Husserl on Abstract Objects

Please do not cite this preliminary version. The final version of this article will appear in the journal Pacific Philosophical Quarterly, DOI: 10.1111/papq.12084 HUSSERL AND THE PROBLEM OF ABSTRACT OBJECTS George Duke Peter Woelert* Abstract One major difficulty confronting attempts to clarify the epistemological and ontological status of abstract objects is determining the sense, if any, in which such entities may be characterised as mind and language independent. Our contention is that the tolerant reductionist position of Michael Dummett can be strengthened by drawing on (usserl s mature account of the constitution of ideal objects and mathematical objectivity. According to the Husserlian position we advocate, abstract singular terms pick out weakly mind-independent sedimented meaning-contents. These meaning-contents serve as the thin referents of abstract singular terms, but are ultimately founded in prior acts of meaning-constitution. Introduction One major difficulty confronting attempts to clarify the epistemological and ontological status of abstract objects like numbers, sets and geometrical shapes is determining the sense, if any, in which such entities may be characterised as mind and language independent. The two extreme positions in the contemporary debate are Platonism and nominalism, with the former upholding, and the latter rejecting, the mind and language independence of mathematical objects. In this paper we argue for an intermediate antinominalist position on abstract objects based upon the application of insights from Husserlian phenomenology to contemporary philosophy of mathematics. Our contention is that the tolerant reductionist position of Michael Dummett can be strengthened by drawing on (usserl s mature account of the constitution of ideal objects and mathematical objectivity.1 (usserl s meaning-constitutional account, we argue, provides the explanatory resources to justify Dummett s attempted middle-way between Platonism and nominalism. Section one provides a critical overview of Although it is well-known within the analytical tradition that the early Husserl was allegedly guilty of a form of psychologism, it is less well-known that his later work on the foundations of the modern natural sciences attributes an important role to linguistic practice in the constitution of mathematical objectivity. The claim that contemporary work on the problem of abstract objects has much to learn from Husserl is hardly new in itself. See, for example, Mohanty (1982), Ortiz Hill (2001), Tieszen (1990) and Willard (1984). For the most part, and with some exceptions (Tieszen 2005; Tieszen, 2010) however, these studies focus upon the Husserl-Frege debate rather than ways in which Husserl can shed light on contemporary debates in the philosophy of mathematics. 1 1 Please do not cite this preliminary version. The final version of this article will appear in the journal Pacific Philosophical Quarterly, DOI: 10.1111/papq.12084 Dummett s intermediate position on abstract objects. Section two then expounds (usserl s later work on ideal objects, with a focus upon the constitution of mathematical objects through intentionality and sedimented meanings. Section three draws together the preceding two sections to argue for a tolerant reductionist position which addresses weaknesses in extant intermediate accounts. (usserl s theory of the constitution of ideal objects explains why the reference of abstract singular terms to objects like numbers can be regarded as both semantically legitimate and referentially thin. According to the Husserlian position we advocate, abstract singular terms pick out weakly mind-independent sedimented meaning-contents. These meaning-contents serve as the thin referents of abstract singular terms, but are ultimately founded in prior acts of meaning-constitution. * Authors are listed alphabetically. Section 1: Dummett and the Problem of Abstract Objects Abstract objects are generally defined in negative terms as objects that are not situated in space and time and not involved in causal relations. Examples are numbers, sets and geometrical shapes. The problem of abstract objects is also generally posed in terms of whether such objects exist in the sense of existence formalised in Quine s doctrine of ontological commitment. Mathematical Platonism is the metaphysical position that abstract entities like numbers possess mind-independent existence. Nominalism, by contrast, asserts that mathematical objects do not exist, a thesis generally associated with the claim that the existence of such objects is not required to explain the success of our best scientific theories. There are a range of intermediate positions on abstract objects, gathered together under the broad rubric of anti-nominalism. In the first section of this paper we present the case for the tolerant reductionist anti-nominalist position defended by Dummett (1991a), whilst also pointing to some areas where it requires supplementation. The core of Dummett s tolerant reductionism a is the thesis that we are justified in ascribing a semantic reference to abstract singular terms like numerals, but that this 2 Please do not cite this preliminary version. The final version of this article will appear in the journal Pacific Philosophical Quarterly, DOI: 10.1111/papq.12084 still allows for a reductive metasemantic analysis of our ontological commitment to mathematical objects. Reference to abstract entities is legitimate insofar as the objects picked out by abstract singular terms like numerals are capable of playing a semantic role in the determination of the truth conditions of sentences. This thin form of reference, however, is disanalogous to that found in the robust model of meaning operative in the case of expressions picking out concrete objects like people, tables and chairs. In the case of terms for concrete objects the identification of constituents of extra-linguistic reality plays a substantive role in the determination of the truth conditions of sentences containing such terms. In the case of abstract singular terms, by contrast, it is mistaken to think of such terms as hitting upon pre-existent entities considered as part of extra-linguistic reality. The details of this position on abstract objects were developed, and are best understood in terms of, Dummett s critiques of nominalism and Platonism. 1.1) Critique of Nominalism Dummett s early critique , and of the reductive nominalism of Goodman and Quine sets out from Fregean insights on the relationship between sentences, truth and meaning. Once we grasp Frege s principle that it is only in the context of a sentence that a word has a meaning, Dummett argues, then we will no longer be susceptible to the nominalist prejudice that we must always be shown the referent of a singular term for it to be legitimate. According to the early Dummett, the context principle can secure a referent for abstract singular terms featuring in true sentences and thus allows us to reject the nominalist superstition that numbers etc. are illegitimate entities because we cannot enter into direct epistemic relations with them. )n Steps towards a Constructive Nominalism Goodman and Quine had sought to establish that statements seemingly about abstract entities can all be rephrased as statements about concrete objects.2 Steps argues that we never need use variables A more recent version of reductive nominalism is found in the work of Hartry Field (1980). Field argues that the nominalist can avoid ontological commitment to abstract objects by denying that statements featuring reference to such objects are literally true. We focus here upon the earlier version of 2 3 Please do not cite this preliminary