2008 Frege on Definitions

Philosophy Compass 3/5 (2008): 992–1012, 10.1111/j.1747-9991.2008.00167.x Frege on Definitions Sanford Shieh* Wesleyan University Abstract This article treats three aspects of Frege’s discussions of definitions. First, I survey Frege’s main criticisms of definitions in mathematics. Second, I consider Frege’s apparent change of mind on the legitimacy of contextual definitions and its significance for recent neo-Fregean logicism. In the remainder of the article I discuss a critical question about the definitions on which Frege’s proofs of the laws of arithmetic depend: do the logical structures of the definientia reflect the understanding of arithmetical terms prevailing prior to Frege’s analyses? Unless they do, it is unclear how Frege’s proofs demonstrate the analyticity of the arithmetic in use before logicism. Yet, especially in late writings, Frege characterizes definitions as arbitrary stipulations of the senses or references of expressions unrelated to predefinitional understanding. I conclude by examining some options for conceiving of the status of Frege’s logicism in light of this apparent tension, and outline a suggestion for a philosophically fruitful way of resolving this tension. Frege discussed definitions throughout his writings. Two prominent themes can be discerned in these discussions. First, Frege thought that a significant number of logical defects in the mathematics of his day stems from inadequacies in definitions. Second, definitions are central to both the execution and the philosophical significance of logicism, Frege’s principal intellectual project. Here I begin with a fairly brief survey of Frege’s criticisms of mathematical definitions. Then, in the bulk of this article, I discuss the philosophical status and significance of definitions in Frege’s logicism. I will not comment on any details of Frege’s derivation(s) of arithmetic from definitions except his use of contextual definitions. 1. Defective Definitions What is a definition for Frege? A fairly neutral account, applicable with some variations to all periods of his philosophical development is: ‘In definition it is always a matter of associating with a sign a sense or a reference. Where sense and reference are missing, we cannot properly speak either of a sign or of a definition’ (FC 139 n. 4; see also FT 115).1 As stated, the account obviously does not apply before Frege made the sense/reference distinction, © 2008 The Author Journal Compilation © 2008 Blackwell Publishing Ltd Frege on Definitions 993 so the required pre-sense/reference account is that a definition associates with a sign a content or meaning (notions which combine sense and reference). There are four main types of criticism that Frege makes of definitions in mathematics. The first type of criticism rules out purported definitions because they conflict with the law of excluded middle. This law of logic grounds Frege’s principle of completeness, one of two principles of definitions Frege sets out in volume II of Grundgesetze der Arithmetik: ‘A definition of a concept . . . must be complete; it must unambiguously determine, as regards any object, whether or not it falls under the concept’ (GG2 §56, 159). Frege habitually uses the metaphor of sharp boundaries for this requirement: ‘[t]he law of excluded middle is . . . just another form of the requirement that the concept should have a sharp boundary’ (GG2 §56, 159).2 The principle as formulated here applies to definitions of concepts – more precisely, of signs for concepts, but a closely related requirement applies to definitions of signs for objects: ‘If we are to use the symbol a to signify an object, we must have a criterion for deciding in all cases whether b is the same as a, even if it is not always in our power to apply this criterion’ (GL §62, 73). The principle of completeness rules out what Frege describes as ‘the mathematicians’ favorite procedure’: First they give the definition for a particular case – e.g. for positive integers – and make use of it; then, many theorems later, there follows a second definition for another case – e.g. for negative integers and zero. (GG2 §57, 159– 60) Frege calls such definitions piecemeal, and the principal problem afflicting them is that at each stage of the procedure nothing is specified about whether the defined concept-sign applies to anything, or the defined objectsign refers to anything, outside the domain of objects for which the definition is given, and this runs afoul of the principle of completeness (see also LM 242–3). The principle of completeness also rules out conditional definitions, which specify the sense or reference of a term given the satisfaction of certain conditions. Piecemeal definitions are a species of conditional definitions, and the problem for the latter is similar: if the conditions are not satisfied, then the signs being defined fail to have sense or reference (see LM 230). The procedure of piecemeal definition leads to two other, at least potential, problems. To begin with, since a piecemeal definition does not cover all cases, one can subsequently advance additional definitions of the definiendum, applying to expanded domains of objects. These definitions might assign different meanings to the definiendum (see SN 265). This leads to the possibility of (apparent) contradictions: ‘[e]ven if in fact [the mathematicians’ definitions] avoid contradictions, in principle their method does not rule them out’ (GG2 §57, 160). For example, [i]f . . . the words ‘square root of 9’ have been defined with a restriction to the domain of positive integers, then we can prove . . . the proposition that there is © 2008 The Author Philosophy Compass 3/5 (2008): 992–1012, 10.1111/j.1747-9991.2008.00167.x Journal Compilation © 2008 Blackwell Publishing Ltd 994 Frege on Definitions only one square root of 9; but this is . . . overthrown when we extend our treatment to negative numbers, (GG2 §61, 164) for there are then two square roots of 9. Thus, ‘[p]iecemeal definition . . . makes the status of theorems uncertain’. The other potential problem arises when mathematicians use an expression defined piecemeal in ‘cases [unspecified by that piecemeal definition], as if they had given it something to stand for’ (GG2 §61, 164). As a result, we might ‘perform calculations with empty signs in the belief that we’re dealing with objects’ (FC 148). Frege does not spell out what exactly happens in such cases, but only says that ‘[p]eople have in the past carried out invalid procedures with divergent infinite series’ (FC 148). Presumably what Frege has in mind are the uses of arithmetical functions defined only for finitely many arguments to compute the results of arithmetical operations on infinitely many numbers, which led even great mathematicians such as Euler to fallacious conclusions about sums of infinite series.3 Note that if definition consists of giving an expression sense or reference, then what Frege’s first type of criticism suggests is that for him logic is more fundamental than sense or reference. Expressions which seem to contribute to the truth conditions of purported statements that seem to conflict with the laws of logic do not provide grounds for limiting the scope of those laws; instead, they do not in fact have any senses or references, and sentences in which they occur have no truth-values, and, at best, express fictional thoughts.4 The second type of criticism is directed at what Frege calls ‘creative’ definitions, purported definitions conceived as ‘able to endow a thing with properties that it has not already got’ (FC 139; see also DRC 69). In the Preface of Grundgesetze Frege traces the formalists’ use of creative definitions to a ‘widespread inclination to acknowledge as existing only what can be perceived by the senses’ (GG1 10). Such a view runs into the problem that numbers seem to be imperceptible, and its proponents respond by claiming that numbers are in fact numerals, signs (at least the tokens of ) which are perceptible. But the properties of perceptible signs are apparently quite different from properties of numbers. To answer this objection, formalists ‘simply ascribe to [the signs] the desired properties [of numbers] by means of what [they] call definitions’ (GG1 10). If the procedure of creative definition were unrestrictedly valid, then it would be possible to confer