## 1. What is Platonism?



Platonism is the view that there exist abstract (that is, non-spatial,
non-temporal) objects (see the entry on
 abstract 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 are *not*, it might be
unclear what they *are*. 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 what *is* the 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 a *place*; 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 of
 properties.
 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 to *exemplify* or *instantiate* redness. Plato
said that they *participate in* redness, 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 isn't 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 of *universals*. It's 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; it's
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 -- as
*particulars*.



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 relation *to 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 of* is instantiated by <San
Francisco, Los Angeles>, and <Edinburgh, London>, but not by
<Los Angeles, San Francisco>, or <London, Edinburgh>). So
according to platonism, the relation *to the north of* is a
*two-place universal*, whereas a property like redness is a
*one-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 the *gave* relation,
which admits of a giver, a givee, and a given -- as in
'Jane gave a CD to Tim'.



Finally, some philosophers claim that *propositions* are
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 of
*senses* of 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,
1910-11) held that propositions are composed of properties,
relations, and objects (e.g., on this view, the proposition that Mars
is red is composed of *Mars* (the planet itself) and the
*property 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
on
 propositions.



(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 of *hybrid* object. It has two components,
namely, Mars (the planet itself) and the property of redness. One of
these components (namely, Mars) is a *concrete* object (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 it's 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, it's still a *set*, 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 isn't 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 called *pure*
abstract 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 in
 section 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.


## 2. A Taxonomy of Positions



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).


1. *Platonism*: This is the view described in
 section 1.
2. *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 we're 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
we're 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 it's 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
(I'll say a bit more about this in
 section 4.1),
 and it's not entirely clear how Mill ought to be classified, relative
to our taxonomy, because it's 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
Maddy's 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.
3. *Conceptualism* (also called *psychologism* and
*mentalism*, 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) is *dependent on us
humans* without 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 some *have* claimed -- 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 (2003-04), 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.
4. *Nominalism* (also called *anti-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 people's 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 don't 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 we're 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 (that *subsists*, or that *is*, 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
don't 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 doesn't 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).


## 3. The One Over Many Argument



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 the *One 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 first *clear* formulation, was given by Frege
(1884, 1892, 1893-1903, 1919); I will call this the *singular
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 in
 section 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 doesn't succeed anyway -- i.e., that it doesn't 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 to *multiple* things
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
'*a* is *F*' without appealing to a property
of
 *F*ness.[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 that *makes 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, or
*brute* facts. The second is that platonists are no better off
than such brute-fact nominalists in terms of real explanatory power.
Now, Quine didn't 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 '*a* is *F*', 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 isn't really a fact at
all. Or, third, we can argue that the fact in question is a
*brute* fact -- 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 we
*can* explain 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 ball's 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 object
*a* is *F*, without appealing to the property of
*F*ness, they can do this by pointing out that (i) *a*
is *G* and (ii) all *F*s are *G*s (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
that *a* is *G*, and if we want to endorse nominalism,
we will have to do this without appealing to the property of
*G*ness.



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 '*a* is *F* because it is
*G*,' or because its parts are *G*s, *H*s,
and *I*s, 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: "There's no reason
why these particles are this way; they just *are*."



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 -- are *G*). 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 are *G*, and who had been
disappointed to hear from scientists and Quineans that this is simply
a brute fact, might get very excited and listen eagerly to what
advocates of the One Over Many have to say. What they say is this:



> 
> Gluons are *G* because they possess the property of
> *G*ness.
> 



This doesn't seem very helpful. The claim that gluons possess
*G*ness seems to do little more than tell us that gluons have
some nature that makes it the case that they are *G*, and so it
seems that no genuine explanation has been given. After all, those who
had been interested in learning why gluons are *G* would not be
very satisfied by this so-called "explanation". Thus, to
use Quine's words, it seems that advocates of the One Over Many are
"no better off, in point of real explanatory power" than
brute-fact nominalists are.



Nominalists might try to push the argument a bit further here,
claiming that the sentence



> 
> (P) Gluons possesses the property of *G*ness
> 



is just a *paraphrase* of the sentence



> 
> (N) Gluons are *G*.
> 



On this view, (P) is *equivalent* to (N). That is, it says the
very same thing. And neither sentence, according to this view, entails
the existence of *G*ness. We can call this a
*paraphrase-nominalist* view of sentences like (P). But
nominalists needn't endorse this view. They can also endorse a
*fictionalist* view of sentences like (P). On this view, (P)
and (N) do not, strictly speaking, say the same thing, because (P)
talks about the property of *G*ness and (N) does not. According
to this fictionalist view, (P) is strictly speaking untrue, because it
talks about the property of *G*ness, and according to
nominalism, there is no such thing as *G*ness. In short, (P) is
strictly speaking untrue, on this view, for the same reason that,
e.g., 'The tooth fairy is nice' is untrue. But while (P)
is not *literally* true on this view, it is
"for-all-practical-purposes true", or some such thing,
because colloquially, it can be used to say what (N) says literally.
This idea is often captured by saying that (P) is just a *manner of
speaking*, or a *facon de parler*. (Notice that the
dispute between fictionalism and paraphrase nominalism is best
understood as a straightforward empirical dispute about the
ordinary-language semantics of sentences like (P); the question is
whether such utterances literally say the same things that the
corresponding sentences like (N) say.)



Whichever view nominalists adopt here, they can respond to the One
Over Many argument -- i.e., to the claim that we can explain (N)
by endorsing (P) -- in the same way, namely, by pointing out that
as an explanation of (N), (P) is completely uninformative. Even if
nominalists endorse a fictionalist view according to which (P) is not
equivalent to (N), they can still say that the above explanation is
uninformative, because it really just says that gluons are *G*
because they possess a nature that makes it the case that they are
*G*.



Having made the point that the platonist explanation of (N) is
uninformative, the nominalist's next move is to appeal to Ockham's
razor to argue that we *shouldn't* believe in Gness (or at
least that we shouldn't believe in *G*ness for any reason that
has anything to do with the need to explain things like (N)). Ockham's
razor is a principle that tells us that we should believe in objects
of a given kind only if they play a genuine explanatory role. This
principle suggests that if *G*ness does not play a genuine role
in an explanation of the fact that gluons are *G*, then we
shouldn't believe in *G*ness -- or, again, we shouldn't
believe in it for any reason having to do with the need to explain the
fact that gluons are *G*.



The Quinean response to the One Over Many argument is often couched in
terms of a *criterion of ontological commitment*. A criterion
of ontological commitment is a principle that tells us when we are
committed to believing in objects of a certain kind in virtue of
having assented to certain sentences. What the above response to the
One Over Many suggests is that we are ontologically committed not by
predicates like 'is red' and 'is a rock', but
by *singular terms*. (A singular term is just a denoting
phrase, i.e., an expression that purports to refer to a specific
object, e.g., proper names like 'Mars' and
'Clinton', certain uses of pronouns like
'she', and on some views, definite descriptions like
'the oldest U.S. senator'.) More specifically, the idea
here seems to be this: if you think that a sentence of the form
'*a* is *F*' is true, then you have to
accept the existence of the object *a*, but you do not have to
accept the existence of a property of *F*ness; for instance, if
you think that 'The ball is red' is true, then you have to
believe in the ball, but you do not have to believe in redness; or if
you think that 'Fido is a dog' is true, then you have to
believe in Fido but not in the property of doghood.



Three points need to be made here. First, the above criterion needs to
be generalized so that it covers the use of singular terms in other
kinds of sentences, e.g., sentences of the form '*a* is
*R*-related to *b*'. Second, on the standard view,
we are ontologically committed not just by singular terms but also by
existential statements -- e.g., by sentences like 'There
are some *F*s', 'There is at least one
*F*', and so on (in first-order logic, such sentences are
symbolized as '([?]*x*)*Fx*', and the
'[?]' is called an *existential quantifier*).
The standard view here is that if you think that a sentence like this
is true, then you are committed to believing in the existence of some
*F*s (or at least one *F*) but you do not have to
believe in *F*ness; for instance, if we assent to 'There
are some dogs', then we are committed to believing in the
existence of some dogs, but we are not thereby committed to believing
in the existence of the property of doghood. (Quine actually thought
that we are committed *only* by existential claims and not by
singular terms; but this is not a widely held view.) Third and
finally, it is usually held that we are ontologically committed by
singular terms and existential expressions (or existential
quantifiers) only when they appear in sentences that we think are
*literally* true and only when we think the singular term or
existential quantifier in question *can't be paraphrased away*.
We can see what's meant by this by returning to the sentence



> 
> (R) The ball possesses the property of redness.
> 



In this sentence, the expression 'the property of redness'
seems to be a singular term -- it seems to *denote* the
property of redness; thus, using the above criterion of ontological
commitment, if we think (R) is true, then it would seem, we are
committed to believing in the property of redness. But there are two
different responses that nominalists can make to this. First, they can
endorse paraphrase nominalism (defined a few paragraphs back) with
respect to (R). If they do this, they will claim that (R) doesn't
really carry an ontological commitment to the property of redness,
because it is really just equivalent to the sentence 'The ball
is red'. This idea is often expressed by saying that in (R), the
singular term 'the property of redness' can be
*paraphrased away* -- which is just to say that (R) can be
paraphrased by (or is equivalent to) a sentence that doesn't contain
the singular term 'the property of redness' (namely,
'The ball is red'). The second view that nominalists can
endorse with respect to (R) is fictionalism. In other words, they can
admit that (R) does commit to the existence of the property of
redness, but they can maintain that because of this (and because there
are no such things as properties), (R) is, strictly speaking, untrue,
even if it is "for-all-practical-purposes true," or some
such thing.