version. The final version of this article will appear in the journal Pacific Philosophical Quarterly, DOI: 10.1111/papq.12084 that call for abstract objects as values , , and hence can avoid ontological commitment to abstract objects by reducing all predicates of abstract entities to predicates of concrete objects (1947, 107). The project of Goodman and Quine is thus to demonstrate that our apparent commitment to abstract entities is a function of our tendency to reify the putative correlates of abstract singular terms based on misleading surface grammar and a faulty analogy with concrete particulars. Dummett s critique of this position assumes both that the correct way to approach the problem of abstract objects is by inquiring whether abstract singular terms (e.g. numerals) genuinely refer to objects and that the context principle is capable of providing an affirmative answer to this question.3 Our experience of reality as divided up into objects, properties etc. is on this account determined by the logical-syntactic categories we employ and linguistic expressions are accorded logical or explanatory priority over the correlative ontological categories. Bob (ale and Crispin Wright, building on Dummett s interpretation of Frege, have characterised this claim as the syntactic priority principle or the priority of syntactic over ontological categories (Wright 1983, 30). Based on these assumptions, Dummett argues that reductive nominalism misses the extent to which our apprehension of objects of any kind is facilitated by our understanding of the meaning of sentences. Frege s context principle teaches us that our commitment to entities should be based on an account of how expressions standing for those entities function in sentential contexts. If a term genuinely fulfils the syntactical function of a proper name in sentences, some of which are true, then we have not only fixed the sense, but also the reference, of that proper name (Dummett 1956, 40). As a result, we do not, as the nominalist suggests, have a right in all cases to demand the possibility of entering into direct epistemic relations with an extra-linguistic correlate of a linguistic expression for that expression to be considered legitimately referential. nominalism proposed by Goodman and Quine on the grounds that our main purpose in this section is to explicate Dummett s position. 3 For Frege, the syntactic category of proper name stands for anything that is not a function and hence lacks an empty place (2002, 13). A Fregean object is thus the referent of a proper name in a quite general sense. Not only concrete things , but also the purported referents of abstract singular terms, such as numbers, truth-values and value-ranges, can be considered as objects, insofar as these can be the semantic correlates of proper names and fall within the range of the variables of a mathematical theory. We here assume that Dummett is correct to attribute an at least inchoate semantics to Frege. For a contrasting view, see Ricketts (2010) and Weiner (2007 and 2008). To enter this debate here would take us too far afield from the main argument of the paper. 4 Please do not cite this preliminary version. The final version of this article will appear in the journal Pacific Philosophical Quarterly, DOI: 10.1111/papq.12084 )t is important to note, however, that Dummett s later work concedes some ground to nominalist assumptions. In his later work, Dummett regards it as essential to the notion of reference in the full-blooded sense that the identification of the bearer of a proper name be a component of the determination of the truth conditions of a sentence containing that name , . This view has repercussions for Dummett s critique of nominalism. )n the case of an abstract singular term, the determination of the truth or falsity of a sentence in which such a term occurs does not involve an identification of an object as the referent of the term Dummett, , xlii , precisely in the sense that we do not enter into direct identifying relations with abstract entities. By insisting on the need to include the identification of the bearer of a proper name within a model of meaning for singular terms, Dummett s later theory manifests a commitment to the epistemic priority of concrete particulars. A semantic approach to the problem of abstract objects now proceeds by an investigation of the transferability to abstract singular terms of a model for the meaning of proper names denoting concrete objects, which, given an associated criterion of identity, can be picked out by an ostensive gesture accompanied by the use of a demonstrative. In the case of names for concrete objects the identification of an extra-linguistic referent is an ingredient in the process of determining the truth-value of a sentence in which that name occurs. The task Dummett now sets an account of abstract objects is to discover whether a model of meaning incorporating an appropriate analogue for the identification of the referent of a concrete singular term can be given some content Dummett b, in the case of abstract singular terms. It is in this context that the later Dummett advocates a thin theory of the reference for abstract singular terms. In order to see how Dummett arrived at this position, however, it is first helpful to consider his critique of the neoFregean Platonism of Hale and Wright. 1.2) Critique of Platonism 5 Please do not cite this preliminary version. The final version of this article will appear in the journal Pacific Philosophical Quarterly, DOI: 10.1111/papq.12084 Despite his rejection of nominalism, Dummett is also critical of what he considers a Platonist mythology, according to which abstract objects like numbers possess robust mind and language-independent existence. Dummett argues that if we are indeed justified in taking abstract singular terms at face-value and attributing reference to them on the grounds that we have provided determinate truth-conditions for the sentences in which they feature, then the means by which such truth-conditions were laid down cannot have involved any appeal to the notion of reference for such names (Dummett 1991b, 83). In other words, contexts determining the truth-conditions of sentences featuring abstract singular terms cannot, at pain of circularity, assume the prior mind-independent existence of abstract objects. This not only undermines the analogy between terms standing for concrete and abstract objects, it could also be taken to entail that we can only permit as legitimate such abstract singular terms as are eliminable . It is in Dummett s critique of the neo-Fregean Platonism of Hale and Wright that clarifies the reductionist component of his theory. Hale and Wright, like Dummett, argue that the problem of abstract objects is one of determining whether our use of abstract singular terms entails objectual reference. Hale and Wright, however, unlike Dummett, hold to a strong version of syntactic priority according to which it is a sufficient (but not necessary) condition for the existence of objects of some kind that there are true statements in which expressions function as singular terms (1983, 14).4 On this basis, the neo-Fregean attempts to prove the existence of numbers and other abstract entities through abstraction principles of the form: (a)(b) ((a) = (b)  E(a,b))5 The emphasis upon syntactic priority is stronger in (ale and Wright s earlier position, but we mention it here for the purposes of explicating Dummett s critique. The more recent focus of debates on neoFregeanism has been the contrast between arrogant and non-arrogant implicit definition of the Numerical Operator (ume s Principle . This principle falls under the schematism set out in the next sentence of the main text. On arrogant implicit definition see, in particular, (ale and Wright , 152). 