The ontological implications of neo-fregeanism

Daimon. Revista Internacional de Filosofía, nº 69, 2016, 159-174 ISSN: 1130-0507 (papel) y 1989-4651 (electrónico) http://dx.doi.org/10.6018/daimon/221831 The ontological implications of neo-Fregeanism Las implicaciones ontológicas del neo-fregeanismo MARÍA DE PONTE* Fecha de recepción: 01/03/2015. Fecha de aceptación: 29/03/2016. * University of Seville. Profesora Ayudante Doctora. [email protected] Areas of Specialization: Philosophy of language, philosophy of mathematics, philosophy of time. Two recent publications: Kepa Korta & María Ponte, 2015. “Tenses, dates and times” Research in Language. Vol. 12, 4: 301-317. María Ponte & Margarita Vázquez, 2012. “Tense and temporal reference. Hybrid temporal logic” Logique & Analyse, 220: 555-578. Thanks to Manuel Liz, Kepa Korta, and Margarita Vázquez for their comments and suggestions on earlier drafts. This work was partially supported by the Spanish Ministry of Economy and Competitivity FFI201563719-P (MINECO/FEDER) and the Andalusian Government (P12-HUM-1216). Abstract: Neo-Fregeanism is a combination of two ideas: logicism, according to which arithmetic can be derived from logic plus definitions, and Platonism, according to which there are mathematical objects (which are abstract). NeoFregeans propose a new interpretation of Frege’s principles of abstraction (mainly the so-called Hume’s Principle) and of the role of reconceptualization and implicit definition for the introduction of numbers into our ontology. I analyze the ontological implications of neo-Fregeanism, not only for mathematics, but for abstract entities in general. After briefly introducing some of the main elements of neo-Fregeanism, I present two possible readings of its ontological implications and I argue that none of them gives the desired results. Keywords: Platonism, logicism, reconceptualization, implicit definitions, priority thesis, nominalism, maximalism, relativism. Resumen: El neo-fregeanismo es una combinación de dos ideas: el logicismo, según el cual se puede derivar la aritmética de la lógica y definiciones, y el platonismo, según el cual hay objetos matemáticos (abstractos). Los neo-fregeanos proponen una interpretación nueva de los principios de abstracción Fregeanos (especialmente del llamado principio de Hume) y del papel de la reconceptualización y las definiciones implícitas para la introducción de los números en nuestra ontología. Analizo las implicaciones ontológicas del neo-fregeanismo, no sólo para las matemáticas, sino también para las entidades abstractas en general. Tras introducir brevemente los principales elementos del neo-fregeanismo, presento dos interpretaciones alternativas de estas implicaciones ontológicas y argumento que ninguna de ella aporta los resultados deseados. Palabras clave: Platonismo, logicismo, reconceptualización, definiciones implícitas, tesis de la prioridad, nominalismo, maximalismo, relativismo. 1. Introduction The literature around neo-Fregeanism (NF, from now on) is very impressive. NeoFregeans promise a solution (or at least the beginnings of one) to some of the most debated philosophical issues in the twentieth century: mathematical existence, truth and knowledge. The goal of NF is, roughly, to combine two ideas: logicism, according to which arithmetic can be derived from logic plus definitions, and Platonism, according to which there are 160 María de Ponte Daimon. Revista Internacional de Filosofía, nº 69, 2016 mathematical objects (which are abstract). Most of the literature focuses on the possibility of reviving Frege’s logicism, on the interpretation of Frege’s principles of abstraction--mainly the so called Hume’s Principle--and on the role of reconceptualization and implicit definition for the introduction of numbers into our ontology. Recently, however, some authors have attempted to analyze the ontological implications of NF, not only for mathematics, but for abstract objects in general. It is within these attempts that this paper belongs. The ontological implications of both Frege’s logicism and NF are far from clear. There are several interpretations of the ontological aspects of NF in the literature. It could be claimed that NF leads to “maximalism” (Eklund, 2006, Hawley, 2007); to “quietism” (MacBride, 2003) or to “quantifier variance” (Sider, 2007).1 These interpretations have important differences among them, but they all have in common a “deflationary” character. That is, according to all of them, NF entails some sort of deflationary or minimal ontology. By deflationary ontology I mean here, roughly, that there are different interpretations of what does and does not exist and none of them is, in principle, better than the others. That is, deflationists believe that none of these possible interpretations is more “natural” than the others or, in Sider’s terms, none “carves logical reality at its joints”. This idea of NF as a kind of ontological deflationism (in which claims about the existence of entities are derived from the analysis of language) is best seen in a quote from Wright: The irresistible metaphor is that pure abstract objects […] are no more than shadows cast by the syntax