Having said all of this, we can summarize by saying that the standard
view of ontological commitment is as follows:



> 
> *Criterion of Ontological Commitment*: We are ontologically
> committed by the singular terms (that can't be paraphrased away) in
> the (simple) sentences that we take to be literally true; and we are
> ontologically committed by the existential quantifiers (that can't be
> paraphrased away) in the (existential) sentences that we take to be
> literally true; but we are not committed by the predicates in such
> sentences. Thus, for instance, if we believe that a sentence of the
> form '*a* is *F*' is literally true, and if
> we think that it cannot be paraphrased into some other sentence that
> avoids reference to *a*, then we are committed to believing in
> the object *a* but not the property of *F*ness; and,
> likewise, if we assent to a sentence of the form '*a* is
> *R*-related to *b*', then we are committed to
> believing in the objects *a* and *b* but not the
> relation *R*; and if we assent to a sentence of the form
> 'There is an *F*' then we are committed to
> believing in an object that is *F* but we are not committed to
> the property of
>  *F*ness.[3]
> 



The One Over Many argument is now widely considered to be a bad
argument. Ironically, though, the above criterion of ontological
commitment -- which Quinean nominalists appeal to in responding
to the One Over Many argument -- is one of the central premises
in what is now thought to be the best argument for platonism. We will
call this argument the *singular term argument*, although one
might just as well call it the
*sucker-punch-on-the-Quinean-nominalist argument*, for as we
will see, the strategy is to accept the above criterion of ontological
commitment and turn it against the Quinean nominalist.


## 4. The Singular Term Argument



The general argument strategy here has roots in the work of Plato, but
its first clear formulation was given by Frege (1884, 1892,
1893-1903, and 1919). We begin with a general formulation of the
argument:


1. If a simple sentence (i.e., a sentence of the form
'*a* is *F*', or '*a* is
*R*-related to *b*', or...) is literally
true, then the objects that its singular terms denote exist.
(Likewise, if an existential sentence is literally true, then there
exist objects of the relevant kinds; e.g., if 'There is an
*F*' is true, then there exist some *F*s.)
2. There are literally true simple sentences containing singular
terms that refer to things that could only be abstract objects.
(Likewise, there are literally true existential statements whose
existential quantifiers range over things that could only be abstract
objects.) Therefore,
3. Abstract objects exist.



Premise (1) follows straightaway from the criterion of ontological
commitment that we discussed in the last section. Again, this is
widely accepted among contemporary philosophers, and for good reason
-- if you think that a sentence of the form '*a* is
*F*' is literally true and that it cannot be paraphrased
into some other sentence, then it's hard to see how you can deny that
there is such a thing as the object *a*. Therefore, since (3)
follows trivially from (1) and (2), the central question we have to
answer, in order to evaluate the above argument, is whether (2) is
true. (Below, I will discuss the possibility of denying (1), but for
now I want to focus on (2).) In any event, in order to motivate (2),
platonists need to provide some examples; that is, they have to
produce some sentences and argue that (i) they contain singular terms
that can only be taken as referring to abstract objects (and that
can't be paraphrased away) and (ii) they are literally true.
Platonists maintain that there are many different kinds of such
sentences. In what follows, we will consider versions of this argument
that attempt to establish the existence of mathematical objects (e.g.,
numbers), propositions, properties, relations, sentence types,
possible worlds, logical objects, and fictional objects.


### 4.1 Mathematical Objects



Platonists about mathematical objects claim that the theorems of our
mathematical theories -- sentences like '3 is prime'
(a theorem of arithmetic) and 'There are infinitely many
transfinite cardinal numbers' (a theorem of set theory) --
are literally true and that the only plausible view of such sentences
is that they are about abstract objects (i.e., that their singular
terms denote abstract objects and their existential quantifiers range
over abstract objects). This general stance toward mathematics goes
back to Plato, but the first clear statement of the argument in this
form was given by Frege (1884); other advocates include Quine (see his
1948 and 1951, though he doesn't explicitly state the argument there),
Godel (1964), Parsons (1965, 1971, 1994), Putnam (1971), Steiner
(1975), Resnik (1981, 1997), Zalta (1983, 1999), Wright (1983),
Burgess (1983), Hale (1987), Shapiro (1989, 1997), the early Maddy
 (1990),[4]
 Katz (1998), Colyvan (2001), McEvoy (2005, 2012), and Marcus
(2015).



Let's begin our discussion of the platonists' argument here by
considering their reasons for thinking that we have to take sentences
like '3 is prime' to be about abstract objects rather than
mental or physical objects of some kind. And let's start with a
discussion of the psychologistic view that mathematics is about mental
objects.



Frege (1884, introduction and section 27; 1893-1903,
introduction; 1894; and 1919) gave several compelling arguments
against psychologism. First, it seems that psychologism is incapable
of accounting for the truth of sentences that are about *all*
natural numbers, because there are infinitely many natural numbers and
clearly, there could not be infinitely many number-ideas in human
minds. Second, psychologism seems to entail that sentences about very
large numbers (in particular, numbers that no one has ever had a
thought about) are not true; for if none of us has ever had a thought
about some very large number, then psychologism entails that there is
no such number and, hence, that no sentence about that number could be
true. Third, psychologism turns mathematics into a branch of
psychology, and it makes mathematical truths contingent upon
psychological truths, so that, for instance, if we all died, '4
is greater than 2' would suddenly become untrue. But this seems
wrong: it seems that mathematics is true independently of us; that is,
it seems that the question of whether 4 is greater than 2 has nothing
at all to do with the question of how many humans are alive. Fourth
and finally, psychologism suggests that the proper methodology for
mathematics is that of empirical psychology; that is, it seems that 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 and ascertain whether there is, in
fact, an idea of such a number in one of our heads; but of course,
this is not the proper methodology for mathematics. As Frege says
(1884, section 27), "Weird and wonderful...are the results
of taking seriously the suggestion that number is an idea."



Platonists do not deny that we have ideas of mathematical objects.
What they deny is that our mathematical sentences are *about*
these ideas. Thus, the dispute between platonism and psychologism is
primarily a semantic one. Advocates of psychologism agree with
platonists that in the sentence '3 is prime',
'3' functions as a singular term (i.e., as a denoting
expression). But they disagree about the referent of this expression.
They think that '3' refers to a mental object, in
particular, an idea in our heads. It is this *semantic* thesis
that platonists reject and that the above Fregean arguments are
supposed to refute. More specifically, they're supposed to show that
the psychologistic semantics of mathematical discourse is not correct
because it has consequences that fly in the face of the actual usage
of mathematical language.



Another argument for the superiority of the platonist semantics of
mathematical discourse over the psychologist semantics is based on the
fact that in ordinary usage, one way to say that something doesn't
exist is to say that "it exists only in your
 head".[5]
 To say that mathematical objects exist only in our heads, it seems,
is just to say that they don't exist. For to say that they (as opposed
to our ideas of them) exist is to say that they exist independently of
us and our thinking. Quine put this point in a very compelling way in
connection with a conceptualistic view of mythical objects like
Pegasus. He writes (1948, p. 2):



> 
> McX [who maintains that Pegasus exists and is an idea in our heads]
> never confuses the Parthenon with the Parthenon-idea. The Parthenon is
> physical; the Parthenon-idea is mental...We cannot easily imagine
> two things more unlike...But when we shift from the Parthenon to
> Pegasus, the confusion sets in -- for no other reason than that
> McX would sooner be deceived by the crudest and most flagrant
> counterfeit than grant the nonbeing of Pegasus.
> 



The same argument can be run against the psychologistic conflation of
3-ideas with 3: you might doubt that there really is such a thing as
the number 3, existing objectively and independently of us, but you
should not for that reason claim that your idea of 3 *is* 3,
for that is just a confusion -- it is like saying that your idea
of Pegasus *is* Pegasus, or that your idea of the Parthenon is
the Parthenon.