5 Where a and b are variables of a given type,  is a term-forming operator denoting a function from items of the given type to objects in the range of the first-order variables, and E is an equivalence over items of the given type (Hale and Wright 2009, 178). 4 6 Please do not cite this preliminary version. The final version of this article will appear in the journal Pacific Philosophical Quarterly, DOI: 10.1111/papq.12084 Principles of this form serve as stipulative implicit definitions of the -operator and in so doing also of the new kind of term introduced by means of it with the corresponding sortal concept (Hale and Wright 2009, 179). Abstraction principles allow us to overcome concerns about our epistemic access to abstract objects by giving an account of the truth conditions of  identities as coincident with those of a kind of statement we already understand , . We can exploit this prior understanding to establish our knowledge of the referents of the -terms, referents whose status as genuine objects is guaranteed by the truth of the identity statements by means of which we gain access to them. One of the advantages of this form of mathematical Platonism is that it seems to provide a response to the generalised version of the well-known objection of Benacerraf (1973), namely that given the classical semantics of mathematical truth it is not possible to explain our epistemic access to objects like numbers. From the neoFregean perspective once we have settled, by syntactic criteria, that an expression fulfils the role of a proper name, and features in statements, some of which are true, then we have done all that we need to do in order to determine that the expression possesses robust objectual reference. Despite the fact that the neo-Fregean position here outlined has many affinities with Dummett s early critique of nominalism, Dummett s later doubts regarding the explanatory role of the context principle in an account of reference to abstract objects would lead him to reject the possibility of a defence of Platonism on neo-Fregean assumptions. For the later Dummett, an account of reference to abstract objects on the basis of the context principle does not justify treating such objects as ontologically robust, rather it presupposes that we are in possession of an understanding of the domain under investigation in advance of the principle s application a. (ale and Wright s resuscitation of Frege s logicism is based upon a rejection of an explicit definition of number in terms of classes of concepts and appropriation of the contextual definitions of directions and numbers abandoned by Frege in the face of the Julius Caesar problem.6 As Dummett argues, however, Wright and (ale s claim to have Frege himself ultimately abandoned a contextual definition of numbers in terms of N= in Grundlagen due to the Julius Caesar problem: The alleged inability of N= to justify the status of numbers as selfsubsistent objects with determinate criteria of identity. According to Frege, the inability of N= to decide 6 7 Please do not cite this preliminary version. The final version of this article will appear in the journal Pacific Philosophical Quarterly, DOI: 10.1111/papq.12084 overcome the Julius Caesar problem disregards the fundamental flaw that led Frege s philosophy of mathematics into contradiction, namely impredicativity. Frege s N= has the same logical form as the fateful abstraction principle embodied in Axiom V regarding the notion of ranges of values. N= is clearly impredicative, insofar as it quantifies over the sortal concept of number in the process of seeking to provide an explanation of that concept.7 Overall, Dummett thus dismisses the neo-Fregean position on abstract objects as exorbitant, denying that the context principle can get us to a genuine, full-blown reference to objects such terms is semantically idle a, a, . The attribution of reference to , insofar as there are no pre-existing referents whose identification goes to determine the truth-conditions of sentences containing terms like numerals. 1.3) Tolerant Reductionism Dummett s characterises his mature intermediate position as a form of tolerant reductionism. The tolerant reductionist recognises that the assertion refers to an object can be construed as the equivalent, in the formal mode, of there is such a number as , and hence as true a, . (e denies, however, that we can take sentences featuring abstract singular terms as having just the semantic structure that they appear to have. This is because the fully-fledged or robust notion of reference is not semantically operative in contextual definitions. In the case of such definitions our grasp of the thought expressed by a sentence containing the term is mediated by our knowledge … of how to arrive at an equivalent sentence not containing that term … so that the notion of the reference of the term, as determined by its sense, plays no role in our conception of what determines the thought as true or false (Dummett 1991a, 193). statements of the form Nx:Fx = Julius Caesar or, more generally, NX:FX = q where q is a term not given in the form Nx:Fx signifies a fatal indeterminacy in the proposed contextual explanation. 7 Dummett diagnoses the fundamental flaw in Frege s philosophy of arithmetic as the unavoidably impredicative character of contextual explanations such as N=. In order to provide a sufficient basis for elementary arithmetic, when combined with second-order logic, N= must be taken to introduce the cardinality operator the number of Fs as applicable to predicates defined over a domain of objects inclusive of the natural numbers themselves; that is, over predicates which themselves contain occurrences of the cardinality operator that are not eliminable. Hale and Wright have attempted to meet the Bad Company objection concerning the legitimacy of appealing to abstraction principles with the same form as the ill-fated Axion V (see Hale and Wright 2001) and elsewhere; it is not possible to assess the success of these attempts here. 8 Please do not cite this preliminary version. The final version of this article will appear in the journal Pacific Philosophical Quarterly, DOI: 10.1111/papq.12084 This suggests a thin theory of the reference of abstract singular terms which attributes such terms with a semantic value but not a bearer. Building on Dummett s work, Øystein Linnebo ; has argued for an anti- nominalist but non-Platonist meta-ontological minimalism about mathematical objects based on the ascription to abstract singular terms of a semantic role in sentences.8 Linnebo employs Stalnaker s distinction between semantics, understood as a theory of how the truth values of sentences are determined by the semantic values of their components, and meta-semantics, which describes how this process takes place with reference to objects of different kinds. The distinctive move, which Linnebo believes overcomes the difficulties in relation to allowing a notion of semantic value for abstract singular terms, is to place the reductionist aspect of the account entirely at the meta-semantic level, i.e. the level of explanation concerned with what is required for the reference relation to obtain between a term and its semantic value. The central tenet of Linnebo s meta-ontological minimalism is that it suffices for a term to be attributed a reference in the sense of semantic value that it make a definite contribution to the truth- value of sentences in which it occurs. Linnebo assumes that the semantic value [[E]] of an expression E in extensional contexts is its referent. His claim that mathematical singular terms have abstract semantic values and that its quantifiers range over the kinds of item that serve as semantic values , allows that mathematical terms have a reference in the sense that they go towards determining the truth-conditions of sentences in which they feature, while still allowing for the possibility of a reductive analysis of their ontological import. The applicability of the semantic notion of reference in a particular case does not necessarily imply that the reference relation is reducible to a semantic account; a description of what the reference relation consists in might point to features that cannot be exhausted by an account given in intra-linguistic terms. Linnebo calls objects of a given kind light-weight if sentences concerning them admit of a meta-semantic reduction to sentences not containing them Dummett , . The meta-semantic analysis operates at the explanatory level of the senses of 8 )n his recent reply to Peter Sullivan s essay Dummett s Case for Constructive Logicism Dummett endorses Linnebo s account, going so far as to say that we may adopt this as a revised formulation of the notion of a thin conception of reference , . 