Abstract
Neo-logicism is, not least in the light of Frege’s logicist programme, an important topic in the current philosophy of mathematics. In this essay, I critically discuss a number of issues that I consider to be relevant for both Frege’s logicism and neo-logicism. I begin with a brief introduction into Wright’s neo-Fregean project and mention the main objections that he faces. In Sect. 2, I discuss the Julius Caesar problem and its possible Fregean and neo-Fregean solution. In Sect. 3, I raise what I take to be a central objection to the position of neo-logicism. In Sect. 4, I attempt to clarify how we should understand Frege’s stipulation that the two sides of an abstraction principle qua contextual definition of a term-forming operator shall be “gleichbedeutend”. In Sect. 5, I consider the options that Frege might have had to establish the analyticity of Hume’s Principle: The number that belongs to the concept F is equal to the number that belongs to the concept G if and only if F and G are equinumerous. Section 6 is devoted to Frege’s two criteria of thought identity. In Sects. 7 and 8, I defend the position of the neo-logicist against an alleged “knock-down argument”. In Sect. 9, I comment on Frege’s description of abstraction in Grundlagen, §64 and the use of the terms “recarving” and “reconceptualization” in the relevant literature on Fregean abstraction and neo-logicism. I argue that Fregean abstraction has nothing to do with the recarving of a sentence content or its decomposition in different ways. I conclude with remarks on global logicism versus local logicisms.
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Notes
Frege defines the relation of equinumerosity in second-order logic in terms of one-to-one correlation. Note that the above formulation of Hume’s Principle is a schematic one; here its two sides are (closed) sentences, that is, “F” and “G” are schematic letters for monadic first-level predicates, not variables for first-level concepts. By contrast, in “∀F∀G(N x F(x) = N x G(x) ↔ Eq x (F(x), G(x)))” “F” and “G” are variables for first-level concepts; here we have the universal closure of the open sentence “N x F(x) = N x G(x) ↔ Eq x (F(x), G(x))”.
In Grundgesetze, §10, Frege encounters a variation of his old Caesar problem from Grundlagen, now clad in formal garb. The metalinguistic stipulation in Frege (1893, §3): “I use the words ‘the function Φ(ξ) has the same course-of-values as the function Ψ(ξ)’ generally as coreferential [gleichbedeutend] with the words ‘the functions Φ(ξ) and Ψ(ξ) always have the same value for the same argument’” is designed to fix the reference of the monadic second-level function-name “the course-of-values of φ”. Yet it does so only incompletely because it gives rise to a pervasive referential indeterminacy of canonical course-of-values names. Such names are introduced a little later in the exposition of the concept-script (cf. §9). By a canonical course-of-values name I understand a name that results from inserting a monadic first-level function-name into the argument-place of the name of the course-of-values function “\( \mathop \varepsilon \limits^{,} \) φ(ε)” (the course-of-values operator as I also call it). Frege is confident that the referential indeterminacy of course-of-values terms can be removed by carrying out a piecemeal process, namely by determining for every first-level function, when introducing it, which values it receives for courses-of-values, just as for all other suitable arguments (that is, objects). (I ignore here names that refer to courses-of-values of dyadic first-level functions; Frege calls these names of double courses-of-values. Such names can be formed by means of the notation available for the designation of “simple” courses-of-values. Thanks to his device of level-reduction regarding functions, which Frege explains in Frege (1893, §34), he need not introduce courses-of-values of second-level functions into his logical system, let alone courses-of-values of third-level functions.) Regarding logical and philosophical aspects of the Caesar problem in Grundgesetze see, for example, the discussion in Heck (1999, 2005), and in Schirn (2014c).
Frege uses the contextual definition of the direction operator “D(x)” (“the direction of line x”) only for the sake of illustration.
It has become common practice in the Frege literature to speak of the Julius Caesar problem and not of the England problem (“Is England the same as the direction of the Earth’s axis?”) when Grundlagen, §66 is discussed.
It is possible that in Frege (1884) Frege had at least an inkling that by providing an explicit definition of the cardinality operator he had by no means solved the Caesar problem, but had only postponed it.
An equation of the form “N x F(x) = N x F(x)” would of course only count as a trivially true instance of the law of identity “a = a”, assuming that both “N x φ(x)” and “F(x)” have a reference.
In Frege (1884), Frege is not remotely as strict about the use of quotation marks as in his later writings. Clearly, only if the symbol “a” is distinguished from the object a by means of quotation marks does Frege’s sentence make real sense.
