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Platonism in Metaphysics

Platonism is the view that there exist such things as abstract objects where an abstract object is an object that does not exist in space or time and which is therefore entirely non-physical and non-mental. Platonism in this sense is aview. It is obviously related to the views of Plato in important ways, but it is not entirely clear that Plato endorsed this view, as it is defined here. In order to remain neutral on this question, the term platonism is spelled with a lower-case p. (See entry onPlato.) The most important figure in the development of modern platonism is Gottlob Frege (1884, 1892, 18931903, 1919). The view has also been endorsed by many others, including Kurt Gdel (1964), Bertrand Russell (1912), and W.V.O. Quine (1948, 1951).

Section 1 will describe the contemporary platonist view in detail. Section 2 will describe the alternatives to platonism namely, conceptualism, nominalism, immanent realism, and Meinongianism. Section 3 will develop and assess the first important argument in favor of platonism, namely, the One Over Many argument. Section 4 will develop and assess a second argument for platonism, namely, the Singular Term argument. This argument emerged much later than the One Over Many argument, but as we will see, it is widely thought to be more powerful. Finally, section 5 will develop and assess the most important argumentagainstplatonism, namely, the epistemological argument.

Platonism is the view that there exist abstract (that is, non-spatial, non-temporal) objects (see the entry onabstract objects). Because abstract objects are wholly non-spatiotemporal, it follows that they are also entirely non-physical (they do not exist in the physical world and are not made of physical stuff) and non-mental (they are not minds or ideas in minds; they are not disembodied souls, or Gods, or anything else along these lines). In addition, they are unchanging and entirely causally inert that is, they cannot be involved in cause-and-effect relationships with other objects.[1]All of this might be somewhat perplexing; for with all of these statements about what abstract objects arenot, it might be unclear what theyare. We can clarify things, however, by looking at some examples.

Consider the sentence 3 is prime. This sentence seems to say something about a particular object, namely, the number 3. Just as the sentence The moon is round says something about the moon, so too 3 is prime seems to say something about the number 3. But whatisthe number 3? There are a few different views that one might endorse here, but the platonist view is that 3 is an abstract object. On this view, 3 is a real and objective thing that, like the moon, exists independently of us and our thinking (i.e., it is not just an idea in our heads). But according to platonism, 3 is different from the moon in that it is not a physical object; it is wholly non-physical, non-mental, and causally inert, and it does not exist in space or time. One might put this metaphorically by saying that on the platonist view, numbers exist in platonic heaven. But we should not infer from this that according to platonism, numbers exist in aplace; they do not, for the concept of a place is a physical, spatial concept. It is more accurate to say that on the platonist view, numbers exist (independently of us and our thoughts) but do not exist in space and time.

Similarly, many philosophers take a platonistic view ofproperties. Consider, for instance, the property of being red. According to the platonist view of properties, the property of redness exists independently of any red thing. There are red balls and red houses and red shirts, and these all exist in the physical world. But platonists about properties believe that in addition to these things, redness the property itself also exists, and according to platonists, this property is an abstract object. Ordinary red objects are said toexemplifyorinstantiateredness. Plato said that theyparticipate inredness, but this suggests a causal relationship between red objects and redness, and again, contemporary platonists would reject this.

Platonists of this sort say the same thing about other properties as well: in addition to all the beautiful things, there is also beauty; and in addition to all the tigers, there is also the property of being a tiger. Indeed, even when there are no instances of a property in reality, platonists will typically maintain that the property itself exists. This isnt to say that platonists are committed to the thesis that there is a property corresponding to every predicate in the English language. The point is simply that in typical cases, there will be a property. For instance, according to this sort of platonism, there exists a property of being a four-hundred-story building, even though there are no such things as four-hundred-story buildings. This property exists outside of space and time along with redness. The only difference is that in our physical world, the one property happens to be instantiated whereas the other does not.

In fact, platonists extend the position here even further, for on their view, properties are just a special case of a much broader category, namely, the category ofuniversals. Its easy to see why one might think of a property like redness as a universal. A red ball that sits in a garage in Buffalo is a particular thing. But redness is something that is exemplified by many, many objects; its something that all red objects share, or have in common. This is why platonists think of redness as a universal and of specific red objects such as balls in Buffalo, or cars in Cleveland asparticulars.

