Introduction
Some platonists truly agonize over the ontological commitments which their platonism demands of them. Peter van Inwagen, for example, confesses candidly,
I am happy to admit that I am uneasy about believing in the existence of ‘causally irrelevant’ objects. The fact that abstract objects, if they exist, can be neither causes or [sic] effects is one of the many features of abstract objects that make nominalism so attractive. I should very much like to be a nominalist, but I don't see how to be one …Footnote 1
In contrast with certain more sanguine platonists, van Inwagen thinks that we should reject platonism if we can: ‘It would be better not to believe in abstract objects if we could get away with it.’Footnote 2 For it is very puzzling that objects should fall into two so radically different and exclusive categories as abstract and concrete:
The platonist must think of objects, of what there is, as falling into two exclusive and exhaustive categories, the abstract and the concrete. If x falls into one of these categories and y into the other, then no two things could be more different than x and y. According to orthodox Christian theology, no two concrete things could differ more than God and an inanimate object. But (assuming for the sake of the illustration that all three things exist) the differences between God and this pen pale into insignificance when they are compared with the differences between this pen and the number 4; indeed, the number seems no more like the pen than like God. The difference between any abstract object and any concrete object would seem to be the maximum difference any two objects could display.Footnote 3
It would be much more appealing, says van Inwagen, to suppose that one of the categories is empty. But concrete objects are indisputably real and well-understood, in contrast to abstract objects. So we should presume that abstract objects do not exist. Nominalism of some sort is thus the default position. Indeed, van Inwagen believes, ‘one should not believe in abstract objects unless one feels rationally compelled by some weighty consideration or argument… my conclusion is that a philosopher should wish not to be a platonist if it's rationally possible for the informed philosopher not to be a platonist.’Footnote 4
Moreover, van Inwagen, as a classical theist, has theological misgivings about the existence of uncreated objects other than God. Van Inwagen tells us that when he recites the Nicene Creed,
I must regard the phrase ‘creator of all things visible and invisible’ as containing a tacitly restricted quantifier… I commit myself only to the proposition that God is the creator of all things (besides himself) that can in some sense be either causes or effects.Footnote 5
Van Inwagen's platonism is thus a sort of agonistic platonism embraced only with great reluctance and struggle.
Other platonists are much more cavalier about postulating the existence of such radically unfamiliar objects. Theirs is an insouciant platonism. They do not seem at all troubled by adding such objects to one's ontology. Why is that? Whence their insouciance about abstract objects? This question has received scant attention in the philosophical literature. But I want to raise the suspicion that there may be, ontologically speaking, less than meets the eye to some platonists' affirmation of the existence of abstract objects.Footnote 6 Consider three examples.
Bob Hale and Crispin Wright
It is not always clear just what sort of ontological status is being ascribed to abstract objects, even by their most ardent defenders. Consider the neo-Fregean view called ‘Abstractionism’ by its proponents Bob Hale and Crispin Wright. The view is so-named because of the crucial role played by so-called abstraction principles of the form (∀a) (∀b) (Σ(a) = Σ(b) ⇔ E(a, b)), where a and b are variables of a given kind, ‘Σ’ is a term-forming operator denoting a function from items of the given kind to objects in the range of the first-order variables, and E is an equivalence relation over items of the given kind. The principles explain the truth conditions of certain Σ-identities as coinciding with the truth conditions of an equivalence statement which we already understand. For example, the direction of a = the direction of b if and only if a is parallel to b. The left-hand side of this biconditional involves reference to abstract objects, namely, directions, even though the right-hand side makes no mention of such entities. The appearance of such abstract objects, as if by magic, has led some ontologists to suggest that Wright and Hale subscribe, or should subscribe, to either quantifier variance, according to which the first-order existential quantifier has more than one meaning (one ontologically committing and another not), or to maximalism, according to which