Let us move on now to immanent-realist, or physicalist, views of
mathematics, and let us concentrate first on views like Mill's (1843,
book II, chapters 5 and 6), i.e., views that maintain that sentences
about numbers are really just general claims about bunches of ordinary
objects. On this view, the sentence '2 + 1 = 3', for
instance, isn't really about specific objects (the numbers 1, 2, and
3). Rather, it says that whenever we add one object to a pile of two
objects, we will get a pile of three objects. Now, in order to account
for contemporary mathematics in this way, a contemporary Millian would
have to take set theory to be about ordinary objects as well. This,
however, is untenable. One argument here is that set theory could not
be about bunches of ordinary objects, or piles of physical stuff,
because corresponding to every physical pile, there are many, many
sets. Corresponding to a ball, for instance, is the set containing the
ball, the set containing its molecules, the set containing its atoms,
and so on. (And we know that these are different sets, because they
have different members, and it follows from set theory that if set
*A* and set *B* have different members, then *A*
is not identical to *B*.) Indeed, the principles of set theory
entail that corresponding to every physical object, there is a huge
infinity of sets. Corresponding to our ball, for instance, there is
the set containing the ball, the set containing that set, the set
containing *that* set, and so on; and there is the set
containing the ball and the set containing the set containing the
ball; and so on and on and on. Clearly, these sets are not just piles
of physical stuff, because (a) there are infinitely many of them
(again, this follows from the principles of set theory) and (b) all of
these infinitely many sets share the same physical base. Thus, it
seems that claims about sets are not claims about bunches of ordinary
objects, or even generalized claims about such bunches. They are
claims about *sets*, which are objects of a different kind.



Another problem with physicalistic views along the lines of Mill's is
that they seem incapable of accounting for the sheer size of the
infinities involved in set theory. Standard set theory entails not
just that there are infinitely large sets, but that there are
infinitely many sizes of infinity, which 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
theory to be about physical stuff.



(These arguments do not refute the kind of immanent realism defended
by the early Maddy (1990). On Maddy's view, sets of physical objects
are located in spacetime, right where their members are. Thus, if you
have two eggs in your hand, then you also have the set containing
those eggs in your hand. Maddy's view has no problem accounting for
the massive infinities in mathematics, for on her view, corresponding
to every physical object, there is a huge infinity of sets that exist
in space and time, right where the given physical object exists. Given
this, it should be clear that while Maddy's view says that sets exist
in spacetime, it cannot be thought of as saying that sets are physical
objects. So Maddy's view is not a physicalist view in the sense that's
relevant here (as was pointed out above, it is better thought of as a
non-standard version of platonism). Thus, in the present context
(i.e., the context in which platonists are trying to undermine views
that take mathematical objects to be physical objects), platonists do
not need to argue against Maddy's view. Of course, in the end, if they
want to motivate the standard version of platonism, then they'll have
to give reasons for preferring their view to Maddy's non-standard
version of platonism, and this might prove hard to do. For some
arguments against Maddy's early view, see, e.g., Lavine (1992),
Dieterle and Shapiro (1993), Balaguer (1998a), Milne (1994), Riskin
(1994), Carson (1996), and the later Maddy (1997).)



If arguments like the ones we've been discussing here are cogent, then
sentences like '3 is prime' are not about physical or
mental objects, and therefore, psychologism and immanent realism are
not tenable views of mathematics. But that is not the end of the
singular term argument for the existence of abstract mathematical
objects, for we still need to consider nominalistic views of sentences
like '3 is prime'. In order for platonists to establish
their view, they need to refute these nominalistic views as well as
psychologistic and physicalistic views. And it should be noted that
this is the hard part. There is a good deal of agreement among
philosophers of mathematics that psychologism and immanent realism are
untenable; that is, most philosophers of mathematics are either
platonists or nominalists; but there is very little agreement as to
whether platonism or nominalism is correct.



How can nominalists proceed in developing an account of sentences like
'3 is prime'? One strategy is to reject premise (1) and
the standard criterion of ontological commitment discussed above. The
most obvious way to do this is to endorse the following view: (a)
simple mathematical sentences like '3 is prime' should be
read as being of the form '*a* is *F*' and as
being about abstract objects (e.g., '3' should be taken as
denoting the number 3, which could only be an abstract object, and
'3 is prime' should be taken as being a claim about that
object); and (b) abstract objects -- in particular, mathematical
objects like the number 3 -- don't exist (and the claim
here is that they don't have any sort of being whatsoever, so
this is not a Meinongian view); but (c) sentences like '3 is
prime' are still literally true. Thus, on this view, a claim
about an object *a* can be true even if that object
doesn't exist at all. Let's call this view
*thin-truth-ism*. Views of this general kind have been endorsed
by Azzouni (1994, 2004), Salmon (1998), and Bueno (2005). 



Thin-truth-ists endorse a similar view of existence claims. For
instance, on their view, the sentence 'There are infinitely many
prime numbers' is literally true, even though there are no such
things as numbers. This might look like a contradiction, but it's not,
because according to thin-truth-ism, existential expressions (or
quantifiers) like 'there is' are ambiguous.



Most philosophers find this view extremely hard to believe. Indeed, a
lot of philosophers would say that it is simply confused, or
incoherent. But, in fact, thin-truth-ism is not incoherent. A better
way to formulate the problem with the view is as follows: in giving up
on the standard criterion of ontological commitment, thin-truth-ists
seem to be using 'true' in a non-standard way. Most of us
would say that if there is no such thing as the number 3, and if
'3 is prime' is read at face value (i.e., as being about
the number 3), then it follows trivially that '3 is prime'
could not be true. Or more generally, most of us would say that if
there is no such thing as the object *a*, then sentences of the
form '*a* is *F*' cannot be literally true.
This, of course, is just to say that most of us accept the standard
criterion of ontological commitment discussed above, but the point
here is that this criterion seems to be built into the standard
meaning of words like 'true'. Indeed, this explains why
the standard criterion of ontological commitment is so widely
accepted.



Another worry one might raise about thin-truth-ism is that it only
differs from fictionalism in a merely verbal way. Let
'thin-true' express the kind of truth that thin-truth-ists
have in mind, and let 'thick-true' express the kind of
truth that everyone else in the debate has in mind (i.e., platonists,
fictionalists, paraphrase nominalists, and so on). Given this,
fictionalists and thin-truth-ists will both endorse all of the
following claims: (a) platonists are right that '3 is
prime' is a claim about the number 3; and (b) there is no such
thing as the number 3; so (c) '3 is prime' isn't
thick-true; but despite this, (d) '3 is prime' is
thin-true. Now, of course, thin-truth-ists and fictionalists will
disagree about whether '3 is prime' is *true*, but
this will collapse into a disagreement about whether thin-truth or
thick-truth is *real* truth, and this is just a disagreement
about what the word 'true' means in ordinary folk English,
and it's hard to see why an empirical question about how the
folk happen to use some word is relevant to the debate about the
existence of mathematical objects.



In any event, if we reject thin-truth-ism -- i.e., if we accept
premise (1) and the standard criterion of ontological commitment
-- then there are two general strategies that nominalists can
adopt in giving a view of mathematical sentences like '3 is
prime'. First, they can endorse a paraphrase view, and second,
they can endorse a fictionalist view. Those who endorse paraphrase
views claim that while sentences like '3 is prime' are
true, they should not be read as platonists read them, because we can
paraphrase these sentences with other sentences that do not commit us
to the existence of abstract objects. One view of this sort, known as
*if-thenism*, holds that '3 is prime' can be
paraphrased by 'If there were numbers, then 3 would be
prime' (for an early version of this sort of view, see the early
Hilbert (1899 and his letters to Frege in Frege (1980)); for later
versions, see Putnam (1967a and 1967b) and Hellman (1989)). A second
version of the paraphrase strategy, which we can call
*metamathematical formalism* (see Curry (1951)), is that
'3 is prime' can be paraphrased by "'3 is
prime' follows from the axioms of
 arithmetic".[6]
 A third version, developed by Chihara (1990), is that mathematical
sentences that seem to make claims about what mathematical objects
exist -- e.g., 'There is a prime number between 2 and
4' -- can be paraphrased into sentences about what it's
possible for us to do (in particular, what it's possible for us to
write down). Others to endorse paraphrase views include Hofweber
(2005), Rayo (2008), Moltmann (2013), and Yi (2002).



One problem with the various paraphrase views (not to put too fine a
point on it) is that none of the paraphrases seems very good. That is,
the paraphrases seem to misrepresent what we actually mean when we say
things like '3 is prime' (and by 'we', I mean
both mathematicians and ordinary folk). What we mean, it seems, is
that 3 is prime -- not that if there were numbers, then 3 would
be prime, or that the sentence '3 is prime' follows from
the axioms of arithmetic, or any such thing. And notice how the
situation here differs from cases where we do seem to have good
paraphrases. For instance, one might try to claim that if we endorse
the sentence



> 
> (A1) The average accountant has two children,
> 



then we are ontologically committed to the existence of the average
accountant; but it is plausible to suppose that, in fact, we are not
so committed, because (A1) can be paraphrased by the sentence



> 
> (A2) On average, accountants have two children.
> 



Moreover, it seems plausible to maintain that this is a *good*
paraphrase of (A1), because it seems clear that when people say things
like (A1), what they *really mean* are things like (A2). But in
the present case, this seems wrong: it does not seem plausible to
suppose that when people say '3 is prime', what they
really mean is 'If there were numbers, then 3 would be
prime'. Again, it seems that what we mean here is, very simply,
that 3 is prime. In short, when people say things like '3 is
prime', they do not usually have any intention to be saying
anything other than what these sentences *seem* to say; and
because of this, it seems that the platonist's face-value semantics
for mathematical discourse is correct.



