Replies

Abstract

I am immensely grateful to the commentators for all of their thoughts and probing questions about TO. My replies will focus on one or two central points from each article. Inevitably, I need to be selective. First, for (§<) to hold, the facts that figure on the two sides of ‘ < ’ need to be distinct. I agree. Indeed, this is what distinguishes my asymmetric approach to abstraction from the more traditional symmetric approach espoused by Frege, the neo-Fregeans and Rayo, where the two sides of an abstraction principle are regarded as merely different ways to ‘carve up’ one and the same fact.2 Second, (§<) raises tricky questions about the individuation of facts. Suppose two distinct specifications are equivalent: α ∼ β . Are § α = § β and § α = § α one and the same fact? The answer will depend on how abstraction terms contribute to the representation of facts. One option is that the abstraction terms make a ‘purely referential’ contribution (in the sense of Quine (1960, 177)): an abstraction term first picks out an object, which is then slotted into the fact. On this analysis, the fact that § α = § β is identical with the fact that § α = § α . Another option is that an abstraction term contributes both a specification and the object thus specified. Then, ‘ § α = § β ’ represents a fact a concerning the two specifications α and β , namely, that they determine one and the same abstract. Since the language for which I try to secure an interpretation is an artificial one, I get to choose. I choose the former option, on the grounds of its greater simplicity (cf. TO, p. 19). Third, suppose my choice is granted, such that ‘ § α = § β ’ represents the fact that a certain object is self-identical. This raises the question of what it would be to ground a fact of self-identity. Understandably, CDP find this question problematic. There is neither room nor need for any grounding here, they think, because self-identity is ‘a kind of “logical fact”, a universal feature of reality’ (p. 12). They conclude that ‘although metaphysical grounding might sound as a natural and interesting connection it is not a viable road’ (ibid.). A better option, they propose, would be to invoke conceptual grounding, understood as ‘a relation amongst truths, which is objective, non-causal and explanatory in nature (as is the case with metaphysical grounding), but which holds in virtue of the concepts these truths contains’. (Sereni and Zanetti make a similar proposal.) While I am agnostic about the prospects of appeals to conceptual grounding, I believe CDP are too quick to give up on metaphysical grounding. (Henceforth, I omit the qualification ‘metaphysical’.) I agree with CDP that it would be obscure to ask of an existing object what more might be required to ground its self-identity. However, following much of the abstractionist tradition, I use a negative free logic where § α = § α is tantamount to the statement that § α exists.3 And I do not find it an obscure idea that α ∼ α , if true, should ground the existence of § α . More importantly, this idea figures at the heart of an account of abstraction and grounding that Louis deRosset and I have recently worked out. We start with the thought that a sufficiency statement φ ⇒ ψ records an explanatory argument (or a ‘grounding potential’, as I put it in TO). We proceed to use such arguments to derive information about grounding.4 This yields an attractive theory of abstraction and grounding, we argue, which solves CDP's problems – as well as some additional problems identified by Donaldson (2017). No sooner have deRosset and I solved some problems due to Donaldson than he points out another. The new problem concerns specifications that have only contingently the properties relevant to some form of abstraction. A clay sphere might instead have been shaped as a cube. A pen pointing north might have pointed east. How, then, might shape or direction abstraction on these specifications work? This is not an idle question. For as Donaldson observes, ‘abstractionism is attractive in large part because it offers us a general theory of types and tokens’ (p. 11). In Section 6.3.3 of TO, I discuss whether it is permissible to abstract on a relation ∼ that is only contingently a partial equivalence. I defend a negative answer on the grounds that the language in which we talk about the resulting abstracts would be too modally fragile: while α ∼ β is in fact a basis for asserting § α = § β , it might easily not have been so. Donaldson's problem is related but different. My problem is that what is a partial equivalence might fail to be so at other possible worlds. His problem is, loosely speaking, that what is a partial equivalence might fail to be so across possible worlds. In our clay example: a as it is fails to be congruent with a as it might have been. Two possible responses suggest themselves. Extending the strict policy of TO, I could prohibit