Platonism in Metaphysics

First published Wed May 12, 2004; substantive revision Tue Dec 24, 2024

Platonism is the view that there exist such things as abstractobjects—where (on one standard definition) an abstract object isan object that’s non-spatial, non-temporal, non-physical,non-mental, and non-causal. Platonism in this sense is acontemporary view. It is obviously related to the views ofPlato in important ways, but it is not entirely clear that Platoendorsed this view, as it is defined here. In order to remain neutralon this question, the term “platonism” is spelled with alower-case “p” (see the entry onPlato). The most important figure in the development of modern platonism isGottlob Frege (1884, 1892, 1893/1903, 1918/19). The view has also beenendorsed by many others, including Kurt Gödel (1947), BertrandRussell (1912), and W.V.O. Quine (1948, 1951).

Section 1 will describe the contemporary platonist view in detail.Section 2 will describe the alternatives to platonism—namely,immanent realism, conceptualism, nominalism, Meinongianism, andnon-factualism. Section 3 will develop and assess the first importantargument in favor of platonism, namely, the One Over Many argument.Section 4 will develop and assess a second argument for platonism,namely, the ontological-commitment argument. This argument emergedmuch later than the One Over Many argument, but as we will see, it iswidely thought to be more powerful. Finally, section 5 will developand assess the most important argumentagainst platonism,namely, the epistemological argument.

1. What is Platonism?

Platonism is the view that there exist abstract objects, where anabstract object is an object that’s non-spatial (i.e.,not spatially extended or located), non-temporal, non-physical (i.e.,not made of physical stuff), non-mental (i.e., not a minds or an ideain a mind or a disembodied soul, or anything else along these lines),and non-causal (i.e., causally inert).

(It’s important to note that there is no consensus in theliterature on how exactly “abstract object” is to bedefined. For instance, some philosophers would want to remove one ormore of the requirements from the above definition; e.g., some peoplewould remove the requirement for causal inertness. If we do removesome of the requirements from the above definition, then more kinds ofviews will count as platonist views—e.g., the view that thereare non-physical, non-mental, non-spatiotemporal objectsthat arecausally efficacious.^[1] The question of how exactly “abstract object” should bedefined won’t matter in what follows; but for more on thisissue, see the entry onabstract objects, as well as Cowling 2017; Plebani 2020; and Liggins 2024.)

The above definition of “abstract object” might besomewhat perplexing; for with all of these claims about what abstractobjects arenot, it might be unclear what theyare.We can clarify things, however, by looking at some examples.

Consider the sentence “3 is prime”. This sentence seems tosay something about a particular object, namely, the number 3. Just asthe sentence “The moon is round” says something about themoon, so too “3 is prime” seems to say something about thenumber 3. But whatis the number 3? There are a few differentviews that one might endorse here, but the platonist view is that 3 isan abstract object. On this view, 3 is a real and objective thingthat, like the moon, exists independently of us and our thinking(i.e., it is not just an idea in our heads). But according toplatonism, 3 is different from the moon in that it is not a physicalobject; it is wholly non-physical, non-mental, and causally inert, andit does not exist in space or time. One might put this metaphoricallyby saying that on the platonist view, numbers exist “in platonicheaven”. But we should not infer from this that according toplatonism, numbers exist in aplace; they do not, for theconcept of a place is a physical, spatial concept. It is more accurateto say that on the platonist view, numbers exist (independently of usand our thoughts) but do not exist in space and time.

Similarly, many philosophers take a platonistic view ofproperties. Consider, for instance, the property of being red. According to theplatonist view of properties, the property of redness existsindependently of any red thing. There are red balls and red houses andred shirts, and these all exist in the physical world. But platonistsabout properties believe that in addition to these things,redness—the property itself—also exists, and according toplatonists, this property is an abstract object. Ordinary red objectsare said toexemplify orinstantiate redness.

Platonists of this sort say the same thing about other properties aswell: in addition to all the beautiful things, there is also beauty;and in addition to all the tigers, there is also the property of beinga tiger. Indeed, even when there are no instances of a property inreality, platonists will typically maintain that the property itselfexists. This isn’t to say that platonists are committed to thethesis that there is a property corresponding to every predicate inthe English language. The point is simply that in typical cases, therewill be a property. For instance, according to this sort of platonism,there exists a property of being a four-hundred-story building, eventhough there are no such things as four-hundred-story buildings. Thisproperty exists outside of space and time along with redness. The onlydifference is that in our physical world, the one property happens tobe instantiated whereas the other does not.

In fact, platonists extend the position here even further, for ontheir view, properties are just a special case of a much broadercategory, namely, the category ofuniversals. It’s easyto 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. Butredness 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 specificred objects—such as balls in Buffalo, or cars inCleveland—asparticulars.

But according to this sort of platonism, properties are not the onlyuniversals; there are other kinds of universals as well, most notably,relations. Consider, for instance, the relationto thenorth of; this relation is instantiated by many pairs of objects(or more accurately, by ordered pairs of objects, since order mattershere—e.g.,to the north of is instantiated by <SanFrancisco, Los Angeles>, and <Edinburgh, London>, but not by<Los Angeles, San Francisco>, or <London, Edinburgh>). Soaccording to platonism, the relationto the north of is atwo-place universal, whereas a property like redness is aone-place universal. There are also three-place relations(which are three-place universals), four-place relations, and so on.An example of a three-place relation is thegave relation,which admits of a giver, a givee, and a given—as in “Janegave Tim a present”.

Finally, some philosophers claim thatpropositions areabstract objects. One way to think of a proposition is as the meaningof a sentence. Alternatively, we can say that a proposition is thatwhich is expressed by a sentence on a particular occasion of use.Either way, we can say that, e.g., the English sentence “Snow iswhite” and the German sentence “Schnee ist weiss”express the same proposition, namely, the proposition that snow iswhite.

There are many different platonistic conceptions of propositions. Forinstance, Frege (1892, 1918/19) held that propositions are composed ofsenses of words (e.g., on this view, the proposition thatsnow is white is composed of the senses of “snow” and“is white”), whereas Russell at one point (1905, 1911)held that propositions are composed of properties, relations, andobjects (e.g., on this view, the proposition that Mars is red iscomposed ofMars (the planet itself) and theproperty ofredness). Others hold that propositions do not have significantinternal structure. The differences between these views will notmatter for our purposes. For more detail, see the entry onpropositions.

(It might seem odd to say that Russellian propositions are abstractobjects. Consider, e.g., the Russellian proposition that Mars is red.This is an odd sort ofhybrid object. It has two components,namely, Mars (the planet itself) and the property of redness. One ofthese components (namely, Mars) is aconcrete object (where aconcrete object is just a non-abstract object (so, presumably, aphysical, spatiotemporal object). Thus, even if redness is an abstractobject, it does not seem that the Russellian proposition is completelyabstract. Nonetheless, philosophers typically lump these objectstogether with abstract objects. Perhaps it’s best to say thatRussellian propositions count as abstract objects on some definitionsbut not others. And similar remarks can be made about various otherkinds of objects. Think, for instance, of impure sets—e.g., theset containing Mars and Jupiter. This seems to be a hybrid object ofsome kind as well, because while it has concrete objects as members,it’s still aset, and on the standard view, sets areabstract objects. If we wanted to be really precise, it would probablybe best to have another term for such objects—e.g.,“hybrid object”, or “impure abstractobject”—but we needn’t worry about this issue here.This essay is almost entirely concerned with what might be calledpure abstract objects—i.e., abstract objects that arecompletely 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 inconnection with linguistic objects (most notably, sentences), possibleworlds, logical objects, and fictional characters (e.g., SherlockHolmes). And it is important to note here that one can be a platonistabout some of these things without being a platonist about theothers—e.g., one might be a platonist about numbers andpropositions but not properties or fictional characters.

Of course, platonism about any of these kinds of objects iscontroversial. Many philosophers do not believe in abstract objects atall. The alternatives to platonism will be discussed insection 2, but it is worth noting here that the primary argument that platonistsgive for their view is that, according to them, there are goodarguments against all other views. That is, platonists think we haveto believe in abstract objects, because (a) there are good reasons forthinking that things like numbers and properties exist, and (b) theonly 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 theexistence of things like numbers and properties altogether. Or one canmaintain that there do exist such things as numbers and properties,and instead of saying that they are abstract objects, one can say thatthey are mental objects of some sort (usually, the claim is that theyare ideas in our heads) or physical objects of some sort (or that theyexist in physical reality, or some such thing). Thus, the fourmainstream views here are as follows (and keep in mind thatanti-platonists can pursue different strategies with respect todifferent kinds of alleged abstract objects, taking one view of, say,numbers, and another view of properties or propositions).

2.1 Platonism

Platonism is the view described insection 1.

2.2 Immanent Realism

Advocates ofImmanent Realism agree with platonists thatthere do exist such things as mathematical objects—orproperties, or whatever category of alleged abstract objectswe’re talking about—and that these things are independentof us and our thinking; but immanent realists differ from platonistsin holding that these objects exist in the physical world. Dependingon the kind of object under discussion—i.e., whether we’retalking about mathematical objects or properties or what haveyou—the details of this view will be worked out differently. Inconnection with properties, the standard immanent-realist view is thatproperties exist in the physical world, in particular, in actualphysical objects; so, e.g., on this view, redness exists in actual redobjects (e.g., in Mars and the Golden Gate Bridge), as nonspatialparts or aspects of those things (this view traces back to Aristotle;in contemporary times, it has been defended by Armstrong 1978). Thereis certainly some initial plausibility to this idea: if you arelooking at a red ball, and you think that in addition to the ball, itsredness exists, then it seems a bit odd to say (as platonists do) thatits redness exists outside of spacetime. After all, the ball issitting right here in spacetime and we can see that it’s red; soit seems initially plausible to think that if the redness exists atall, then it exists in the ball. As we will see below, however, thereare serious problems with this view.

Another kind of immanent realism says that properties aretropes, where a trope is aparticular, not auniversal. So, e.g., on this view, one redness-trope exists in Marsand adifferent redness-trope exists in the Golden GateBridge, whereas on an Aristotelean version of immanent realism, oneand the same thing (namely,the property of redness) existsin both Mars and the Golden Gate Bridge—and, indeed, in all redthings. (Note that while trope theory counts as an immanent-realistview ofproperties, it’s ananti-realist viewofuniversals—because it says that there are no suchthings as universals.)^[2]

In connection with numbers, one immanent-realist strategy is to takenumbers to be properties of some sort—e.g., one might take themto be properties of piles of physical objects, so that, for instance,the number 3 would be a property of, e.g., a pile of threebooks—and to take an immanent realist view of properties. (Thissort of view has been defended by Armstrong 1978.) But views of thiskind have not been very influential in the philosophy of mathematics.A more prominent strategy for taking number talk to be about thephysical world is to take it to be about actual piles of physicalobjects, rather than properties of piles. Thus, for instance, onemight maintain that to say that \(2 + 3 = 5\) is not really to saysomething about specific entities (numbers); rather, it is to say thatwhenever we push a pile of two objects together with a pile of threeobjects, we will wind up with a pile of five objects—orsomething along these lines. Thus, on this view, arithmetic is just avery general natural science. A view of this sort was developed byMill (1843) and, more recently, a similar view has been defended byPhilip Kitcher (1984). It should be noted, however, that while thereare certainly physicalist themes running through the views of Mill andKitcher, it is not clear that either of them should be interpreted asan immanent realist. Kitcher is probably best classified as a kind ofanti-realist (I’ll say a bit more about this insection 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 wouldanswer the question, “Are there numbers, and if so, what arethey?”

Finally, Penelope Maddy (1990) has also developed a sort of immanentrealist view of mathematics. Concentrating mainly on set theory, Maddymaintains that sets of physical objects are located in space and time,right where their members are located. But Maddian sets cannot beidentified with the physical matter that constitutes their members. OnMaddy’s view, corresponding to every physical object, there is ahuge infinity of sets (e.g., the set containing the given object, theset containing that set, and so on) that are all distinct from oneanother but which all share the same matter and the samespatiotemporal location. Thus, on this view, there is more to a setthan the physical stuff that makes up its members, and so Maddy mightbe better interpreted as endorsing a nonstandard version ofplatonism.