9 Please do not cite this preliminary version. The final version of this article will appear in the journal Pacific Philosophical Quarterly, DOI: 10.1111/papq.12084 expressions and sentences, whereas the semantic level works at the level of reference by giving a compositional account of how the semantic values of expressions go towards determining the semantic value of larger linguistic units, such as sentences, of which they are a part. This still leaves open the question, however, as to the ontological status of objects that are thin and amenable to meta-semantic reduction. Linnebo grants that his meta-ontological minimalism is suggestive of a form of reductionism, whilst seeking, like Dummett, to avoid the extremes of an intolerant nominalism. As Linnebo s own account of truth-value realism suggests, however, it is possible to align his views concerning the semantic values of abstract singular terms with nominalism. By making a distinction between the language LM in which mathematicians make their claims and the language LP in which nominalists and other philosophers make theirs , , the nominalist can assert, like Field, that the statement there are prime numbers between and is true while simultaneously arguing that there are no numbers. This is because the nominalist s statement about prime numbers is made in LM whereas the nominalist s statement that there are no numbers is made in LP. As a result, it would appear that that the nominalist s assertion regarding the existence of numbers is coherent provided that the sentence about primes is translated non-homophonically from LM into LP , . The lesson to draw from this seems to be that a fully worked out anti-nominalist position needs to confront the question of the qualified mind and language independence of mathematical objects, if it is not to culminate in a form of sophisticated nominalism. )n order to consider Dummett s thin theory of reference for mathematical objects as a genuinely intermediate position, therefore, it would be necessary to demonstrate how abstract singular terms are more than mere shadows of syntax . As such, it is a desideratum of an intermediate position to give a more complete explanation of the way in which the referents of abstract singular terms can play a semantic role in determining the truth of the sentences of mathematics. What we require is thus an account which can explain the qualified sense in which mathematical objects possess mind and language independent existence. (usserl s later genetic-phenomenological work on the constitution of geometrical and mathematical ideal objects, we argue, can point us in the 10 Please do not cite this preliminary version. The final version of this article will appear in the journal Pacific Philosophical Quarterly, DOI: 10.1111/papq.12084 direction of such an account and it is to this work that we now turn in the second section of this paper. 11 Please do not cite this preliminary version. The final version of this article will appear in the journal Pacific Philosophical Quarterly, DOI: 10.1111/papq.12084 Section 2: Husserl on the Constitution of Ideal Objects In the previous section we argued that a well-motivated intermediate theory of abstract entities needs to explain the qualified mind and language independence possessed by objects such as numbers, sets and shapes. It is in this regard, we argue in what follows, that the work of the later Husserl is particularly instructive. Whilst Husserlian phenomenology obviously differs from Dummett s approach in many of its fundamental assumptions,9 his philosophy of mathematics nonetheless suggests an intermediate position whereby abstract or ideal objects are constituted through intentional acts and yet also possess a qualified mind and language independence. The position we defend in the next two sections has many affinities with Richard Tieszen s characterisation of (usserl s position in the philosophy of mathematics as a form of constituted platonism , . Tieszen s argument involves two core claims: a) that knowledge of mathematical objects is impossible without intentionality (2010, , and b that the objects that are intended by consciousness are to be thought of as founded invariants in mathematical experience , . Constituted platonism thus regards mathematical objects as constituted non-arbitrarily , through the intentional activity of consciousness but as nonetheless subject to grammatical, formal and meaning-theoretic constraints. A position which emphasises the importance of intentional acts of meaning-constitution for the existence of mathematical objects is obviously inconsistent with the robust mind-independence associated with mathematical Platonism. It is consistent, however, we will argue, with an acknowledgment that mathematical objects possess mind-independence in the weaker sense that their existence (and the objective truth of statements referring to them) is not attributable to any private cognition or speech act. Dummett himself gives an instructive overview of the differences between his own approach and Husserlian phenomenology in (1993). Dummett, who concentrates in his analysis upon the Husserl of the Ideas 1 period, rejects in particular the phenomenological emphasis upon the transcendental ego in the constitution of meaning and the role of intentionality in the theory of reference. Another important difference in this context is the complete absence within the Dummettian account of anything equivalent to (usserl s earlier theory of categorial intuition as put forward in his Logical Investigations (1970c), although a full discussion of this difficult theme is outside the scope of this paper (see for instructive discussions of (usserl s theory of categorial intuition Lohmar and Cobb-Stevens (1990)). 9 12 Please do not cite this preliminary version. The final version of this article will appear in the journal Pacific Philosophical Quarterly, DOI: 10.1111/papq.12084 Building upon Tieszen s account, in what follows we aim to show that the thesis of such qualified mind independence, and more generally that of a constituted realism with regard to mathematical objects, can be qualified further in important respects by paying heed to (usserl s later genetic-phenomenological writings. These writings and the meaning-constitutional considerations contained in them are not directly considered in Tieszen s account. More specifically, we think that (usserl s genetic phenomenology makes it possible to a) specify the nature of and role played by the grammatical, formal and meaning-theoretic constraints Tieszen refers to, and b) to clarify and highlight the role played by intersubjectivity in the constitution of abstract mathematical objects. 