of our discourse. And the aptness of the metaphor is enhanced by the reflection that shadows are, after their own fashion, real. (Wright, 1992, p. 181-2). My plan for this paper is the following. I begin by introducing the main theses of NF, paying special attention to its defense of the priority of syntax over ontology and to its notion of reconceptualization. I then explore the existential commitments derivable from NF and its relation with the notion of analyticity. Afterwards I present two alternative interpretations of these commitments, a nominalist and a maximalist one. I claim that both of them are viable interpretations of NF and so that, being as it is an attempt to defend a realist position in mathematics, maximalism seems to be the best reading (perhaps the only possible reading) of NF. Maximalism, though, faces serious problems, which, I claim, NF cannot face unless it adopts some kind of relativization. 2. Neo-Fregeanism, a brief introduction NF is to be considered mainly as a particular philosophy of mathematics, a proposal regarding mathematical existence, but also mathematical truth and mathematical knowledge. But it presupposes some substantive assumptions about the nature of reality and language in general. More specifically, it presupposes a particular view about the relation between language and reality, a view according to which “the structure of reality mirrors the contours of our speech” (MacBride, 2003, p. 108). 1 Sider does not defend quantifier variance himself; on the contrary, he devotes most of his efforts to combating it, but he does claim that it is the best interpretation for a defender of NF. 161 The ontological implications of neo-Fregeanism Daimon. Revista Internacional de Filosofía, nº 69, 2016 As is well known, Frege’s philosophy of mathematics was founded upon two basic elements: Platonism and logicism.2 It is also well known that his system failed when Russell proved it to be inconsistent. Shortly stated, NF accepts most of the philosophical framework underlying Frege’s project, mainly its Platonism, its notion of analyticity of mathematics (i.e. its reduction to logic plus definitions) and the thesis that epistemological problems that affect platonist theses in mathematics can be solved through this reduction. But they only accept part of Frege’s formal apparatus, thus avoiding Russell’s paradox.3 Following the lead of Frege, NF argues that we gain knowledge of mathematical reality through the so called “abstraction principles”, in particular the so-called “Hume’s principle” (HP, from now on), which functions as an implicit definition of the concept of (cardinal) number. In other words, our knowledge of mathematical objects arises from principles that are analytic or true by meaning alone, and from our ability to derive mathematical truths from those analytic principles. One central idea is the Fregean notion of the context principle, according to which only in the context of a proposition does a word mean anything. This, plus the idea, also Fregean, of the priority of syntax over ontology and the defense of the so called “abstraction principles” as an explanation of the way in which we reason, will be the core of the argument of NF. According to the neo-Fregeans, a proper understanding of the context principle and the priority thesis will not only prove that numbers exist, but it will also provide us with an explanation of our knowledge of them, so that it won’t be necessary to postulate something like Gödel’s idea of mathematical intuition to explain our access to them. In order to apply all this to mathematical terms, and specifically (from now on) to number terms, it is necessary to determine the sense of the statements in which number terms occur. Besides, taking -as Platonists dothat number terms stand for self-sufficient objects, what is needed is an explanation of the sense of identity statements connecting terms with numbers. Very roughly, once we can assume that one such identity statement is true, and taken that number terms are singular terms, we will have to accept that they have a reference. Therefore, if number terms refer, number objects must exist. According to NF, knowledge of the basic arithmetical laws (essentially, the DedekindPeano axioms), and hence of the existence of a range of objects which satisfy them -that is, knowledge of the concept of (cardinal) numbercan be explained (a priori) by fixing the truth-conditions of identity statements between (canonical) singular terms and their instances; that is, by means of Hume’s Principle, a second order abstraction principle according to which, informally: (HP) The number of Fs is identical to the number of Gs if and only if there is a oneone correspondence between the Fs and the Gs Formally: (HP) ∀F∀G (nx:Fx = nx:Gx ↔ F 1-1 G) 2 Frege (1884). 