2.3 Conceptualism

Conceptualism (also calledpsychologism andmentalism, depending on the sorts of objects underdiscussion) is the view that there do exist numbers—orproperties, or propositions, or whatever—but that they do notexist independently of us; instead, they are mental objects; inparticular, the claim is usually that they are something like ideas inour heads. As we will see below, this view has serious problems andnot very many people endorse it. Nonetheless, it has had periods ofpopularity in the history of philosophy. It is very often thought thatLocke held a conceptualistic view of universals, and prior to thetwentieth century, this was the standard view of concepts andpropositions. In the philosophy of mathematics, psychologistic viewswere popular in the late nineteenth century (the most notableproponent being the early Husserl (1891)) and even in the first partof the twentieth century with the advent of psychologisticintuitionism (Brouwer 1913 and 1949, and Heyting 1956). Finally, NoamChomsky (1965) has endorsed a mentalistic view of sentences and otherlinguistic objects, and he has been followed here by others, mostnotably, Fodor (1975, 1987).

It should also be noted here that one can claim that the existence ofnumbers (or propositions or whatever) isdependent on ushumans without endorsing a psychologistic view of the relevantentities. For one can combine this claim with the idea that theobjects in question are abstract objects. In other words, one mightclaim—and somehave claimed—that numbers (orpropositions 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 humanbeings. Liston (2004), Cole (2009), and Bueno (2009) endorse views ofthis general kind in connection with mathematical objects; Schiffer(2003: chapter 2), Soames (2014), and King (2014) endorse views likethis of propositions; and Salmon (1998) and Thomasson (1999) endorseviews like this of fictional objects.

2.4 Nominalism

Nominalism (also calledanti-realism) is the viewthat there are no such things as numbers, or universals, or whateversort of alleged abstract objects are under discussion. Thus, forinstance, a nominalist about properties would say that while there aresuch things as red balls and red houses, there is no such thing as theproperty of redness, over and above the red balls and red houses. Anda nominalist about numbers would say that while there are such thingsas piles of three stones, and perhaps “3-ideas” existingin people’s heads, there is no such thing as the number 3. As wewill see below, there are many different versions of each of thesekinds of nominalism, but for now, we don’t need anything morethan this general formulation of the view. (Sometimes“nominalism” is used to denote the view that there are nosuch things as abstract objects; on this usage,“nominalism” is synonymous with“anti-platonism”, and views like immanent realism count asversions of nominalism. In contrast, on the usage employed in thisessay, “nominalism” is essentially synonymous with“anti-realism”, and so views like immanent realism willnot 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 andconceptualism are for the simple reason that the latter two viewsadmit that there do exist such things as numbers (or universals, orwhatever). It is important to note, however, that nominalists agreewith platonists on an important point that immanent realists andconceptualists reject; in particular, nominalists (in agreement withplatonists) endorse the following thesis:

(S): If there were such things as numbers (or universals, or whateversort of alleged abstract objects we’re talking about), then theywould 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 thereare some very compelling arguments (which we will discuss) in favor of(S). As a result, there are very few advocates of immanent realism andconceptualism, especially in connection with mathematical objects andpropositions. There is wide-spread agreement about what numbers andpropositions would be if there were such things (namely, abstractobjects) but very little agreement as to whether there do exist suchthings. Thus, today, the controversial question here is a purelyontological one: Are there any such things as abstract objects (e.g.,mathematical objects, propositions, and so on)?

In addition to the four views discussed so far (i.e., platonism,immanent realism, conceptualism, and nominalism), two other views areworth mentioning…

2.5 Meinongianism

According toMeinongianism (see Meinong 1904), every singularterm—e.g., “Obama”, “3”, and“Sherlock Holmes”—picks out an object that has somesort of being (thatsubsists, or thatis, in somesense) but only some of these objectsexist. According toMeinongianism, sentences that platonists take to be about abstractobjects—sentences like “3 is prime” and “Redis a color”—express truths about objects that don’texist.

Meinongianism has been almost universally rejected by philosophers.The standard argument against it (see, e.g., Quine 1948: 3 and Lewis1990) is that it does not provide a view that is clearly distinct fromplatonism and merely creates the illusion of a different view byaltering the meaning of the term “exist”. The idea here isthat on the standard meaning of “exist”, any object thathas any being at all exists, and so according to standard usage,Meinongianism entails that numbers and universals exist; but this viewclearly doesn’t take such things to exist in spacetime and so,the argument concludes, Meinongianism entails that numbers anduniversals are abstract objects—just as platonism does.

It is worth noting, however, that while Meinongianism has mostly beenrejected, it does have some more contemporary advocates, most notably,Routley (1980), Terence Parsons (1980), and Priest (2003, 2005[2016]).

2.6 Non-Factualism

All the views considered so far can be thought of as either platonistviews or anti-platonist views. A final view,Non-Factualism,is that there’s no fact of the matter whether platonism is truebecause there’s no fact of the matter whether abstract objectsexist. Views of this kind have been endorsed by Balaguer (1998a,2021), Yablo (2009), and Warren (2016). OnBalaguer’s version of the view, “abstract objectsexist” iscatastrophically imprecise—i.e., soimprecise that it lacks truth conditions and truth value. Roughlyspeaking, the idea here is that

it’s built into the meaning of “abstract objectsexist” that some substantive fact is required of reality to makethat sentence true, but
it’s catastrophically unclear what this substantive extrafact could be.

A different sort of non-factualist view, developed by Carnap (1950)and Maddy (2011), says that

sentences like “3 is prime” and “Numbersexist” are trivially true (or analytic) in some languages (whichwe can calltrivialist languages); and
these sentences are false in other languages (non-trivialistlanguages); and
there are no other facts to discover here.

But while this view implies that there’s no language-independentfact of the matter whether “Numbers exist” is true, itisn’t happily thought of as implying that there’s no factof the matter whether platonism is true (or whether abstract objectsexist); for on this view, sentences like “Numbers exist”aretrivially true in trivialist languages—and sonothing is required of reality to make these sentences true in thoselanguages.

3. The One Over Many Argument

Section 4 will be concerned with what is widely considered to be the bestargument (or kind of argument) for platonism, namely, what we can calltheontological-commitment argument. But first, in thepresent section, we will consider theOne Over Many argument,which goes back to Plato and, prior to the twentieth century, was themost prominent argument for platonism.

(There are, of course, many other arguments for platonism in theliterature, aside from these two arguments. For instance, to name justtwo recent arguments, De Cruz (2016) argues that realism about numbersprovides the best explanation of the numerical cognition of humanbeings and other animals, and Tugby (2022) argues for a platonisticview of properties by showing how fruitful the view is—i.e., byshowing how platonism can do theoretical work for us. But this essaycovers just the One Over Many argument and the ontological-commitmentargument.)

The One Over Many argument can be formulated as follows:

I have in front of me three red objects (a ball, a hat, and a rose).These objects resemble one another. Therefore, they have something incommon. 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 threeobjects resemble each other. The explanation is that they all possessa single property, namely, redness. Thus, platonists argue, if thereis 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 toproperties), then we are justified in believing in properties.

Notice that as the argument has been stated here, it is not anargument for a platonistic view of properties; it is an argument forthe thesis that properties exist, but not for the thesis thatproperties are abstract objects. Thus, in order to use this argumentto motivate platonism, one would have to supplement it with somereason for thinking that the properties in question here could not beideas in our heads or immanent properties existing in particularphysical objects. There are a number of arguments that one might usehere, and insection 4.2, we will discuss one such argument. But there is no need to pursuethis here, because there is good reason to think that the One OverMany argument doesn’t succeed anyway—i.e., that itdoesn’t provide a good reason for believing in properties of anysort. In other words, the One Over Many argument fails to refutenominalism about properties.

Before proceeding, it is worth pointing out that the One Over Manyargument described above can be simplified. As Michael Devitt (1980)points out, the appeal to resemblance, or tomultiple thingshaving a given property, is a red herring. On the traditionalformulation, nominalists are challenged to account for the followingfact: the ball is red and the hat is red. But if nominalists canaccount for the fact that the ball is red, then presumably, they cansimply 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 simplepredicative facts, e.g., the fact that the ball is red, withoutappealing to properties, e.g., redness. More generally, they need toshow how we can account for the truth of sentences of the form“a isF” without appealing to a property ofFness.^[3]

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

There is a very well-known nominalist response to the One Over Manyargument. The heart of the response is captured by the followingremark from Quine (1948: 10):

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

There are two different ideas here. The first is that nominalists canrespond to the One Over Many with an appeal to irreducible facts, orbrute facts. The second is that platonists are no better offthan such brute-fact nominalists in terms of real explanatory power.Now, Quine didn’t say very much about these two ideas, but bothideas have been developed by Devitt (1980, 2010a), whose exposition wefollow here.

The challenge to nominalists is to provide an explanation of facts ofa certain kind, namely, predicative facts expressed by sentences ofthe form “a isF”, e.g., the fact that agiven ball is red. Now, whenever we are challenged to provide anexplanation of a fact, or alleged fact, we have a number of options.The most obvious response is simply to provide the requestedexplanation. But we can also argue that the alleged fact isn’treally a fact at all. Or, third, we can argue that the fact inquestion is abrute fact—i.e., a fact that does nothave an explanation. Now, in the present case, nominalists cannotclaim that all predicative facts are brute facts, because it is clearthat wecan explain at least some facts of this sort. Forinstance, it seems that the fact that a given ball is red can beexplained very easily by saying that it is red because it reflectslight in such and such a way, and that it reflects light in this waybecause its surface is structured in thus and so a manner. Sonominalists should not claim that all predicative facts are brutefacts. But as Devitt points out, there is a more subtle way to appealto bruteness here, and if Quinean nominalists make use of this, theycan block the One Over Many argument.

The Quine-Devitt response to the One Over Many begins with the claimthat we can account for the fact that the ball is red, withoutappealing to the property of redness, by simply using whateverexplanation scientists give of this fact. Now, by itself, thisexplanation will not satisfy advocates of the One Over Many argument.If we explain the fact that the ball is red by pointing out that itssurface is structured in some specific way, then advocates of the OneOver Many argument will say that we have only moved the problem back astep, because nominalists will now have to account for the fact thatthe ball’s surface is structured in the given way, and they willhave to do this without appealing to the property of being structuredin the given way. More generally, the point is this: it is of coursetrue that if nominalists are asked to account for the fact that someobjecta isF, without appealing to the property ofFness, they can do this by pointing out that (i)a isG and (ii) allFs areGs (this is the sort ofexplanation they will get if they borrow their explanations fromscientists); but such explanations only move the problem back a step,for they leave us with the task of having to explain the fact thata isG, and if we want to endorse nominalism, we willhave to do this without appealing to the property ofGness.

This is where the appeal to bruteness comes in. Nominalists can saythat

we can keep giving explanations of the above sort (i.e.,explanations of the sort “a isF because it isG”, or because its parts areGs,Hs, andIs, or whatever) for as long as we can, and
when explanations of this sort cannot be given, no explanation atall can be given.

The thought here is that at this point, we will have arrived atfundamental facts that do not admit of explanations—e.g., factsabout the basic physical natures of elementary particles. When wearrive at facts like this, we will say: “There’s no reasonwhy these particles are this way; they justare”.