But according to this sort of platonism, properties are not the only universals; there are other kinds of universals as well, most notably,relations. Consider, for instance, the relationto the north of; this relation is instantiated by many pairs of objects (or more accurately, by ordered pairs of objects, since order matters here e.g.,to the north ofis instantiated by San Francisco, Los Angeles, and Edinburgh, London, but not by Los Angeles, San Francisco, or London, Edinburgh). So according to platonism, the relationto the north ofis atwo-place universal, whereas a property like redness is aone-place universal. There are also three-place relations (which are three-place universals), four-place relations, and so on. An example of a three-place relation is thegaverelation, which admits of a giver, a givee, and a given as in Jane gave a CD to Tim.

Finally, some philosophers claim thatpropositionsare abstract objects. One way to think of a proposition is as the meaning of a sentence. Alternatively, we can say that a proposition is that which is expressed by a sentence on a particular occasion of use. Either way, we can say that, e.g., the English sentence Snow is white and the German sentence Schnee ist weiss express the same proposition, namely, the proposition that snow is white.

There are many different platonistic conceptions of propositions. For instance, Frege (1892, 1919) held that propositions are composed ofsensesof words (e.g., on this view, the proposition that snow is white is composed of the senses of snow and is white), whereas Russell at one point (1905, 191011) held that propositions are composed of properties, relations, and objects (e.g., on this view, the proposition that Mars is red is composed ofMars(the planet itself) and theproperty of redness). Others hold that propositions do not have significant internal structure. The differences between these views will not matter for our purposes. For more detail, see the entry onpropositions.

(It might seem odd to say that Russellian propositions are abstract objects. Consider, e.g., the Russellian proposition that Mars is red. This is an odd sort ofhybridobject. It has two components, namely, Mars (the planet itself) and the property of redness. One of these components (namely, Mars) is aconcreteobject (where a concrete object is just a spatiotemporal object). Thus, even if redness is an abstract object, it does not seem that the Russellian proposition is completely non-spatiotemporal. Nonetheless, philosophers typically lump these objects together with abstract objects. And its not just Russellian propositions; similar remarks can be made about various other kinds of objects. Think, for instance, of impure sets--e.g., the set containing Mars and Jupiter. This seems to be a hybrid object of some kind as well, because while it has concrete objects as members, its still aset, and on the standard view, sets are abstract objects. If we wanted to be really precise, it would probably be best to have another term for such objects--e.g., hybrid object, or impure abstract object--but, again, this isnt how philosophers typically talk; they usually just treat these things as abstract objects. None of this will matter very much in what follows, however, because this essay is almost entirely concerned with what might be calledpureabstract objects--i.e., abstract objects that are completely non-spatiotemporal.)

Numbers, propositions, and universals (i.e., properties and relations) are not the only things that people have taken to be abstract objects. As we will see below, people have also endorsed platonistic views in connection with linguistic objects (most notably, sentences), possible worlds, logical objects, and fictional characters (e.g., Sherlock Holmes). And it is important to note here that one can be a platonist about some of these things without being a platonist about the others e.g., one might be a platonist about numbers and propositions but not properties or fictional characters.

Of course, platonism about any of these kinds of objects is controversial. Many philosophers do not believe in abstract objects at all. The alternatives to platonism will be discussed insection 2, but it is worth noting here that the primary argument that platonists give for their view is that, according to them, there are good arguments against all other views. That is, platonists think we have to believe in abstract objects, because (a) there are good reasons for thinking that things like numbers and universals exist, and (b) the only tenable view of these things is that they are abstract objects. We will consider these arguments in detail below.

There are not very many alternatives to platonism. One can reject the existence of things like numbers and universals altogether. Or one can maintain that there do exist such things as numbers and universals, and instead of saying that they are abstract objects, one can say that they are mental objects of some sort (usually, the claim is that they are ideas in our heads) or physical objects of some sort. Thus, the four mainstream views here are as follows (and keep in mind that anti-platonists can pursue different strategies with respect to different kinds of alleged abstract objects, taking one view of, say, numbers, and another view of properties or propositions).

Immanent Realism: Advocates of this view agree with platonists that there do exist such things as mathematical objects or universals, or whatever category of alleged abstract objects were talking about and that these things are independent of us and our thinking; but immanent realists differ from platonists in holding that these objects exist in the physical world. Depending on the kind of object under discussion i.e., whether were talking about mathematical objects or properties or what have you the details of this view will be worked out differently. In connection with properties, the standard immanent-realist view is that properties like redness exist only in the physical world, in particular, in actual red things, as nonspatial parts or aspects of those things (this view traces back to Aristotle; in contemporary times, it has been defended by Armstrong (1978)). There is certainly some initial plausibility to this idea: if you are looking at a red ball, and you think that in addition to the ball, its redness exists, then it seems a bit odd to say (as platonists do) that its redness exists outside of spacetime. After all, the ball is sitting right here in spacetime and we can see that its red; so it seems initially plausible to think that if the redness exists at all, then it exists in the ball. As we will see below, however, there are serious problems with this view.