3 See Cook (2009) for an exposition of the differences and similarities between the original Fregean project and the NF one. 162 María de Ponte Daimon. Revista Internacional de Filosofía, nº 69, 2016 Where “F1-1G” is (an abbreviation of) a second order formula expressing that there is a one-one correspondence between the object falling under the concept F and the objects falling under the concept G.4 Frege did not consider HP as a definition of the concept of number. Taken as a definition, HP faces the so-called “Caesar Problem”, that is, it is incapable of stating whether Julius Caesar is a number or not (the definition is not applicable to sentences such as “the number of Fs = Julius Caesar”). For this reason, Frege introduced the notion of numbers as the extensions of concepts and with it the infamous basic law V which, as is well known, leads directly to Russell’s paradox and thus to the collapse of Frege’s logicism. Now, it is well known that the Peano Axioms (including the second-order induction axiom) can be derived from HP plus suitable definitions of zero, successor and the like. This result is known as Frege’s Theorem. It is the contention of NF that this theorem is enough for a proper definition of numbers. Further, it is NF’s contention that such a definition is enough to ensure the existence of numbers. Be that as it may, it is not my intention to get into these (otherwise amply discussed) technical issues; rather, I will just offer a brief description of the program and focus on the ontological implications of HP and NF.5 According to NF, HP is an implicit definition of –cardinal– number and that is what is needed to get both the concept of number and to derive the very existence of numbers. This raises many questions. To begin with, one might wonder: how can a definition determine the existence of the entities it attempts to define? Neo-Fregeans claim that HP is an implicit definition of the concept of cardinal number and that it is possible to derive the existence of cardinal numbers through the (truth of the) identity with equinumerosity. But, it might be objected, when we talk about equinumerosity we are not talking about numbers: equinumerosity does not require the existence of numbers. How can HP support their existence then? How can a definition introduce new objects? 3. Implicit Definitions The answer to these questions requires understanding first the notion of “implicit definition” and its role in explaining mathematical knowledge. An implicit definition, roughly, is a definition in which the meaning of a term may be given by the assertion of statements containing it, by imposing some sort of constraint on the use of longer expressions containing the term in question. So if we consider, for example, a sentence #f to be true, we can provide “f” with a meaning (a meaning that would make #f true). According to Wright and Hale, so long as we have asserted that concepts are one-one correspondent (so long as we have established Hume’s Principle to be true) “there need to be no further problem about our knowledge of certain basic kinds of truths about numbers” (2002). Provided that we accept that Hume’s Principle is an implicit definition of the concept of number we can know 4 The second order formula of equinumerosity can be formalized as: $R∀x (Fx →$y (Gy ∧ ∀z (Rxz ↔z=y)) ∧ (Gx → $y (Fy ∧ ∀z (Rzy → z=y))). 5 Wright, 1983, is the locus classicus of NF. See also Hale and Wright, 2001, a compilation of their writings on this subject for the last decade or so. 163 The ontological implications of neo-Fregeanism Daimon. Revista Internacional de Filosofía, nº 69, 2016 [S]tatements of [...] numerical identity to be true just by knowing the truth of the appropriate statements of [...] one-one correspondence among concepts. We can do so for the unremarkable reason that the truth conditions of the former are fixed by stipulation to coincide with those of the latter. (Hale and Wright, 2002) There are many problems to be solved, however, before this can be a compelling account of a priori knowledge of mathematical statements. I focus here on one of them, in my opinion a very relevant one since it begs the very problem NF wanted to solve: to prove that there are abstract independent objects and that we have access to them. According to implicit definitions, then, taking #f to be true, and knowing the meaning of ‘#’, we can know the meaning of ‘f’: the one which makes ‘#f’ true. But an obvious question arises here for, if we can state a priori that #f is true only by knowing the meaning of # and the syntactic structure of the sentence, then the meaning of ‘f’ can’t add anything substantial to the semantic value of the statement; the meaning of ‘f’ must be, in this sense, conservative. But actually, in the case of HP the “number of F” asserts, on a platonist reading, something quite substantial, it implies an ontological commitment not present, prima facie, in the one-one correspondence. This becomes clearer if we analyze the two sides of the biconditional separately: The number of Fs is identical to the number of Gs There is a one-one correspondence between the Fs and the Gs Is easy to see that a) requires the existence of numbers to be considered as a literal truth6 while b) does not. Wright answers this stating that even though speakers don’t refer to numbers in b), their existence is implicit in the concept of equinumerosity. He claims that we can obtain knowledge of abstract objects (numbers or directions, in the other example used by him) through our knowledge of concrete ones (relation of equinumerosity or relation of parallelism between two lines). Moreover, and this leads us to the point to be discussed here, he claims that we can infer the existence of the former from the existence of the latter, but how can this be possible? According to him, the existence of abstract objects such as numbers is conceptually necessary and follows logically from the existence of equinumerosity (as the existence of directions follows logically from the existence of parallel lines). And this is so, he claims, because HP is to be considered a “reconceptualization”. 