contradictory properties on any object by means of such definitions (CES 223). So at the very least the procedure must be constrained by consistency (FT 117). How, though, can we be sure that the properties we have put together in a definition are consistent? Only, Frege claims, ‘by proving that there is an object with all these properties together’ (GG2 §143, 158). But if such a proof is required to vindicate a creative definition, then, in the words of a later writer, all its advantages over honest toil are lost. © 2008 The Author Philosophy Compass 3/5 (2008): 992–1012, 10.1111/j.1747-9991.2008.00167.x Journal Compilation © 2008 Blackwell Publishing Ltd Frege on Definitions 995 The third type of criticism, advanced in Frege’s disputes with Hilbert and Korselt, rejects Hilbert’s procedure of ‘defining’ basic terms of geometry by groups of axioms, and then proving the (relative) consistency or independence of an axioms from others by interpreting the axioms in (often non-geometrical) mathematical structures. Frege’s initial criticism, in the first of the first series of essays, is that Hilbert’s conceptions of definition and of axiom do not conform to traditional conceptions. The key points of the traditional conceptions as Frege articulates them are, that definition is stipulation of the meanings of signs, and that no other types of mathematical propositions, including axioms, can contain signs whose references have ‘not previously been established’ (FG1 274). On this conception axioms presuppose definitions, and cannot be used to define signs. This criticism is obviously not very persuasive absent an examination of how Hilbert’s definitions-by-axioms are supposed to work. In the second essay of the first series Frege does this, and comes to acknowledge that Hilbert’s axioms can constitute definitions. What these sets of axioms define, however, are not first-level geometrical concepts such as point, line, parallel, etc., but a second-level concept within which sets of first-level concepts can fall. Frege thus construes Hilbert’s claims about the interpretation of the geometrical axioms by various classes of mathematical objects, such as pairs of numbers as, e.g., the claim that ‘[t]he first-level concept is a pair of numbers of the domain Ω, just like the Euclidean concept of a point, is supposed to fall within Mr. Hilbert’s second-level concept’ (FG1 284). Any set of firstlevel concepts that fall within Hilbert’s second-level one can be termed a geometry, and ‘in every one of these geometries there will be a (first-level) concept of a point’ (FG1 284), of a straight line, of being parallel, etc. But the concepts of point in distinct geometries are distinct from one another; similarly there are distinct concepts of line, parallelism, etc. So, not only would it be equivocal to use a single sign to express them, but, most significantly for Hilbert’s project, there only seem to be axioms common to these geometries. In fact, ‘[o]ne could not simply say “the axiom of parallels”, for the different geometries would have distinct axioms of parallels’ (FG1 284). But then it’s not clear that the logical relation of a parallels axiom to the remaining axioms in one geometry has anything to do with the logical relations of other parallels axioms in different geometries. Frege’s critique is not conclusive, but the complexity of the issues preclude further discussion here.5 Finally, in mostly unpublished writings starting late in 1890s, Frege makes two closely related criticisms. First, he rejects so-called definitions that are ‘never adduced in the course of a proof ’ (NLD 256); they are like ‘those stucco-embellishments on buildings which look as though they supported something whereas in reality they could be removed without the slightest detriment to the building’ (LM 212). Second, if definitions are never used, then a mathematician may fail to realize that he ‘uses a word in a way that conflicts with his own definition’ (LDM 157), or that his definition conflicts with other mathematicians’ definitions of the same word, so that ‘definitions, © 2008 The Author Philosophy Compass 3/5 (2008): 992–1012, 10.1111/j.1747-9991.2008.00167.x Journal Compilation © 2008 Blackwell Publishing Ltd 996 Frege on Definitions which seem to be utterly irreconcilable, lie peacefully alongside one another like animals in paradise’ (LM 217). I turn now to the nature of definitions in Fregean logicism. 2. Definitions in Grundlagen der Arithmetik In Grundlagen der Arithmetk one of Frege’s explicitly stated aims is to present a prima facie anti-Kantian case for the analyticity of arithmetical truths. Frege’s conception of analyticity is not Kant’s, but it is recognizable as a generalization thereof. For Kant a subject-predicate judgment is analytic if the concept of the predicate is contained in the concept of the subject (see, e.g., Kant A6/B10). For Frege a proposition is analytic if its proof involves only general logical laws and definitions (GL 4).6 For Kant we come to know that an analytic proposition is true by (a) analyzing the subjectconcept to see that it contains the predicate-concept, and (b) inferring that denying the predicate of the subject would violate the principle of noncontradiction (see, e.g., Kant A151/B190). That is to say, knowledge of an analytic proposition rests on conceptual analysis and logic. That the same is true for Frege emerges from his characterization of the ‘task’ of Grundlagen. In sections 1 to 3 he sets out mathematical and philosophical reasons for demanding that ‘the fundamental propositions of arithmetic should be proved, if in any way possible’ (GL 4). In trying to meet this demand, we very soon come to propositions which cannot be proved so long as we do not succeed in analysing concepts which occur in them into simpler ones or in reducing them to something of greater generality. Now here it is above all Number which has to be either defined or recognized as indefinable. This is the task which the present work is meant to settle. (GL 5) Thus, the key definitions of Grundlagen express analyses of arithmetical concepts (GL 4 –5). There is another property of definitions in Grundlagen whose role in Frege’s logicist project has only recently received extended attention. Frege takes Kant to have an impoverished view of how a concept can be formed from, and so analyzed into, other concepts: Kant ‘seems to think of concepts as defined by giving a simple list of characteristics in no special order’ (GL 100). Such concept formation and analysis is inadequate because from concepts with such structures ‘[n]othing essentially new . . . emerges’, and, the ‘conclusions we draw’ using such concepts do not ‘extend our knowledge’ (GL 101). More generally, concepts formed by Boolean operations alone fail to extend knowledge (BLC 33–4). This failure is the reason why Kant took analytic propositions to be obvious or uninformative. In contrast, the logical resources available to Frege for the analysis and formation of concepts results in concepts that Frege calls ‘fruitful’ (fruchtbar) in the sense of making possible extensions of knowledge.7 Hence Frege rejects Kant’s association of analyticity with uninformativeness.8 It is plausible that Frege thought © 2008 The Author Philosophy Compass 3/5 (2008): 992–1012, 10.1111/j.1747-9991.2008.00167.x Journal Compilation © 2008 Blackwell Publishing Ltd Frege on Definitions 997 fruitfulness is explained by quantificational structure.9 In particular, when the analysis of a concept initially taken to be logically unstructured reveals it as possessing quantificational structure, it becomes possible to establish deductive relations to and from propositions involving that concept that could not be established prior to the analysis. One example of this that is particularly important for logicism is Frege’s analysis of the concept following in a series using what we now call the ancestral of a relation. Given that analysis, his analysis of the relation of being a successor, and his definition of the number zero, Frege could prove the principle of mathematical induction from definitions and second-order logic.10 The basic picture of definition in Grundlagen is this. We start with an expression that has been in use for some time, and in which we have not discerned any logical structure – call such expressions ‘simple’. A definition of this expression rests on an analysis of the concept that it expresses as logically structured from simpler concepts; the definiens reflects this logical structure by including occurrences of truth-functional connectives and quantifiers – call such expressions ‘complex’. Using such a definition, one can import additional logical structure into those propositions which have been in use prior to the analysis and in which the definiendum occurs. In virtue of the presence of the additional logical structure, we can discern deductive relationships to and from these propositions that we could not discern prior to the analysis-based definition. In this sense, the definition enables us to prove things we could not prove without it. But one crucial issue not mentioned in Grundlagen is the correctness of definitions. We have seen earlier a number of negative requirements on definitions: they must not be piecemeal, conditional, creative, or unused. We can now add fruitfulness as a positive requirement. But is it the case that any definition that satisfies these constraints is as good as any other? 