need to be satisfied in order for it to be true that there are abstract objects. This view of the debate is extremely controversial, however.Mark BalaguerAdditional ReadingEdward A. Maziarz and Thomas Greenwood, Greek Mathematical Philosophy (1968, reissued 1995), discusses the evolution of mathematical philosophy from Thales of Miletus and the Pythagoreans through Plato and Aristotle.Paul Benacerraf and Hilary Putnam (eds.), Philosophy of Mathematics: Selected Readings, 2nd ed. (1983), is the standard anthology of early and mid20thcentury writings. Bertrand Russell, Introduction to Mathematical Philosophy, 2nd ed. (1920, reissued 1993), is perhaps the most famous introductory book on the subject, though it is mainly dedicated to developing Russell's own view. Stephan Körner, The Philosophy of Mathematics (1960, reissued 1986), is a classic introductory overview of the debate between logicists, intuitionists, and formalists during the first half of the 20th century.Stewart Shapiro, Thinking About Mathematics (2000), is a very good recent book that provides a more general introduction to the philosophy of mathematics. Mark Balaguer, Platonism and AntiPlatonism in Mathematics (1998, reissued 2001), provides an overview of various theories in the philosophy of mathematics, while arguing for a very specific, original view of its own. Finally, two works written by mathematicians are G.H. Hardy, A Mathematician's Apology, rev. ed. (1969, reissued 1999); and Reuben Hersh, What Is Mathematics, Really? (1997).Mark Balaguer* * *
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