2.1) The Genetic Constitution of Ideal Objectivity: From Intentionality to Sedimentation Let us begin by sketching some of the key concepts of (usserl s phenomenology and his phenomenological theory of constitution more narrowly. Regardless of its variations, (usserl s use of the concept of constitution is intrinsically connected to reflections on the nature of meaning.10 The point of departure for these reflections is the question of how meaning manifests itself, for example, in and through perceptual acts and linguistic expressions. )n regard to (usserl s concept of constitution, one can distinguish between a static and genetic variety of this concept (see Bernet, Kern and Marbach 1993, 195204); and it is the latter, genetic variety that is most relevant for our discussion. What defines (usserl s analysis of static constitution is that it has stable objects, a stable ontology as its guide Bernet, Kern and Marbach , . By contrast, in an analysis of genetic constitution, the major questions concern the ways in which these objects are constituted. Thus, as Bernet, Kern and Marbach (1993, 201) stress, in the transition from static to genetic phenomenology, the concept of constitution, in term of (usserl s understanding of the concept of constitution changed significantly along with the major shifts in his philosophy, in particular the transition from static to genetic phenomenology (Sokolowski 1964; Bernet, Kern and Marbach 1993). The most comprehensive discussion of (usserl s concept of constitution and its formation is Sokolowski (1964). For a more recent discussion that relates Husserl s concept of constitution to contemporary philosophy of mind see Huemer (2003). 10 13 Please do not cite this preliminary version. The final version of this article will appear in the journal Pacific Philosophical Quarterly, DOI: 10.1111/papq.12084 its direction of enquiry, undergoes a significant shift: The object is no longer the guidepost as it is in static phenomenology. )t is rather something that has come to be . Importantly, and as we will see in more detail below, this does not entail, however, that (usserl s genetic phenomenology denies the objectivity of abstract objects. The phenomenological concept of constitution can only be understood properly if considered against the backdrop of the concept of intentionality. The main function of this concept is to properly reflect the apparent peculiarity that individual consciousness, in its various cognitive and affective manifestations, essentially is directed towards objects, that is, it is consciousness of something (usserl , . Consciousness intentionality becomes most evidently manifest in the domain of senseperception, where an object can be directly intended by an individual and embodied cognitive agent in its concrete bodily presence (see Zahavi 2003a, 28-29). However, there is of course also directedness of consciousness involving abstraction, idealization, reflection, formalization, and other higher-order cognitive activities 185). Tieszen , Precisely such higher-order cognitive activities are central to the constitution of those objects, like numbers and geometrical shapes, to which (usserl attributes objectivity (usserl b, ideal . For (usserl, such ideal objectivity entails a that the intended objects always remain the same, regardless of particular spatial situations and historical contexts (see Husserl 1970b, 356- , and b that they do not exist as something personal within the personal sphere of consciousness but rather are exactly the same objects accessible by all actual and potential minds (Husserl 1970b, 356). The particular task of a phenomenological theory of constitution then is, as Husserl describes it in his Formal and Transcendental Logic, to reveal the sense in which ideal objects are essentially products of the correlative structures of productive cognitive life , without this compromising the ideal Objectivity of those objects discussions Hartimo 2012; Tieszen 2010). , 63; see for further In his later phenomenological account, which foregrounds the genetic dimensions of the constitution of ideal objects, Husserl singles out Euclidean geometry as having for the first 14 Please do not cite this preliminary version. The final version of this article will appear in the journal Pacific Philosophical Quarterly, DOI: 10.1111/papq.12084 time constituted a totality formed of pure rationality (usserl a, . For (usserl the constitution of ideal objects conceived of as belonging to an infinite world of idealities within a rational-mathematical science (Husserl 1970a, 22) only becomes possible, however, with the modern algebraisation of geometry. What characterises the modern algebraisation of geometry is that algebra is effectively used in scientific practice as a universal formal language by means of which both geometrical and arithmetical problems can be solved in equal measure. For Husserl, such a systematised employment of formal symbols in mathematical thinking, where the symbols employed are progressively emptied from all intuitive content and associations and solely related to each other through a rational canon of rules (see Husserl 1970a, 46), coincides with the systematic development of a truly symbolic, abstract conception of number (Husserl 1970a, 43-48, and more comprehensively, Klein 1968, chapter 12). What characterises such a conception, Jacob Klein observes, is that a number thus conceptualised no longer represents something that has a concrete reference, for example, a definite number of material bodies. Rather, a number now essentially constitutes a symbolic abstraction, where a number symbol signifies the concept of the number as a multitude of units (Klein 1985, 62-63). As a consequence, the (symbolic) concept of number is thus understood to exist independent from any certain, countable objects, while at the same time, in algebraic practice and thinking, it is increasingly taken for granted as having its own real and objective existence Klein , . For Husserl then, the development of such a symbolic form of mathematical thinking is a crucial step in to the process leading to the constitution of mathematical ideal objectivity. (usserl speaks in this regard of the transition from an originally evident conceptuality to a symbolic substitute-conceptuality (usserl , , our translation . This transition is directly related to the overarching process whereby material mathematics is put into formal-logical form (usserl completely universal formalization a, (usserl , and which ultimately culminates in a a, of the science of mathematics. Husserl notes that from a genetic-phenomenological perspective on meaning constitution, the symbolic formalisation of mathematics goes hand in hand with a peculiar shift or displacement of meaning (Sinnverschiebung). A displacement or shift of meaning, Husserl notes, because in formalised mathematical practice, for example in 15 Please do not cite this preliminary version. The final version of this article will appear in the journal Pacific Philosophical Quarterly, DOI: 10.1111/papq.12084 algebraic calculation, one lets the geometric signification recede into the background as a matter of course, indeed drops it altogether; one calculates, remembering only at the end that the numbers signify magnitudes (usserl a, . Thus, in such a mathematical practice, the numbers attain, as it were, a displaced, symbolic meaning , where this displacement is however considered to be crucial for the accomplishment of the aforementioned project of a completely universal formalization (usserl a, . Husserl further argues that such a displacement of meaning, if considered from a genetic-phenomenological