everything possible actually exists.Footnote 7 Hale and Wright, however, repudiate such interpretations of their view.Footnote 8 They concede, ‘If it looks as if the truth of abstraction principles may turn on substantial metaphysical hostages, or as if there are special problems about knowing that they are true, or can be stipulated to be true, this appearance needs to be disarmed before the abstractionist can expect much sympathy for his proposals.’Footnote 9 Wright and Hale, however, insist that no metaphysical assistance is needed in defense of their principles. ‘There is no metaphysical hostage to redeem. A (good) abstraction itself has the resources to close off the alleged (epistemic metaphysical) possibility [that the denoted abstracts do not exist]… the truth of the right-hand side of an instance of a good abstraction is conceptually sufficient for the truth of the left. There is no gap for metaphysics to plug, and in that sense no ‘metaontology’ to supply.’Footnote 10 This view of the matter, they muse, is essential not only to abstractionism but also to the quantifier variantist ‘rescue’ of abstractionism. Quantifier variance ensures the conservation of truth conditions right to left across the biconditional by taking the right-hand side to be sufficient for existential generalizations of the left-hand side ‘at a purely conceptual level, without collateral metaphysical assumption’.Footnote 11 This ‘is a thesis about what meanings-concepts-there are, not about the World of the metaphysician.’Footnote 12 If the metaphysician spurns this ‘minimalist conception of objects and singular reference’, they warn, then he will have to deny that abstractions can ever be said to be stipulatively true.Footnote 13
It remains quite unclear just to what sort of ontology Wright and Hale think themselves committed by singular reference to abstract objects. Elsewhere Hale differentiates sharply between objects in the ordinary sense and objects in the peculiarly Fregean sense relevant to their view. For Frege, Hale explains, objects constitute one category of entities alongside others, like concepts (properties), relations, and functions. Objects are complete entities, while concepts, relations, and functions are incomplete entities. Hale says that this categorization of non-linguistic entities presupposes a prior, logical categorization of types of linguistic expressions into singular terms and various incomplete expressions, like predicates. In Hale's view we have little insight into the nature of these non-linguistic entities apart from the linguistic expressions which distinguish them: ‘there is no fully general explanation of what it is to be an entity of one of those types save by reference to the type of expressions of which entities of that type are the non-linguistic correlates.’Footnote 14 An object, then, just seems to be the potential referent of a singular term: ‘To be an object, in the sense intended, is just to be the sort of thing that can be referred to by means of a singular term…’Footnote 15 Indeed, in Hale's view once one understands that some expression is functioning as a singular term and one understands what is being referred to, it makes no sense to ask whether the referent is an object: ‘It is, in consequence, simply unintelligible to suppose that someone fully equipped with the relevant notions of object, property, etc., might understand a certain expression, and know which entity it stood for, and yet be in serious doubt about whether that entity is, say, an object or not.’Footnote 16 On this view it would seem fair to say that the weather, for example, is an object, since it may be referred to in true sentences like ‘Today the weather in Atlanta will be stormy.’
Now lest we have reservations about including the weather in our ontological inventory as an existent object, Hale hastens to add that it is crucial that it is the Fregean notion of object that is at issue here. Even if something is an object in the peculiarly Fregean sense, we might still wonder, he says, whether it is an object in the common use of that word. Hale gives the example of a perforated sheet of postage stamps. One might wonder whether the singular term ‘the third from the left in the fourth row’ refers to an object in the ordinary sense of that word. But anyone who understands this expression cannot doubt that it refers to an object in the Fregean sense if it refers to anything at all. It seems to be a conceptual truth that the referent of a singular term is an object in the Fregean sense. Even though Frege uses ‘object’ (Gegenstand) to distinguish certain entities from other entities like concepts, relations, and functions, Hale emphasizes that the Fregean use of the word is, in contrast to the common use of ‘object,’ not a sortal term at all (not even a very general one).