3. Interlude: Contextual Definitions One of the central mathematical, and, many would argue, philosophical ideas underlying Frege’s logicism is nowadays, following George Boolos, known as Hume’s Principle (hereafter HP): the number of Fs is the same as the number of Gs just in case there is a one-to-one correlation between the Fs and the Gs. Much of the mathematical significance of HP lies in the following facts. In both Grundlagen and Grundgesetze, Frege in essence uses HP together with definitions of zero and of successor to prove the Dedekind-Peano axioms for arithmetic.11 In Grundgesetze the only ineliminable uses Frege makes of the inconsistent Basic Law V are to derive HP.12 Finally, HP is equi-consistent with second-order arithmetic.13 These mathematical facts suggest that a Fregean form of logicism might be developed on the basis of some philosophical account of the status of HP. The most well-known attempt to follow out such a suggestion is the neoFregean logicism of Crispin Wright and Bob Hale (see especially Hale and © 2008 The Author Philosophy Compass 3/5 (2008): 992–1012, 10.1111/j.1747-9991.2008.00167.x Journal Compilation © 2008 Blackwell Publishing Ltd 998 Frege on Definitions Wright). It is, of course, not the business of this article to delve into the complex mathematical and philosophical issues surrounding this brand of logicism. But these issues connect with our topic because HP first appears, in the critical sections 62 to 67 of Grundlagen, as what we would call a (schema for) contextual definitions of what Frege calls ascriptions of number – expressions of the form ‘the number of Fs’ – by stipulating that all instances of HP hold. In Grundlagen section 63 Frege characterizes this contextual definition as ‘determining’ the concept of cardinal number by constructing ‘the content of a judgment that can be regarded as an identity on either side of which a number stands’, and ‘by means of the already known concept of identity, to attain that which is to be regarded as identical’. Since a definition associates content with a sign, we may understand this opaque characterization as claiming that it is possible to assign meanings to ascriptions of number by setting out conditions under which identities flanked by ascriptions of number are true, taking the meaning of the sign of identity as already known.14 One may read Frege as taking this procedure of contextual definition as underwritten by the enigmatic Context Principle enunciated in Grundlagen: ‘never ask for the meaning of a word in isolation, but only in the context of a proposition’ (GL x). That is, so long as all statements in which an expression occurs have determinate truth-conditions, and some of these conditions are fulfilled, that expression is a candidate for being a referring expression.15 The philosophical core of Fregean neo-logicism consists of two ideas. The Context Principle licenses claiming that principles like HP, not known to be inconsistent, provide part of a justification for reference to mathematical objects. Second, taken as contextual definitions, principles like HP express analyses of the concept of number, and so proofs of the axioms of arithmetic on their basis have a claim to demonstrating the analyticity of arithmetic. One complication for the neo-logicist program is that, apart from the criticisms canvassed in Section 1 above, Frege also rejects, in Grundgesetze II, the procedure of defining ‘a symbol . . . by defining an expression in which it occurs, the remaining parts of which are already known’ (§66). This plausibly is precisely Frege’s procedure in contextually defining ascriptions of number by HP. In Grundgesetze II Frege’s reason for rejecting such definitions is that ‘the reference of an expression and of one of its parts do not always determine the reference of the remaining part’, and so, ‘an enquiry would first be necessary whether any solution for the unknown [reference of the definiendum] is possible, and whether the unknown is uniquely determined’ (§66). But ‘it is impracticable [untunlich] to make the justifiability of a definition depend upon the outcome of such an enquiry, which, moreover, may perhaps not even be feasible [durchführbar]’ (§66). So it appears that there is a change in Frege’s view of the legitimacy of contextual definitions from the Grundlagen to the Grundgesetze. And this raises among others the questions whether the change is due to an abandonment of the Context Principle, and whether © 2008 The Author Philosophy Compass 3/5 (2008): 992–1012, 10.1111/j.1747-9991.2008.00167.x Journal Compilation © 2008 Blackwell Publishing Ltd Frege on Definitions 999 Frege’s later reasons for rejecting contextual definitions also go against the neo-logicist program. However, it is not clear, on closer examination, just how much of a change there really is. For one thing, in Grundlagen already Frege rejects HP as a definition of cardinal number. The reason for this rejection is a version of the notorious Caesar Problem: from HP there is no obvious way of deriving truth-conditions for identities flanked by an ascription of number and a non-numerical singular term such as ‘Julius Caesar’.16 This problem leads Frege to give an explicit definition of the numerical operator, ‘the number of ’ in terms of extensions of concepts, and, eventually, in Grundgesetze, to adopt Basic Law V to govern inferences involving extensions. The intricacies of the Caesar Problem are naturally not in our purview. The important point for our concerns is the following way of thinking of the Caesar Problem. Since HP by itself fails to settle whether Caesar is identical to any given number, it is, for all we know, consistent to accept any one of the infinity of statements equating Caesar with a natural number. Thus the contextual definition by itself fails to satisfy the principle of completeness for definitions described in Section 1 above. But, this situation might also be understood as precisely one in which, to continue Frege’s metaphor, we do not have a solution to the unknown meaning of any numerical expression. If so, then in fact the same consideration moves Frege, in Grundlagen and Grundgesetze, against contextual definitions. Note also the term of Frege’s criticism in Grundgesetze II section 66 quoted above: it is ‘impracticable’ to make the justifiability of a definition depend upon a proof of the existence and uniqueness of the reference assigned to the definiendum. Thus it is not clear that the problem is a logical one. Indeed, the sentence from which the quotation is taken begins with ‘as I have said above’, and it is clear that Frege is referring to Grundgesetze II section 60, where he writes, In general, we must reject a way of defining that makes the correctness of a definition depend on our having first to carry out a proof; for this makes it extraordinarily difficult to check the rigor of the deduction, since it is necessary to inquire, as regards each definition, whether any propositions have to be proved before laying it down. Prima facie, Frege’s objection to contextual definitions is a practical one for the construction of rigorous deductions, rather than any principled objection to their possibility. The situation, then, is this. Frege always held that a contextual definition is defective if it fails to determine