Some paraphrase nominalists (e.g., Chihara 1990, 2004) maintain that
it doesn't matter what we *really mean*, that paraphrase
nominalists aren't committed to the thesis that their paraphrases
capture the real ordinary-language meanings of our mathematical
sentences. But this is false. If paraphrase nominalists admit that
platonists are right about the ordinary-language meanings of
mathematical sentences like '3 is prime', then their view
will collapse into a fictionalist view, according to which sentences
like '3 is prime' are not literally true. For since
paraphrase nominalists don't believe in the existence of mathematical
objects, if they admit that ordinary utterances of '3 is
prime' are best interpreted as being about mathematical objects,
or purporting to be about such objects, then they will have to admit
that such sentences are literally untrue, as fictionalists maintain.
Thus, if the paraphrase nominalist view is going to be a genuine
alternative to fictionalism, it has to involve the thesis that the
paraphrases that nominalists are offering capture the real meanings of
ordinary mathematical sentences.



On the other hand, paraphrase nominalists might try to argue that they
*are* right about the ordinary-language meanings of sentences
like '3 is prime', even though this is not obvious or
transparent to ordinary speakers. This stance, however, would be
extremely controversial and difficult to motivate.



One paraphrase view that has become somewhat popular recently holds
that sentences that seem to be about numbers are best read as being
about plurals. For instance, we might read '2 + 2 = 4' as
really saying something like this: *two and two are four* (or
*two objects and two (more) objects are four objects*, or some
such thing). Views of this general kind have been endorsed or defended
by, e.g., Yi (2002, forthcoming), Hofweber (2005), and Moltmann
(2013). This view fits better with ordinary usage than some of the
other paraphrase-nominalist views, and for sentences like '2 + 2
= 4' they can seem plausible. But when we switch to sentences
like '3 is prime' -- and, even worse, 'There
are infinitely many primes' -- they can start to seem
cumbersome and less plausible.



(For a good in-depth discussion and critique of some of the
paraphrase-nominalist views, see Burgess and Rosen (1997).)



Let's move on now to a discussion of fictionalism, which is the last
option for nominalists. Unlike paraphrase nominalists, fictionalists
admit that the platonist's face-value semantics for mathematical
discourse is correct; but because fictionalists don't believe in
abstract objects, they think that mathematical sentences like '3
is prime' are not true. In other words, fictionalists maintain
that (a) platonists are right that sentences like '3 is
prime' do purport to be about abstract objects, but (b) there
are no such things as abstract objects, and so (c) these sentences
-- and, indeed, our mathematical theories -- are untrue.
Thus, on this view, just as *Alice in Wonderland* is not true
because there are no such things as talking rabbits, hookah-smoking
caterpillars, and so on, so too our mathematical theories are not true
because there are no such things as numbers and sets and so
 on.[7].
 (Fictionalism has been developed by Field (1980, 1989, 1998),
Balaguer (1998a, 2009), Rosen (2001), and Leng (2005a, 2005b, 2010).
One might also interpret Melia (2000), Yablo (2002a, 2002b, 2005), and
Bueno (2009) as fictionalists. Finally, Hoffman (2004) endorses a kind
of fictionalism, but her view is very different from the one under
discussion here; for a bit more on her view, see the entry on
 fictionalism in the philosophy of mathematics.)



There are a few different ways that platonists might try to argue
against fictionalism. The most famous and widely discussed argument
against fictionalism is the *Quine-Putnam indispensability
argument* (see Quine (1948, 1951), Putnam (1971, 2012), and
Colyvan (2001)). This argument (or at any rate, one version of it)
proceeds as follows: it cannot be that mathematics is untrue, as
fictionalists suggest, because (a) mathematics is an indispensable
part of our physical theories (e.g., quantum mechanics, general
relativity theory, evolutionary theory, and so on) and so (b) if we
want to maintain that our physical theories are true (and surely we do
-- we don't want our disbelief in abstract objects to force us to
be anti-realists about natural science), then we have to maintain that
our mathematical theories are true.



Fictionalists have developed two different responses to the
Quine-Putnam argument. The first, developed by Field (1980) and
Balaguer (1998a), is based on the claim that mathematics is, in fact,
not indispensable to empirical science -- i.e., that our
empirical theories can be *nominalized*, or reformulated in a
way that avoids reference to abstract objects. The second response,
developed by Balaguer (1998a), Rosen (2001), Yablo (2005), Bueno
(2009), Leng (2010), and perhaps Melia (2000), is to grant the
indispensability of mathematics to empirical science and to simply
account for the relevant applications from a fictionalist point of
view. (A counterresponse to this second response has been given by
Colyvan (2002) and Baker (2005, 2009), who argue that fictionalists
can't account for the *explanatory* role that mathematics
plays in science; responses to the explanatory version of the
indispensability argument have been given by Melia (2002), Leng
(2005b), Bangu (2008), and Daly and Langford (2009).)



There is no consensus on whether the fictionalist responses to the
Quine-Putnam argument are successful. But even if they are, there are
other objections that platonists might raise against fictionalism. For
instance, one might argue that fictionalists cannot account for the
*objectivity* of mathematics (for responses to this, see Field
(1980, 1989, 1998) and Balaguer (2009)). Or, second, one might argue
that fictionalism is not a nominalistically acceptable view because
formulations of it invariably involve tacit reference to various kinds
of abstract objects, such as sentence types, or stories, or possible
worlds (for responses to this, see Field (1989), Balaguer (1998a), and
Rosen (2001)). For other objections to fictionalism, see, e.g.,
Malament (1982), Shapiro (1983a), Resnik (1985), Chihara (1990,
chapter 8, section 5), Horwich (1991), O'Leary-Hawthorne (1997),
Burgess and Rosen (1997), Katz (1998), Thomas (2000, 2002), Stanley
(2002), Bueno (2003), Szabo (2003), Hoffman (2004), and Burgess
(2004). For responses to these objections, see the various
fictionalist works cited above, as well as Daly (2008) and Liggins
(2010). And for a discussion of all these objections, as well as
fictionalist responses to them, see the entry on
 fictionalism in the philosophy of mathematics.)



In the end, it's not obvious whether platonists can successfully
refute fictionalism, and more generally, it's not obvious whether the
version of the singular term argument rehearsed in this subsection
provides a good reason for believing in abstract mathematical
objects.


### 4.2 Propositions



We turn now to a version of the singular term argument aimed at
establishing the existence of propositions. Once again, the most
important figure in the development of this argument is Frege (1892,
1919). Other relevant figures (who wouldn't all endorse an argument
like the one sketched below) include Russell (1905, 1910-1911),
Church (1950, 1954), Quine (1956), Kaplan (1968-69, 1989),
Kripke (1972, 1979), Schiffer (1977, 1987, 1994), Perry (1979), Evans
(1981), Peacocke (1981), Barwise and Perry (1983), Bealer (1982,
1993), Zalta (1983, 1988), Katz (1986), Salmon (1986), Soames (1987,
2014), Forbes (1987), Crimmins and Perry (1989), Richard (1990),
Crimmins (1998), Recanati (1993, 2000), King (1995, 2014), Braun
(1998), and Saul (1999). 



The relevant sentences here are belief ascriptions, i.e., sentences
like 'Clinton believes that snow is white' and
'Emily believes that Santa Claus is fat'. The first point
to note about these sentences is that they involve
'that'-clauses, where a 'that'-clause is
simply the word 'that' added to the front of a complete
sentence -- e.g., 'that snow is white'. The second
point to be made is that 'that'-clauses, in English, are
singular terms. A common way to illustrate this point -- see,
e.g., Bealer (1982 and 1993) and Schiffer (1994) -- is to appeal
to arguments like the following:



> 
> 
> 
> |  |  |
> | --- | --- |
> | I. | Clinton believes that snow is white.
> 
> Therefore, Clinton believes something (namely, that snow is
> white). |
> 
> 
> 



This argument seems to be valid, and platonists claim that the best
and only tenable explanation of this fact involves a commitment to the
idea that the 'that'-clause in this argument, i.e.,
'that snow is white', is a singular term.



But if 'that'-clauses are singular terms, what sorts of
objects do they refer to? Well, it might seem that they refer to
facts, or states of affairs. For instance, it might seem that
'that snow is white' refers to the fact that snow is
white. This, however, is a mistake (at least in connection with the
'that'-clauses that appear in belief reports). For since
beliefs can be false, it follows that the 'that'-clauses
in our belief reports refer to things that can be false. E.g., if
Sammy is seven years old, then the sentence 'Sammy believes that
snow is powdered sugar' could be true; but if this sentence is
true, then (by our criterion of ontological commitment) its
'that'-clause refers to a real object; but then it cannot
refer to a fact, because (obviously) there is no such thing as the
fact that snow is powdered sugar.