Now if abstract objects are just the referents of certain abstract singular terms but not objects in the ordinary sense of the word, it is far from clear, I think, what the affirmation that abstract objects exist really amounts to ontologically. Hale avers,
If it is taken as invoking the everyday notion of object, the question whether there are abstract objects is devoid of philosophical interest; its answer is quite certainly that there are not, but that is trivial-a great many kinds of thing beside those whose title to be recognized as abstract objects has been taken seriously by philosophers fail to count as objects in that sense. Vague though the common notion is, it is evidently outrageous to suggest that numbers, classes, directions and shapes, say, are objects in that sense. But the same goes for hurricanes, speeches (i.e., the actual historical events) and holes in the ground.Footnote 17
I find Hale's disclaimers to be very puzzling, in large part because it is difficult to understand the distinction he insists on between objects in the ordinary sense and objects in his Fregean sense. Hale does not explain what the ordinary sense of ‘object’ is, other than to say that the word is a sortal term, in contrast to Frege's usage. This is not very helpful, since ‘object’, ‘concept’, ‘relation’, and ‘function’ do, in fact, seem to be functioning for Frege as sortal terms used to classify various entities.Footnote 18
So what is the contrast between Fregean objects and objects in the ordinary sense? The most help we get from Hale is the contrast of ordinary objects with hurricanes, speech events, and holes. This contrast suggests that objects in the ordinary sense are what are classically called substances, things which exist in and of themselves.Footnote 19 If this is right, then I am puzzled by two things. First, if for the platonist abstract objects are not substances, then what are they? On platonism it seems that some abstracta, at least, should be classed as objects in the sense of substances. Numbers seem clearly to be stand-alone objects, if they exist, and the shape of a ball is supposed to be an entity that exists independently of the ball. Even directions could seem to stand alone, as when someone says, ‘He fled in that direction.’ If these are not objects in the ordinary sense, then what are they?
Second, if abstract objects are taken to be merely the sort of things that can be referred to via an abstract singular term, do they exist at all, even for the platonist? The answer to that question will depend on one's theory of reference.Footnote 20 A Gegenstand can be merely the subject of conversation, what one is talking about. Prima facie we talk all the time about things that do not exist. Are abstract objects among these? Hale's examples of things besides abstract objects which are not objects in the ordinary sense does not inspire confidence that abstract objects, even on platonism, truly exist, for hurricanes, speech events, and holes are precisely the sorts of thing that many metaphysicians plausibly deny exist.Footnote 21 These things are real in the sense that they are not illusory, but they are not, properly speaking, existents. If abstract objects have no more reality than holes, then perhaps Hale is right in thinking that the question of their existence is of no philosophical interest, for then the affirmation that they exist may be trivial, having no ontological significance.
Hale goes on to say,
When, as I intend, the question is understood in terms of the Fregean notion of object, it is inseparable from questions belonging to the philosophy of language-asking whether there are objects of a certain general kind is tantamount to asking whether there are, or at least could be, expressions functioning as non-vacuous singular terms of a certain kind. When the domain of objects is understood as including at least the referents of all genuine singular terms, it is anything but obvious that it is does not include abstract objects of various sorts; rather there is a quite strong prima facie case for believing that it does. For there can be no serious doubt that we frequently find ourselves employing what at least appear to be singular terms for entities of many kinds-numbers, classes, shapes, to mention some of the more obvious examples-which, in advance of any deep account of the abstract/concrete distinction, we would have little hesitation in classifying as abstract.Footnote 22
Writing in the aftermath of ‘the linguistic turn’ introduced into philosophy by Frege,Footnote 23 Hale and Wright speak of abstract objects in terms of semantic categories rather than classical metaphysical categories.Footnote 24 Hence, it is hard to know what is the ontological import of non-vacuous singular terms. We may agree without hesitation that we employ singular terms to talk about entities which, if they exist, are abstract; but the problem is that we similarly employ singular terms to talk about things like Wednesdays, shortages, holes, and events, which plausibly do not exist. It might be said that in such cases the relevant singular terms are vacuous and the sentences containing them false. But as mentioned, whether non-vacuous singular terms have real world ontological correlates is going to depend upon one's theory of reference, which at this point remains yet to be enunciated.Footnote 25
Michael Dummett
Michael Dummett, whose earlier work inspired Hale and Wright, is another philosopher whose remarks on the existence of abstract objects are far from perspicuous. For example, noting that informal discourse is permeated by abstract terms, Dummett cities as an illustration the following paragraph from a London daily:
Margaret Thatcher yesterday gave her starkest warning yet about the dangers of global warming caused by air pollution. But she did not announce any new policy to combat climate change and sea level rises, apart from a qualified commitment that Britain would stabilize its emissions of carbon dioxide-the most important ‘greenhouse’ gas altering the climate-by the year 2005. Britain would only fulfill that commitment if other, unspecified nations promised similar restraint.