uniquely the reference of the definienda, and so he never thought that justifications of such definitions are not needed. But, in the period of Grundlagen Frege did not think that there were any general considerations against the possibility providing such justifications, while by the time of Grundgesetze, he came to see that considerations of system construction provide a general practical reason against the incorporation © 2008 The Author Philosophy Compass 3/5 (2008): 992–1012, 10.1111/j.1747-9991.2008.00167.x Journal Compilation © 2008 Blackwell Publishing Ltd 1000 Frege on Definitions of contextual definitions in a system of science. Thus Frege’s later reasons against contextual definitions bears neither positively nor negatively on neoFregean logicism. This is not, of course, a defense of that philosophical program, or of the irrelevance of the Caesar problem for it. I claim simply that the reasons underlying the small change in Frege’s view of contextual definition have little to do with the viability of that program. This should indeed be fairly obvious, since a version of the Caesar problem arises again for Frege in Grundgesetze volume I, after he abandons contextual definitions, when he attempts in sections 10 and 33 to elucidate or justify Basic Law V. 4. Frege’s Review of Husserl’s Philosophie der Arithmetik Between Grundlagen and the execution of the logicist project in Grundgesetze Frege formulated the Sinn/Bedeutung distinction, and a natural question is how the distinction affects the Grundlagen conception of definition based on analysis of concepts. There are two main sources for addressing this question; unfortunately they apparently support incompatible answers. The first source is Frege’s review of Husserl’s The Philosophy of Arithmetic (hereafter RH ), in particular, Frege’s reply to Husserl’s criticism of the definition in Grundlagen sections 70 to 71 of one-to-one correspondence (RH 199–200). Frege represents Husserl as arguing that a ‘definition is . . . incapable of analyzing the sense’ of the definiens, on the basis of a dilemma for such analyses that bears some obvious affinities with the paradox of analysis: In using the word to be explained, I either think clearly everything I think when I use the defining expression: we then have [an] ‘obvious circle’; or the defining expression has a more richly articulated sense, in which case I do not think the same thing in using it as I do in using the word to be explained: the definition is then wrong. (RH 199–200; emphases mine) Frege clearly takes Husserl to assume that a sense-analyzing definition is correct just in case the sense of the definiens is identical to the sense of the definiendum; let’s call this the sense-identity requirement. The problem for sense-analyzing definitions is supposed to be that in order for such a definition to be correct it would have to be obviously circular. It is hardly obvious what this obvious circle is, but we can understand how Frege saw the purported problem for his project as follows. The definitional analyses of Grundlagen, as noted above, apply to concepts and terms that are in use and are generally taken to be unstructured. So, prior to Frege’s analysis, in using the definiendum people did not grasp a structured, i.e., ‘articulated’, sense. So in using the definiendum pre-analytically we did not already ‘think clearly everything’ we now post-analytically think in using the explicitly structured definiens. Hence Frege’s definition fails the sense-identity requirement and is incorrect. © 2008 The Author Philosophy Compass 3/5 (2008): 992–1012, 10.1111/j.1747-9991.2008.00167.x Journal Compilation © 2008 Blackwell Publishing Ltd Frege on Definitions 1001 Frege replies by rejecting Husserl’s concern with ‘the sense of the words’ (RH 200) in favor of the mathematicians’ concern with ‘the thing itself: the references of the words’: For the mathematician, it is no more right and no more wrong to define a conic as the line of intersection of a plane with the surface of a circular cone than to define it as a plane curve with an equation of the second degree in parallel coordinates. His choice of one or the other of these expressions or of some other one is guided solely by reasons of convenience and is made irrespective of the fact that the expressions have neither the same sense nor evoke the same ideas. (RH 200) In effect Frege both rejects the assumption that his definition is based on an analysis of sense and insists that in order for a definition of a mathematical predicate to be legitimate it merely has to be true of the same objects as the predicate being defined. Call this the reference-matching requirement.17 In light of this reply, how are we to understand Frege’s conceptions of definition and of analysis? Has there been a change from Grundlagen? Here’s one possibility. Let’s take Frege’s talk of concepts in Grundlagen to refer to concepts in Frege’s technical sense after the Sinn/Bedeutung distinction, i.e., the Bedeutungen of predicates. Analysis of concepts applies to concepts expressed by simple predicates. To analyze such a concept is to provide a complex predicate true of exactly the same objects as the simple predicate being analyzed; this complex predicate of course is the definiens. The correctness of such a definition requires satisfaction of only the referencematching, not the sense-identity, requirement. Clearly the legitimacy of alternative analyses of a concept follows from this view, e.g., alternative analyses of the concept of conic section. Frege claims that choice among alternative mathematical definitions ‘is guided solely by reasons of convenience’, but surely he would not object to fruitfulness as a basis of choice. Is this a sustainable account of Grundlagen? Michael Dummett argues that it is not, because Frege’s argument for his definition of equinumerosity requires more than reference-identity; it requires that the definiens – stating the existence of a one-to-one correspondence between two concepts – be conceptually prior to the definiendum – stating the identity of the number ascribed to these two concepts (see 1991a, 148–54, and 1991b.) In addition, if the requirement for correctness of definitions is reference-matching, then in order to evaluate the correctness of a definition we would have to be in the position of knowing what is the reference of the definiendum. So this requirement presupposes that the term being defined has at least a reference independently of the definition(s) being offered. But it’s not clear that this presupposition is satisfied for Grundlagen. After all, the book begins by asking the question, ‘what is the number one?’, and much of the polemic in the earlier parts of the book seems to be directed at incorrect conceptions of the references of numerical expressions. This suggests that Frege in fact believes that many people, including mathematicians, don’t © 2008 The Author Philosophy Compass 3/5 (2008): 992–1012, 10.1111/j.1747-9991.2008.00167.x Journal Compilation © 2008 Blackwell Publishing Ltd 1002 Frege on Definitions know what are the references of numerical expressions. How then would one attempt to satisfy the requirement of reference-matching? 