perspective, is closely linked to a process he refers to as an emptying of meaning Sinnentleerung) – an emptying since in formal-mathematical practice, the system of formal symbols is semantically emptied from any intuitive and intuitively evident associations, as is the ultimately the formalised thinking itself (see, e.g., Husserl 1970a, 46). This displacement and emptying of meaning can be aligned with what (usserl more generally refers to in his later writings as the process of a sedimentation of meanings b . Generally speaking, (usserl s concept of sedimentation refers to a linguistically mediated process of a consolidation of meanings. Such consolidation entails for Husserl that in human thinking, those meaning-constitutive cognitive structures that have their base in the human individual s own, perceptual-intuitive activity, are progressively superimposed by persisting linguistic acquisitions (usserl b, . (usserl s take on sedimentation is at times negative insofar as he associates sedimentation both with cognitive passivity and forgetfulness (see Klein 1940, 155-156). At the same time, however, it also appears as if Husserl considers the sedimentation of meanings, and in particular the sedimentation of meaning through writing, to play an essential role in the constitution of more abstract and complex modes of thinking and that of the objects associated with them. This applies most of all to those scientific modes of thinking whose underlying, constitutive form of intentionality necessarily involves systematic use of the technique of formalisation. Closely related to this concession, (usserl also admits that the emptying of the meaning of mathematical natural science through the technique of formalisation (Husserl 1970a, 46), if considered from a 16 Please do not cite this preliminary version. The final version of this article will appear in the journal Pacific Philosophical Quarterly, DOI: 10.1111/papq.12084 practically-minded scientific point of view, is perfectly legitimate, indeed necessary (Husserl 1970a, 47). To spell this out, Husserl seems to acknowledge, either directly or indirectly, that processes of sedimentation centrally participate in the constitution of scientific thought and that of its objects in at least two ways. First, processes of sedimentation, in stabilising and conserving thought through persisting linguistic acquisitions , facilitate the liberation of the individual mind from the impossible, and ultimately unproductive task of thinking everything and constantly anew. This is crucial, for as a matter of fact, any form of scientific practice incessantly questioning and reflecting upon its own concepts and their status and limitations, and thus constantly starting anew, would be intolerably constrained from a practical perspective. Of particular importance in this regard is that processes of sedimentation, and those being related to the use of formalising symbolic technologies in particular, allow to significantly detach thinking from the individual mind and the latter s particular, subjective dispositions and motivations. This ensures that the results produced by such thinking are reliable, reproducible, translatable and exact. Given that sedimentation in some sense stabilises and objectifies thought, and also frees up cognitive resources in this process, it can even be plausibly argued that the more that thought is sedimented, the more potential for scientific progress there is (see Buckley 1992, 91). Second, Husserl appears to recognise that with particular regard to the constitution of ideal objects, processes of sedimentation, specifically those processes that involve the symbolic or linguistic embodiment of ideal meanings through writing, guarantee that the original idealities of geometrical thinking retain their genuine, original meaning (Husserl 1970b, 366). This is not to say that such idealities are originally constituted linguistically. Even in his later thought, Husserl indeed is careful to maintain a distinction between ideal geometrical objects – which are constituted by intentional acts of consciousness – and the idealities of geometrical words, sentences, theories – considered purely as linguistic structures (usserl b, . From a meaning-constitutional perspective, however, the symbolic embodiment and sedimentation of original ideal 17 Please do not cite this preliminary version. The final version of this article will appear in the journal Pacific Philosophical Quarterly, DOI: 10.1111/papq.12084 meanings can nevertheless be considered crucial in that it makes such meanings accessible, at least potentially, to their reactivation by a plurality of individual, embodied minds, regardless of concrete spatio-temporal settings, where moreover in all such reactivation such meanings manifest themselves intersubjectively as remaining always one and the same.11 According to Husserl, such sedimentation and subsequent reactivation of constituted meanings is important for it ensures that ideal objects come to exist, in enduring fashion, objectively in the world see (usserl b, . Viewed in this light, it can plausibly be said that, from the perspective of meaning-constitution, the external, material medium of writing constitutes a necessary condition for the lasting existence of the ideal objects that are constituted through intentional acts, ultimately establishing these objects perfect objectivity Klein , . Finally, there is also a third, more radical and unorthodox line of argument to be made for the crucial role processes of sedimentation plays in the constitution of ideal objects, one which builds upon and yet also departs from (usserl s position presented in his late writings. This is the argument that the constitution of more complex sorts of ideal objects is from the very outset only accomplishable by way of the embodiment of ideal meanings through writing. The reasoning is that the sensible embodiment (Husserl 1970a, 26) of geometrical and mathematical significations through writing makes such significations first of all accessible to the sort of repeated procedures of mental manipulation (usserl 1970a, 27) that are crucial for the scientific constitution of ideal objects. This applies for instance clearly to the algebraic, formal-symbolic sort of mathematics that facilitates the overarching mathematisation of geometrical bodies. This would however mean, contrary to what Husserl insists in his late writings, that the constitution of ideal objectivity can no longer be based exclusively on an act of pure thinking that would be internal to intentional consciousness (Husserl 1970b, 377; see similarly also Tieszen 1989, 116117). Importantly, the enduring symbolic embodiment of scientific concepts and contents makes it not only possible that they can be repeatedly and continuously accessed by a theoretically unlimited number of individual, embodied minds, but it also makes it considerably easier to subject complex theoretic-scientific constructs to a sustained form of critical examination, and at the same time, to progressively building upon and systematically refining them (see Donald 1991, 316). 