This gives us a way of understanding how nominalists can plausibly usean appeal to bruteness to respond to the One Over Many argument. Butthe appeal to bruteness is only half of the Quinean remark quotedabove. What about the other half, i.e., the part about the platonistbeing no better off than brute-fact nominalists in terms of realexplanatory power? To appreciate this claim, let us suppose that wehave arrived at a bottom-level fact that Quinean nominalists take tobe a brute fact (e.g., the fact that physical particles of someparticular kind—say, gluons—areG). Advocates ofthe One Over Many would say that their view is superior to Quineannominalism because they can provide an explanation of the fact inquestion. Now, when they announce this, people who were interested inthe question of why gluons areG, and who had been disappointedto hear from scientists and Quineans that this is simply a brute fact,might get very excited and listen eagerly to what advocates of the OneOver Many have to say. What they say is this:

Gluons areG because they possess the property ofGness.

This doesn’t seem very helpful. The claim that gluons possessGness seems to do little more than tell us that gluons havesome nature that makes it the case that they areG, and so itseems that no genuine explanation has been given. After all, those whohad been interested in learning why gluons areG would not bevery satisfied by this so-called “explanation”. Thus, touse Quine’s words, it seems that advocates of the One Over Manyare “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 ofGness

is just aparaphrase of the sentence

[N]: Gluons areG.

On this view,[P] isequivalent to[N]. That is, it says the very same thing. And neither sentence, accordingto this view, entails the existence ofGness. We can call thisaparaphrase-nominalist view of sentences like [P]. Butnominalists needn’t endorse this view. They can also endorse afictionalist 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 ofGness and [N] does not. Accordingto this fictionalist view, [P] is strictly speaking untrue, because ittalks about the property ofGness, and according to nominalism,there is no such thing asGness. In short, [P] is strictlyspeaking untrue, on this view, for the same reason that, e.g.,“The tooth fairy is nice” is untrue. But while [P] is notliterally 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 amanner ofspeaking, or afaçon de parler. (Notice that thedispute between fictionalism and paraphrase nominalism is bestunderstood as a straightforward empirical dispute about theordinary-language semantics of sentences like [P]; the question iswhether such utterances literally say the same things that thecorresponding sentences like [N] say.)

Whichever view nominalists adopt here, they can respond to the OneOver 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 afictionalist 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 areG because theypossess a nature that makes it the case that they areG.

Having made the point that the platonist explanation of[N] is uninformative, the nominalist’s next move is to appeal toOckham’s razor to argue that weshouldn’t believeinGness (or at least that we shouldn’t believe inGness for any reason that has anything to do with the need toexplain things like [N]). Ockham’s razor is a principle thattells us that we should believe in objects of a given kind only ifthey play a genuine explanatory role. This principle suggests that ifGness does not play a genuine role in an explanation of thefact that gluons areG, then we shouldn’t believe inGness—or, again, we shouldn’t believe in it for anyreason having to do with the need to explain the fact that gluons areG.

The Quinean response to the One Over Many argument is often couched interms of acriterion of ontological commitment. A criterionof ontological commitment is a principle that tells us when we arecommitted to believing in objects of a certain kind in virtue ofhaving assented to certain sentences. What the above response to theOne Over Many suggests is that we are ontologically committed not bypredicates like “is red” and “is a rock”, butbysingular terms. (A singular term is just a denotingphrase, i.e., an expression that purports to refer to a specificobject, e.g., proper names like “Mars” and“Obama”, 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: ifyou think that a sentence of the form “a isF” is true, then you have to accept the existence of theobjecta, but you do not have to accept the existence of aproperty ofFness; for instance, if you think that “Theball is red” is true, then you have to believe in the ball, butyou do not have to believe in redness; or if you think that“Fido is a dog” is true, then you have to believe in Fidobut not in the property of doghood.

Three points need to be made here. First, the above criterion needs tobe generalized so that it covers the use of singular terms in otherkinds of sentences, e.g., sentences of the form “a isR-related tob”. Second, on the standard view, weare ontologically committed not just by singular terms but also byexistential statements—e.g., by sentences like “There aresomeFs”, “There is at least oneF”,and so on (in first-order logic, such sentences are symbolized as“\(\exists xFx\)”, and the“\(\exists\)” is called anexistential quantifier). The standard view here is that ifyou think that a sentence like this is true, then you are committed tobelieving in the existence of someFs (or at least oneF) but you do not have to believe inFness; forinstance, if we assent to “There are some dogs”, then weare committed to believing in the existence of some dogs, but we arenot thereby committed to believing in the existence of the property ofdoghood. (Quine actually thought that we are committedonlyby existential claims and not by singular terms; but this is not awidely held view.) Third and finally, it is usually held that we areontologically committed by singular terms and existential expressions(or existential quantifiers) only when they appear in sentences thatwe think areliterally true and only when we think thesingular term or existential quantifier in questioncan’t beparaphrased away. We can see what’s meant by this byreturning 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 todenote theproperty of redness; thus, using the above criterion of ontologicalcommitment, if we think[R] is true, then it would seem, we are committed to believing in theproperty of redness. But there are two different responses thatnominalists can make to this. First, they can endorse paraphrasenominalism (defined a few paragraphs back) with respect to [R]. Ifthey do this, they will claim that [R] doesn’t really carry anontological commitment to the property of redness, because it isreally 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 beparaphrasedaway—which is just to say that [R] can be paraphrased by(or is equivalent to) a sentence that doesn’t contain thesingular term “the property of redness” (namely,“The ball is red”). The second view that nominalists canendorse with respect to [R] is fictionalism. In other words, they canadmit that [R] does commit to the existence of the property ofredness, but they can maintain that because of this (and because thereare no such things as properties), [R] is, strictly speaking, untrue,even if it is “for-all-practical-purposes true”, or somesuch thing.

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

The Standard Criterion of Ontological Commitment: We areontologically committed by the singular terms (that can’t beparaphrased away) in the (simple) sentences that we take to beliterally true; and we are ontologically committed by the existentialquantifiers (that can’t be paraphrased away) in the(existential) sentences that we take to be literally true; but we arenot committed by the predicates in such sentences. Thus, for instance,if we believe that a sentence of the form “a isF” is literally true, and if we think that it cannot beparaphrased into some other sentence that avoids reference toa, then we are committed to believing in the objectabut not the property ofFness; and, likewise, if we assent to asentence of the form “a isR-related tob”, then we are committed to believing in the objectsa andb but not the relationR; and if we assentto a sentence of the form “There is anF” then weare committed to believing in an object that isF but we arenot committed to the property ofFness.^[4]

The One Over Many argument is now widely considered to be a badargument. Ironically, though, the standard criterion of ontologicalcommitment—which Quinean nominalists appeal to in responding tothe One Over Many argument—is one of the central premises inwhat is now thought to be the best argument (or the bestkindof argument) for platonism. We can call arguments of this kindontological-commitment arguments.

4. Ontological-Commitment Arguments for Platonism

Very roughly, an ontological-commitment argument for platonism is anargument that proceeds by locating an ordinary sentenceS andclaiming that

S is true, and
S has a logical form that makes it the case that it has anontological commitment to certain objects—or certain kinds ofobjects—that could only be abstract objects.

The three most important logical forms here—i.e., the threelogical forms that feature most often in these arguments—are

“a isF” (as in, e.g., “3 isprime”), and
“a isR-related tob” (as in,e.g., “4 is greater than 3”), and
“There are someFs” (as in, e.g., “Thereare some prime numbers between 10 and 20”).

Ontological-commitment arguments for platonism are arguably alreadypresent in the works of Plato, but the first really clear formulationof an ontological-commitment argument was given by Frege (see, e.g.,his 1884, 1892, 1893/1903, 1918/19).

We will begin, insection 4.1, by constructing an ontological-commitment argument for the existenceofnumbers. After that, we will discussontological-commitment arguments for the existence of propositions,properties, relations, sentence types, possible worlds, logicalobjects, and fictional objects.

4.1 Mathematical Objects

Here’s an ontological-commitment argument for a platonistic viewof numbers:

[1]: The sentence “3 is prime” is true.
[2]: The sentence “3 is prime” should be read at facevalue; that is, it should be read as being of the form “aisF” and, hence, as making straightforward claims aboutthe nature of a certain object—namely, the number 3. But
[3]: If [1] and [2] are both true, then the number 3 exists. (Thisfollows from the standard criterion of ontological commitment that wasarticulated insection 3.) Therefore,
[4]: The number 3 exists. But
[5]: If the number 3 exists, then all of the natural numbers exist.And
[6]: If the natural numbers exist, then they’re either abstractobjects, or physical objects, or mental objects. But
[7]: The natural numbers aren’t physical objects. And
[8]: The natural numbers aren’t mental objects. Therefore, from[4]–[8], it follows that
[9]: The natural numbers exist, and they are abstract objects.But
[10]: If [9] is true, then platonism is true. Therefore,
[11]: Platonism is true.

The first clear statement of an argument of this general kind wasgiven by Frege (1884); other advocates include Quine (see his 1948 and1951, though he doesn’t explicitly state the argument there),Gödel (1947), C. Parsons (1965, 1971, 1994), Putnam (1972),Steiner (1975), Resnik (1981, 1997), Zalta (1983, 1999), Wright(1983), Burgess (1983), Hale (1987), Shapiro (1989, 1997), the earlyMaddy (1990),^[5] Katz (1998), Colyvan (2001), McEvoy (2005, 2012), and Marcus(2015).

This version of the argument is formulated much morecarefully—using more premises—than is usual in theliterature. The reason for this is to isolate all of the separateclaims that the argument rests on, and to lay bare the reasoningbehind the argument. The argument is clearly valid, and so the onlyquestion is whether the basic premises (i.e.,[1]–[3],[5]–[8], and[10]) are true. But[5] and[6] are both trivially true, and I don’t know of anyone who woulddeny either of them, so we can assume that they’re true.^[6] So that leaves six basic premises, namely,[1]–[3],[7]–[8], and[10]. But,prima facie, all six of those premises seem extremelyplausible, if not downright obvious, and so the argument in[1]–[11] has considerable force.

The nice thing about the way this argument is formulated is that thereare six main varieties of anti-platonism in the literature, and eachof these six views involves a rejection of a different premise in theargument in[1]–[11]. In the remainder of this section, I’ll discuss these sixdifferent responses to the argument.

4.1.1 Psychologism and the Rejection of Premise [8]

Advocates of psychologism reject premise[8] and claim that the natural numbers are mental objects—and, moregenerally, that our mathematical theories are theories of mentalobjects.

Frege (1884: introduction and §27; 1893/1903: introduction; 1894;and 1918/19) gave several compelling arguments against psychologism.First, it seems that psychologism is incapable of accounting for thetruth of sentences that are aboutall natural numbers,because there are infinitely many natural numbers and clearly, therecould 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) arenot true; for if none of us has ever had a thought about some verylarge number, then psychologism entails that there is no such numberand, hence, that no sentence about that number could be true. Third,psychologism turns mathematics into a branch of psychology, and itmakes mathematical truths contingent upon psychological truths, sothat, for instance, if we all died, “4 is greater than 2”would suddenly become untrue. But this seems wrong: it seems thatmathematics is true independently of us; that is, it seems that thequestion of whether 4 is greater than 2 has nothing at all to do withthe question of how many humans are alive. Fourth and finally,psychologism suggests that the proper methodology for mathematics isthat of empirical psychology; that is, it seems that if psychologismwere true, then the proper way to discover whether, say, there is aprime number between 10,000,000 and 10,000,020, would be to do anempirical study of humans and ascertain whether there is, in fact, anidea of such a number in one of our heads; but of course, this is notthe proper methodology for mathematics. As Frege says (1884:§27), “Weird and wonderful…are the results of takingseriously 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 areaboutthese ideas. Thus, the dispute between platonism and psychologism isat least partly asemantic one. Advocates of psychologismagree with platonists that in the sentence “3 is prime”,“3” functions as a singular term (i.e., as a denotingexpression). But they disagree about the referent of this expression.They think that “3” refers to a mental object, inparticular, an idea in our heads. It is thissemantic thesisthat platonists reject and that the above Fregean arguments aresupposed to refute. More specifically, they’re supposed to showthat the psychologistic semantics of mathematical discourse is notcorrect because it has consequences that fly in the face of the actualusage of mathematical language.