These considerations suggest that the referents of the
'that'-clauses that appear in belief ascriptions are
things that can be true or false. But if this is right, then it seems
that the objects of belief must be either sentences or propositions.
The standard platonist view is that they are propositions. Before we
consider their arguments for this claim, we need to say a few words
about the different kinds of sentential views that one might
endorse.



To begin with, we need to distinguish between sentence *types*
and sentence *tokens*. To appreciate the difference, consider
the following indented sentences:



> 
> Cats are cute.
>   
> 
> Cats are cute.
> 



We have here two different tokens of a single sentence type. Thus, a
token is an actual physical thing, located at a specific place in
spacetime; it is a pile of ink on a page (structured in an appropriate
way), or a sound wave, or a collection of pixels on a computer screen,
or something of this sort. A type, on the other hand, can be tokened
numerous times but is not identical with any single token. Thus, a
sentence type is an abstract object. And so if we are looking for an
anti-platonist view of what 'that'-clauses refer to, or
what belief reports are about, we cannot say that they're about
sentence types; we have to say they're about sentence tokens.



A second distinction that needs to be drawn here is between sentence
tokens that are *external*, or public, and sentence tokens that
are *internal*, or private. Examples of external sentence
tokens were given in the last paragraph -- piles of ink, sound
waves, and so on. An internal sentence token, on the other hand,
exists inside a particular person's head. There is a wide-spread view
-- due mainly to Jerry Fodor (1975 and 1987) but adopted by many
others, e.g., Stich (1983) -- that we are able to perform
cognitive tasks (e.g., think, remember information, and have beliefs)
only because we are capable of storing information in our heads in a
neural language (often called *mentalese*, or the *language
of thought*). In connection with beliefs, the idea here is that to
believe that, say, snow is white, is to have a neural sentence stored
in your head (in a belief way, as opposed to a desire way, or some
other way) that means in mentalese that snow is white.



This gives us two different anti-platonist alternatives to the view
that belief reports involve references to propositions. First, there
is the conceptualistic (or mentalistic) view that belief reports
involve references to sentences in our heads, or mentalese sentence
tokens. And, second, there is the physicalistic view that belief
reports involve references to external sentence tokens, i.e., to piles
of ink, and so on (versions of this view have been endorsed by Carnap
(1947), Davidson (1967), and Leeds (1979)).



There are a number of arguments that suggest that ordinary belief
reports cannot be taken to be about (internal or external) sentences
and that we have to take them to be about propositions. We will
rehearse one such argument here, an argument that goes back at least
to Church (1950). Suppose that Boris and Jerry both live in cold
climates and are very familiar with snow. Thus, they both believe that
snow is white. But Boris lives in Russia and speaks only Russian,
whereas Jerry lives in Minnesota and speaks only English. Now,
consider the following argument:



> 
> 
> 
> |  |  |
> | --- | --- |
> | II. | Boris believes that snow is white.
> 
> Jerry believes that snow is white.
> 
> Therefore, there's at least one thing that Boris and Jerry both
> believe, namely, that snow is white. |
> 
> 
> 



This argument seems clearly valid; but this seems to rule out the idea
that the belief reports here are about sentence tokens. For (a) in
order to account for the validity of the argument, we have to take the
two 'that'-clauses to refer to the same thing, and (b)
there is no sentence token that they could both refer to. First of
all, they couldn't refer to any external sentence token (or, for that
matter, any sentence type associated with any natural language),
because (i) if the first 'that'-clause refers to such a
sentence, it would presumably be a Russian sentence, since Boris
speaks only
 Russian;[8]
 and (ii) if the second 'that'-clause refers to such a
sentence, it would presumably be an English sentence, since Jerry
speaks only English; and so (iii) the two 'that'-clauses
cannot both refer to the same external sentence token (or
natural-language sentence type). And second, they cannot refer to any
mentalese sentence token, because (i) if the first
'that'-clause refers to such a sentence, it would
presumably be in Boris's head; and (ii) if the second
'that'-clause refers to such a sentence, it would
presumably be in Jerry's head; and so (iii) the two
'that'-clauses cannot both refer to the same mentalese
sentence token. Therefore, it seems to follow that the
'that'-clauses in the above argument do not refer to
sentence tokens of any kind. And since these are ordinary belief
ascriptions, it follows that, in general, the
'that'-clauses that appear in ordinary belief ascriptions
do not refer to sentence tokens.



Now as it's formulated here, this argument doesn't rule out the view
that 'that'-clauses refer to mentalese sentence types, but
the argument can be extended to rule out that view as well (e.g., one
might do this by talking not of an American and a Russian but of two
creatures with different internal languages of thought). I won't run
through the details of this here, since, as we've seen,
anti-platonists can't claim that 'that'-clauses refer to
types anyway, because types are abstract objects. But if we assume
that that version of the argument is cogent as well, then it follows
that 'that'-clauses don't refer to sentences of any kind
at all and, hence, that they must refer to
 propositions.[9]



Now, notice that the issue so far has been purely semantic. What the
above argument suggests is that regardless of whether there are any
such things as propositions, our 'that'-clauses are best
interpreted as purporting to refer to such objects. Platonists then
claim that if this is correct, then there must be such things as
propositions, because, clearly, many of our belief ascriptions are
true. For instance, 'Clinton believes that snow is white'
is true; thus, if the above analysis of 'that'-clauses is
correct, and if our criterion of ontological commitment is correct, it
follows that there is such a thing as the proposition that snow is
white.



This version of the singular term argument might seem even more
powerful than the mathematical-object version of the singular term
argument sketched in
 section 4.1,
 because in this case, it doesn't seem that there is as much room for
paraphrase nominalism. We saw in
 section 4.1
 that there are a number of programs for paraphrasing the statements
of mathematics, but there are no obvious strategies for paraphrasing
ordinary belief ascriptions. One might think this could be done by
taking sentences of the form '*S* believes that
*p*' to mean 'If there existed propositions, then
*S* would believe that *p*'; but this sort of view
is even less plausible here than it is in the mathematical case. It is
just wildly implausible to suppose that when common folk say things
like 'Clinton believes that his presidency was
successful', they mean to be making hypothetical claims about
what the person in question *would* believe in some alternate
situation.



However, while paraphrase nominalism seems hopeless in the case of
propositions, Balaguer (1998b) has argued that fictionalistic
nominalism carries over very well to the case of propositions. More
specifically, fictionalists can say that 'Clinton believes that
snow is white' is strictly speaking not true (because its
'that'-clause is supposed to refer to a proposition, and
there are no such things as propositions) but that we can still use it
to say something essentially accurate about Clinton's belief state,
because there are facts about Clinton that make it the case that if
there existed propositions, then it would be true that he believes
that snow is white.


### 4.3 Properties and Relations



One way to argue for a platonistic view of properties and relations is
first to use the argument of
 section 4.2
 to argue for a platonistic view of propositions, and then to claim
that this argument already contains an argument for properties and
relations, because properties and relations are components of
propositions. If we adopt a Russellian view of propositions, this is
straightforward, because it is built into the Russellian view that
propositions are composed of objects, properties, and relations (see
 section 1).
 If we adopt a Fregean view of propositions, however, the situation is
different. On the Fregean view, propositions are composed of senses,
which we can think of as meanings, or concepts. Now, Frege himself did
not use the word 'concept' to talk about these things
-- he used the German 'sinn', which is usually
translated as 'sense' -- but we can use the word
'concept' here. On this way of talking, we can say that on
a Fregean view, if we have good reason to countenance the existence
of, e.g., the proposition that roses are red, then we also have good
reason to countenance the existence of the concept *red*. Now,
some Fregeans might want to say that the property of redness just
*is* the concept of redness, and if so, then they could
maintain with Russellians that if there exist propositions, then there
also exist properties and relations. But most Fregeans would want to
deny that properties are concepts, so in order to motivate a
platonistic view of properties and relations, they would need an
entirely different argument. And since we have already found that the
One Over Many argument for properties and relations is not cogent,
this other argument would presumably be a version of the singular term
argument, one that was aimed specifically at establishing the
existence of properties and relations.



The most obvious way to formulate an independent property-and-relation
version of the singular term argument would be to appeal to sentences
like



> 
> (P1) Mars possesses the property of redness
> 



and



> 
> (R1) San Francisco stands in the north-of relation to Los Angeles.
> 



In order to make out a version of the singular term argument here,
platonists would need to begin by arguing that these sentences commit
us to the existence of the property of redness and the north-of
relation, respectively, because (a) they have singular terms that
denote those things, and (b) they are true. In order to motivate claim
(a), platonists would have to refute the paraphrase-nominalist claim
that sentences like (P1) and (R1) are equivalent to sentences like
'Mars is red' and 'San Francisco is north of Los
Angeles' and that none of these sentences entails the existence
of properties or relations. And in order to motivate claim (b),
platonists would have to refute the fictionalist view that sentences
like (P1) and (R1) are untrue because they do entail the existence of
properties and relations and because there are no such things as
properties or relations (of course, fictionalists maintain that while
(P1) and (R1) aren't literally true, they can still be used
colloquially to say things that about the world that are essentially
accurate -- see
 section 3).[10]



If platonists managed to establish the existence of properties and
relations in this way, they would still need to argue that such things
could only be abstract objects. That is, they would have to argue that
properties and relations are not ideas (as conceptualists claim) or
universals inhering in physical things (as immanent realists
claim).