11 18 Please do not cite this preliminary version. The final version of this article will appear in the journal Pacific Philosophical Quarterly, DOI: 10.1111/papq.12084 Overall, from a genetic-phenomenological perspective, it can thus be concluded that ideal objects belong to a peculiar class of constituted objects whose own constitution is based on intentional acts and yet is inseparable from specific constraints imposed by external symbolic practices and related processes of sedimentation on such acts. In a minimal sense, it has been argued that these practices and processes make it possible that such objects are intersubjectively accessible, by a plurality of cognitive agents, in persisting form, and hence become manifest to these agents as being objectively one and the same. A more radical argument presented here was that symbolic practices also can be considered to play a crucial role in the original scientific constitution of specific, mathematical-formal sorts of ideal objects and their meanings. In either case, this does not imply that the meanings associated with ideal objects are arbitrary or subjective. In the final section of this paper we will now explain the relevance of this meaningconstitutional account for debates on the referential status of abstract singular terms and the objectivity of mathematics. 19 Please do not cite this preliminary version. The final version of this article will appear in the journal Pacific Philosophical Quarterly, DOI: 10.1111/papq.12084 Section 3: Husserl and Abstract Objects Tolerant reductionism upholds mathematical truth and the legitimacy of reference to objects like numbers and sets, but rejects Platonist assumptions about the mindindependent existence of abstract entities. Extant tolerant reductionist accounts, however, including those of Dummett and Linnebo, have failed adequately to justify the claim that the referents of abstract singular terms can play a semantic role in determining the truth conditions of sentences. In the final section of this paper we draw together the preceding analyses to argue that (usserl s theory of meaning-constitution and sedimentation can (a) explain the qualified mind-independence of mathematical objects (b) provide a framework for justifying the attribution of a thin notion of reference to terms for mathematical objects, without this entailing that abstract objects exist in an ontologically robust sense which renders our epistemic access to them mysterious. Just as Dummett s tolerant reductionism seeks a position intermediate between Platonism and nominalism, (usserl s meaning-constitutional account attempts to avoid the extremes of Platonism and psychologism.12 For the later Husserl, ideal objects are independent of the psychological activity of any particular agent, but are nonetheless dependent upon meaning-constitutional acts of intentional consciousness. As Tieszen s (20 appeal to the notion of a constituted Platonism makes clear, (usserl s phenomenological approach thus mediates between the two extreme positions that tend to dominate contemporary debates on abstract entities. Tieszen s account of constituted Platonism – and the solution it provides to the problem of the qualified mind and language independence of abstract entities – can be further clarified, however, by taking into consideration the later (usserl s work on the meaning-constitutional import of the sedimentation of meanings. The sedimentation of constituted ideal objects through writing makes such objects intersubjectively accessible to a plurality of This is not the place to enter into debates about (usserl s alleged early psychologism. )f we define psychologism as the failure to distinguish between subjective psychological experiences and the objective meaning-content that is instantiated in such experiences, however, then it is clear that (usserl s later work is not vulnerable to this charge. (usserl s middle and later phenomenology demonstrates a clear recognition of the distinctions between acts of meaning-constitution considered from the perspective of intentionality and the ideal meaning content that is instantiated in such acts (see, e.g., Husserl 1970b, 356). 12 20 Please do not cite this preliminary version. The final version of this article will appear in the journal Pacific Philosophical Quarterly, DOI: 10.1111/papq.12084 linguistically competent agents and provides the basis for subsequent reactivations of meaning-content. It is through such secondary reactivation that the enduring objectivity of ideal objects first of all manifests itself to such agents. We are now in a position to demonstrate the significance of this development in (usserl s thought for debates on abstract entities. The emphasis that the later Husserl places upon symbolic processes of the sedimentation of meaning as the condition for the enduring, intersubjectively validated objectivity of ideal mathematical objects explains why the reference relation in the case of abstract singular terms differs from that operative in the case of names for concrete objects. In the former case, constituted meanings, in combination with processes of sedimentation, are able to achieve an intersubjectively accessible and demonstrable objectivity that goes beyond any particular psychological act. Those higher-order cognitive practices necessary for the constitution of abstract objects presuppose that we can refer to and re-activate constituted meanings, on the basis of their prior sedimentation, but this clearly does not entail a naïve metaphysical platonism Tieszen, , , where the objects referred to are conceived of as existing in complete independence from human intentionality, thought and practice.13 Husserl s meaning-constitutional account of ideal objects provides a more complete account of the qualified mind and language independence of abstract objects than is found in Frege or the analytic tradition more generally. As we saw in section one, the traditional Platonist upholds a strong counterfactual notion of mind and language independence according to which abstract entities would exist even if there were no intelligent agents to think or talk about them. Although such a position is difficult to justify without recourse to Platonist mythology, it nonetheless remains incumbent on 13 It has been claimed by Zahavi (2003b) that Husserl himself in his earlier writings, notably the Logical Investigations, originally upheld a view concerning philosophical questions regarding the (mindindependent reality of intentional objects that can be referred to as metaphysical neutrality . The question whether the earlier Husserl s position regarding abstract objects really can be subsumed under the principle of metaphysical neutrality lies beyond the scope of this paper; The prevailing view in the literature continues to be that (usserl s earlier view of abstract objects may indeed be classified as Platonist (see for a discerning discussion Hartimo 2012). In regard to Husserl later, genetic writings, which we have focused upon in our discussion, (usserl s key insight is that questions concerning reality and objectivity, including the reality of ideal objectivity, cannot be considered apart from questions concerning (transcendental) intersubjectivity and its constitutive function (see for a range of supporting references from (usserl s work Zahavi a, -125). 