In this connection, one important point should be clear. Freges logicism requires that an abstraction principle that it designed to play a key role in the logical construction of the theories of cardinal, real and complex numbers must be of a purely logical nature.
I trust that Frege had at least an inkling that this assumption was, from a methodological point of view, of debatable merit. No doubt, at the time he was writing Grundlagen he could not rely on a commonly accepted view of the nature of extensions of concepts, let alone of the nature of numbers. However, towards the end of his book (§107) Frege informs us that he does not place decisive weight on the introduction of the extension of a concept anyway. To my mind, this remark mirrors only his wavering attitude towards the role that extensions of concepts ought to play in the logical construction of number theory and other branches of classical mathematics. What is even worse, the remark in §107 is flatly at odds with what Frege says and does elsewhere in Grundlagen. In particular, it strikes me as outlandish (to say the least) that, on the one hand, he stresses the key role of extensions of concepts for the envisaged definitions of fractions, irrational and complex numbers and, on the other, contends that they can eventually be dispensed with. In my opinion, it is fairly obvious that in a number of crucial places in part IV “The concept of cardinal number” and part V “Conclusion” of Grundlagen Frege refrains from putting all his cards on the table. It is this evasive attitude that makes it sometimes difficult for the reader to take him at his word or to find out what he really has in mind or aims at when making certain remarks. Either he did not clearly recognize that his exposition gives rise to some fundamental difficulties or he fell short of thinking through them and of coming to grips with them. Here are some important questions that Frege leaves after all unanswered: In what specific sense is the explicit definition of the cardinality operator supposed to solve the Julius Caesar problem? The second and third questions derive directly from what I have said above: Is the transsortal identification of cardinal numbers with extensions of concepts essential for the execution of the logicist programme or can it be dispensed with? If Frege considered it to be essential, what is it then to mean that he does not attach decisive importance to the introduction of extensions of concepts?
In Frege (1884), Frege considers the first-order domain to be both homogeneous and all-encompassing. By “homogeneous” I mean that he does not distinguish between categories or types of objects.
Roughly speaking, Benacerraf (1965) argues as follows (note that my choice of phrasing differs from his): Numbers cannot be sets because there is no compelling reason to hold that any particular number is some particular set. Plainly, there is no unique set-theoretic characterization of, say, the natural or the real numbers. The best we can achieve is to determine the numbers of either kind up to isomorphism. Not only does the identification of numbers with sets appear to be undermined by the randomness of our actual choices, but also by the arbitrariness of our taking sets to be the fundamental objects of mathematics. It is possible, for instance, to regard ordinal numbers as basic and to define sets in terms of them. By way of extending his argument from sets to objects in general, Benacerraf concludes that numbers cannot be objects at all.
See the detailed analysis in Schirn (2014c).
Hale (2000) pursues the aim of providing an informal axiomatic characterization of quantitative domains, on the basis of which it will be possible to introduce the real numbers by means of an appropriate abstraction principle. Following this idea, he introduces objects—cuts—corresponding to cut-properties by the following abstraction principle:
Cut: #F = #G ↔ ∀a(Fa ↔ Ga)
where F, G are any cut-properties on RN+ and a ranges over RN+.
Informally, a cut-property is a non-empty property whose extension is a proper subset of RN+ and which is downwards closed and has no greatest instance (cf. p. 411). Like HP and Axiom V, Cut is a second-order abstraction principle. Yet unlike the former two principles, Cut is a restricted abstraction principle: the domain for the abstraction embraces only cut-properties on a certain specified underlying domain of objects. See Hale’s discussion of cut-abstraction on pp. 414 ff. If I am right, then Cut would not have been an option for Frege had he thought about the prospects of saving the intended purely logical foundation of analysis in the aftermath of Russell’s Paradox. First, to use it as a contextual definition is ruled out from the outset. Second, in the light of Frege’s conception of primitive laws of logic, Cut can hardly claim to be such a law. Hence, it could not be laid down as a logical axiom in the theory of real numbers. Moreover, I doubt that Cut could be shown to be analytic in any plausible sense of analyticity. See also Wright’s reflections concerning a possible extension of the neo-logicist programme to analysis in Wright (1997). Regarding Frege’s approach to the foundations of analysis see von Kutschera (1966), Currie (1986), Simons (1987), Dummett (1991, chapter 22), Schirn (2013, 2014a). Concerning the answer to the hypothetical question whether in the aftermath of Russell’s Paradox Frege could have introduced the real numbers, in principle, via an appropriate abstraction principle without offending against his logicist credo see Schirn (2014d).