Once we appreciate the fact that psychologism involves a semanticthesis, the view starts to seem implausible in the extreme. Toappreciate this, imagine someone rejecting Euclid’s theorem thatthere are infinitely many prime numbers on the grounds that there areonly finitely many human ideas in the universe, and so therecouldn’t be infinitely many numbers, let alone prime numbers.Anyone who objected in this way could be accused of not understandingwhat “There are infinitely many prime numbers”means in our language. In our language, this sentence issimplynot a claim about things that exist in our heads. Andso psychologism is just false—it involves an empirically falsetheory of what mathematical sentences actually mean in ordinaryEnglish.

You might think that the above arguments target an implausible versionof psychologism; more specifically, you might think that advocates ofpsychologism could say not that our mathematical theories are aboutactual mental things that really exist in really human heads,but rather that our mathematical theories are claims about what sortsof ideas it would bepossible to construct in our heads. Butnow we have to ask what apossible idea is. Assuming that wedon’t endorse a Lewisian (1986) modal realism according to whichevery possible idea really exists in a real human head in some realpossible world, then it would seem that either

possible ideas are abstract objects of some kind (thus, makingthis possible-idea view a version of platonism, rather thanpsychologism), or
there are really no such things as possible ideas (thus, makingthis view a version of nominalism—in particular, this view wouldinvolve a rejection of premise[2], rather than premise[8]).

4.1.2 Physicalism/Immanent Realism and the Rejection of Premise [7]

A similar argument can be run against the view that numbers arephysical object and, hence, that premise[7] is false. Imagine someone rejecting Euclid’s theorem that thereare infinitely many prime numbers on the grounds that there are onlyfinitely many physical objects in the universe, and so therecouldn’t be infinitely many numbers, let alone prime numbers.Regardless of whether this person is correct in their claim that thereare only finitely many physical objects in the universe, they aredeeply confused about what the sentence “There are infinitelymany prime numbers” means in our language. In ordinary English,the truth of “There are infinitely many prime numbers”doesn’t depend on the claim that there are infinitely manyphysical objects because, in our language, that sentence isn’tabout physical objects. And so, like psychologism, thisphysicalistic view of mathematics is clearly false—because itinvolves an obviously false empirical theory of what mathematicalsentences like “There are infinitely many prime numbers”mean in ordinary English.

Now, once again, you might think that this argument targets the wrongkind of physicalistic view. Consider, e.g., Mill’s (1843: bookII, chapters 5 and 6) view that sentences about numbers are reallyjust general claims about bunches of ordinary objects. Morespecifically, on this view, the sentence “2 + 1 = 3”, forinstance, isn’t really about specific objects (the numbers 1, 2,and 3). Rather, it says that whenever we add one object to a pile oftwo objects, we will get a pile of three objects. You might complainthat the argument of the preceding paragraph doesn’t undermineMill’s version of physicalism because Mill’s viewdoesn’t entail that there are infinitely many physicalobjects.

But while Mill’s view clearly has physicalistic leanings, itdoesn’t really involve a rejection of premise[7]. It is better thought of as involving a rejection of premise[2]—because it rejects the face-value reading of sentences like “2 + 1 =3”. But as we’ll see insection 4.1.4, if anti-platonists are going to reject premise [2], there are betterways to do it. One problem with Mill’s view is that it is noteasily generalizable to other branches of mathematics. For instance,it’s hard to see how Mill’s view could be extended to settheory. Now, it might seem that we could do this by taking set theoryto be about bunches of ordinary objects, or piles of physical stuff.This, however, is untenable. For the principles of set theory entailthat corresponding to every physical object, there is a huge infinityof sets. Corresponding to the ball on my table, for instance, there isthe set containing the ball, the set containing that set, the setcontainingthat set, and so on; and there is the setcontaining the ball and the set containing the set containing theball; and so on and on and on. Clearly, these sets are not just pilesof physical stuff, because (a) there are infinitely many of them(again, this follows from the principles of set theory) and (b) all ofthese infinitely many sets share the same physical base. Thus, itseems that claims about sets are not claims about bunches of ordinaryobjects, or even generalized claims about such bunches. They areclaims aboutsets, which are objects of a different kind.

Another problem with physicalistic views along the lines ofMill’s is that they seem incapable of accounting for the sheersize of the infinities involved in set theory. Standard set theoryentails not just that there are infinitely large sets, but that thereare infinitely many sizes of infinity, which get larger and largerwith no end, and that there actually exist sets of all of thesedifferent sizes of infinity. There is simply no plausible way to takethis theory to be about physical stuff.

These arguments do not refute the kind of immanent realism defended bythe early Maddy (1990). On Maddy’s view, sets of physicalobjects are located in spacetime, right where their members are. Thus,if you have two eggs in your hand, then you also have the setcontaining those eggs in your hand. Maddy’s view has no problemaccounting for the massive infinities in mathematics, for on her view,corresponding to every physical object, there is a huge infinity ofsets that exist in space and time, right where the given physicalobject exists. Given this, it should be clear that while Maddy’sview says that sets exist in spacetime, it cannot be thought of assaying that sets are physical objects. So Maddy’s view is not aphysicalist view in the sense that’s relevant here (as waspointed out above, it is better thought of as a non-standard versionof platonism). Thus, in the present context (i.e., the context inwhich platonists are trying to undermine views that take mathematicalobjects to be physical objects), platonists do not need to argueagainst Maddy’s view. Of course, in the end, if they want tomotivate the standard version of platonism, then they’ll have togive reasons for preferring their view to Maddy’s non-standardversion of platonism, and this might prove hard to do. For somearguments against Maddy’s early view, see, e.g., Lavine (1992),Dieterle and Shapiro (1993), Milne (1994), Riskin (1994), Carson(1996), and the later Maddy (1997).

(Similar remarks can be made in connection with other immanent realistviews, e.g., Armstrong’s (1978) view, but we won’t pursuethis here.)

If the arguments of this subsection and the preceding one are cogent,then sentences like “3 is prime” are not about physical ormental objects, and so numbers aren’t physical or mentalobjects, and premises[7] and[8] are true. But if this is right, then if we want to resist platonism,we’re going to have to defend a nominalistic philosophy ofmathematics. There are four different nominalistic views that we needto consider; these four views correspond to the rejection of premises[1]–[3] and[10].

4.1.3 Fictionalism/Error Theory and the Rejection of Premise [1]

One way for nominalists to respond to the argument in[1]–[11] is to reject premise[1] and endorse anerror theory—or, as it’s alsooften called, afictionalist view—of mathematicaldiscourse. We can define this view as follows:

Mathematical Error Theory (aka,MathematicalFictionalism):

The platonist’s face-value semantics of mathematicaldiscourse is correct (i.e., sentences like “3 is prime” dopurport to be about abstract objects); but
there are no such things as abstract objects; and so
these sentences (and, indeed, our mathematical theories) are nottrue.

Thus, on this view, just asAlice in Wonderland is not truebecause there are no such things as talking rabbits, hookah-smokingcaterpillars, and so on, so too our mathematical theories are not truebecause there are no such things as numbers and sets and so on.^[7] (Fictionalist views have been developed by numerous philosophers, aswe will see in discussing the objections to fictionalism and theresponses to those objections that fictionalists have mounted.)

There are a few different ways that platonists might try to argueagainst fictionalism. The most famous and widely discussed argumentagainst fictionalism is theQuine-Putnam indispensabilityargument (see Quine 1948, 1951; Putnam 1972, 2012; and Colyvan2001). This argument (or at any rate, one version of it) proceeds asfollows: it cannot be that mathematics is untrue, as fictionalistssuggest, because

mathematics is an indispensable part of our physical theories(e.g., quantum mechanics, general relativity theory, evolutionarytheory, and so on) and so
if we want to maintain that our physical theories are true (andsurely we do—we don’t want our disbelief in abstractobjects to force us to be anti-realists about natural science), thenwe have to maintain that our mathematical theories are true.

Fictionalists have developed two different responses to theQuine-Putnam argument.The hard-road response, developed byField (1980, 1989) and Balaguer (1998a), is based on the claim thatmathematics is, in fact, not indispensable to empiricalscience—i.e., that our empirical theories can benominalized, or reformulated in a way that avoids referenceto abstract objects. Theeasy-road response, developed byBalaguer (1998a), Rosen (2001), Yablo (2001, 2002, 2005), Bueno(2009), Leng (2005a, 2010), and perhaps Melia (2000), is to grant theindispensability of mathematics to empirical science and to simplyaccount for the relevant applications from a fictionalist point ofview. (A counterresponse to the easy-road response has been given byColyvan (2002), Baker (2005, 2009), and Baron (2020), who argue thatfictionalists can’t account for theexplanatory rolethat mathematics plays in science; responses to the explanatoryversion of the indispensability argument have been given by Melia(2002), Leng (2005b), Bangu (2008), Daly and Langford (2009),Martínez-Vidal and Rivas-de-Castro (2020), and Boyce(2021).)

A second objection that one might raise against fictionalism is thatit’s incompatible with theobjectivity of mathematics.In response to this, Field (1980, 1989, 1998) argues thatfictionalists can account for the objectivity of mathematics byappealing to objective facts about which mathematical sentences aretrue in the story of mathematics, where (somewhat roughly) asentence is true in the story of mathematics if and only if it followsfrom the axioms of our mathematical theories. And Balaguer (2009,2021) responds by appealing to objective facts about whichmathematical sentences arefor-all-practical-purposes true(orFAPP-true), where (somewhat roughly) a sentence isFAPP-true if and only if it would have been true if platonism had beentrue).

A third objection to fictionalism is based on Lewis’s (1991:59) claim that it would be laughablypresumptuous to suggest that we should reject our mathematicaltheories for philosophical reasons, given the track records of the twodisciplines. Balaguer (2021) responds to this by claiming thatfictionalists aren’t suggesting that we should reject ourmathematical theories, or that there’s somethingwrongwith these theories; on the contrary, on this version of fictionalism,there’snothing wrong with our mathematical theoriesbecause (a) they’re FAPP-true, and (b) the mark of goodness inmathematics is FAPP-truth, not literal truth.

A fourth objection that one might raise against fictionalism is thatit’s not a nominalistically acceptable view because it involvestacit reference to various kinds of abstract objects, e.g., sentencetypes, or stories, or possible worlds. For responses to thisobjection, see, e.g., Field (1989) 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 (1994),Burgess and Rosen (1997), Katz (1998),Thomas (2000, 2002), Stanley (2001), Bueno (2003), Szabó(2005), Hoffman (2004), and Burgess (2004). For responses to theseobjections, see the various fictionalist works cited above, as well asDaly (2008) and Liggins (2010). And for a discussion of all theseobjections, as well as fictionalist responses to them, see the entryonfictionalism in the philosophy of mathematics.

Prima facie, it might have seemedobvious that ourmathematical theories are true—and, hence, that fictionalism isfalse. But on reflection, it’s not clear that we have any goodargument for the claim that our mathematical theories arestrictly and literally true. Nonetheless, nominalists might not wantto give up on the truth of mathematics. If so, then they’ll haveto reject either[2], or[3], or[10] in the argument in[1]–[11].

4.1.4 Paraphrase Nominalism and the Rejection of Premise [2]

One very traditional way to respond to the argument in[1]–[11] is to reject premise[2] and endorse aparaphrase nominalist view. According to viewsof this kind, sentences like “3 is prime” are true, butthey should not be read as platonists read them; on the contrary,these sentences should be interpreted as being equivalent toother sentences—what are calledparaphrases ofthese sentences—that donot involve ontologicalcommitments to abstract objects. One view of this sort, known asif-thenism, holds that, e.g., “3 is prime” can beparaphrased by—i.e., isequivalent to—thesentence “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, seePutnam 1967a, 1967b and Hellman 1989.) A second version of theparaphrase strategy, which we can callmetamathematicalformalism (see Curry 1951), is that “3 is prime” canbe paraphrased by “‘3 is prime’ follows from theaxioms of arithmetic”.^[8] A third paraphrase view holds that sentences that seem to be aboutnumbers are best read as being aboutplurals. For instance,we might read “2 + 2 = 4” as really saying something likethis:two objects and two (more) objects are four objects.(Views of this general kind have been defended by, e.g., Yi 2002,2016; Hofweber 2005; and Moltmann 2013a.) (For a fourth version ofparaphrase nominalism, see Rayo 2008.)