4. Reconceptualization Basically, considering HP as a reconceptualization means claiming that the two sides of the biconditional have the same content, the difference being only in the way this content is conceptualized. Hence it is relatively easy to see how HP guarantees the existence of numbers. The idea is to go from the truth of the right-hand-side of the biconditional (there is a one-one correspondence between the Fs and the Gs) to the truth of the left-hand-side (the number of Fs is identical to the number of Gs). 6 Literal in contrast with fictional or metaphorical truths. NF requires singular terms to refer to “real” entities, and not fictional or imaginary ones. Further on, I specify how they claim that if a statement is determined to be true, following the established criteria, then it is “really” true. That is, literally true. 164 María de Ponte Daimon. Revista Internacional de Filosofía, nº 69, 2016 Now, if both sides of the biconditional have the same content, numbers would have to be somehow present in talk of equinumerosity. But surely, one might object, this cannot be so, since the left-hand side quantifies over numbers and the right hand side doesn’t. Against this, neo-Fregeans argue that, contrary to appearances, when we talk about equinumerosity we are implicitly assuming the existence of numbers (even though we are not quantifying over them and thus we are not making any reference to them). This is a crucial, and problematic, aspect of NF. We have to be able to use concepts without knowing which entities fall under it, or at least, without knowing all the entities that fall under it. Wright establishes a useful distinction here; he claims that even though talk about equinumerosity implicitly entails a commitment to the existence of numbers, it does not entail reference to numbers. Using his example, talk about “aunt” implicitly entails a commitment to the existence of parent and a sister (irrespective of whether we know it or not), but it does not necessarily entail referring to any of those. One aspect worth mentioning, for it will be relevant for our later discussion, is that by HP and other abstraction principles, and by the idea of reconceptualization, Neo-Fregeanists do not attempt to claim that numbers are introduced or created in some way. What is introduced is the concept of number, but the objects (numbers) were part of the underlying ontology presupposed by talk of equinumerosity. I think it might be helpful to quote Wright’s exposition of the idea of reconceptualization (through the use of another example involving the concept of direction): Consider again the abstraction for directions: Da=Db if and only if a//b The dilemma was that we either regard the left hand side simply as a definitional transcription of the right, and thereby forfeit the possibility of taking its syntax at face value, of treating it as a genuine identity statement linking genuine singular terms in existentially generalizable position; or we take the principle as a substantial claim, to the effect that certain abstract objects –directionsare associated with lines in the way it describes, in which case we have no right simply to lay the principle down as a definition. But the key to Frege’s view is that the dilemma is a false one – it is the thought, roughly, that we have the option of laying down the Direction abstraction, of reconceptualizing, as it were, the type of state of affairs which is described on the right. […] The concept of direction is thus so introduced that that two lines are parallel constitutes the identity of their directions. […] It is important to be clear that it would be a misrepresentation of this idea to view it as involving the notion that abstract objects are creations of the human mind, brought into being by a kind of stipulation. What is formed –createdby such an abstraction is rather a concept: the effect is merely to fix the truth-conditions of identity statements concerning a new kind of thing, and it is quite another question whether those truthconditions are ever realized. (Wright, 1997, pp. 277-278). 165 The ontological implications of neo-Fregeanism Daimon. Revista Internacional de Filosofía, nº 69, 2016 So, NF claims that the existence of numbers (or directions) is already present (albeit implicitly) when we talk about equinumerosity relations (or parallelism). Hence neoFregeans do not want to defend the conditional assertion according to which “if numbers exist, then HP”. Such conditional is not necessary, they claim, for numbers are part of the underlying ontology. Now, even though the idea of reconceptualization is central for the NF program, it ultimately rests on a general conception of the relation between language and reality and therefore on the thesis of the priority of syntax over ontology and the context principle. These two principles (or theses) will also allow us to explore the possibilities of generalizing the NF proposal beyond the mathematical realm and to explore thus its ontological implications. 