Dummett then observes,
Save for ‘Margaret Thatcher’, ‘air’ and ‘sea’, there is not a noun or noun phrase in this paragraph incontrovertibly standing for or applying to a concrete object (is a nation a concrete object, or a gas?). Ordinary literate people readily understand such paragraphs; few would be easily able to render them in words involving reference only to concrete objects, if indeed they can be so rendered, or even to understand such a rendering if presented with it. An ordinary reader's comprehension of the abstract terms does not consist in the grasp of any such procedure of translation, but in a knowledge of how those terms function in sentences: no reason whatsoever exists for supposing him to attach a reference to ‘Margaret Thatcher’, but not to ‘the climate’ or ‘air pollution’.
The notion of reference to an object is employed to mark the difference in linguistic function between a singular term and a predicate or relational expression; and that difference is as salient in the sentence ‘Carbon dioxide is a compound’ as in ‘Margaret Thatcher is a woman’. One can know a great deal about Margaret Thatcher without ever having had to identify her; but, to understand a personal name, one has to know that there is a such a thing as identifying its bearer. There being such a thing is what constitutes it as referring to its bearer, and explains our understanding of its use in predicating something of its bearer. Identification of a county, say as that in which one is, of a gas, say as being emitted from an exhaust pipe, of a political group, say as holding a meeting, all differ greatly from identifying a person, because counties, gases and political groups are things of very different kinds from people: but such identifications occur, and play the same fundamental role in our discourse about such things as the identification of people plays in our discourse about them. To deny those things the status of objects, and to the corresponding expressions the function of referring to them, is to fall into nominalist superstition, based ultimately on the myth of the unmediated presentation of genuine concrete objects to the mind.Footnote 26
This is an extraordinarily puzzling passage. Dummett holds that almost all the noun phrases in the cited paragraph do not refer to concrete objects but do refer to objects. Do they refer, then, to abstract objects? Clearly not, for not only are things like air pollution and sea level rises and gas not abstract, but Dummett accepts the characterization of abstract objects as objects having no causal powers,Footnote 27 yet here it is explicitly stated that global warming is caused by air pollution, that carbon dioxide gas alters the climate, and that nations must promise restraint. On pain of contradiction Dummett must be using the word ‘object’ in two different senses. The things mentioned are not objects in what Hale calls the ordinary sense of the term, but they are objects in the semantic sense of being the referents of singular terms. As Dummett says, the notion of reference to an object in the latter sense is just a way of marking the difference between a singular term and a predicate or relational expression. The metaphysician can happily grant the things in question the status of objects in this sense without thinking that his ontology will have to include in it things like dangers, sea level rises, commitments, and restraints.Footnote 28
If this interpretation is correct, it explains Dummett's otherwise strange insouciance about embracing an object so bizarre as the Equator. In response to a sort of dispensability argument for nominalism-what Dummett calls ‘the nominalist challenge’-to the effect that abstract objects, lacking as they do all causal powers, explain nothing and so may be dispensed with, Dummett writes,
In Grundlagen, Frege's examples of objective but non-actual objects are the Equator and the centre of mass of the solar system. The existence of the Equator is certainly an a posteriori truth. It depends on the fact that the Earth has poles, which in turn depends on the unquestionably contingent fact that it spins about an axis. Yet, if someone argued that to assume the existence of the Equator explains nothing, that, moreover, since it has no causal powers, everything would be exactly the same if it did not exist, and that therefore we have no reason to accept the hypothesis of its existence, we should gape at the crudity of his misunderstanding.