the offending forms of abstraction on the grounds of excessive modal fragility. A promising alternative, though, would be to permit the abstraction but acknowledge its modal fragility. One way to do so would be to let the abstraction work, not on ordinary objects such as our piece of clay, but on objects qua shaped in a particular way. It is entirely plausible that a -qua-thus-shaped should necessitate the existence of s . On this proposal, the specifications would be precisely as modally fragile as the abstraction that they underwrite. By contrast, it would be a bad idea to respond by denying (1). For I wish to use principles of this form to explain why cardinal numbers are necessary objects, thus meeting a request by Donaldson. Letting e e be the empty plurality, we obtain □ ( e e ≈ e e → E ( 0 ) ) .5 Since the antecedent holds of (logical) necessity, it follows that □ E ( 0 ) . This, in turn, ensures that the singleton-plurality consisting of 0 only is self-equinumerous – indeed necessarily so. Since this self-equinumerosity necessitates the existence of 1, we easily derive □ E ( 1 ) . Continuing in this way, we can prove that every natural number exists necessarily. An analogous argument shows that every pure set exists necessarily. A second manifestation of Donaldson's new problem remains to be addressed. While a (necessitated) abstraction principle enables comparisons of the shapes of objects within any one possible world, its ability to underwrite comparisons across possible worlds is unclear. But such comparisons seem possible. To borrow an example from Donaldson, we can meaningfully say that a projected church would have had exactly the same shape regardless of which of two competing proposals, differing only with respect to their choice of materials, had prevailed. Moreover, I agree with him it would be unwise for abstractionists to secure such cross-world comparisons by embracing Lewisian modal realism, according to which actual and merely possible specifications exist on a par and so are straightforwardly comparable. To illustrate my alternative proposal, consider cardinal numbers. Suppose we followed Frege and the neo-Fregeans in abstracting cardinal numbers from (Fregean) concepts. This would result in modally fragile abstraction, with the associated problems discussed above. Let F be true of all and only my children. Although cardinality abstraction on F in fact yields the number 2, it might easily have given a different number. It is better, therefore, to proceed as in TO and do cardinality abstraction on pluralities, which have their numbers essentially. For example, let c c be my children. Then necessarily, if c c exist, they are two. To enable cross-world comparisons of cardinal numbers, all that remains is to find pluralities that exist across all possible worlds. Pluralities of pure sets or of cardinal numbers are natural options. These pluralities can serve as rigid ‘measuring rods’ that enable the desired comparisons. For example, while c c are equinumerous with the plurality of 0 and 1, it might easily have been the case that there are some objects who are all and only my children and who are not equinumerous with said measuring rod. In the case of shapes, our rigid measuring rods might be regions of space, as opposed to physical objects located at these regions. For a region plausibly has its shape essentially and can – modulo familiar (and serious) Leibnizian worries – be compared across possible worlds. Sereni and Zanetti (henceforth, S & Z) explain how two versions of ‘lightweight’ platonism – namely, Rayo's trivialism and my account of thin objects – are naturally seen as forms of Aristotelianism. On both views, abstract mathematical objects exist but are somehow derivative from non-mathematical facts. I thank them for this astute observation. They proceed to formulate a good challenge. As discussed, it is natural to explicate this Aristotelianism in terms of grounding (by which I continue to mean metaphysical grounding). Is this appeal to grounding compatible with the flexible conception of reality that both Rayo and I embrace?6 ‘[M]ost proponents of grounding’, S & Z claim, ‘seem to think of relations of determination as metaphysically rigid’ (p. 10). If correct, this poses an obvious danger: ‘if abstraction principles are interpreted as claims of grounding, then even if the relevant principle is coherent, the world might fail to respond to our stipulation’ (ibid.). I am not in a position to assess S & Z's claim about ‘most proponents of grounding’. Let me instead explain why I believe the flexible conception offers all the metaphysical realism that we need, both in general and in order to make sense of grounding. The flexible conception, as I define it, is first and foremost a metasemantic thesis to the effect that there is no uniquely right way to apply the apparatus of first-order logic to reality. We choose which objects to pick out. But, I claim, the objects that we pick out are real and typically exist independently of us. Let me first make some remarks on realism.7 As I understand it, realism about some domain has to do with reality providing an objective answer to every meaningful question about the domain. The flexible conception does not restrict reality's ability to provide answers to meaningful questions. Rather, it is a thesis to the effect that more work than what one might have expected is required to formulate a meaningful question, which we can then put to reality. Consider the question ‘How many objects are there in my office?’ As it stands, this is not a fully meaningful question: we additionally need to know what kind of objects the questioner has in mind. On the flexible conception, it takes more work than naively expected to formulate a meaningful question. Yet once we have formulated a meaningful question, reality provides an objective answer. In short, we must not conflate reality's inability to answer questions that are not even fully meaningful with some form of anti-realism. Next, there is the concern that on the flexible conception the objects we pick out will somehow be counterfactually dependent on being thus picked out. That is, had we (or someone else) not picked out the objects, they would not have existed. I do not see why any such thing should follow. Consider my account of reference to numbers or physical bodies. In both cases, we obtain an account of what it takes to ground the existence of the relevant referents: the existence of 2, for example, is grounded in the existence of any pair, while the existence of a stick is grounded in the existence of parcels of matter appropriately arranged. In neither case, then, do the objects depend on us for their existence. Numbers and physical bodies are therefore fundamentally different from objects whose existence is counterfactually dependent on us and our activities. Contracts and marriages provide examples. By withholding your assent or vow, you could have prevented a contract or a marriage from coming into existence. Fine's hylomorphism appears to be a way of emphasising the role of conceptual elements in the constitution of objects. To a certain extent, it may be argued that principles of embodiment are provided by concepts, and thus that objects are constituted also in terms of the concepts that ‘we bring to bear’. (p. 12) Pantsar first discusses my approach to arithmetic before he concludes with some remarks about the metaphysics of numbers. There are two different abstractionist conceptions of the natural numbers. The Fregean tradition has almost universally regarded the numbers as cardinals, in the sense that they are obtained by abstraction on equinumerosity, as discussed above. In ch. 10 of TO, I develop an alternative abstractionist conception, which regards the natural numbers as ordinals, obtained by abstraction on numerals under the equivalence of occupying matching positions in their respective numeral sequences. While I regard both conceptions as legitimate, I argue that the ordinal conception better matches our actual arithmetical thought and language.8 As Pantsar notes, this latter claim is in large part empirical. This means that ‘we should study the trajectory of cognitive development of arithmetical cognition. In some parts, our best understanding of this trajectory is potentially in conflict with Linnebo's account of the epistemology of arithmetic' (p. 4). I accept this observation and enthusiastically welcome work, such as Pantsar's own, that can help us integrate the philosophical and psychological aspects of the problem. I also agree with much of what he writes, for example when he emphasises the importance of the Object Tracking System for our most basic arithmetical capacities and its conceptual priority to any explicit understanding of equinumerosity or its ordinal equivalent. In my defense, though, I did signal some distance from empirical claims, for example when I wrote that ‘it is generally good methodology to begin by articulating and exploring various “pure” analyses of some phenomenon, even while admitting that the phenomenon might turn out to be too messy to be fully captured by any single pure analysis’ (p. 178). In retrospect, I admit I may have gone a bit overboard in my enthusiasm for the ordinal conception and downplaying of cardinal aspects of our arithmetical competence. But I was fed up with the cardinal conception's hegemony within abstractionism. And I still regard it as important to develop an alternative, “pure” ordinal conception – even though this purity is bound to be compromised when we seek a better, empirically informed match with our actual abilities. Deeper disagreements emerge when we turn to the metaphysics of numbers. In particular, Pantsar disagrees with my claim that numbers are counterfactually independent of us. His argument begins with an innocuous observation: ‘Had there been no intelligent agents, the cognitive trajectory outlined in [his paper] would never have taken place’ (p. 12). But then things take a problematic turn. ‘[W]hat reason is there to believe in the existence of numbers independently of the existence of number concepts?’ (p. 13). Whereas I take the existence of numbers to make no substantial demand on the world, Pantsar writes: ‘My account makes one such demand, namely, that there are agents that possess number concepts’ (p. 14). Why make this demand? Pantsar appears to be conflating representations with things represented. It is certainly true that relations of representation found in our thought and language are constituted, at least in part, by facts about us. But this does not imply that the objects represented are thus constituted. In short, whereas I ascribe to thin objects an Aristotelian dependence profile, where they depend on the existence of appropriate specifications, Pantsar ascribes to them a Kantian dependence profile, where they depend on being represented by appropriate agents (cf. the Précis, Section 8). Plebani, San Mauro and Venturi (henceforth PSV) investigate the epistemology of abstraction, making the important observation that ‘the doctrine of thin objects might make it easy to prove that mathematical objects a and b exist but hard to know whether they are the same object or not’ (pp. 1–2). Let me explain this observation and place it in context. As discussed in the Précis, Section 5, the ground for asserting an identity ‘ § α = § β ’ is the equivalence of the associated specifications, namely α ∼ β . In many of the cases discussed in TO, this equivalence is comfortably within our epistemic reach; examples include the parallelism of two lines and the ‘co-typicality’ of two tokens. Not all cases are that straightforward, however. In some cases, the specifications α and β may be infinite pluralities of objects, which are not in fact within our epistemic reach, but figure only in a highly idealised account of how reference can be constituted. In other cases, it takes some work to determine whether the specifications are equivalent. A case in point are physical bodies, which are specified by parcels of matter (cf. the Précis, Section 4). Whether two parcels of matter stand in the appropriate unity relation is not intrinsic to these two parcels but turns on the physical environment to which they belong. This explains the epistemic gulf that separates physical bodies from abstract objects (cf. TO, Sections 11.3 and 11.4). To decide whether the bodies specified by two parcels of matter are identical, it is not sufficient to examine these two specifications. By contrast, to decide whether the cardinal numbers specified by two pluralities are identical, it is sufficient to examine these two pluralities; for whether or not they are equinumerous is intrinsic to the two pluralities in question.10 PSV identify a new and different type of case, namely, where the specifications in question are finite and the objects that they specify are abstract, yet where there is no effective procedure to decide whether two specifications are equivalent. Here is a nice example. ‘[C]onsider the case where D is the set of the Turing machines' programs (conceived here as concrete tokens, say, sequences of marks on a surface) and the equivalence relation is the one that holds between two programs just in case for any (code of a) numerical input they return the same output' (p. 4). As PSV observe, the resulting unity relation is not decidable, which means it can be hard to tell whether two mathematical objects thus specified are identical or not. What are we to make of this? The example shows that abstraction does not on its own guarantee an easy epistemology. This need not be a problem, though. As PSV observe (quoting, Linnebo (2017, p. 128)), ‘abstract objects introduced via an abstraction principle “don't pose any additional epistemological problem” ' (p. 8). The equivalence α ∼ β can be more or less epistemically tractable. But the identity § α = § β poses no additional epistemological problem. Moreover, we can diagnose the source of the problem. The example involves a unity relation where the equivalence of two specifications is not intrinsic to the specifications but rather involves quantification over a countable infinity of numerical inputs. This gives the example an interesting hybrid character. As in the case of physical bodies, the unity relation is extrinsic. But it is still mathematical in character, being concerned with the functions computed by Turing machines. Gareth Pearce sees two arguments in TO for the thesis of Reference by Abstraction (‘RBA’, cf. the Précis, Section 5): one deductive and another abductive. Finding both arguments wanting, he instead defends the nominalist view that there are no abstract objects. The former argument seeks a deductive path from the flexible conception of reality