5. The priority thesis According to Wright, the priority thesis not only asserts that there is a particular relation between reference and truth, i.e., that if certain statements (paradigmatically identity statements) are true then singular terms that occur in them (in the right way) do refer to determinate objects (through the context principle), and thus these objects exist. It also states that truth is prior to reference. That is, if, by ordinary criteria, a statement is true, then it is really true (beyond any reasonable doubt). Hence, if by ordinary criteria7 we establish the truth of a mathematical statement, we can establish that the singular terms occurring in it refer and, thus, the existence of mathematical entities (to which these terms refer) would be proved beyond any doubt. Wright, in one of his most quoted paragraphs, explains it so: According to [the thesis of the priority of syntactic over ontological categories], the question whether a particular expression is a candidate to refer to an object is entirely a matter of the sort of syntactic role which it plays in whole sentences. If it plays that sort of role, then the truth of appropriate sentences in which it so features will be sufficient to confer on it an objectual reference; and questions concerning the character of its reference should then be addressed by philosophical reflection on the truth-conditions of sentences of the appropriate kind. If, therefore, certain expressions in a branch of our language function syntactically as singular terms, and descriptive and identity contexts containing them are true by ordinary criteria, there is no room for any ulterior failure of ‘fit’ between these contexts and the structure of the states of affairs which make them true. So there can be no philosophical science of ontology, no well-founded attempt to see past our categories of expression and glimpse the way in which the world is truly furnished. (Wright, 1983, pp. 51-52) 7 What is meant by ordinary criteria is, precisely, one of the most problematic issues of the proposal. We’ll try to see the alternatives later, when we talk about the possible limitations to NF’s overinflated ontology. Field, 1989, expresses the difference between a trivial reading of the context principle and the more radical one by differentiating between a “weak” and a “strong” priority thesis. He then argues that it is not possible to develop a proper account of what is meant by “ordinary criteria” and thus, that the strong priority thesis (needed to ensure Platonism) is not viable. 166 María de Ponte Daimon. Revista Internacional de Filosofía, nº 69, 2016 So, NF reverses the usual, realist, way of understanding the relation between language and reality, in which it is generally assumed that the nature of reality is fixed independently of language. Neo-Fregeans believe that language and reality are very closely related; so closely in fact that, as MacBride puts it, “the structure of reality inevitably mirrors the contours of our language” (2003, p. 108). 6. Existential commitments and analyticity So far so good, but it is far from clear how the notion of implicit definition, the priority thesis and the subsequent notion of reconceptualization could bear the weight of the existential commitments required by platonists. I present two possible interpretations of NF. One leads to triviality, making the claim that numbers exist superfluous. The second leads to an overgeneralization, entailing the existence of counterintuitive, exceptional and even incompatible entities. Ultimately, I claim, the only way available for NF is to accept the introduction of a certain level of relativization. This should not be a surprise - it follows directly from some basic aspects of the neoFregeans’ philosophical framework, i.e., from their belief in the preeminence of language over ontology. But it would certainly be a problem for them. It would mean, that their main goal, defending and justifying Platonism, accounting for mathematical knowledge and mathematical truth, has failed. An apparent problem with NF, we hinted, was understanding how it was possible to logically derive the existence of certain kind of entities from the existence of some other –completely differententities. That is, how to make sense of the claim that numbers are independent objects, completely different from equinumerosity and yet their existence and properties follow logically from the existence and properties of equinumerous objects. Actually, nothing would prevent constructing the identity in nominalistic terms. Or, more precisely, a nominalist reading of the meaning of “the number of F” will not change the truth-value of the whole. It will not make it false, but only “fictionally” true. The “number of