One problem with the various paraphrase views (not to put too fine apoint on it) is that none of the paraphrases seems very good. That is,the paraphrases seem to misrepresent what we actually mean when we saythings like “3 is prime” (and by “we”, I meanboth mathematicians and ordinary folk). What we mean, it seems, isthat 3 is prime—not that if there were numbers, then 3 would beprime, or that the sentence “3 is prime” follows from theaxioms of arithmetic, or any such thing. And notice how the situationhere differs from cases where we do seem to have good paraphrases. Forinstance, 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 averageaccountant; but it is plausible to suppose that, in fact, we are notso 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 agoodparaphrase of[A1] because it seems clear that when people say things like [A1], whattheyreally mean are things like[A2]. But in the present case, this seems wrong: it does not seem plausibleto suppose that when ordinary people utter “3 is prime”,what they really mean is that if there were numbers, then 3 would beprime. Again, it seems that what we mean here is, very simply, that 3is prime. In short, when people say things like “3 isprime”, they do not usually have any intention to be sayinganything other than what these sentencesseem to say; andbecause of this, it seems that the platonist’s face-valuesemantics for mathematical discourse is correct.

If we focus on sentences like “2 + 2 = 4”, then theYi-Hofweber-Moltmann view—i.e., the view that these sentencesare about plurals—might seem to fit better with ordinary usage.But when we switch to sentences like “3 isprime”—and, even worse, “There are infinitely manyprimes”—they start to seem cumbersome and lessplausible.

You might think that paraphrase nominalists don’t need to committo the thesis that their paraphrases capture the realordinary-language meanings of our mathematical sentences. One way todevelop this idea would be to maintain that the so-called“paraphrases” of our mathematical claims should be thoughtof asreplacements for those claims—i.e., as the claimsthat we should make when we’re being careful to say exactly whatwe mean, or some such thing. Philosophers who endorse replacementviews of this general kind include Chihara (1990, 2004), Dorr (2008),Boyce (2020), and Himelright (2023a)—and perhaps some of theparaphrase nominalists listed above.

It is of course true that we can endorse a replacement view of thiskind. But it’s important to note that if “paraphrasenominalism” is defined in terms of an acceptance of premise [1]and a rejection of premise[2]—i.e., if paraphrase nominalism is supposed to give nominalists a way tosalvage thetruth of mathematics—then replacementnominalism isnot a version of paraphrase nominalism. It is,rather, a version offictionalism, orerror theory.For if replacement nominalists don’t claim that their“paraphrases” capture the real ordinary-language meaningsof our mathematical sentences, then they’ll presumably begranting that platonists are right about the ordinary-languagemeanings of mathematical sentences like “3 is prime”. Butif so—if replacement nominalists admit that ordinary utterancesof “3 is prime” are best interpreted as being aboutabstract mathematical objects, or aspurporting to be aboutsuch objects—then since replacement nominalists don’tbelieve in the existence of abstract objects, they’ll have toadmit that such sentences arenot literally true, and sotheir view will collapse into a version of mathematical fictionalism,or error theory. So replacement theorists would not be saving thetruth of our mathematical theories. And given this, their so-called“paraphrases” would be best thought of as giving them away of accounting for theobjectivity of mathematics, and thegoodness of mathematics; and so these paraphrases would beplaying roughly the same role in replacement nominalism thatField’struth in the story of mathematics (orBalaguer’sFAPP-truth) plays in mathematical errortheory.

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

4.1.5 Thin-Truth-ism and the Rejection of Premise [3]

A third nominalist strategy is to reject premise[3] and the standard criterion of ontological commitment. The mostobvious way to do this is to endorse the following view:

simple mathematical sentences like “3 is prime”should be read as being of the form “a isF”and as being about abstract objects (e.g., “3” should betaken as denoting the number 3, which could only be an abstractobject, and “3 is prime” should be taken as being a claimabout that object); and
abstract objects—in particular, mathematical objects likethe number 3—don’t exist (and the claim here is that theydon’t have any sort of being whatsoever, so this is not aMeinongian view); but
sentences like “3 is prime” are still literally true.

Thus, on this view, a claim about an objecta can be true evenif that object doesn’t exist at all. Let’s call this viewthin-truth-ism. Views of this general kind have been endorsedby Azzouni (1994, 2004), Salmon (1998), and Bueno (2005).

Thin-truth-ists endorse a similar view of existence claims. Forinstance, on their view, the sentence “There are infinitely manyprime numbers” is literally true, even though there are no suchthings as numbers. This might look like a contradiction, butit’s not, because according to thin-truth-ism, existentialexpressions (or quantifiers) like ‘there is’ areambiguous.

Most philosophers find this view extremely hard to believe. Indeed, alot of philosophers would say that it’s simply confused, orincoherent. But, in fact, thin-truth-ism is not incoherent. A betterway to formulate the problem with the view is as follows: in giving upon the standard criterion of ontological commitment, thin-truth-istsseem to be using “true” in a non-standard way. Most of uswould 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 aboutthe number 3), then it follows trivially that “3 is prime”could not be true. Or more generally, most of us would say that ifthere is no such thing as the objecta, then sentences of theform “a isF” cannot be literally true.This, of course, is just to say that most of us accept the standardcriterion of ontological commitment articulated insection 3; but the point here is that this criterion seems to be built into thestandard meaning of words like “true”. Indeed, thisexplains why the standard criterion of ontological commitment is sowidely accepted.

Here’s a quick argument for thinking that thin-truth-ists areusing “true” in a non-standard way:

sentences like “Mars doesn’t exist at all, butit’s red” seem contradictory to us (ordinary speakers ofEnglish); but
thin-truth-ism implies that they’re not contradictory; andso
thin-truth-ism is out of step with ordinary usage.

Another worry one might raise about thin-truth-ism is that it onlydiffers from fictionalism in a merely verbal way. Let“thin-true” express the kind of truth that thin-truth-istshave in mind, and let “thick-true” express the kind oftruth 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 thefollowing claims:

platonists are right that “3 is prime” is a claimabout the number 3; and
there is no such thing as the number 3; so
“3 is prime” isn’t thick-true; but despitethis,
“3 is prime” is thin-true.

Now, of course, thin-truth-ists and fictionalists will disagree aboutwhether “3 is prime” istrue, but this willcollapse into a disagreement about whether thin-truth or thick-truthisreal truth, and this is just a disagreement about what theword “true” means in ordinary folk English, and it’shard to see why an empirical question about how the folk happen to usesome word is relevant to the debate about the existence ofmathematical objects.

(Meinongians would also reject premise[3]; but Meinongianism is not happily thought of as a version ofnominalism. Indeed, as we saw insection 2, it seems that Meinongianism is a non-standard version of platonism;for it implies that thereare numbers and that they arenon-physical, non-mental, non-spatiotemporal, and so on.)

4.1.6 Carnapian Conventionalism and the Rejection of Premise [10]

A final version of anti-platonism, which we can callconventionalism, involves the rejection of premise[10]. According to this view, sentences like “3 exists”, and“There are numbers” are true because they’reanalytic—i.e., because they follow from the semanticrules of the language. In essence, according to this view, we’vejust collectivelystipulated—through our use ofmathematical sentences like “There are some prime numbersbetween 10 and 20”—that these sentences are to be true.Thus, nothing is required of reality to make these sentences true, andso it doesn’t follow from the truth of “There arenumbers” that there are abstract objects that actually exist inreality. Views of this general kind have been endorsed by Carnap(1950), Rayo (2013), Warren (2020), and perhaps Hale and Wright(2009). Jenkins (2008) also endorses aview in this general vicinity, although her view is a bitdifferent.

It can be hard to pin down the difference between this view andfictionalism. Warren seems to think that the difference boils down toametasemantic disagreement. In particular, borrowing somelingo from Warren, we can say that whereas fictionalists endorse abottom-up metasemantics (i.e., a view that says that semanticfacts about sentences are determined by semantic facts about (a) wordsand (b) compositional semantic rules), conventionalists endorse atop-down metasemantics (i.e., a view that says that semanticfacts about subsentential expressions are determined by semantic factsabout sentences). More specifically, on a Warren-style top-downmetasemantics, the principle of charity (which tells us to interpretpeople, as far as possible, as endorsing truths) more or less dictatesthat sentences like “3 is prime” are true.

The standard view since Quine (1948), in analytic philosophy, is thatconventionalist views of the kind under discussion here aren’ttrue because sentences of the form “a isF”and “There are someFs”—aren’tanalytic. The reason these sentences aren’t analytic accordingto this view is that

the standard criterion of ontological commitment articulated insection 3 is true, and so sentences of the form “a isF” and “There are someFs” can’tbe true unless objects of the relevant kinds actually exist inreality; and
we can’t define objects into existence.

4.2 Properties and Relations

We can construct an ontological-commitment argument for a platonisticview of properties and relations—analogous to the argument in[1]–[11]—by locating some ordinary sentences that seem to be both true and aboutproperties and/or relations. We will focus on sentences aboutproperties, but exactly parallel remarks could be made aboutrelations. You might begin here by considering sentences like thefollowing:

[P1]: Mars possesses the property of redness.

But,prima facie, it seems that we can plausibly take [P1] tobe equivalent to the following:

[P2]: Mars is red.

And if[P1] is equivalent to[P2], then we can endorse a paraphrase-nominalist view of [P1] by pointingout that, according to the standard criterion of ontologicalcommitment, [P2] doesn’t commit us to the existence of aproperty. But there are other sentences about properties thatdon’t seem so easily paraphrased—e.g., sentences like

[P3]: There’s at least one property that Mars and Venus bothpossess.

It’s at least initially plausible to suppose that[P3] is true, and that it commits to the existence of a property, and thatproperties are abstract objects. If we wanted to, we could constructan argument here—an argument for a platonistic view ofproperties that began with the claim that [P3] is true—thatexactly paralleled the argument in[1]–[11], i.e., the argument for a platonistic view of numbers articulated insection 4.1. We won’t bother to do this here, but it’s important tonote that ontological-commitment arguments for the existence ofproperties rely on premises that are exactly analogous to the premisesin the argument in [1]–[11]. And this means that all the sameavenues of response are available to anti-platonists—and, hence,that platonists would need to undermine all of the variousanti-platonist views.

For example, platonists would need to undermineconceptualism—the view that properties are mentalobjects that exist in our heads, rather than abstract objects. Many ofthe arguments listed insection 4.1.1 against psychologistic views of numbers also tell against theconceptualistic view of properties and relations. For instance, asRussell (1912: chapter IX) points out, property claims and relationalclaims seem to beobjective; e.g., the fact that MountEverest is taller than Mont Blanc is a fact that holds independentlyof us; but conceptualism about universals entails that if we all died,it would no longer be true that Mount Everest bears the taller thanrelation to Mont Blanc, because that relation would no longer exist.And, second, conceptualism seems simply to get the semantics of ourproperty discourse wrong, for it seems to confuse properties with ourideas of them. The English sentence “Red is a color” doesnot seem to be about anybody’s idea of redness; it seems to beabout redness, the actual color, which, it seems, is somethingobjective.