21 Please do not cite this preliminary version. The final version of this article will appear in the journal Pacific Philosophical Quarterly, DOI: 10.1111/papq.12084 any philosophy of mathematics to explain the ideality of mathematical objects. What is required, therefore, is an account that acknowledges both the independence of mathematical objects from discrete psychological acts and the objective truth of mathematical statements, without this entailing complete independence from the mind and its meaning-constituting activity. (usserl s genetic-phenomenological account achieves this balance because it explains mind and language independence in terms of the abiding intersubjective validity of meaning-content that is rationally constituted through processes of idealisation and formalisation, and which can be re-activated as a result of prior processes of sedimentation. It is worthwhile recalling at this point the Dummett-Linnebo thesis that reference to abstract objects is language-internal or thin. In the case of an expression picking out a medium-sized concrete object, Dummett argues, the identification of that object plays a role in the determination of the truth conditions of a sentence containing reference to it. In the case of abstract singular terms, by contrast, the identification of an abstract object – at least considered as a constituent of external reality – does not play such a role. This is what leads Dummett to express serious doubts about the attribution of a robust notion of reference to abstract singular terms, and ultimately to adopt a position that has strong nominalist tendencies. Linnebo s analysis of the capacity of abstract singular terms to play a semantic role within sentences is similarly suggestive of a sophisticated version of nominalism. Linnebo s analysis of the notion of semantic role, moreover, does not adequately address how the thinness of reference to abstract objects is compatible with their qualified mind and language independence. (usserl s later account of ideal objects, by contrast, allows for a more cogent explanation of the thinness of abstract singular terms than is found in extant intermediate accounts. In his later work, Husserl emphasises the role played by symbolic forms that embody sedimented meaningcontents based on earlier acts of meaning-constitution (even if the authentic, original meaning of such contents is often left dormant). This account accordingly explains the thinness of reference to abstract objects in terms of the capacity of abstract singular terms, such as numerals, to refer to sedimented meaning contents that, in turn, presuppose prior acts of meaning-constitution. 22 Please do not cite this preliminary version. The final version of this article will appear in the journal Pacific Philosophical Quarterly, DOI: 10.1111/papq.12084 Construing the reference of abstract singular terms as thin in the above sense explains why framing the problem of abstract objects in terms of existence is potentially misleading. As Tieszen s analysis suggests, the primary problem of abstract objects is that of adequately characterising their qualified mind-and language independence. Once we adopt (usserl s meaning-constitutional perspective, however, there is no conflict between regarding numbers as constituted entities from a phenomenological-genetic perspective whilst also regarding them as extensional from the semantic perspective. This returns us to the discussion in section one regarding the reductionist aspect of Dummett s intermediate position. The interpretation one puts upon the quantifier from a meta-semantic perspective is decisive here and a few comments are therefore needed at this point on the meta-semantic position suggested by tolerant reductionism. This involves a clarification, if not rejection, of Quine s doctrine of ontological commitment. Nominalistically inclined philosophers of mathematics such as Jody Azzouni (2004) have recently argued that the claim that the objectual quantifier quantifies over a domain of real objects is a postulate and is not guaranteed by the semantic condition for the quantifier. Even if we assume a standpoint where there are Fs commits us to Fs, and that there are Fs is thus true, we cannot conclude that there really are Fs. The conclusion is that we cannot move from reference to ontology in the full-blooded sense of telling us what there really is. We can talk about immanent ontology if we wish – the presupposition here is that the object language is part of the metalanguage – but this is really just to talk about what makes our statements true within a particular language.14 Carnap s account of existential quantification is germane here. Carnap suggests that it is at the level of statements presupposing (in the background semantics of the interpreted language) that numbers exist that philosophical argument about the existence of such entities needs to be carried out (see Tennant 1997, 310). Pace Quine, statements such as x x = (1997, 3 do not, of themselves, convey commitment to numbers, unless taken jointly . This is because only the stance or attitude of mind involved in whole- heartedly adopting the relevant form of discourse conveys commitment to the things whereof one speaks (1997, 310). So, from a Carnapian perspective, we can quantify 14 Our analysis here is indebted to Brogaard (2008) and Eklund (2010). 23 Please do not cite this preliminary version. The final version of this article will appear in the journal Pacific Philosophical Quarterly, DOI: 10.1111/papq.12084 over numbers and express commitment to the truth of a mathematical theory whilst still maintaining that numbers are ontologically thin. One of the advantages of a Carnapian account (see 1950) of ontological commitment is that it allows for the adoption of a tolerant attitude towards abstract objects. The rationale for such a tolerant stance, however, is best understood in terms of the Husserlian meaning-constitutional account we have outlined. Reference to mathematical objects is not only possible; it is a presupposition of the meaningfulness of a science such as arithmetic. In referring to numbers as objects, however, we still leave open the possibility of a metasemantic reduction of their ontological import. Such a reduction is indeed strongly suggested by the fact that we are dealing with sedimented meaning-contents rather than pre-existing constituents of extra-linguistic reality. In closing, the merits of a tolerant reductionist position incorporating Husserlian insights on meaning-constitution can be seen in the way it accommodates the partial truths and avoids the extremes of both Platonism and nominalism. The truth implicit in Platonism is that mathematical objects have a qualified mind and language independence. But this is not to say that one can simply establish that they are independent of the minds and linguistic practice of agents simpliciter, or that such objects are discovered. Rather, what it means is that the objects of mathematics, while constituted, have an intersubjectively accessible, enduring ideal objectivity, which is validated through the reactivation of sedimented meanings. The nominalist, like Husserl, rejects the Platonist image of a pre-existing realm of strongly mindindependent mathematical objects. Yet unsophisticated variants of nominalism contradict our capacity to engage in meaningful discourse about mathematical objects. More sophisticated variants are not necessarily incompatible with the position defended here, but lack the explanatory power of (usserl s genetic-phenomenological account due to their neglect of meaning-constitutional considerations. School of Humanities and Social Sciences Deakin University 24 Please do not cite this preliminary version. The final version of this article will appear in the journal Pacific Philosophical Quarterly, DOI: 10.1111/papq.12084 Melbourne Graduate School of Education The University of Melbourne 25 Please do not cite this preliminary version. The final version of this article will appear in the journal Pacific Philosophical Quarterly, DOI: 10.1111/papq.12084 References Azzouni, J. (2008). Deflating existential consequence: A case for nominalism. Oxford: Oxford University Press. Balaguer, M. (1998). Platonism and anti-platonism in mathematics. New York: Oxford University Press. Benacerraf, P. . Mathematical truth . The Journal of Philosophy, 70, 661-679. Bernet, R., Kern, I. and Marbach, E. (1993). An introduction to Husserlian phenomenology. Evanston, Ill.: Northwestern University Press. (Original work published 1989). Brogaard, B. (2008). Inscrutability and Ontological Commitment. Philosophical Studies, 141, 21-42. Buckley, P. (1992). 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