The arguments listed above against conceptualistic or psychologistic
views of numbers also tell against conceptualism about properties and
relations. For instance, as Russell (1912, chapter IX) points out,
property claims and relational claims seem to be objective; e.g., the
fact that Mount Everest is taller than Mont Blanc is a fact that holds
independently of us; but conceptualism about universals entails that
if we all died, it would no longer be true that Mount Everest bears
the taller than relation to Mont Blanc, because that relation would no
longer exist. And, second, conceptualism seems simply to get the
semantics of our property discourse wrong, for it seems to confuse
properties with our ideas of them. The English sentence 'Red is
a color' does not seem to be about anybody's idea of redness; it
seems to be about redness, the actual color, which, it seems, is
something objective.



There are also some very famous arguments against the immanent realist
view of properties and relations. First, it is not clear that it is
coherent to say that there is such a thing as redness and that this
*one* thing exists in many different objects at the same time.
Second, it is not clear what it is for an object to possess a property
on the immanent realist view. Most immanent realists would not say
that property possession is a full-blown *relation*, for this
would just be another universal, and it is commonly thought that if
immanent realists adopted this view, it would lead to an unacceptable
infinite regress. (If we're told that an object *a* possesses
*F*ness iff *a* stands in the possession relation to
*F*ness, then one might ask, "What is it for an object
and a property to stand in the possession relation to one
another?", and so on. For more on this, see the entry on
 properties.)
 In light of this, many immanent realists maintain that when an object
*a* possesses a property *F*ness, *a* and
*F*ness are "linked together" in some
non-relational way, e.g., a way that is more intimate, or primitive,
than ordinary relational connections. But it is not clear what this
really amounts to. (Immanent realists might respond that platonists
also have a problem here -- i.e., that platonists also have to
provide an account of the relation, or "connection",
between objects and properties. But some platonists might argue that
the problem isn't as bad for them because platonistic properties are
causally inert, and so they are not *responsible* in any way
for objects having the natures they have, and they do not play any
important role in our explanations of *why* objects have the
natures they have. For instance, if *a* is *F*,
*F*ness is not responsible in any way for *a* having the
nature that it has. Thus, platonists might claim that *a* is
simply an *example* of *F*ness and that there is no more
to their relation than
 that.[11]
 Immanent realists, however, think that ordinary physical objects are
the way they are *because* they possess the properties they do.
Thus, they seem committed to the thesis that there is some sort of
physically substantial connection, or link, between objects and their
properties, and it is not at all clear what this could be. There has
been a lot of philosophy dedicated to this problem, but there is no
consensus on how (or whether) it can be solved.)



It is worth noting that platonists who argue for properties and
relations in conjunction with propositions -- i.e., by first
arguing for propositions and then claiming that properties and
relations are components of propositions -- will have an easier
time arguing that properties and relations couldn't exist in our minds
(as conceptualists say) or in things (as immanent realists say). In
connection with conceptualism, platonists of this sort could claim
that the argument given in
 section 4.2
 for thinking that 'that'-clauses don't refer to mentalese
sentence tokens suggests that propositions (which are the referents of
'that'-clauses) could not be made up of properties that
exist in our heads. And in connection with immanent realism,
platonists of this sort could argue that propositions couldn't be
composed of immanent-realist properties, because people can believe
propositions that are composed of properties that are not instantiated
in the physical world. For instance, it seems that sentences like
'Johnny believes that there is a four-hundred-story building in
Sally's backyard' can be true, and so according to the above
platonist argument for propositions, there must be such a thing as the
proposition that there is a four-hundred-story building in Sally's
backyard. But if propositions have properties as components, then this
proposition has as a component the property of being a
four-hundred-story building. But if properties exist only in physical
things, as immanent realists suggest, then there is no such thing as
the property of being a four-hundred-story building, since presumably,
nothing in the universe has this property. Thus, the conclusion here
is that if propositions have properties as components, then the
properties in question have to be transcendent, platonist properties,
not immanent
 properties.[12]


### 4.4 Sentence Types



Linguistics is a branch of science that tells us things about
sentences. For instance, it says things like



> 
> (A) 'The cat is on the mat' is a well-formed sentence of
> English,
> 



and



> 
> (B) 'Visiting relatives can be boring' is structurally
> ambiguous.
> 



The quoted sentences that appear in (A) and (B) are singular terms;
e.g., "'The cat is on the mat'" refers to the
sentence 'The cat is on the mat', and (A) says of this
sentence that it has a certain property, namely, that of being a
well-formed English sentence. Thus, if sentences like (A) are true
-- and it certainly seems that they are -- then they commit
us to believing in the existence of sentences. Now, one might hold a
physicalistic view here according to which linguistics is about actual
(external) sentence tokens, e.g., piles of ink and verbal sound waves.
(This view was popular in the early part of the 20th
century -- see, e.g., Bloomfield (1933), Harris (1954), and Quine
(1953).) Or alternatively, one might hold a conceptualistic view,
maintaining that linguistics is essentially a branch of psychology;
the main proponent of this view is Noam Chomsky (1965, chapter 1), who
thinks of a grammar for a natural language as being about an ideal
speaker-hearer's knowledge of the given language, but see also Sapir
(1921), Stich (1972), and Fodor (1981). But there are reasons for
thinking that neither the physicalist nor the conceptualist approach
is tenable and that the only plausible way to interpret linguistic
theory is as being about sentence types, which of course, are abstract
objects (proponents of the platonistic view include Katz (1981),
Soames (1985), and Langendoen and Postal (1985)). Katz constructs
arguments here that are very similar to the ones we considered above,
in connection with mathematical objects
 (section 4.1).
 One argument here is that linguistic theory seems to have
consequences that are (a) true and (b) about sentences that have never
been tokened (internally or externally), e.g., sentences like
'Green Elvises slithered unwittingly toward Arizona's favorite
toaster'. (Of course, now that I've written this sentence down,
it has been tokened, but it seems likely that before I wrote it down,
it had never been tokened.) Standard linguistic theory entails that
many sentences that have never been tokened (internally or externally)
are well-formed English sentences. Thus, if we want to claim that our
linguistic theories are true, then we have to accept these
consequences, or theorems, of linguistic theory. But these theorems
are clearly not true of any sentence tokens (because the sentences in
question have never been tokened) and so, it is argued, they must be
true of sentence types.


### 4.5 Possible Worlds



It is a very widely held view among contemporary philosophers that we
need to appeal to entities known as *possible worlds* in order
to account for various phenomena. There are dozens of phenomena that
philosophers have thought should be explained in terms of possible
worlds, but to name just one, it is often argued that semantic theory
is best carried out in terms of possible worlds. Consider, for
example, the attempt to state the truth conditions of sentences of the
form 'It is necessary that *S*' and 'It is
possible that *S*' (where *S* is any sentence). It
is widely believed that the best theory here is that a sentence of the
form 'It is necessary that *S*' is true if and only
if *S* is true in all possible worlds, and a sentence of the
form 'It is possible that *S*' is true if and only
if *S* is true in at least one possible world. Now, if we add
to this theory the premise that at least one sentence of the form
'It is possible that *S*' is true -- and this
seems undeniable -- then we are led to the result that possible
worlds exist.



Now, as was the case with numbers, properties, and sentences, not
everyone who endorses possible worlds thinks that they are abstract
objects; indeed, one leading proponent of the use of possible worlds
in philosophy and semantics -- namely, David Lewis (1986) --
maintains that possible worlds are of the same kind as the actual
world, and so he takes them to be concrete objects. However, most
philosophers who endorse possible worlds take them to be abstract
objects (see, e.g., Plantinga (1974, 1976), Adams (1974), Chisholm
(1976), and Pollock (1984)). It is important to note, however, that
possible worlds are very often not taken to constitute a new kind of
abstract object. For instance, it is very popular to maintain that a
possible world is just a set of propositions. (To see how a set of
propositions could serve as a possible world, notice that if you
believed in full-blown possible worlds -- worlds that are just
like the actual world in kind -- then you would say that
corresponding to each of these worlds, there is a set of propositions
that completely and accurately describes the given world, or is true
of that world. Many philosophers who don't believe in full-blown
possible worlds maintain that these sets of propositions are good
enough -- i.e., that we can take them to *be* possible
worlds.) Or alternatively, one might think of a possible world as a
*state of affairs*, or as a *way things could be*. In so
doing, one might think of these as constituting a new kind of abstract
object, or one might think of them as properties -- giant,
complex properties that the entire universe may or may not possess.
For instance, one might say that the actual universe possesses the
property of being such that snow is white and grass is green and San
Francisco is north of Los Angeles, and so on.