What should we say to correct the objector's misunderstanding? He is trying to conceive of the Equator as an actual object that has been stripped of its causal powers; naturally, then, he cannot see what grounds we can have for believing in such an object. We have to teach him that it is an altogether different kind of object. We can do that only by patiently explaining to him the use, or the truth-conditions, of sentences containing the term ‘the Equator’; such an object as the Equator is given to us only by means of our grasp of what can be meaningfully said about it and when it is true to say it. When we have given these explanations, he will grasp that there is nothing problematic about the existence of the Equator; that its existence is not a hypothesis, but stands or falls with the proposition that the Earth rotates about an axis. Or, if he does not, we may abandon him to self-congratulation on his resistance to platonistic superstition.Footnote 29
According to Dummett, Frege's distinction between objects which are or are not actual (wirklich) does not contrast real objects with fictitious objects, but ‘serves, rather, as his manner of distinguishing between what present-day philosophers usually call ‘concrete’ and ‘abstract’ objects’.Footnote 30 The Equator is not wirklich because it has no Wirkungen (effects) upon anything. Now I understand that the platonist does not postulate the Equator as an explanatory hypothesis. But I do not understand the platonist's insouciance about permitting so strange an object into his ontology. The reason Dummett thinks that there is nothing problematic about so weird an abstract object as the Equator is that, I suspect, he himself accepts the Equator's existence merely in the sense of being what ‘the Equator’ refers to in true sentences like ‘The Equator bisects the Congo’ rather than in the sense of the platonistic superstition that one's ontological inventory includes this circular, spatio-temporal, contingent, 25,000 mile long, abstract object currently girdling the earth. It is an object only in the semantic sense of being a referent of a singular term. If this understanding is correct, rejecting the ‘platonistic superstition’ does not entail falling into the ‘nominalist superstition’ of denying that the function of singular terms is to refer to objects.
John Burgess and Gideon Rosen
My final example of insouciant platonists is John Burgess and Gideon Rosen. In criticizing imagined ‘content-hermeneutic nominalism’, they entertain the suggestion that mathematical existence theorems are not ontologically committing to mathematical objects because ‘there is simply an ambiguity in the word ‘exists’, between a strong and a weak sense, which we may write as ‘exists’ and ‘exists.’ (A) [‘There exist prime numbers greater than a thousand’] is supposed to be quite true if by ‘exists’ one means ‘exists’, and to become false only if one takes ‘exists’ to mean ‘exists’’.Footnote 31 Burgess and Rosen rejoin that the nominalist will need to explain these two senses of existence and show that philosophical claims involve one sense while internal mathematical and scientific claims involve the other. Then they add,
Suppose it is said … that for a thing to exist is for it to [sic; supply ‘be’] part of the ultimate furniture of the universe. However this last phrase is interpreted, it seems quite plausible that large, composite objects like the Eiffel Tower do not exist in this sense. But an anti-nominalist may be perfectly willing to grant that the Euler function may not exist in this sense either. The most the anti-nominalist wishes to claim [sic; supply ‘is’] that the Euler function exists in the same sense that the Eiffel Tower does.Footnote 32
They conclude, ‘the genuinely controversial question is whether or not numbers, functions, and the like exist in the sense in which the planet Venus does and the planet Vulcan doesn't.’Footnote 33
It is difficult to know what to make of this. Burgess and Rosen seem to agree with metaphysicians like van Inwagen that chairs, tables, and other alleged composite objects do not exist and therefore are not to be included with, for example, fundamental particles in one's ontological inventory. If mathematical functions are like those things, then they do not exist, which is the nominalistic position. Van Inwagen insists that on his view chairs-and, by extension, the Eiffel Tower and Venus-are real in the sense that they are not illusory.Footnote 34 But they do not exist as unified objects. There literally are no such things. Burgess and Rosen seem to think likewise.