Second, platonists would need to undermine the immanent realist viewthat properties existin the objects that instantiatethem—either asuniversals (so that, e.g., there isone property of roundness that exists in Mars and Venus andso on), or astropes (so that, e.g., there is aroundness-trope that exists in Mars, and adifferentroundness-trope, and so on). There are many different arguments thatplatonists might run against immanent realist views of properties. Forexample, they might argue that (a) immanent realism is incompatiblewith the existence ofuninstantiated properties, and (b) weneed to countenance the existence of such properties. One reason tothink that we need to endorse the existence of uninstantiatedproperties is that we need them to be the semantic contents ofuninstantiated predicates; e.g., you might think that we need toendorse the existence of the propertybeing a witch toaccount for the meaningfulness of sentences like “There are nowitches”—and for the truth of sentences like “Janebelieves that the property of being a witch isuninstantiated”.

Third, platonists would need to respond to paraphrase nominalistviews. There are a few different recent nominalization programs worthconsidering here. First, Moltmann (2013b) argues that words that seemto refer to universals (e.g., “redness”) actually referplurally to all the possible tropes of the given kind; so, e.g., onthis view, “redness” refers plurally to all possibleredness tropes. One problem with this view is that it commits to theexistence ofpossible objects; but if these aren’tabstract objects, then they’re presumably Lewisian possibleobjects; and one might reasonably want to avoid committing to theexistence of Lewisian possible objects.

Another paraphrase-type view, endorsed by Himelright (2021), is toparaphrase sentences like[P3] with counterfactual sentences like the following:

[P3a]: If the entire plenitude of Lewisian worlds had existed, then itwould have been the case that there was at least oneone-place-predicate token that applied to both Mars and Venus.

One problem with[P3a] is that it clearly doesn’t capture the actual ordinary-languagemeaning of[P3]. Himelright acknowledges this and offers [P3a] as areplacement—in the sense ofsection 4.1.4. But given this, Himelright’s view isn’t really aparaphrase view; it is rather a fictionalist/error-theoretic viewaccording to which [P3] is literally false. In particular, itisn’t all that different from Balaguer’s (2021)error-theoretic view, according to which sentences like [P3] are falsebutfor-all-practical-purposes true, where a sentence isfor-all-practical-purposes true if and only if it would have been trueif the entire plenitudinous realm of properties had existed.

A final paraphrase-type view of our talk of properties and relationscan be given in ahigher-order logical language. For example,we could paraphrase[P3] with the following sentence:

[P3b]: \(\exists X(Xm \amp Xv)\),

where “m” denotes Mars, “v”denotes Venus, “\(\exists\)” is a second-order quantifierthat quantifies over predicate positions, and “X”is a second-order variable. Now, on one way of indicating what[P3b] means—and until recently, this was the standard reading of suchsentences—[P3b] is no help to anti-platonists because it commitsto the existence of properties. But in recent years, higher-ordermetaphysicians have pointed out that we don’t have to interprethigher-order sentences like [P3b] as being about properties. Rather,we can just learn higher-order languages on their own terms,without translating them into English, and we can take [P3b]as saying something that isn’t about properties at all.It’s simply a higher-order way of generalizing from[P3]. (Dorr forthcoming develops an argument for a view of this kind, butsee also Bacon 2024 and Fritz & Jones 2024b.)The most interesting version of thehigher-order view, in my opinion, is the view that commits to the ideathat the higher-order sentences in question are trueparaphrases, capturing the actual meanings ofordinary-language claims about properties, as opposed to replacements.But the paraphrase version of the view is a controversial empiricalclaim, and it’s not clear that there’s very good evidencefor it.

Another view that employs higher-order resources, developed by Trueman(2024) and Button and Trueman (2024), combines afictionalist/error-theoretic view ofuniversals with arealist view ofproperties, where

universals are (or aresupposed to be) objects of a kindthat are picked out by words like “wisdom” and“redness”, and
properties aren’t objects at all and are, rather,higher-type entities picked out by expressions like “…iswise” and “…is red”.

It might seem that this view is just a non-standard version ofplatonism—because it might seem that properties of this kindcould only be abstract entities. And this is especially true if weendorse the existence ofuninstantiated properties, as Buttonand Trueman do. Now, higher-order-ists of the Button-Trueman kindcould respond here by claiming that the idea that these properties areabstract is unintelligible—because if we tried to articulatethis idea, we’d end up saying things like “Is red isabstract”, which isn’t even grammatical. But there mightbe ways to press the issue here and force higher-order-ists to choosebetween

a non-standard version of platonism according to which propertiesare non-physical, non-mental entities, and
a nominalistic view according to which properties don’treally exist in an “out-there-in-reality” kind ofway.

4.3 Propositions

We turn now to ontological-commitment arguments for the existence ofpropositions. To construct an argument of this kind, we need to findsome sentences that involve ontological commitments to propositions.Probably the most widely discussed sentences here arebeliefascriptions, i.e., sentences like the following:

[O]: Obama believes that snow is white.

The idea, then, is to argue for a platonistic view of propositions byarguing that

sentences like[O] are true, and
they involve ontological commitments to propositions, and
propositions are abstract objects.

(The most important figure in the development of arguments of thiskind is Frege [1892, 1918/19]. Other relevant figures [whowouldn’t all endorse an argument of this kind] include Russell[1905, 1911], Church [1950, 1954], Quine [1956], Kaplan [1968, 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].)

Why think that sentences like[O] involve ontological commitments to propositions? Well, the firstpoint to note here is that these sentences contain“that”-clauses—where a“that”-clause is simply the word “that” addedto the front of a complete sentence, as in, e.g., “that snow iswhite”. And the second point to note is that, in English,“that”-clauses are singular terms. A common wayto illustrate this second point—see, e.g., Bealer (1982 and1993) and Schiffer (1994)—is to appeal to arguments like thefollowing:

I.: Obama believes that snow is white.
Therefore, Obama believes something (namely, that snow is white).

This argument seems to be valid, and platonists claim that the bestand only tenable explanation of this fact involves a commitment to theidea that the “that”-clause in this argument—i.e.,“that snow is white”—is a singular term. So,returning to[O], the idea here is that the logical form of [O] is “o isB-related to t”, where “B” expresses atwo-place belief relation, “o” denotes Obama, and“t” denotes the thing that Obama is being said tobelieve—namely,that snow is white. And so if [O] istrue (and it sure seems to be), then according to thestandard criterion of ontological commitment, the thing that Obama isbeing said to believe—namely,that snow iswhite—has to actuallyexist.

But what kinds of object is this? What kinds of objects do our“that”-clauses refer to? It might seem that they refer tofacts, or tostates of affairs. For instance, itmight seem that “that snow is white” refers to the factthat snow is white. This, however, is a mistake (at least inconnection with the “that”-clauses that appear in beliefreports). For since beliefs can be false, it follows that the“that”-clauses in our belief reports refer to things thatcan be false. For example, if Sammy is seven years old, then thesentence “Sammy believes that snow is powdered sugar”could be true; but if this sentence is true, then its“that”-clause refers to doesn’t refer to a fact,because (obviously) there is no such thing as the fact that snow ispowdered sugar.

These considerations suggest that the referents of the“that”-clauses that appear in belief ascriptions arethings that can be true or false. But if this is right, then it seemsthat the objects of belief—i.e., the things that the“that”-clauses in our belief sentences refer to—areeithersentences orpropositions. The standardplatonist view is that they are propositions. Before we consider theirarguments for this claim, we need to say a few words about thedifferent kinds of sentential views that one might endorse.

To begin with, we need to distinguish between sentencetypesand sentencetokens. To appreciate the difference, considerthe following indented sentences:

Cats are cute.
Cats are cute.

We have here two different tokens of a single sentence type. Thus, atoken is an actual physical thing, located at a specific place inspacetime; it is a pile of ink on a page (structured in an appropriateway), 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 tokenednumerous times but is not identical with any single token. Thus, asentence type is an abstract object. And so if we are looking for ananti-platonist view of what “that”-clauses refer to, orwhat belief reports are about, we cannot say that they’re aboutsentence types; we have to say they’re about sentencetokens.

A second distinction that needs to be drawn here is between sentencetokens that areexternal, or public, and sentence tokens thatareinternal, or private. Examples of external sentencetokens were given in the last paragraph—piles of ink, soundwaves, and so on. An internal sentence token, on the other hand,exists inside a particular person’s head. There is a wide-spreadview—due mainly to Jerry Fodor (1975 and 1987) but adopted bymany others, e.g., Stich (1983)—that we are able to performcognitive tasks (e.g., think, remember information, and have beliefs)only because we are capable of storing information in our heads in aneural language (often calledmentalese, or thelanguageof thought). In connection with beliefs, the idea here is that tobelieve that, say, snow is white, is to have a neural sentence storedin your head (in a belief way, as opposed to a desire way, or someother way) that means in mentalese that snow is white.

This gives us two different anti-platonist alternatives to the viewthat belief reports involve references to propositions. First, thereis the conceptualistic (or mentalistic) view that belief reportsinvolve references to sentences in our heads, or mentalese sentencetokens. And, second, there is the physicalistic view that beliefreports involve references to external sentence tokens, i.e., to pilesof 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 beliefreports cannot be taken to be about (internal or external) sentencesand that we have to take them to be about propositions. We willrehearse one such argument here, an argument that goes back at leastto Church (1950). Suppose that Boris and Jerry both live in coldclimates and are very familiar with snow. Thus, they both believe thatsnow 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 bothbelieve, namely, that snow is white.

This argument seems clearly valid; but this seems to rule out the ideathat the belief reports here are about sentence tokens. For (a) inorder to account for the validity of the argument, we have to take thetwo “that”-clauses to refer to the same thing, and (b)there is no sentence token that they could both refer to. First ofall, they couldn’t refer to any external sentence token (or, forthat matter, any sentence type associated with any natural language),because

if the first “that”-clause refers to such a sentence,it would presumably be a Russian sentence, since Boris speaks only Russian;^[9] and
if the second “that”-clause refers to such asentence, it would presumably be an English sentence, since Jerryspeaks only English; and so
the two “that”-clauses cannot both refer to the sameexternal sentence token (or natural-language sentence type).

And second, they cannot refer to any mentalese sentence token, because

if the first “that”-clause refers to such a sentence,it would presumably be in Boris’s head; and
if the second “that”-clause refers to such asentence, it would presumably be in Jerry’s head; and so
the two “that”-clauses cannot both refer to the samementalese sentence token.

Therefore, it seems to follow that the “that”-clauses inthe above argument do not refer to sentence tokens of any kind. Andsince these are ordinary belief ascriptions, it follows that, ingeneral, the “that”-clauses that appear in ordinary beliefascriptions do not refer to sentence tokens.

Now, as it’s formulated here, this argument doesn’t ruleout the view that “that”-clauses refer to mentalesesentence types, but the argument can be extended to rule out that viewas well (e.g., one might do this by talking not of an American and aRussian but of two creatures with different internal languages ofthought). We 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 areabstract objects. But if we assume that that version of the argumentis cogent as well, then it follows that “that”-clausesdon’t refer to sentences of any kind at all and, so it seemsthat they must refer to propositions.^[10]

It’s important to note that the issue so far has been purelysemantic. What the above argument suggests is that regardlessof whether there are any such things as propositions, our“that”-clauses are best interpreted as referring (orpurporting to refer) to such objects. Platonists then claim that ifthis is correct, then there must be such things as propositions,because, clearly, many of our belief ascriptions aretrue.For instance, “Obama believes that snow is white” is true;thus, if the above analysis of “that”-clauses is correct,and if the standard criterion of ontological commitment is correct, itfollows that there is such a thing as the proposition that snow iswhite.

Finally, it seems that propositions could only be abstract objects.Indeed, the above arguments against the idea that“that”-clauses refer to sentences already seems to ruleout physicalistic and psychologistic views of propositions.

So that is how the ontological-commitment argument for a platonisticview of propositions proceeds.

How might anti-platonists respond to this argument? Given theimplausibility of physicalistic and psychologistic views ofpropositions, it seems that the only hope for anti-platonists is todevelop a nominalistic views. One way to do this would be to develop afictionalistic/error-theoretic view. Balaguer (1998b) develops a viewof this kind that mirrors his fictionalistic view of mathematics. Onthis view, sentences like[O] are strictly speaking false (because their “that”-clauseare supposed to refer to propositions, and there are no such things aspropositions) but they’re for-all-practical-purposes truebecause they would have been true if the entire plenitude ofpropositions existed.