5. Definitions in ‘Logik in der Mathematik’ The second main source of Frege’s views of definition and analysis after the Sinn/Bedeutung distinction is the set of unpublished lecture notes titled ‘Logic in Mathematics’ (hereafter LM ). Here a definition provides ‘a simple sign to replace . . . a group of signs with the stipulation that this simple sign is always to take the place of that group of signs’ (see also FG1 274), and is introduced because ‘[i]n constructing a system [of mathematics] the same group of signs . . . may occur over and over again’ (LM 207). Since the definiens is a group of signs observed as already repeatedly occurring in propositions, this account differs sharply from Grundlagen in which the definiens is a complex expression that one has to discover by conceptual analysis, and by which one introduces logical complexity into propositions that did not originally contain such complexity. In addition, Frege here takes definitions ‘considered from a logical point of view [to be] wholly inessential and dispensable’, because a stipulative abbreviation ‘adds nothing to the content [of sentences, but] only makes for ease and simplicity of expression’ (LM 208). Indeed, ‘it is not possible to prove something new from a definition alone that would be unprovable without it’ (LM 208). This again seems to differ sharply from Frege’s Grundlagen view, since, as noted, there the additional logical structure introduced by definitions is indispensable to establishing deductive relations that could not be established without those definitions. I want to pause to emphasize that the change from Grundlagen to LM that we have just described centers on the issue of whether a definition captures existing logical structure in concepts in use, or imports additional logical structure into propositions. This issue is independent of two others. First, it is independent of whether definitions are abbreviations or stipulations. This is evidenced by the fact that Frege characterized definitions as stipulations in as early a work as Begriffsschrift: ‘Our sole purpose in introducing is to bring about an extrinsic simplification by stipulating an abbreviation’ (B 55). The sense in which, according to LM, a definition is inessential is that all the deductive power of the definition lies in the logical structure of the definiens. This feature holds also of definitions in Grundlagen, once they have been laid down. Second, the change is independent of the notion of fruitfulness. A number of commentators have taken Frege’s claim, in ‘Foundations of Geometry’ and LM, that nothing new can be proven from a definition that cannot be proved without it to imply that either, on Frege’s late conception, genuine definitions are not fruitful, or the fruitfulness of a definition is not a matter of its making proofs possible that are impossible in its absence.18 But Frege ascribes fruitfulness to definitional abbreviations as well as to definitions resulting from analysis.19 We © 2008 The Author Philosophy Compass 3/5 (2008): 992–1012, 10.1111/j.1747-9991.2008.00167.x Journal Compilation © 2008 Blackwell Publishing Ltd Frege on Definitions 1003 can make sense of this if we take fruitfulness to be an intrinsic property of concepts or senses expressed by definientia, for example, the property of having quantificational structure. Whether this structure is already present in propositions in use or is introduced into propositions makes no difference to the fruitfulness of the concept or sense. The LM account of definitions also seems to differ from the one in RH. In particular, it is unclear how, on the LM account, one would arrive at alternative definitions. If one observed that the group of signs ‘the line of intersection of a plane with the surface of a circular cone’ occurs over and over again, one might decide to abbreviate it, and similarly if one observed repeated occurrences of ‘a plane curve with an equation of the second degree in parallel coordinate’. But why would one choose the same abbreviation for both sets of observed repeated occurrences? Why does Frege abandon definitions based on conceptual analysis? LM contains a discussion of what Frege calls ‘logical analysis’: In the development of science it can indeed happen that one has used . . . an expression over a long period whose sense one had regarded as simple until one succeeds in analyzing it into simpler logical constituents. By means of such an analysis, we may hope to reduce the number of axioms; for it may not be possible to prove a truth containing a complex constituent so long that constituent remains unanalyzed; but it may be possible, given an analysis, to prove it from truths in which the elements of the analysis occur. (209) Except for the fact that Frege here speaks of analyzing senses rather than concepts, this account seems to fit Grundlagen’s conceptual analysis perfectly.20 But here Frege argues that definitions cannot be based on logical analyses. First a bit of terminology: purported definitions based on logical analyses are called ‘analytic’; definitions proper, i.e., stipulative abbreviations, are called ‘constructive’. Frege argues that these so-called analytic definitions are not really definitions at all. For any logical analysis, either we recognize that the sense of the complex analysans ‘agrees with the sense of the long established simple sign’ (LM 209), or we don’t; call these, respectively, α-analyses and β-analyses. The first possibility can obtain only if our recognition is achieved ‘through unmediated self-evidence [unmittelbares Einleuchten]’ (LM 209). But then, Frege claims, the analytic definition ‘is really to be regarded as an axiom’ (LM 209).21 If we don’t recognize through unmediated evidence an agreement of sense, then ‘we are not certain whether the analysis is successful’ (LM 209), a situation which can arise only if ‘we do not have a clear grasp of the sense of the simple sign’ (LM 221). When it obtains, if ‘it is our intention to put forward a definition proper, we are not entitled to choose the [simple] sign . . . , which already has a sense’ (LM 210), and so, once again, the logical analysis does not yield a genuine definition of the simple sign. This argument presents a dilemma reminiscent of Husserl’s dilemma. Here the second horn, concerning β-analyses, shows that Frege now accepts the © 2008 The Author Philosophy Compass 3/5 (2008): 992–1012, 10.1111/j.1747-9991.2008.00167.x Journal Compilation © 2008 Blackwell Publishing Ltd 1004 Frege on Definitions sense-identity requirement on definitions. The first horn, concerning αanalyses, is different from RH: here our recognition of sense identity results, not in a circle, but in an axiom rather than a genuine definition. But the upshot of the dilemma is the same. If a logical analysis has the same sense as a simple sign then an attempted definition based on it is not a definition; if it has a different sense then the attempted definition cannot be correct. In neither case can there be a definition based on logical analysis. The dilemma also seems applicable to Grundlagen. If we recognize through unmediated self-evidence that the expressions of Frege’s analyses coincide in sense with ‘number’, then the supposed definitions of Grundlagen are axioms. It follows that Frege’s derivation of arithmetic would consist of proofs of arithmetical propositions from the laws of logic augmented by a set of axioms. It is then not clear how Frege can correctly take himself to have shown arithmetic to be a branch of logic.22 In fact, Frege himself in Grundlagen section 69 expresses a suspicion that the correctness of his definition of ‘the number of Fs’ ‘will perhaps be hardly evident at first’ (GL 80). This suggests that we do not recognize with unmediated evidence an agreement in sense between Frege’s analyses and ordinary arithmetical expressions, and so are not entitled to take Frege’s definitions to be correct. Frege does not completely reject β-analyses as a basis for definitions. Instead of taking a β-analysis to define an existing simple sign A, ‘we must choose a fresh sign B, say’, and use the analysis to give a proper, constructive definition of B (LM 210). Frege recommends that we ‘evade’ the question whether A and B have the same sense by ‘constructing a new system from the bottom up’ in which ‘we shall make no further use of the sign A – we shall only use B’ (LM 210). Frege goes on to say, [I]t may be felt expedient to use sign A instead of sign B. But if we do this, we must treat it as an entirely new sign which had no sense prior to the definition. We must therefore explain that the sense in which this sign was used before the new system was constructed is no longer of any concern to us . . . . In constructing the new system we can take no account, logically speaking, of anything in mathematics that existed prior to the new system. (LM 210) It seems that Frege is recommending that we view, e.g., his work in Grundgesetze as the construction of two complex expressions from logically simple terms, abbreviating them with two arbitrarily chosen signs, say ‘H ’ and ‘G’, and then proving statements using these abbreviations and the laws of logic. By substituting the numeral ‘0’ for ‘G’ and the predicate ‘ξ is a successor of ζ’ for ‘H ’ in all these statements, we would obtain statements syntactically identical to the propositions of arithmetic that have been in use until Frege came along.23 We can use the system of mathematics constituted by these Doppelgänger of pre-Fregean arithmetic, but we have to explain that they, or