In connection with numbers, one strategy is to take numbers to be universals of some sort e.g., one might take them to be properties of piles of physical objects, so that, for instance, the number 3 would be a property of, e.g., a pile of three books and to take an immanent realist view of universals. (This sort of view has been defended by Armstrong (1978).) But views of this kind have not been very influential in the philosophy of mathematics. A more prominent strategy for taking number talk to be about the physical world is to take it to be about actual piles of physical objects, rather than properties of piles. Thus, for instance, one might maintain that to say that 2 + 3 = 5 is not really to say something about specific entities (numbers); rather, it is to say that whenever we push a pile of two objects together with a pile of three objects, we will wind up with a pile of five objects or something along these lines. Thus, on this view, arithmetic is just a very general natural science. A view of this sort was developed by Mill (1843) and, more recently, a similar view has been defended by Philip Kitcher (1984). It should be noted, however, that while there are certainly physicalist themes running through the views of Mill and Kitcher, it is not clear that either of them should be interpreted as an immanent realist. Kitcher is probably best classified as a kind of anti-realist (Ill say a bit more about this insection 4.1), and its not entirely clear how Mill ought to be classified, relative to our taxonomy, because its not clear how he would answer the question, Are there numbers, and if so, what are they?

Finally, Penelope Maddy (1990) has also developed a sort of immanent realist view of mathematics. Concentrating mainly on set theory, Maddy maintains that sets of physical objects are located in space and time, right where their members are located. But Maddian sets cannot be identified with the physical matter that constitutes their members. On Maddys view, corresponding to every physical object, there is a huge infinity of sets (e.g., the set containing the given object, the set containing that set, and so on) that are all distinct from one another but which all share the same matter and the same spatiotemporal location. Thus, on this view, there is more to a set than the physical stuff that makes up its members, and so Maddy might be better interpreted as endorsing a nonstandard version of platonism.

Conceptualism(also calledpsychologismandmentalism, depending on the sorts of objects under discussion): This is the view that there do exist numbers or properties, or propositions, or whatever but that they do not exist independently of us; instead, they are mental objects; in particular, the claim is usually that they are something like ideas in our heads. As we will see below, this view has serious problems and not very many people endorse it. Nonetheless, it has had periods of popularity in the history of philosophy. It is very often thought that Locke held a conceptualistic view of universals, and prior to the twentieth century, this was the standard view of concepts and propositions. In the philosophy of mathematics, psychologistic views were popular in the late nineteenth century (the most notable proponent being the early Husserl (1891)) and even in the first part of the twentieth century with the advent of psychologistic intuitionism (Brouwer 1912 and 1948, and Heyting 1956). Finally, Noam Chomsky (1965) has endorsed a mentalistic view of sentences and other linguistic objects, and he has been followed here by others, most notably, Fodor (1975, 1987).

It should also be noted here that one can claim that the existence of numbers (or propositions or whatever) isdependent on us humanswithout endorsing a psychologistic view of the relevant entities. For one can combine this claim with the idea that the objects in question are abstract objects. In other words, one might claim and somehaveclaimed that numbers (or propositions or whatever) are mind-dependent abstract objects, i.e., objects that exist outside of the mind, and outside of space and time, but which only came into being because of the activities of human beings. Liston (200304), Cole (2009), and Bueno (2009) endorse views of this general kind in connection with mathematical objects; Schiffer (2003, chapter 2), Soames (2014), and King (2014) endorse views like this of propositions; and Salmon (1998) and Thomasson (1999) endorse views like this of fictional objects.

Nominalism(also calledanti-realism): This is the view that there are no such things as numbers, or universals, or whatever sort of alleged abstract objects are under discussion. Thus, for instance, a nominalist about properties would say that while there are such things as red balls and red houses, there is no such thing as the property of redness, over and above the red balls and red houses. And a nominalist about numbers would say that while there are such things as piles of three stones, and perhaps 3-ideas existing in peoples heads, there is no such thing as the number 3. As we will see below, there are many different versions of each of these kinds of nominalism, but for now, we dont need anything more than this general formulation of the view. (Sometimes nominalism is used to denote the view that there are no such things as abstract objects; on this usage, nominalism is synonymous with anti-platonism, and views like immanent realism count as versions of nominalism. In contrast, on the usage employed in this essay, nominalism is essentially synonymous with anti-realism, and so views like immanent realism will not count as versions of nominalism here.)