Burgess elsewhere confirms this impression by looking at the dispute in a theological light:
One very traditional sort of way to try to make sense of the question of the ultimate metaphysical existence of numbers would be to turn the ontological question into a theological question: Did it or did it not happen, on one of the days of creation, that God said, ‘Let there be numbers!’ and there were numbers, and God saw the numbers, that they were good? According to Dummett, and according to Nietzsche-or my perspective on Nietzsche-this is the only way to make sense of questions of ontological metaphysics… I myself believe, like Russell, that analytic atheism [the thesis that theological language is meaningless] is false, and suspect, contrary to the Australians, that the Nietzsche–Dummett thesis is true. If as I believe the theological question does make sense, and if as I suspect it is the only sensible question about the italics-added real or capital-R Real existence of numbers, then I would answer that question in the negative; but then I would equally answer in the negative the question of the Real existence of just about anything.Footnote 35
It is clear that Burgess rejects what he calls ‘capital-R Realism’ in favor of a much weaker ‘realism’ which amounts to just ‘a willingness to repeat in one's philosophical moments what one says in one's scientific moments, not taking it back, explaining it away, or otherwise apologizing for it.’Footnote 36 This weak realism does not presume to tell us ‘just what God was saying to Himself when He was creating the universe’.Footnote 37
Conclusion
I earlier suggested that the platonist could be understood as affirming that abstract objects are just as real as hurricanes, holidays, and events-and, we may add, the Eiffel Tower-and yet be no more committed to the existence of abstract objects than we are to the existence of objects serving as the referents of the relevant terms. Now obviously, abstract objects cannot be said to be real in virtue of the fact that they are not visual or auditory illusions. But it seems to me that a distinction can be drawn with respect to abstracta which is analogous to the distinction between real and illusory for concrete objects.
We are all familiar with the paradoxes of naïve set theory, such as Russell's Paradox, which destroyed Frege's edifice and eventually catapulted him into nominalism in late life. To Frege there was every appearance that impredicative sets were real sets; but if set theory was to be salvaged, then, as W. V. Quine put it, ‘such illusory combinations of entities’ had to be ruled out, and the reality of such pseudo-sets denied.Footnote 38 Other would-be mathematical objects-like a proof of the consistency of arithmetic or a derivation of the Continuum Hypothesis from the axioms of Zermelo–Fraenkel set theory-have proven to be just mathematical pipedreams. With respect to properties, van Inwagen similarly notes that, despite every appearance, there is no real property unable to be said truly of itself, for this leads inevitably to the analogue for properties of Russell's Paradox for sets.Footnote 39 D. M. Armstrong goes further. He distinguishes between ‘real properties’ and mere linguistic predicates truly said of a thing.Footnote 40 Certain terms may appear to ascribe properties, but in fact they do not. For example, being self-identical and exists are not, in his view, real properties. Now if properties and sets are, for the platonist, real only in the sense that they are not like these pseudo-entities, but they, like holes, hurricanes, and the Eiffel Tower, do not exist, then the platonist has not ventured beyond nominalism, since the nominalist can similarly distinguish real from pseudo-properties, sets, and other mathematical objects.
So it is not entirely clear just what ontological commitments are being affirmed by certain platonists, especially by linguistic philosophers. Could that be why they are so insouciant?