their senses, have nothing to do with arithmetic as we have used it previously. © 2008 The Author Philosophy Compass 3/5 (2008): 992–1012, 10.1111/j.1747-9991.2008.00167.x Journal Compilation © 2008 Blackwell Publishing Ltd Frege on Definitions 1005 6. The Nature of Frege’s Logicism: Some Options Let’s take stock of our discussion. We have seen that after Grundlagen Frege considers two criteria for the correctness of definitions based on analyses of terms in use. In both cases satisfaction of the criterion by arithmetical vocabulary is blocked by deficiencies in our understanding of arithmetical language. It’s unclear that Frege is aware of the difficulty in RH; in LM he explicitly rejects analysis-based definitions for failing to satisfy the criterion of correctness. The consequence of this rejection is that logical analysis of arithmetic can result only in the replacement of arithmetic with a new mathematical science, not in revealing previously existing but somehow concealed deductive structures in arithmetical statements. From this last formulation it should be clear that the fundamental issue is how to square this replacement view with the view of the Grundlagen in which definitions based on conceptual analyses play a key role in demonstrating the analyticity arithmetic, and so in vindicating logicism. Before going into this issue, I pause to take up a suggestion sometimes made that much of the apparent inconsistencies among Frege’s claims about definitions can be resolved by distinguishing Frege’s view of the role of definitions in the systematic exposition of a science from his view of the philosophical work that must precede such a presentation. There is no doubt that, especially in later writings such as LM, Frege made such a distinction. But it’s not clear that this distinction does anything to resolve the tensions that we have been examining. For example, consider the apparent tension between the sufficiency of reference-matching for definitions in RH and the claim, in LM, that sense-identity is required as well of definitions based on logical analyses. How does our distinction help resolve this tension? Is it that in the systematic exposition of a science, definitions have to match reference, but in pre-systematic philosophical work sense-identity is required as well? Or is it the other way around? In either case, one would ask, why? Moreover, we might ask, what references must intra-systematic definitions match? It’s hard to see that, in this case, they can be anything other than the references of the terms used pre-systematically. But then this runs into the problem we noted above that it’s not clear that on Frege’s view, before his account of the nature of arithmetic, we knew to what our arithmetical terms refer. Four main positions have appeared in recent work on the relationship between Frege’s later replacement view and his conception of logicism in Grundlagen. Perhaps the most traditionalist position is advanced by Dummett (Frege; ‘Frege and the Paradox of Analysis’) and Eva Picardi. On their account the overarching project of Frege’s logicism is to settle the epistemological status of arithmetic. This requires determining ‘the status of the arithmetical laws we already have, involving the arithmetical concepts we already grasp’ (Dummett, Frege 20), and it ‘would be incomprehensible’ how Frege’s proofs tells us anything about the arithmetic we already have unless his ‘definitions [are] somehow responsible to the meaning of [arithmetical] sentences as © 2008 The Author Philosophy Compass 3/5 (2008): 992–1012, 10.1111/j.1747-9991.2008.00167.x Journal Compilation © 2008 Blackwell Publishing Ltd 1006 Frege on Definitions these are understood’ (Picardi 228). Clearly this view implies the sense-identity requirement, at least on arithmetical definitions. Thus on this view the criterion of correctness in RH is inadequate, since many distinct senses can present the same referent. No more adequate is the view of LM, according to which the proofs of Grundgesetze are of mathematical statements unrelated in sense to existing arithmetical statements. So on this reading Frege fails to achieve a coherent account of how his execution of the logicist program can fulfill his philosophical aims. Since I’m loath to saddle Frege with such a substantial incoherence, I take this to be a reading of last resort, even if it might be most straightforwardly supported by the texts. Weiner proposes a partially revisionary reading. A crucial move in the Dummett-Picardi argument is that logicism can only make sense if it shows, on the basis of the senses that we have always attached to arithmetical terms and propositions, that these propositions are analytic, or a branch of logic. But, as I have mentioned above, it is plausible that Frege in fact believes that many people, including mathematicians, don’t know what are the references of numerical expressions before his logical analysis. If that is so, it is not obvious that he accepts that arithmetical terms and propositions, as we have always used them, express any determinate, or indeed any, senses at all. Weiner exploits this idea to argue that even in the Grundlagen Frege had a ‘hidden agenda’ (‘Philosopher’ 263) namely, replacing existing arithmetic with a new science based on stipulative definitions that assign new senses to key arithmetical terms. I’m reluctant to ascribe unstated doctrines to Frege; moreover, on this reading Dummett’s and Picardi’s question remains: what’s the epistemological significance of showing that this entirely new science is analytic? A considerably more non-traditional reading is due to Paul Benacerraf. Frege’s aims, on his reading, are primarily, perhaps even exclusively, mathematical. He was interested in traditional philosophical issues such as the source and nature of arithmetical knowledge only to the extent that they are relevant to, or can be exploited for, the mathematical ends of providing proofs of arithmetical statements for which there previously have been no proofs. It is then at least unclear that Frege would have any interest in or need for showing that his definitions satisfy the sense-identity requirement; this perhaps accounts for his relatively cursory discussion of the correctness of definitions. A refinement of Benacerraf ’s reading can be derived from Jamie Tappenden’s account of fruitfulness of definitions. Tappenden takes Frege as proposing the existence of quantificational structure as an explanation of what makes definitions mathematically fruitful, and hence as having philosophical motivations that are clearly separable from mathematical ones. If this is Frege’s main interest in definitions (a claim that Tappenden does not make) then, as I pointed out above, it’s independent of the issue of sense-identity. While I agree with Benacerraf that Frege’s mathematical interests must be taken into consideration, and with Tappenden that the nature of successful mathematical concept formation is a central concern of Frege’s, I don’t think it’s philosophically fruitful to conclude that Frege © 2008 The Author Philosophy Compass 3/5 (2008): 992–1012, 10.1111/j.1747-9991.2008.00167.x Journal Compilation © 2008 Blackwell Publishing Ltd Frege on Definitions 1007 has no interest in, or anything substantial to say on, more traditional epistemological issues concerning mathematics. Some of what Frege says in LM is not clearly consistent with the claim that the logical analysis of a term whose sense is not clearly grasped can result in statements with entirely new senses to replace those statements containing the terms that we have been using. For example, he writes that ‘[t]he effect of . . . logical analysis . . . will . . . be precisely. . . to work the sense out clearly [der Sinn deutlich herausgearbeitet]’ (LM 221). This description suggests that logical analysis does something more like provide a clarification of the sense that we do attach to the existing term but which we do not grasp clearly. This picture of logical analysis fits well with Tyler Burge’s reading. He claims that Frege tacitly distinguishes between expressing senses, which is relatively easy to do, and discovering what senses one has expressed, which is relatively difficult (‘Frege on Senses’ §6). In order to develop this view, one would have to answer a number of questions. How do we express senses that we do not fully