Prima facie, it might seem that nominalism, or anti-realism, is further from the platonist view than immanent realism and conceptualism are for the simple reason that the latter two views admit that there do exist such things as numbers (or universals, or whatever). It is important to note, however, that nominalists agree with platonists on an important point that immanent realists and conceptualists reject; in particular, nominalists (in agreement with platonists) endorse the following thesis:

(S) If there were such things as numbers (or universals, or whatever sort of alleged abstract objects were talking about), then they would be abstract objects; that is, they would be non-spatiotemporal, non-physical, and non-mental.

This is an extremely important point, because it turns out that there are some very compelling arguments (which we will discuss) in favor of (S). As a result, there are very few advocates of immanent realism and conceptualism, especially in connection with mathematical objects and propositions. There is wide-spread agreement about what numbers and propositions would be if there were such things (namely, abstract objects) but very little agreement as to whether there do exist such things. Thus, today, the controversial question here is a purely ontological one: Are there any such things as abstract objects (e.g., mathematical objects, propositions, and so on)?

It is worth noting that while there are only four mainstream views here (viz., platonism, immanent realism, conceptualism, and nominalism) a fifth view deserves mention, namely,Meinongianism(see Meinong (1904)). On this view, every singular term e.g., Clinton, 3, and Sherlock Holmes picks out an object that has some sort of being (thatsubsists, or thatis, in some sense) but only some of these objects have full-blown existence. According to Meinongianism, sentences that platonists take to be about abstract objects sentences like 3 is prime and Red is a color express truths about objects that dont exist.

Meinongianism has been almost universally rejected by philosophers. The standard argument against it (see, e.g., Quine (1948), p. 3, and Lewis (1990)) is that it does not provide a view that is clearly distinct from platonism and merely creates the illusion of a different view by altering the meaning of the term exist. The idea here is that on the standard meaning of exist, any object that has any being at all exists, and so according to standard usage, Meinongianism entails that numbers and universals exist; but this view clearly doesnt take such things to exist in spacetime and so, the argument concludes, Meinongianism entails that numbers and universals are abstract objects just as platonism does.

It is worth noting, however, that while Meinongianism has mostly been rejected, it does have some more contemporary advocates, most notably, Routley (1980), Parsons (1980), and Priest (2003, 2005).

There are two mainstream arguments for platonism. The first, which goes back to Plato, is an argument for the existence of properties and relations only; this is theOne Over Many argument. The second is also present in some sense in the works of Plato (at least on some readings of those works), but its first modern formulation, and certainly the firstclearformulation, was given by Frege (1884, 1892, 18931903, 1919); I will call this thesingular term argument, and unlike the One Over Many, it can be used in connection with all the different kinds of abstract objects, i.e., numbers, properties, propositions, and so on. In the present section, we will discuss the One Over Many argument, and in the next section, we will discuss the singular term argument.

The One Over Many argument can be formulated as follows:

I have in front of me three red objects (say a ball, a hat, and a rose). These objects resemble one another. Therefore, they have something in common. What they have in common is clearly a property, namely, redness; therefore, redness exists.

We can think of this argument as an inference to the best explanation. There is a fact that requires explanation, namely, that the three objects resemble each other. The explanation is that they all possess a single property, namely, redness. Thus, platonists argue, if there is no other explanation of this fact (i.e., the fact of resemblance) that is as good as their explanation (i.e., the one that appeals to properties), then we are justified in believing in properties.

Notice that as the argument has been stated here, it is not an argument for a platonistic view of properties; it is an argument for the thesis that properties exist, but not for the thesis that properties are abstract objects. Thus, in order to use this argument to motivate platonism, one would have to supplement it with some reason for thinking that the properties in question here could not be ideas in our heads or immanent properties existing in particular physical objects. There are a number of arguments that one might use here, and insection 4.3, we will discuss some of these. But there is no need to pursue this here, because there is good reason to think that the One Over Many argument doesnt succeed anyway i.e., that it doesnt provide a good reason for believing in properties of any sort. In other words, the One Over Many argument fails to refute nominalism about properties.