In any event, if possible worlds are indeed abstract objects, and if
the above argument for the existence of possible worlds is cogent,
then this would give us another argument for platonism.


### 4.6 Logical Objects



Frege (1884, 1893-1903) appealed to sentences like the
following:



> 
> (D) The number of *F*s is identical to the number of
> *G*s if and only if there is a one-to-one correspondence
> between the *F*s and the *G*s.
> 
> 
> (E) The direction of line *a* is identical to the direction of
> line *b* if and only if *a* is parallel to
> *b*.
> 
> 
> 
> (F) The shape of figure *a* is identical to the shape of figure
> *b* if and only if *a* is geometrically *similar*
> to *b*.
> 
> 
> 



On Frege's view, principles like these are true, and so they commit us
to the existence of numbers, lines, and shapes. Now, of course, we
have already gone through a platonistic argument -- indeed, a
Fregean argument -- for the existence of numbers. Moreover, the
standard platonist view is that the argument for the existence of
mathematical objects is entirely general, covering all branches of
mathematics, including geometry, so that on this view, we already have
reason to believe in lines and shapes, as well as numbers. But it is
worth noting that in contrast to most contemporary platonists, Frege
thought of numbers, lines, and shapes as logical objects, because on
his view, these things can be identified with *extensions of
concepts*. What is the extension of a concept? Well, simplifying a
bit, it is just the set of things falling under the given concept.
Thus, for instance, the extension of the concept *white* is
just the set of white
 things.[13]
 And so the idea here is that since logic is centrally concerned with
predicates and their corresponding concepts, and since extensions are
tied to concepts, we can think of extensions as logical objects. Thus,
since Frege thinks that numbers, lines, and shapes can be identified
with extensions, on his view, we can think of these things as logical
objects.



Frege's definitions of numbers, lines, and shapes in terms of
extensions can be formulated as follows: (i) the number of *F*s
is the extension of the concept *equinumerous with F* (that is,
it is the set of all concepts that have exactly as many objects
falling under them as does *F*); and (ii) the direction of line
*a* is the extension of the concept *parallel to a*; and
(iii) the shape of figure *a* is the extension of the concept
*geometrically similar to a*. A similar approach can be used to
define other kinds of logical objects. For instance, the truth value
of the proposition *p* can be identified with the extension of
the concept *equivalent to p* (i.e., the concept *true if
and only if *p* is true*).



It should be noted that contemporary neo-Fregeans reject the
identification of directions and shapes and so on with extensions of
concepts. They hold instead that directions and shapes are *sui
generis* abstract objects.



For contemporary work on this issue, see, e.g., Wright (1983), Boolos
(1986-87), and Anderson and Zalta (2004).


### 4.7 Fictional Objects



Finally, a number of philosophers (see, most notably, van Inwagen
(1977), Wolterstorff (1980), and Zalta (1983, 1988)) think that
fictional objects, or fictional characters, are best thought of as
abstract objects. (Salmon (1998) and Thomasson (1999) also take
fictional objects to be abstract, but their views are a bit different;
they maintain that abstract fictional objects are created by humans.)
To see why one might be drawn to this view, consider the following
sentence:



> 
> (G) Sherlock Holmes is a detective.
> 



Now, if this sentence actually appeared in one of the Holmes stories
by Arthur Conan Doyle, then that token of it would not be true --
it would be a bit of fiction. But if you were telling a child about
these stories, and the child asked, "What does Holmes do for a
living?", and you answered by uttering (G), then it seems
plausible to suppose that what you have said is true. But if it is
true, then it seems that its singular term, 'Sherlock
Holmes', must refer to something. What it refers to, according
to the view in question, is an abstract object, in particular, a
fictional character. In short, present-day utterances of (G) are true
statements about a fictional character; but if Doyle had put (G) into
one his stories, it would not have been true, and its singular term
would not have referred to anything.



There is a worry about this view that can be put in the following way:
if there is such a thing as Sherlock Holmes, then it has arms and
legs; but if Sherlock Holmes is an abstract object, as this view
supposes, then it does not have arms and legs (because abstract
objects are non-physical); therefore, it cannot be the case that
Sherlock Holmes exists and is an abstract object, for this leads to
contradiction. Various solutions to this problem have been proposed.
For instance, Zalta argues that in addition to *exemplifying*
certain properties, abstract objects also *encode* properties.
The fictional character Sherlock Holmes encodes the properties of
being a detective, being male, being English, having arms and legs,
and so on. But it does not exemplify any of these properties. It
exemplifies the properties of being abstract, being a fictional
character, having been thought of first by Arthur Conan Doyle, and so
on. Zalta maintains that in English, the copula 'is'
-- as in '*a* is *F*' -- is
ambiguous; it can be read as ascribing either property exemplification
or property encoding. When we say 'Sherlock Holmes is a
detective', we are saying that Holmes *encodes* the
property of being a detective; and when we say 'Sherlock Holmes
is a fictional character', we are saying that Holmes
*exemplifies* the property of being a fictional character. (It
should be noted that Zalta employs the device of encoding with respect
to all abstract objects -- mathematical objects, logical objects,
and so on -- not just fictional objects. Also, Zalta points out
that his theory of encoding is based on a similar theory developed by
Ernst Mally (1912).)



Those who endorse a platonistic view of fictional objects maintain
that there is no good paraphrase of sentences like (G), but one might
question this. For instance, one might maintain that (G) can be
paraphrased by a sentence like this:



> 
> 'Sherlock Holmes is a detective' is
> true-in-the-Holmes-stories.
> 



If we read (G) in this way, then it is not about Sherlock Holmes at
all; rather, it is about the Sherlock Holmes stories. Thus, in order
to believe (G), so interpreted, one would have to believe in the
existence of these stories. Now, one might try to take an
anti-platonistic view of the nature of stories, but there are problems
with such views, and so we might end up with a platonistic view here
anyway -- a view that takes sentences like (G) to be about
stories and stories to be abstract objects of some sort, e.g., ordered
sets of
 propositions.[14]
 Which of these platonistic views is superior can be settled by
determining which (if either) captures the correct interpretation of
sentences like (G) -- i.e., by determining whether ordinary
people who utter sentences like (G) are best interpreted as talking
about stories or fictional characters.



It should be noted that some people who take fictional characters to
be abstract objects (e.g., Thomasson 1999) would actually agree with
the idea that (G) should be read in the above way -- i.e., as a
claim about the Sherlock Holmes stories and not about Sherlock Holmes
himself. Thomasson's main argument for believing in fictional
characters is based not on sentences like (G) but rather on sentences
like the following:



> 
> (H) Some 19th century heroines are better developed than any 18th
> century heroines.
> 



It's hard to see how to paraphrase this as being about a story,
or even a bunch of stories. But, of course, one could still endorse a
fictionalist (i.e., an error-theoretic) view of sentences like (H). In
other words, one could admit that (H) is a claim about fictional
characters and then one could claim that since there are no such
things as fictional characters, (H) is simply not true, although of
course it might be *true-in-the-story-of-fictional-characters*,
where this just means that it would have been true if there had been a
realm of fictional characters of the sort that platonists believe in.
(Brock (2002) endorses a fictionalist view of fictional characters
that's similar in spirit to the view alluded to here.) 


## 5. The Epistemological Argument Against Platonism



Over the years, anti-platonist philosophers have presented a number of
arguments against platonism. One of these arguments stands out as the
strongest, namely, the epistemological argument. This argument goes
all the way back to Plato, but it has received renewed interest since
1973, when Paul Benacerraf presented a version of the argument. Most
of the work on this problem has taken place in the philosophy of
mathematics, in connection with the platonistic view of mathematical
objects like numbers. We will therefore discuss the argument in this
context, but all of the issues and arguments can be reproduced in
connection with other kinds of abstract objects. The argument can be
put in the following way:


1. Human beings exist entirely within spacetime.
2. If there exist any abstract mathematical objects, then they do not
exist in spacetime. Therefore, it seems very plausible that:
3. If there exist any abstract mathematical objects, then human
beings could not attain knowledge of them. Therefore,
4. If mathematical platonism is correct, then human beings could not
attain mathematical knowledge.
5. Human beings have mathematical knowledge. Therefore,
6. Mathematical platonism is not correct.



The argument for (3) is everything here. If it can be established,
then so can (6), because (3) trivially entails (4), (5) is beyond
doubt, and (4) and (5) trivially entail (6). Now, (1) and (2) do not
strictly entail (3), and so there is room for platonists to maneuver
here -- and as we'll see, this is precisely how most platonists
have responded. However, it is important to notice that (1) and (2)
provide a strong *prima facie* motivation for (3), because they
seem to imply that mathematical objects (if there are such things) are
totally inaccessible to us, i.e., that information cannot pass from
mathematical objects to human beings. But this gives rise to a
*prima facie* worry (which may or may not be answerable) about
whether human beings could acquire knowledge of mathematical objects.
Thus, we should think of this argument not as *refuting*
platonism but as issuing a challenge to platonists. The challenge is
simply to explain how human beings could acquire knowledge of abstract
mathematical objects.