to RBA. I doubt there is any viable such path. As explained in the Précis, Section 3, the flexible conception was introduced only in order to locate my view in a broader philosophical landscape. The task of being ‘entirely clear and specific’ (p. 4) falls, not to the flexible conception, but to the later parts of the book. Nor can a rough first approximation of a view be expected to entail a precise statement of it. The abductive argument seeks a path from my example of the constitution of reference to physical bodies to the full thesis of RBA: ‘the best explanation of why the principle holds in the case of physical bodies is that RBA holds generally’ (p. 6). Again, this was not my intention. My strategy was to begin with an easily grasped example, before proceeding to the general thesis. Pearce's discussion does, however, raise the interesting question of whether philosophers of a nominalist pursuasion can accept my account of reference to physical bodies but resist its extension to the realm of the abstract. To answer the question, let us revisit the three steps of my defense of RBA, summarised in the Précis, Section 5. First: it is permissible to start speaking as if there are F s. This is defensible because our language has reductive assertibility conditions, formulated solely in terms of antecedently accepted objects. This reducibility requirement applies to physical bodies and Fregean abstracta alike. Second: if available, a semantic interpretation that takes the apparent reference to F s at face value would be preferable. Again, the reasons I adduce play out in similar ways for bodies and Fregean abstracta. Third: such an interpretation is available. My defense of this claim turns on permitting the disputed form of reference in the metalanguage in which the interpretation is formulated. Once again, there is no relevant difference between the two cases. My conclusion is that, while the specific example of physical bodies does not lend much abductive support to the general thesis of RBA, the best defense of the former generalises naturally to a defense of the latter. Here is an example. Suppose we defined ‘ x x ∼ a y y ’ as ‘every mereological atom is part of one of x x just in case it is part of one of y y ’. The resulting criterion of identity is quite attractive — but performs miserably if there are objects that do not decompose into atoms. For example, any two pieces of atomless gunk would be identified. I agree with Lando, not only on this option, but also on the others he considers. My only complaint is that he gives up the search for a suitable criterion too quickly. Proposition 1.Let ≤ be a partial order on a domain D that satisfies the mereological principle of Strong Supplementation, namely x ≤ y → ∃ z ( z ≤ x ∧ ¬ z ○ y ) . Then there is a partial order ≤ + extending ≤ on a domain D + extending D , on which a summing operation ∑ is defined, such that: (CI- ∑ o ) holds for all pluralities of members of D + ; when x x consist of x only, then ∑ ( x x ) = x ; for every object y of D + there are x x of D such that ∑ ( x x ) = y ; ≤ + and D + satisfy the axioms of Classical Extensional Mereology. Moreover, ≤ + and D + are unique up to isomorphism. I think this moral also answers Lando's question of ‘Why bother looking for abstraction principles?’ (p. 13). One important reason is that we obtain a systematic and well-motivated defense of the existence of arbitrary sums. Eklund asks a perceptive question: ‘what is the big deal [with thin objects]?’ (p. 2). ‘Even before numbers and other mathematical objects […] have entered the stage, Linnebo already has in his metaphysics whatever the predicates employed stand for, in particular given that he, like neo-Fregeans before him, employs a second-order framework with quantification into predicate position’ (ibid.). The worry, then, is that my defense of thin objects relies on a prior commitment to ‘the existence of some thin entities, the predicables’ (ibid.). An unexciting answer would be that TO is almost exclusively concerned with what we may call extensional abstraction, that is, roughly, abstraction on either single objects or pluralities thereof.12 Unlike Frege and the neo-Fregeans, for example, I base cardinality abstraction on pluralities of objects, not on Fregean concepts (cf. Section 2 of these Replies). Granted the widely held assumption that plural logic introduces no ontological commitments beyond those incurred by its first-order variables, my account avoids any commitment to thin entities and instead progresses directly to thin objects. I want to give Eklund a more interesting answer as well. Although TO focuses on extensional abstraction, I am also friendly to what we may call intensional abstraction, understood as abstraction on Fregean concepts or the like. So even though TO offers only some brief remarks on intensional abstraction (Appendix 3.A.2), let me attempt a more direct answer to Eklund's