grasp? How do know what senses these are? What are the criteria for determining that a purported analysis has worked out an unclearly grasped sense, rather than suggested a replacement of it? Most importantly for understanding Frege, why, if this is his picture of logical analysis, does he nevertheless insists that the outcome of logical analysis should be a stipulative definition, rather than a definition that purports to state the existing sense? I’ll conclude with a brief sketch of how the distinction between senseexpression and grasp of expressed sense can be further spelt out that answers some of these questions. My notion of sense-expression is developed from the idea of logical segmentation due to Thomas Ricketts. Ricketts claims that for Frege patterns of inference that we acknowledge as valid are bases for discerning (logical) structures in the sentences that we use in these inferences. For example, that we use a sentence as the conclusion of an inference that has the pattern of universal instantiation and also as a premise in an argument that has the form of one of Leibniz’s laws gives us reason to regard the sentence as composed of a singular term and a (simple or complex) predicate. Moreover, discerning this logical structure in a sentence (as used in certain ways) is a ground for regarding the component expressions as having certain types of references or semantic values.24 Thus in our example seeing the sentence as composed of a singular term and a predicate is a ground for taking the expression with the role of the singular term as referring to an object and the expression with the role of the predicate as referring to a concept. Going now further than Ricketts, I claim that once we’re in a position to see this sentence as logically segmented into an object-name and a concept-name, we are also in a position to characterize the condition for that sentence to be true in terms of these component expression, namely, in terms of whether the referent of the object-name falls under the referent of the concept-name. Since for Frege the thought expressed by a sentence is its truth condition, and the senses of sub-sentential © 2008 The Author Philosophy Compass 3/5 (2008): 992–1012, 10.1111/j.1747-9991.2008.00167.x Journal Compilation © 2008 Blackwell Publishing Ltd 1008 Frege on Definitions expressions are the contributions they make to this truth condition, we are also now in a position to take the sentence to be expressing a thought, and its component expressions as expressing senses. I hold that, for Frege, the expression of thoughts and senses that are not thoughts requires no more than the use of sentences in accordance to (syntactically characterized) inferential norms. Thus, the ascription of thoughts expressed to sentences and senses to their parts is supervenient on logical segmentation, and so ultimately on the occurrence of those sentences in patterns of inference. There is no basis for thinking that the senses expressed play any role in fixing the standards of correctness to which those who use the sentences in question hold themselves accountable. My notion of grasp of sense is developed from Dummett’s view that, for Frege, understanding a sentence requires knowledge of the thought it expresses that is derived from knowledge of the senses of its parts and the mode of their composition into the sentence.25 On my account, Dummett is not wrong to ascribe this account of understanding to Frege, but I hold that this conception of understanding is an ideal, rather than a description of what is always involved in our actual practices. We can express senses simply by using language in rule-governed practices, but in order to grasp clearly or fully the senses of expressions we express, we would have to derive our uses of those expressions from our knowledge of their senses. The former is relatively an easier accomplishment than the latter. We can now explicate Frege’s notion of an unclear grasp of sense. To grasp the sense of an expression E unclearly is to use E in sentences in rule-governed ways, but without being able to derive (some of ) those uses from a grasp of the rules governing the use of E. There is then, in the established use of E, no such thing as the way in which we understand E. And, so, there is no such thing as coming up with a complex expression whose sense matches that of our understanding of E. There was no ‘concept (or sense) that we already grasp’ for the analysis to be faithful to; but this doesn’t mean that no senses were expressed. So we need not take Frege to hold that his analysis and derivation of arithmetic is a replacement of existing arithmetic with a new theory. On my account, Frege thinks of his analyses and proofs as giving us back exactly the arithmetical assertions and proofs that we have always made. It’s just that with these analyses we’re (finally) in a position to make those very assertions and to recognize the correctness of those very proofs on the basis of rules governing the use of the component expressions of the sentences in question. That is, on the basis of a full grasp of the senses that we have expressed all along.26 Short Biography Sanford Shieh works primarily in philosophy of logic and mathematics, and the history of analytic philosophy. His writings center on three topics: © 2008 The Author Philosophy Compass 3/5 (2008): 992–1012, 10.1111/j.1747-9991.2008.00167.x Journal Compilation © 2008 Blackwell Publishing Ltd Frege on Definitions 1009 the clarification and criticism of arguments against classical logic and mathematics based on anti-realism in the theory of meaning, the interpretation of Frege’s logic and metaphysics, and the logic and metaphysics of modality in twentieth-century analytic philosophy. His articles have appeared in such journals as Synthese, Philosophia Mathematica, and Pacific Philosophical Quarterly. He is co-editor, with Juliet Floyd, of Future Pasts: The Analytic Tradition in Twentieth-Century Philosophy (Oxford University Press, 2000), with Alice Crary, of Reading Cavell (Routlege, 2006), and editor of The Limits of Logical Empiricism (Kluwer, 2006). At present he is working on a conceptual history of the development of modal logic from C. I. Lewis to Saul Kripke, forthcoming from Oxford University Press. Shieh received an A.B. from Cornell University, a B.A. and an M.A. from the University of Oxford, and a Ph.D. from Harvard University. He is currently Associate Professor of Philosophy at Wesleyan University. Notes * Correspondence and address: Wesleyan University, 350 High Street, Middletown, CT 06459, USA. Email: sanford.shieh@wesleyan.edu. 1 The relevant writings of Frege will be cited by the following abbreviations. Original date of publication, or date of composition if unpublished, is in brackets. B Begriffschrift [1879]. BLC ‘Boole’s Logical Calculus and the Concept-Script’, in Posthumous Writings, 9–46 [1880– 81]. CES ‘A Critical Elucidation of some Points in E. Schröder, Vorlesungen über die Algebra der Logik’, in Collected Papers, 210–28 [1895]. DRC Draft towards a review of Cantor’s Gesammelte Abhandlungen zur Lehre vom Transfiniten, in Posthumous Writings, 68–71 [1890–92]. FC ‘Function and Concept’, in Collected Papers, 137–56 [1891]. FG1 ‘On the Foundations of Geometry: First Series’, in Collected Papers, 273– 84 [1903]. FT ‘On Formal Theories of Arithmetic’, in Collected Papers, 112–21 [1885]. GL Die Grundlagen derArithmetik Breslau, translated by J. L. Austin as Foundations [1884]. GG1 Grundgesetze der Arithmetik, begriffsschriftlich abgeleitet, vol. 1, Pohle, Jena, partial English translation as Basic Laws [1893]. GG2 Grundgesetze der Arithmetik, begriffsschriftlich abgeleitet, vol. 2, Pohle, Jena, selections translated into English by various hands in Translations, 139–224 [1903]. LDM ‘Logical Defects in Mathematics’, in Posthumous Writings, 157– 66 [1898–99]. LM ‘Logic in Mathematics’, in Posthumous Writings, 203–50 [1914]. NLD ‘Notes for Ludwig Darmsteader’, in Posthumous Writings, 253–7 [1919]. PCN ‘On Mr. Peano’s Conceptual Notation and My Own’, in Collected Papers, 234–48 [1897]. RC Review of Hermann Cohen, Das Prinzip der Infinitesimal-Methode und seine Geschichte, in Collected Papers, 108–11 [1885]. RH Review of E. G. Husserl, Philosophie der Arithmetik I, in Collected Papers, 195–209 [1894]. SN ‘On Mr. H. Schubert’s Numbers’, in Collected Papers, 249–72 [1899]. T ‘Thoughts’, in Collected Papers, 351–72 [1918]. In quoting from Frege I have very occasionally altered the translations slightly; I use ‘reference’ and its cognates throughout for Frege’s ‘Bedeutung’ and its cognates. Unless specifically noted, all emphases in citations are in the original. 