A second way for nominalists to proceed would be to develop aparaphrase view. The paraphrase views of mathematical discoursedon’t carry over very well to the case of propositions, but someof the paraphrase views of properties seem more applicable. Perhapsmost notably, the higher-order views of properties discussed insection 4.2 seems to carry over seamlessly to the case of propositions. E.g.,Dorr’s (forthcoming) paraphrase view seems to work in the sameway; for just as we can quantify over predicate positions withoutclaiming that we’re quantifying over properties, so too we canquantify over sentence positions without claiming that we’requantifying over propositions.

Finally, the Moltmann (2013b) view of properties discussed insection 4.2 carries over to the case of propositions as well, and so does theButton-Trueman view of properties discussed insection 4.2 (see Trueman 2020 and Button & Trueman 2024).

4.4 Sentence Types

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

[A]: “The cat is on the mat” is a well-formed sentence ofEnglish,

and

[B]: “Visiting relatives can be boring” is structurallyambiguous.

The quoted sentences that appear in [A] and [B] are singular terms;e.g., “‘The cat is on the mat’” refers to thesentence “The cat is on the mat”, and [A] says of thissentence that it has a certain property, namely, that of being awell-formed English sentence. Thus, sentences like [A] commit to theexistence of sentences. And so if [A] is true, thensentencesexist.

Now, one might hold a physicalistic view here according to whichlinguistics is about actual (external) sentence tokens, e.g., piles ofink and verbal sound waves. (This view was popular in the early partof the twentieth century—see, e.g., Bloomfield 1933, Harris1954, and Quine 1953.) Or alternatively, one might hold aconceptualistic view, maintaining that linguistics is essentially abranch of psychology; the main proponent of this view is Noam Chomsky(1965: chapter 1), who thinks of a grammar for a natural language asbeing about an ideal speaker-hearer’s knowledge of the givenlanguage, but see also Sapir (1921), Stich (1972), and Fodor (1981).But there are reasons for thinking that neither the physicalist northe conceptualist approach is tenable and that the only plausible wayto interpret linguistic theory is as being about sentence types, whichof course, are abstract objects (proponents of the platonistic viewinclude Katz [1981], Soames [1985], and Langendoen & Postal[1984]). Katz constructs arguments here that are very similar to theargument for a platonistic view of numbers discussed insection 4.1). One argument here is that linguistic theory seems to haveconsequences that are (a) true and (b) about sentences that have neverbeen tokened (internally or externally), e.g., sentences like“Green umbrellas slithered unwittingly toward Arizona’sfavorite toaster”. (Of course, now that we’ve written thissentence down, it has been tokened, but it seems likely that before wewrote it down here, it had never been tokened.) Standard linguistictheory entails that many sentences that have never been tokened(internally or externally) are well-formed English sentences. Thus, ifwe want to claim that our linguistic theories are true, then we haveto accept these consequences, or theorems, of linguistic theory. Butthese theorems are clearly not true of any sentence tokens (becausethe sentences in question have never been tokened) and so, it isargued, they must be true of sentence types.

4.5 Possible Worlds

It is a very widely held view among contemporary philosophers that weneed to appeal to entities known aspossible worlds in orderto account for various phenomena. There are dozens of phenomena thatphilosophers have thought should be explained in terms of possibleworlds, but to name just one, it is often argued that semantic theoryis best carried out in terms of possible worlds. Consider, forexample, the attempt to state the truth conditions of sentences of theform “It is necessary thatS” and “It ispossible thatS” (whereS is any sentence). It iswidely believed that the best theory here is that a sentence of theform “It is necessary thatS” is true if and onlyifS is true in all possible worlds, and a sentence of the form“It is possible thatS” is true if and only ifS is true in at least one possible world. Now, if we add tothis theory the premise that at least one sentence of the form“It is possible thatS” is true—and thisseems undeniable—then we are led to the result that possibleworlds exist.

Now, as was the case with numbers, properties, and sentences, noteveryone who endorses possible worlds thinks that they are abstractobjects; indeed, one leading proponent of the use of possible worldsin philosophy and semantics—namely, David Lewis(1986)—maintains that possible worlds are of the same kind asthe actual world, and so he takes them to be concrete objects.However, most philosophers who endorse possible worlds take them to beabstract 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 kindof abstract object. For instance, it is very popular to maintain thata possible world is just a set of propositions. (To see how a set ofpropositions could serve as a possible world, notice that if youbelieved in full-blown possible worlds—worlds that are just likethe actual world in kind—then you would say that correspondingto each of these worlds, there is a set of propositions thatcompletely and accurately describes the given world, or is true ofthat world. Many philosophers who don’t believe in full-blownpossible worlds maintain that these sets of propositions are goodenough—i.e., that we can take them tobe possibleworlds.) Or alternatively, one might think of a possible world as astate of affairs, or as away things could be. In sodoing, one might think of these as constituting a new kind of abstractobject, or one might think of them as properties—giant, complexproperties that the entire universe may or may not possess. Forinstance, one might say that the actual universe possesses theproperty of being such that snow is white and grass is green and SanFrancisco is north of Los Angeles, and so on.

In any event, if possible worlds are indeed abstract objects, and ifthe 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 ofFs is identical to the number ofGsif and only if there is a one-to-one correspondence between theFs and theGs.
[E]: The direction of linea is identical to the direction oflineb if and only ifa is parallel tob.
[F]: The shape of figurea is identical to the shape of figureb if and only ifa is geometricallysimilar tob.

On Frege’s view, principles like these are true, and so theycommit us to the existence of numbers, lines, and shapes. Now, ofcourse, we have already gone through a platonisticargument—indeed, a Fregean argument—for the existence ofnumbers. Moreover, the standard platonist view is that the argumentfor the existence of mathematical objects is entirely general,covering all branches of mathematics, including geometry, so that onthis view, we already have reason to believe in lines and shapes, aswell as numbers. But it is worth noting that in contrast to mostcontemporary platonists, Frege thought of numbers, lines, and shapesas logical objects, because on his view, these things can beidentified withextensions of concepts. What is the extensionof a concept? Well, simplifying a bit, it is just the set of thingsfalling under the given concept. Thus, for instance, the extension ofthe conceptwhite is just the set of white things.^[11] And so the idea here is that since logic is centrally concerned withpredicates and their corresponding concepts, and since extensions aretied to concepts, we can think of extensions as logical objects. Thus,since Frege thinks that numbers, lines, and shapes can be identifiedwith extensions, on his view, we can think of these things as logicalobjects.

Frege’s definitions of numbers, lines, and shapes in terms ofextensions can be formulated as follows:

the number ofFs is the extension of the conceptequinumerous withF (that is, it is the set of allconcepts that have exactly as many objects falling under them as doesF); and
the direction of linea is the extension of the conceptparallel to a; and
the shape of figurea is the extension of the conceptgeometrically similar to a.

A similar approach can be used to define other kinds of logicalobjects. For instance, the truth value of the propositionp canbe identified with the extension of the conceptequivalent top (i.e., the concepttrue if and only ifp istrue).

It should be noted that contemporary neo-Fregeans reject theidentification of directions and shapes and so on with extensions ofconcepts. They hold instead that directions and shapes aresuigeneris abstract objects.

For contemporary work on this issue, see, e.g., Wright (1983), Boolos(1987), 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 thatfictional objects, or fictional characters, are best thought of asabstract objects. (Salmon [1998], Thomasson [1999], and Voltolini[2020] also take fictional objects to be abstract, but their views area bit different; they maintain that abstract fictional objects arecreated 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 storiesby Arthur Conan Doyle, then that token of it would not betrue—it would be a bit of fiction. But if you were telling achild about these stories, and the child asked, “What doesHolmes 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 refersto, according to the view in question, is an abstract object, inparticular, a fictional character. In short, present-day utterances of[G] are true statements about a fictional character; but if Doyle hadput [G] into one his stories, it would not have been true, and itssingular 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 andlegs; but if Sherlock Holmes is an abstract object, as this viewsupposes, then it does not have arms and legs (because abstractobjects are non-physical); therefore, it cannot be the case thatSherlock Holmes exists and is an abstract object, for this leads tocontradiction. Various solutions to this problem have been proposed.For instance, Zalta argues that in addition toexemplifyingcertain properties, abstract objects alsoencode properties.The fictional character Sherlock Holmes encodes the properties ofbeing a detective, being male, being English, having arms and legs,and so on. But it does not exemplify any of these properties. Itexemplifies the properties of being abstract, being a fictionalcharacter, having been thought of first by Arthur Conan Doyle, and soon. Zalta maintains that in English, the copula“is”—as in “a isF”—is ambiguous; it can be read as ascribing eitherproperty exemplification or property encoding. When we say“Sherlock Holmes is a detective”, we are saying thatHolmesencodes the property of being a detective; and when wesay “Sherlock Holmes is a fictional character”, we aresaying that Holmesexemplifies the property of being afictional character. (It should be noted that Zalta employs the deviceof encoding with respect to all abstract objects—mathematicalobjects, logical objects, and so on—not just fictional objects.Also, Zalta points out that his theory of encoding is based on asimilar theory developed by Ernst Mally [1912].)

Those who endorse a platonistic view of fictional objects maintainthat 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” istrue-in-the-Holmes-stories.

If we read[G] in this way, then it is not about Sherlock Holmes at all; rather, itis about the Sherlock Holmes stories. Thus, in order to believe [G],so interpreted, one would have to believe in the existence of thesestories. Now, one might try to take an anti-platonistic view of thenature of stories, but there are problems with such views, and so wemight end up with a platonistic view here anyway—a view thattakes sentences like [G] to be about stories and stories to beabstract objects of some sort, e.g., ordered sets of propositions.^[12] Which of these platonistic views is superior can be settled bydetermining which (if either) captures the correct interpretation ofsentences like [G]—i.e., by determining whether ordinary peoplewho utter sentences like [G] are best interpreted as talking aboutstories or fictional characters.

It should be noted that some people who take fictional characters tobe abstract objects (e.g., Thomasson 1999) would actually agree withthe idea that [G] should be read in the above way—i.e., as aclaim about the Sherlock Holmes stories and not about Sherlock Holmeshimself. Thomasson’s main argument for believing in fictionalcharacters is based not on sentences like [G] but rather on sentenceslike the following:

[H]: Some 19th century heroines are better developed than any 18thcentury 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 afictionalist (i.e., an error-theoretic) view of sentences like[H]. In other words, one could admit that [H] is a claim about fictionalcharacters and then one could claim that since there are no suchthings as fictional characters, [H] is simply not true, although ofcourse it might betrue-in-the-story-of-fictional-characters,orfor-all-practical-purposes true, where this just meansthat it would have been true if there had been a realm of fictionalcharacters of the sort that platonists believe in. (Brock [2002]endorses a fictionalist view of fictional characters that’ssimilar in spirit to the view alluded to here.)

5. The Epistemological Argument Against Platonism

In this section, we will consider what is widely considered thestrongest argument against platonism, namely, theepistemological argument.

(Before discussing the epistemological argument, it’s worthnoting that there are many other important arguments againstplatonism. Perhaps most obviously, there is an argument againstplatonism based on Ockham’s razor—an argument that goesback to William of Ockham himself. In the second half of the twentiethcentury, the multiple-reductions argument developed by Benacerraf[1965] received a lot of attention. Andeven more recently, anti-platonist arguments have been developed byBuiles [2022], Himelright [2023b], and Goodman [2024].)