challenge. Yes, I regard Fregean concepts as thin entities. (I am not attracted to a ‘neutralist’ conception of higher-order quantification, on which such quantification avoids any new ontological commitments, for roughly the reasons Eklund lays out.) By characterising a Fregean concept as ‘thin’, I mean that all it takes to define such a concept is to specify a function from objects to truth-values. (This is much like the view of Hale & Wright (2009b), expressed in the passage quoted by Eklund.) This thinness is then ‘inherited’ by any abstracta whose specifications are Fregean concepts. I do not see why this view would trivialise the thinness of the resulting abstract objects. Many philosophers appear to think that first-order (or ontological) commitments are a bigger deal than higher-order (or ideological) commitments. I disagree. I find the two kinds of commitment broadly on a par. Thus, I am happy to make Eklund's words my own: ‘there is a kind of parity between claims that entail the existence of predicables and claims that entail the existence of objects that are not predicables’ (p. 6). I hasten to add, though, that my view does not render the definition of a Fregean concept a trivial matter. True, all that it takes is to specify a function from objects to truth-values. But doing so can be surprisingly hard! The problem stems from the dynamic character of my account, where domains successively expand, as described in the Précis, Section 6. These expansions may disrupt an attempted definition.13 Suppose we try to use a condition φ ( x ) to define a concept. Suppose an object a satisfies the condition when evaluated relative to one of the expanding domains. But a may fail to satisfy the condition relative to a later, expanded domain. My proposed solution is that the definition of Fregean concepts be subjected to restrictions of a broadly predicativist character. Since Studd's is a very rich paper, I will briefly comment on some of the questions it raises, before focusing on what I take to be its most weighty challenge. First: Like Donaldson, Studd requests a better defense of the necessary existence of the objects of pure mathematics. Fair enough: I outline such a defense in Section 2 of these Replies. Second: Studd claims that ‘Linnebo's restriction to predicative abstraction requires him to eschew familiar one-sorted abstraction principles, such as HP’ (p. 5). I note that my use of two-sorted abstraction is limited and only a means to an end. I first use a two-sorted analogue of the familiar abstraction principle, before merging the sorts to return to the ordinary one-sorted setting – with a domain that may have expanded (TO, p. 58). In TO, I develop the resulting dynamic account of expanding one-sorted domains by means of modal logic (see the Précis, Section 6). I factorise each familiar abstraction principle into separate criteria of existence and identity, which are then transposed to the modal setting. The resulting principles are admittedly a bit less familiar, but there is nothing two-sorted here. Regardless, if readers prefer to retain the familiar abstraction principles rather than the factorise-and-transpose manoeuvre of TO, I can cater to that too, namely, by adopting an alternative development of dynamic abstraction based on a ‘critical’ plural logic rather than modality (cf. the Précis, fn. 11). Third: While I argue that each round of dynamic abstraction is epistemically ‘for free’, Studd observes that this falls short of justifying infinite iterations of such expansions. I agree. Quoting Studd quoting TO, my view is that ‘our justification for theories that result from infinitely iterated abstraction is “more indirect and less conclusive” (p. 202)’ (p. 15). This seems to me unsurprising. There is more robust evidence for elementary mathematics than for infinitary. Fourth: Might we postpone the need for infinite iterations by allowing abstract objects to exist solely on the basis of the possible existence of appropriate specifications? I appeal to such specificationless abstracts some places in TO (pp. 45, 188 and 190). While I admit that TO offers no worked-out account of such abstraction, I believe an account can be developed – say, by abstracting on actually existing concepts that uniquely characterise the relevant merely possible specifications. For now, I rest content with TO's primary route into the infinite, namely via the long iterations. Studd's most weighty challenge concerns the possibility of absolutely general quantification. My view here is intermediate between generality relativism and traditional generality absolutism.14 I call a domain extensional if it can be given as a plurality and merely intensional if not. On the one hand, I argue, with the relativists, that every extensional domain can be surpassed by a larger such domain. On the other hand, I argue, with the absolutists, that there is an absolute domain. But I break with tradi