There are three ways for platonists to respond. First, they can argue
that (1) is false and that the human mind is capable of somehow
forging contact with abstract mathematical objects and thereby
acquiring information about such objects. This strategy has been
pursued by Plato in *The Meno* and *The Phaedo*, and by
Godel (1964). Plato's idea is that our immaterial souls acquired
knowledge of abstract objects before we were born and that
mathematical learning is really just a process of coming to remember
what we knew before we were born. On Godel's version of the view,
we acquire knowledge of abstract objects in much the same way that we
acquire knowledge of concrete physical objects; more specifically,
just as we acquire information about physical objects via the faculty
of sense perception, so we acquire information about abstract objects
by means of a faculty of *mathematical intuition*. Now, other
philosophers have endorsed the idea that we possess a faculty of
mathematical intuition, but Godel's version of this view --
and he seems to be alone in this -- involves the idea that the
mind is non-physical in some sense and that we are capable of forging
contact with, and acquiring information from, non-physical
mathematical
 objects.[15]
 This view has been almost universally rejected. One problem is that
denying (1) doesn't seem to help. The idea of an immaterial mind
receiving information from an abstract object seems just as mysterious
and confused as the idea of a physical brain receiving information
from an abstract object.



The second strategy that platonists can pursue in responding to the
epistemological argument is to argue that (2) is false and that human
beings can acquire information about mathematical objects via normal
perceptual means. The early Maddy (1990) pursued this idea in
connection with set theory, claiming that sets of physical objects can
be taken to exist in spacetime and, hence, that we can perceive them.
For instance, on her view, if there are two books on a table, then the
set containing these books exists on the table, in the same place that
the books exist, and we can see the set and acquire information about
it in this way. This view has been subjected to much criticism,
including arguments from the later Maddy (1997). Others to attack the
view include Lavine (1992), Dieterle and Shapiro (1993), Balaguer
(1998a), Milne (1994), Riskin (1994), and Carson (1996).



It may be objected that according to the definitions we've been using,
views like Maddy's are not versions of platonism at all, because they
do not take mathematical objects to exist outside of spacetime.
Nonetheless, there is some rationale for thinking of Maddy's view as a
sort of non-traditional platonism. For since Maddy's view entails that
there is an infinity of sets associated with every ordinary physical
object, all sharing the same spatiotemporal location and the same
physical matter, she has to allow that these sets differ from one
another in some sort of non-physical way and, hence, that there is
something about these sets that is non-physical, or perhaps abstract,
in some sense of these terms. Now, of course, the question of whether
Maddy's view counts as a version of *platonism* is purely
terminological; but whatever we say about this, the view is still
worth considering in the present context, because it is widely thought
of as one of the available responses to the epistemological argument
against platonism, and indeed, that is the spirit in which Maddy
originally presented the view.



The third and final strategy that platonists can pursue is to accept
(1) and (2) and explain why (3) is nonetheless false. This strategy is
different from the first two in that it doesn't involve the
postulation of an information-transferring contact between human
beings and abstract objects. The idea here is to grant that human
beings do not have such contact with abstract objects and to explain
how they can nonetheless acquire knowledge of such objects. This has
been the most popular strategy among contemporary platonists. Its
advocates include Quine (1951, section 6), Steiner (1975, chapter 4),
Parsons (1980, 1994), Katz (1981, 1998), Resnik (1982, 1997), Wright
(1983), Lewis (1986, section 2.4), Hale (1987), Shapiro (1989, 1997),
Burgess (1990), Balaguer (1995, 1998a), Linsky and Zalta (1995),
Burgess and Rosen (1997), and Linnebo (2006). There are several
different versions of this view; we will look very briefly at the most
prominent of them.



One version of the third strategy, implicit in the writings of Quine
(1951, section 6) and developed by Steiner (1975, chapter four,
especially section IV) and Resnik (1997, chapter 7), is to argue that
we have good reason to believe that our mathematical theories are
true, even though we don't have any contact with mathematical objects,
because (a) these theories are embedded in our empirical theories, and
(b) these empirical theories (including their mathematical parts) have
been confirmed by empirical evidence, and so (c) we have empirical
evidence for believing that our mathematical theories are true and,
hence, that abstract mathematical objects exist. Notice that this view
involves the controversial thesis that *confirmation is
holistic*, i.e., that entire theories are confirmed by pieces of
evidence that seem to confirm only parts of theories. One might doubt
that confirmation is holistic in this way (see, e.g., Sober (1993),
Maddy (1992), and Balaguer (1998a)). Moreover, even if one grants that
confirmation is holistic, one might worry that this view leaves
unexplained the fact that mathematicians are capable of acquiring
knowledge of their theories before these theories are applied in
empirical science.



A second version of the third strategy, developed by Katz (1981, 1998)
and Lewis (1986, section 2.4), is to argue that we can know that our
mathematical theories are true, without any sort of
information-transferring contact with mathematical objects, because
these theories are *necessarily* true. The reason we need
information-transferring contact with ordinary physical objects in
order to know what they're like is that these objects could have been
different. For instance, we have to look at fire engines in order to
know that they're red, because they could have been blue. But we don't
need any contact with the number 4 in order to know that it is the sum
of 3 and 1, because it is necessarily the sum of 3 and 1. (For
criticisms of this view, see Field (1989, pp. 233-38) and
Balaguer (1998a, chapter 2, section 6.4).)



A third version of the third strategy has been developed by Resnik
(1997) and Shapiro (1997). Both of these philosophers endorse
(platonistic) *structuralism*, a view that holds that our
mathematical theories provide true descriptions of mathematical
structures, which, according to this view, are abstract. Moreover,
Resnik and Shapiro both claim that human beings can acquire knowledge
of mathematical structures (without coming into any sort of
information-transferring contact with such things) by simply
constructing mathematical axiom systems; for, they argue, axiom
systems provide *implicit definitions* of structures. One
problem with this view, however, is that it does not explain how we
could know which of the various axiom systems that we might formulate
actually pick out structures that exist in the mathematical realm.



A fourth and final version of the third strategy, developed
independently (and somewhat differently) by Balaguer (1995, 1998a) and
Linsky & Zalta (1995), is based on the adoption of a particular
version of platonism called *plenitudinous platonism* (Balaguer
also calls it *full-blooded platonism*, or FBP, and Linsky and
Zalta call it *principled platonism*). Balaguer defines
plenitudinous platonism (somewhat roughly) as the view that there
exist mathematical objects of all possible kinds, or the view that all
the mathematical objects that possibly could exist actually do exist.
But, in general, Balaguer would define a different plenitude principle
for every different kind of abstract object. Linsky & Zalta
develop plenitudinous platonism by proposing a distinctive plenitude
principle for each of three basic domains of abstracta: abstract
individuals, relations (properties and propositions), and contingently
nonconcrete individuals (1995, 554). For example, on their view, the
plenitude principle for abstract individuals asserts (again, somewhat
roughly) that every possible description of an object characterizes an
abstract object that encodes -- and, thus, in an important sense,
*has* -- the properties expressed in the description. 



Balaguer and Linsky & Zalta then argue that if platonists endorse
plenitudinous platonism, they can solve the epistemological problem
with platonism without positing any sort of information-transferring
contact between human beings and abstract objects. Balaguer's version
of the argument proceeds as follows. Since plenitudinous platonism, or
FBP, says that there are mathematical objects of all possible kinds,
it follows that if FBP is true, then every purely mathematical theory
that could possibly be true (i.e., that's internally consistent)
accurately describes some collection of actually existing mathematical
objects. Thus, it follows from FBP that in order to attain knowledge
of abstract mathematical objects, all we have to do is come up with an
internally consistent purely mathematical theory (and know that it is
consistent). But it seems clear that (i) we humans *are*
capable of formulating internally consistent mathematical theories
(and of knowing that they are internally consistent), and (ii) being
able to do this does not require us to have any sort of
information-transferring contact with the abstract objects that the
theories in question are about. Thus, if this is right, then the
epistemological problem with platonism has been solved.



One might object here that in order for humans to acquire knowledge of
abstract objects in this way, they would first need to know that
plenitudinous platonism is true. Linsky & Zalta respond to this by
arguing that plenitudinous platonism (or in their lingo, principled
platonism) is knowable *a priori* because it is required for
our understanding of any possible scientific theory: it alone is
capable of accounting for the mathematics that could be used in
empirical science no matter what the physical world was like.
Balaguer's response, on the other hand, is based on the claim that to
demand that platonists explain how humans could know that FBP is true
is exactly analogous to demanding that external-world realists (i.e.,
those who believe that there is a real physical world, existing
independently of us and our thinking) explain how human beings could
know that there is an external world of a kind that gives rise to
accurate sense perceptions. Thus, Balaguer argues that while there may
be some sort of Cartesian-style skeptical argument against FBP here
(analogous to skeptical arguments against external-world realism), the
argument in (1)-(6) is supposed to be a different kind of
argument, and in order to respond to that argument, FBP-ists do not
have to explain how humans could know that FBP is true.