Replies

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