2 See also PCN, ‘We can also argue for this requirement on the ground that the law of excluded middle must hold’ (155). © 2008 The Author Philosophy Compass 3/5 (2008): 992–1012, 10.1111/j.1747-9991.2008.00167.x Journal Compilation © 2008 Blackwell Publishing Ltd 1010 Frege on Definitions 3 For details about Euler’s computations, see Kline. For Frege’s conception of fictional thoughts see T 362f. 5 See Blanchette. 6 Note first that this account does not settle whether the laws of logic are analytic, since Frege does not allow a single proposition to constitute a proof of itself from itself. Beaney argues that this lacuna partly motivated Frege to develop the sense-reference distinction. I’m inclined to agree with Dummett (Frege 24–5) and Burge (‘Frege on Knowing the Foundation’ 310) that it is a mere oversight on Frege’s part not to characterize logic as analytic. Note also that Frege adds that one has to ‘take account also of all propositions upon which the admissibility of any of the definitions depends’ (GL 4). Unfortunately Frege does not here provide any criteria for the admissibility of definitions, nor does he specify how we’re to take the propositions in question into account. 7 See Benaceraf; Weiner, ‘Philosopher’; Tappenden. 8 Horty (ch. 3.1, note 1) points out that in Prolegomena to Any Future Metaphysics Kant characterizes analytic propositions as uninformative ones, and then goes on to argue for the conceptual containment account of their truth as an explanation of their uninformativeness. 9 Tappenden makes a forceful case for this claim. 10 Frege himself characterizes this result as showing that the inference from n to n+1 is based on the general laws of logic (GL iv). For references on the details of Frege’s proofs of this result see note 12 below. 11 See Heck; Boolos and Heck. As far as I know, Charles Parsons was the first to observe that by adopting HP as an assumption, Frege’s proofs of the axioms of arithmetic can be accomplished. The details of various versions of these proofs are presented in Wright; Boolos; Heck. 12 See Heck 259–64. 13 See Boolos. The standard reference on the mathematical logic of Frege’s logicism is Burgess. 14 Another text from the period of Grundlagen in which Frege appears to endorse such a procedure of contextual definitions is his review of Hermann Cohen’s history of infinitesimals, in which he refers to Grundlagen as indicating how on the explanation of differentiation in terms of limits ‘the differential can preserve a certain self-subsistence’ (RC 111). 15 This way of understanding the Context Principle in Grundlagen is spelt out in detail in Wright; Hale; Hale and Wright. Note that I write ‘a candidate’, because it is plausible that Frege requires more: the expression has also to play a set of syntactic roles in valid forms of inference – in particular, to qualify as a singular referring term, an expression has to occur in (first- and higher-order) quantifier inferences of instantiation and generalization, and Leibniz’s Law inferences governing identity. On this last point see especially Dummett (Frege ch. 4); Hale. 16 In fact Frege uses the example of Caesar not in the context of rejecting HP as a definition but rather in Grundlagen section 56 where he rejects on its basis a recursive definition of numerically definite quantifiers – ‘there are n Fs’, for each n – as definitions of the cardinal numbers (GL 67–8). Moreover, in Grundlagen section 66 Frege actually rejects the analogue of HP for contextually defining direction terms on the basis of parallelism for lines. But the problem for this contextual definition is its failure to provide a basis for fixing the truth-value of identities of direction terms with ‘England’ (GL 78), which is clearly constructed by analogy with the earlier question about the identification of the number one with the conqueror of Gaul. 17 Not reference-identity because for Frege identity strictly speaking holds only of objects. 18 According to Grossmann and Proust, only analytic definitions can be fruitful, so when in LM Frege no longer accepted analytic definitions as definitions, he also was forced to drop the fruitfulness requirement. According to Benacerraf there is an unresolved tension between Grundlagen’s requirement that definitions be fruitful and the absence of the fruitfulness requirement on definitions in LM. Weiner (‘Philosopher’; Frege in Perspective) holds that the fruitfulness of a definition merely amounts to its being usable in proofs, so that conventional abbreviations can be fruitful; but she also argues that even in Grundlagen Frege took definitions to be merely conventions of abbreviation. 19 See Tappenden 456. 20 There are of course differences. For one thing, in the present text Frege makes no mention of any epistemological significance for analysis. 21 Why are such so-called definitions to be regarded as resulting in axioms? The reason, I take it, is has to do with the kind of epistemic grounding that our recognition of the identity of sense 4 © 2008 The Author Philosophy Compass 3/5 (2008): 992–1012, 10.1111/j.1747-9991.2008.00167.x Journal Compilation © 2008 Blackwell Publishing Ltd Frege on Definitions 1011 between the analysans and the simple term yields. The phrase I translate here as ‘unmediated self-evidence’ is ‘unmittelbares Einleuchten’, and Robin Jeshion has argued convincingly that Frege uses this phrase systematically to indicate the kind of non-inferential epistemic grounding required by axioms. 22 Of course, as we saw above, neo-Fregean logicism proposes a foundation for arithmetic by augmenting second-order logic with HP as a non-logical axiom that expresses an analysis of the concept of number. So much depends on the status of the axioms; if they all express conceptual analyses, we might still take the resulting theory to demonstrate the analyticity of arithmetic. In addition, it should be noted that, as Burge argues (‘Ferge on Knowing the Foundation’ §1), Frege distinguishes between axioms and basic truths. 23 See Weiner, ‘Philosopher’; Frege in Perspective. 24 Clearly some form of this view also underlies neo-Fregean logicism. Note also that it’s open to question whether these are conclusive reasons; Dummett and Hale have argued that involvement in a number of other, higher-order, patterns of inference would be required to justify ascription of this logical structure. On both points see the references in note 16 above. 25 See especially Dummett, Interpretation ch. 15. 26 I develop these claims further in work in progress. I would like to thank an anonymous reviewer for helpful criticism of an earlier version of this article. Works Cited Beaney, Michael. ‘Sinn, Bedeutung and the Paradox of Analysis’. Frege: Critical Assessments, Vol. 4, Frege’s Philosophy of Thought and Language. Eds. M. Beaney and E. Reck. London: Routledge, 2005. 288–310. Benacerraf, Paul. ‘Frege: The Last Logicist’. Midwest Studies in Philosophy. Vol. 6. Eds. P. French, T. Uehling, and H. Wettstein. Minneapolis, MN: U of Minnesota P, 1981. 17–35. Blanchette, Patricia. ‘Frege and Hilbert on Consistency’. Journal of Philosophy 93 (1996): 317– 36. Boolos, George. ‘The Consistency of Frege’s Foundations of Arithmetic’. On Being and Saying: Essays for Richard Cartwright. Ed. J. J. Thomson. Cambridge, MA: MIT Press, 1987. 3 –20. —— and Richard Heck. ‘Die Grundlagen der Arithmetik, §§82–83’. Philosophy of Mathematics Today. Ed. Matthias Schirn. Oxford: Oxford UP, 1997. Burge, Tyler. ‘Frege on Knowing the Foundation’. Mind 107 (1998): 305– 47. ——. ‘Frege on Sense and Linguistic Meaning’. The Analytic Tradition. Eds. D. Bell and N. Cooper. Oxford: Blackwell, 1990. 30– 60. Burgess, John. Fixing Frege. Princeton, NJ: Princeton UP, 2005. Dummett, Michael. ‘Frege and the Paradox of Analysis’. Frege and other Philosophers. Oxford: Oxford UP, 1991. ——. Frege: Philosophy of Language. 2nd ed. London: Duckworth, 1981. ——. Frege: Philosophy of Mathematics. Cambridge, MA: Harvard UP, 1991. ——. The Interpretation of Frege’s Philosophy. London: Duckworth, 1981. Frege, Gottlob. 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