The epistemological argument goes all the way back to Plato, but ithas received renewed interest since 1973, when Paul Benacerrafpresented a version of the argument. Most of the work on this problemhas taken place in the philosophy of mathematics, in connection withthe platonistic view of mathematical objects like numbers. We willtherefore discuss the argument in this context, but all of the issuesand arguments can be reproduced in connection with other kinds ofabstract objects. The argument can be put in the following way:

[B1]: Human beings exist entirely within spacetime.
[B2]: If there exist any abstract mathematical objects, then they donot exist in spacetime. Therefore, it seems very plausible that:
[B3]: If [B1] and [B2] are true, then if there exist any abstractmathematical objects, then human beings could not attain knowledge ofthem. Therefore,
[B4]: If there exist any abstract mathematical objects, then humanbeings could not attain knowledge of them. Therefore,
[B5]: If mathematical platonism is true, then human beings could notattain mathematical knowledge. But
[B6]: Human beings have mathematical knowledge. Therefore,
[B7]: Mathematical platonism is not true.

(Before discussing possible responses to the argument, it’sworth noting that there’s a sizable literature on how exactlythe epistemological argument should be formulated. Field [1989]famously articulated the argument as a challenge to explainthereliability of our mathematical beliefs. Donaldson [2014] arguesthat Field’s version of the argument shouldn’t beunderstood in terms of our mathematical beliefs counterfactuallydepending on the mathematical facts. Benacerraf’s originalversion of the argument relied on a causal theory of knowledge; thatreliance has been almost universally rejected; but Nutting [2016]provides a causal version of the epistemological argument thatdoesn’t rely on a causal theory of knowledge. Finally, worriesabout the epistemological argument are raised by Linnebo [2006] andClarke-Doane [2016, 2020a], and a response is given in Berry2020.)

There seem to be only three ways for platonists to respond to thisargument—they have to reject[B1],[B2], or[B3]. For there is no reasonable way for them to reject—and theypresumably wouldn’twant to reject—[B6] or any of the three inferences in the argument. Also, it’s notenough for platonists to simply reject one of these three premises andleave it at that. The epistemological argument is best thought of as achallenge to explain how we humans could acquire knowledge of abstractobjects; and it seems that a satisfying response to the argument wouldhave to deliver such an explanation.

The first way for platonists to respond to the epistemologicalargument is to reject[B1] and to argue that the human mind is capable of somehow forgingcontact with abstract mathematical objects and thereby acquiringinformation about such objects. This strategy has been pursued byPlato inThe Meno andThe Phaedo, and by Gödel(1947). Plato’s idea is that our immaterial souls acquiredknowledge of abstract objects before we were born and thatmathematical learning is really just a process of coming to rememberwhat we knew before we were born. On Gödel’s version of theview, we acquire knowledge of abstract objects in much the same waythat we acquire knowledge of concrete physical objects; morespecifically, just as we acquire information about physical objectsvia the faculty of sense perception, so we acquire information aboutabstract objects by means of a faculty ofmathematicalintuition. Now, other philosophers have endorsed the idea that wepossess a faculty of mathematical intuition, but Gödel’sversion of this view—and he seems to be alone inthis—involves the idea that the mind is non-physical in somesense and that we are capable of forging contact with, and acquiringinformation from, non-physical mathematical objects.^[13] This view has been almost universally rejected. One problem is thatdenying [B1] doesn’t seem to help. The idea of an immaterialmind receiving information from an abstract object seems just asmysterious and confused as the idea of a physical brain receivinginformation from an abstract object.

The second strategy that platonists can pursue in responding to theepistemological argument is to argue that[B2] is false and that human beings can acquire information aboutmathematical objects via normal perceptual means. The early Maddy(1990) pursued this idea in connection with set theory, claiming thatsets of physical objects can be taken to exist in spacetime and,hence, that we can perceive them. For instance, on her view, if thereare two books on a table, then the set containing these books existson the table, in the same place that the books exist, and we can seethe set and acquire information about it in this way. This view hasbeen subjected to much criticism, including arguments from the laterMaddy (1997). Others to attack the view include Lavine (1992),Dieterle and Shapiro (1993), Milne (1994), Riskin (1994), and Carson(1996).

It may be objected that according to the definitions we’ve beenusing, views like Maddy’s are not versions of platonism at all,because they do not take mathematical objects to exist outside ofspacetime. Nonetheless, there is some rationale for thinking ofMaddy’s view as a sort of non-traditional platonism. For sinceMaddy’s view entails that there is an infinity of setsassociated with every ordinary physical object, all sharing the samespatiotemporal location and the same physical matter, she has to allowthat these sets differ from one another in some sort of non-physicalway and, hence, that there is something about these sets that isnon-physical, or perhaps abstract, in some sense of these terms. Now,of course, the question of whether Maddy’s view counts as aversion ofplatonism is purely terminological; but whateverwe say about this, the view is still worth considering in the presentcontext, because it is widely thought of as one of the availableresponses to the epistemological argument against platonism, andindeed, that is the spirit in which Maddy originally presented theview.

The third and final strategy that platonists can pursue is to reject[B3]. This has been the most popular strategy among contemporaryplatonists. Its advocates include Quine (1951: §6), Steiner(1975: chapter 4), C. Parsons (1980, 1994), Katz (1981, 1998), Resnik(1982, 1997), Wright (1983), Lewis (1986: §2.4), Hale (1987),Shapiro (1989, 1997), Burgess (1990), Balaguer (1995), Linsky andZalta (1995), Burgess and Rosen (1997), and Linnebo (2006).

The strategy of rejecting[B3] presumably goes hand-in-hand with an assumption that[B1] and[B2] are true—and, hence, that mathematical objects (if there aresuch things) are totally inaccessible to us and that informationcannot pass from mathematical objects to human beings. So the ideabehind this third strategy is to grant that human beings do not haveany information-transferring contact with abstract objects, and toattempt to explain how human beings could nonetheless acquireknowledge of such objects. Now, platonists might try toclaim—and Collard (2007)does claim—that even if [B1] and [B2] are true, therecould still be an information-transferring contact between abstractobjects and human beings, because it could be that abstract objectsare not causally inert. But in order for this response to be at allsatisfying, it would need to come with an explanation ofhowabstract objects could cause us to believe things, and no one has evercome close to giving such an explanation.

(Bengson [2015] has proposed a view according to which ourmathematical intuitions areconstituted by the abstractobjects that they’re about. It’s not clear whether hethinks that this involves an information transfer between abstractobjects and human beings, but either way, this view seems to beundermined by the fact that we don’t have any explanation ofhow it could be that our intuitions are constituted byabstract objects in a way that makes it the case that our intuitionsare reliable indicators of the facts about the relevant abstractobjects.)

In any event, most of the philosophers who have rejected[B3] have been happy to grant that human beings do not have anyinformation-transferring contact with such objects; their aim has beento explain how we could acquire knowledge of abstract objects withoutthe aid of any such contact. We will briefly consider the mostprominent of these attempted explanations.

One version of the reject-[B3] strategy, implicit in the writings ofQuine (1951: §6) and developed by Steiner (1975: chapter four,especially section IV) and Resnik (1997: chapter 7), is to argue thatwe have good reason to believe that our mathematical theories aretrue, even though we don’t have any contact with mathematicalobjects, because

these theories are embedded in our empirical theories, and
these empirical theories (including their mathematical parts)have been confirmed by empirical evidence, and so
we have empirical evidence for believing that our mathematicaltheories are true and, hence, that abstract mathematical objectsexist.

Notice that this view involves the controversial thesis thatconfirmation is holistic, i.e., that entire theories areconfirmed by pieces of evidence that seem to confirm only parts oftheories. 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 thatthis view leaves unexplained the fact that mathematicians are capableof acquiring knowledge of their theories before these theories areapplied in empirical science.

A second version of the reject-[B3] strategy, developed by Katz (1981,1998) and Lewis (1986: §2.4), is to argue that we can know thatour mathematical theories are true, without any sort ofinformation-transferring contact with mathematical objects, becausethese theories arenecessarily true. The reason we needinformation-transferring contact with ordinary physical objects inorder to know what they’re like is that these objects could havebeen different. For instance, we have to look at fire engines in orderto know that they’re red, because they could have been blue. Butwe don’t need any contact with the number 4 in order to knowthat it is the sum of 3 and 1, because it is necessarily the sum of 3and 1. (For criticisms of this view, see Field [1989: 233–38]and Balaguer [1998a: chapter 2, section 6.4].)

A third version of the reject-[B3] strategy has been developed byResnik (1997) and Shapiro (1997). Both of these philosophers endorse(platonistic)structuralism, a view that holds that ourmathematical theories provide true descriptions of mathematicalstructures, which, according to this view, are abstract. Moreover,Resnik and Shapiro both claim that human beings can acquire knowledgeof mathematical structures (without coming into any sort ofinformation-transferring contact with such things) by simplyconstructing mathematical axiom systems; for, they argue, axiomsystems provideimplicit definitions of structures. Oneproblem with this view, however, is that it does not explain how wecould know which of the various axiom systems that we might formulateactually pick out structures that exist in the mathematical realm.

A fourth and final version of the reject-[B3] strategy, developedindependently (and somewhat differently) by Balaguer (1995, 1998a) andLinsky & Zalta (1995), is based on the adoption of a particularversion of platonism calledplenitudinous platonism (Balagueralso calls itfull-blooded platonism, or FBP, and Linsky andZalta call itprincipled platonism). Balaguer definesplenitudinous platonism (somewhat roughly) as the view that thereexist mathematical objects of all possible kinds, or the view that allthe mathematical objects that possibly could exist actually do exist.But, in general, Balaguer would define a different plenitude principlefor every different kind of abstract object. Linsky & Zaltadevelop plenitudinous platonism by proposing a distinctive plenitudeprinciple for each of three basic domains of abstracta: abstractindividuals, relations (properties and propositions), and contingentlynonconcrete individuals (1995: 554). For example, on their view, theplenitude principle for abstract individuals asserts (again, somewhatroughly) that every possible description of an object characterizes anabstract 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 endorseplenitudinous platonism, they can solve the epistemological problemwith platonism without positing any sort of information-transferringcontact between human beings and abstract objects. Balaguer’sversion of the argument proceeds as follows. Since plenitudinousplatonism, or FBP, says that there are mathematical objects of allpossible kinds, it follows that if FBP is true, then every purelymathematical theory that could possibly be true (i.e., that’sinternally consistent) accurately describes some collection ofactually existing mathematical objects. Thus, it follows from FBP thatin order to attain knowledge of abstract mathematical objects, all wehave to do is come up with an internally consistent purelymathematical theory (and know that it is consistent). But it seemsclear that (i) we humansare capable of formulatinginternally consistent mathematical theories (and of knowing that theyare internally consistent), and (ii) being able to do this does notrequire us to have any sort of information-transferring contact withthe abstract objects that the theories in question are about.^[14] Thus, if this is right, then the epistemological problem withplatonism has been solved.

One might object here that in order for humans to acquire knowledge ofabstract objects in this way, they would first need to know thatplenitudinous platonism is true. Linsky & Zalta respond to this byarguing that plenitudinous platonism (or in their lingo, principledplatonism) is knowablea priori because it is required forour understanding of any possible scientific theory: it alone iscapable of accounting for the mathematics that could be used inempirical science no matter what the physical world was like.Balaguer’s response, on the other hand, is based on the claimthat to demand that platonists explain how humans could know that FBPis true is exactly analogous to demanding that external-world realists(i.e., those who believe that there is a real physical world, existingindependently of us and our thinking) explain how human beings couldknow that there is an external world of a kind that gives rise toaccurate sense perceptions. Thus, Balaguer argues that while there maybe some sort of Cartesian-style skeptical argument against FBP here(analogous to skeptical arguments against external-world realism), theargument in[B1]–[B7] is supposed to be a different kind of argument, and in order torespond to that argument, FBP-ists do not have to explain how humanscould know that FBP is true.^[15]

The FBP-based response to the epistemological argument has probablybeen the most widely accepted response in the literature. But somepushback against this response has been given by McSweeney (2020) andClarke-Doane (2020b).

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