F” could be read as a singular term syntactically speaking, but functioning semantically like fictional terms such as “Hamlet”, and this will not change the semantic value of the identity. So, what prevents us for making this nominalist reading of the identity, apart from some independent considerations about the convenience of not doing so? That is, the choice between a platonist or a nominalist reading does not seem to be determined by anything contained in NF, failing thus in one of its main goals: justifying and accounting for Platonism. This ontological neutrality follows from the required analyticity of HP and of the abstraction principles in general. The problem is explaining how an analytic principle, such as HP, can guarantee the existential commitment required by the platonists. This is a doubt similar to the one raised by Boolos in his article “Is Hume’s principle analytic?” HP possesses too much content to be analytic. Boolos suggests that there is a clear analogy between HP and “the present king of France is royal” in that “we have no analytic guarantee that for every value of “F”, there is an object that the open definite singular description “the number belonging 167 The ontological implications of neo-Fregeanism Daimon. Revista Internacional de Filosofía, nº 69, 2016 to F” denotes” (1998, p. 306).8 In other words, there seems to be “no reason at all to believe that it is analytic that for every F, there is such a (unique) object x”. (Boolos, 1998, p. 308).9 Platonists could argue that implicit definitions do indeed require existential commitment since, as we have seen, the context principle requires that number terms within it have a referent. But this is not straightforward. Following Hartry Field (1989) we could differentiate between weak and strong priority theses (priority of syntax). According to the weak priority thesis number terms function syntactically as singular terms and therefore also function semantically as such. This ensures that number terms can’t be reduced to terms about oneone correspondence (it keeps the semantical independence of the former and so avoids what Field calls “ontological reductionism”). This weak thesis is strong enough to grant the viability of existential commitment of statements of the form: 2 + 1 = 3 The number of books in this room is greater than or equal to 3 But it doesn’t rule out completely the possibility of denying it. It is not enough, in order to defend Platonism, to develop a way of stating that by Hume’s Principle together with the context principle, we can infer that number terms refer and that number objects exist. What is needed is a way of asserting that it is necessarily true that numbers exist, that every question about the existence of numbers is “vacuous”. And the weak priority thesis certainly does not grant this. Consider a staunch nominalist. She could accept, in the light of the arguments given so far, that (2) is true and thus that the term “3”, being a singular term, implies an existential commitment. But that would not prevent her from arguing (as Field in fact does, or at least did) that (2) can and should be re-formulated so that “3” doesn’t function as a singular term any more, but rather appears as part of a numerical quantifier. That is, (2) could, and perhaps should, be read as, 2*. There are at least 3 books in the room, which does not entail the existence of numbers or any other abstract entity. Of course, the nominalist who re-formulates sentences this way will have to give reasons as to why we should prefer (2*) over (2) and show that we can make similar re-formulation for the relevant cases. But this is not the important point here. The relevant point is to notice that implicit definition plus context principle does not guarantee the existence of numbers. Or rather, that implicit definitions plus the context principle are compatible with a nominalist reading. If what we want is to justify our knowledge of mathematical entities and account for mathematical truth, we need to show that the existence of those objects is necessary; we need to prove, by implicit definition, that those objects exist. Otherwise it won’t be possible to state, as NF does, that just by knowing the truth of statements about 8 Boolos also develops in this article what has been called “the bad company argument” according to which “there may be some analytic truths in the vicinity of HP with which it is being confused”. More on this later. 9 Ibid, pp. 308. Recall that in this quote another problem is being mentioned: the problem of uniqueness. How can we know that the concept F denotes one, and only one, object x?. 174 María de Ponte Daimon. Revista Internacional de Filosofía, nº 69, 2016 NF can only satisfy those who are already convinced that abstract entities exist and that numbers are indeed abstract entities. If taken as an attempt to prove Platonism, I have claimed that NF fails, because its ontological implications are not the desired ones. A defense of Platonism can only come from considerations external to NF. References Balaguer, M. (1998) Platonism and Anti-Platonism in Mathematics. Boolos, G. 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