Before proceeding, it is worth pointing out that the One Over Many argument described above can be simplified. As Michael Devitt (1980) points out, the appeal to resemblance, or tomultiplethings having a given property, is a red herring. On the traditional formulation, nominalists are challenged to account for the following fact: the ball is red and the hat is red. But if nominalists can account for the fact that the ball is red, then presumably, they can simply repeat the same sort of explanation in connection with the hat, and they will have accounted for the fact that both things are red. Thus, the real challenge for the nominalist is to explain simple predicative facts, e.g., the fact that the ball is red, without appealing to properties, e.g., redness. More generally, they need to show how we can account for the truth of sentences of the form aisF without appealing to a property ofFness.[2]

(One might also think of the argument as asking not for an explanation of the fact that, say, Mars is red, but rather for an account of what it is about the world thatmakes the sentence Mars is red true. See Peacock (2009) in this regard.)

There is a very well-known nominalist response to the One Over Many argument. The heart of the response is captured by the following remark from Quine (1948, p. 10):

That the houses and roses and sunsets are all of them red may be taken as ultimate and irreducible, and it may be held that...[the platonist] is no better off, in point of real explanatory power, for all the occult entities which he posits under such names as redness.

There are two different ideas here. The first is that nominalists can respond to the One Over Many with an appeal to irreducible facts, orbrutefacts. The second is that platonists are no better off than such brute-fact nominalists in terms of real explanatory power. Now, Quine didnt say very much about these two ideas, but both ideas have been developed by Devitt (1980, 2010), whose exposition we follow here.

The challenge to nominalists is to provide an explanation of facts of a certain kind, namely, predicative facts expressed by sentences of the form aisF, e.g., the fact that a given ball is red. Now, whenever we are challenged to provide an explanation of a fact, or alleged fact, we have a number options. The most obvious response is simply to provide the requested explanation. But we can also argue that the alleged fact isnt really a fact at all. Or, third, we can argue that the fact in question is abrutefact i.e., a fact that does not have an explanation. Now, in the present case, nominalists cannot claim that all predicative facts are brute facts, because it is clear that wecanexplain at least some facts of this sort. For instance, it seems that the fact that a given ball is red can be explained very easily by saying that it is red because it reflects light in such and such a way, and that it reflects light in this way because its surface is structured in thus and so a manner. So nominalists should not claim that all predicative facts are brute facts. But as Devitt points out, there is a more subtle way to appeal to bruteness here, and if Quinean nominalists make use of this, they can block the One Over Many argument.

The Quine-Devitt response to the One Over Many begins with the claim that we can account for the fact that the ball is red, without appealing to the property of redness, by simply using whatever explanation scientists give of this fact. Now, by itself, this explanation will not satisfy advocates of the One Over Many argument. If we explain the fact that the ball is red by pointing out that its surface is structured in some specific way, then advocates of the One Over Many argument will say that we have only moved the problem back a step, because nominalists will now have to account for the fact that the balls surface is structured in the given way, and they will have to do this without appealing to the property of being structured in the given way. More generally, the point is this: it is of course true that if nominalists are asked to account for the fact that some objectaisF, without appealing to the property ofFness, they can do this by pointing out that (i)aisGand (ii) allFs areGs (this is the sort of explanation they will get if they borrow their explanations from scientists); but such explanations only move the problem back a step, for they leave us with the task of having to explain the fact thataisG, and if we want to endorse nominalism, we will have to do this without appealing to the property ofGness.

This is where the appeal to bruteness comes in. Nominalists can say that (a) we can keep giving explanations of the above sort (i.e., explanations of the sort aisFbecause it isG, or because its parts areGs,Hs, andIs, or whatever) for as long as we can, and (b) when explanations of this sort cannot be given, no explanation at all can be given. The thought here is that at this point, we will have arrived at fundamental facts that do not admit of explanations e.g., facts about the basic physical natures of elementary particles. When we arrive at facts like this, we will say: Theres no reason why these particles are this way; they justare.

This gives us a way of understanding how nominalists can plausibly use an appeal to bruteness to respond to the One Over Many argument. But the appeal to bruteness is only half of the Quinean remark quoted above. What about the other half, i.e., the part about the platonist being no better off than brute-fact nominalists in terms of real explanatory power? To appreciate this claim, let us suppose that we have arrived at a bottom-level fact that Quinean nominalists take to be a brute fact (e.g., the fact that physical particles of some particular kind say, gluons areG). Advocates of the One Over Many would say that their view is superior to Quinean nominalism because they can provide an explanation of the fact in question. Now, when they announce this, people who were interested in the question of why gluons areG, and who had been disappointed to hear from scientists and Quineans that this is simply a brute fac.

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