Hale’s analysis of Frege (1884, §64) in Hale (1997), plus the extensive new postscript in Hale and Wright (2001, pp. 105–116), is a thorough one. Yet after having read it again while I was composing the present essay I think that he occasionally makes a little too much of the issue. Several passages of Hale’s paper are, in my view, unnecessarily complicated and might give rise to confusion. It is unfortunate that he does not provide Ariadne’s thread to emerge from the labyrinth of some of his reflections. The postscript is of little help in this respect.
The primitive truths of logic qua axioms are unprovable only relative to T, as Frege certainly knew; cf., for example, Frege (1969, pp. 221 f). Even a primitive truth of logic may be provable in a theory T*, but due to the self-evidence it is supposed to possess it does not require proof. While in Frege (1879) “⊢a = a” is taken as an axiom, in Frege (1893) it appears as a theorem. In Frege 1893, §50, Frege comments on “⊢a = a” qua theorem of his logical calculus as follows: “Although this sentence is by our explanation of the equality-sign obvious [selbstverständlich], it is nonetheless worth seeing how it can be developed out of (III).” (III) is Basic Law III: ⊢g(a = b) → g(∀f(f(a) → f(b))), in words: the truth-value ∀f(f(a) → f(b)) falls under every concept under which the truth-value a = b falls. “a = a” does not need proof (= deductive justification), because it is obvious.
In Schirn (2014e), I analyze Frege’s use of ‘analytic’ in the period 1879–1892. I do this by considering also Leibniz and Kant’s notion of analyticity.
In Frege (1903, §66), Frege rejects contextual definitions as follows (all translations from Frege’s works into English are my own):
…we may not define [erklären] a sign or word by defining an expression in which it occurs, while the remaining parts are known. For it would first be necessary to examine whether a solution for the unknown—to use a readily understandable image from algebra—is possible, and whether the unknown is uniquely determined. Yet, as was already said above [§60], it is inappropriate to make the legitimacy of a definition dependent on the result of such an examination, which, moreover, would perhaps be quite impracticable. Rather, the definition must have the character of an equation that is solved for the unknown, and on the other side of which nothing unknown occurs. It is even less admissible to define two things by using a single definition. Every definition must, on the contrary, contain a single sign whose reference [Bedeutung] it stipulates. After all we cannot determine two unknowns by means of a single equation.
The point of this objection can be appreciated by considering, for example, the following contextual definition. One may define a dyadic function x * y for fractions by means of
x 1/y 1 * x 2/y 2 := x 1 + x 2/y 1 + y 2.
According to this definition, it holds, for example, that 6/4 * 2/3 = 8/7; since 6/4 = 3/2 holds, we have 6/4 * 2/3 = 3/2 * 2/3. However, 3/2 * 2/3 = 1 holds. Thus, from the definition of x * y it follows that 8/7 = 1 and, consequently, we are facing a contradiction. The definition contains “/” as a sign that has already been defined, and the stipulation above is incompatible with the definition of these fractions (cf. von Kutschera 1967, pp. 369 f.). Generally speaking, a contextual definition requires a proof that there exists exactly one entity that satisfies the definition. In Russell (1903, pp. 114 f.), Russell argues in this vein: “Now this definition by abstraction … suffers from an absolutely fatal formal defect: it does not show that only one object satisfies the definition. Thus instead of obtaining one common property of similar classes, which is the number of the classes in question, we obtain a class of such properties, with no means of deciding how many terms this class contains.” Here is an example of a contextual definition that meets the requirement of existence and uniqueness (cf. von Kutschera 1967, p. 370); I use the symbol “#” as a class operator. We may define, for example, fractions x/y as equivalence classes of ordered pairs of integers <x, y> with respect to the equivalence relation <x, y> ≈ <u, v> := u · y = x · v ∧ y ≠ 0 ∧ v ≠ 0 such that x/y = #z∃u∃v(z = <u, v> ∧ <u, v> ≈ <x, y>. The sum x ⊕ y of rational numbers is then defined by means of the contextual definition
x/y ⊕ u/v := x · v + u · y/y · v,
where “+” denotes addition defined for the integers. In order to establish that there exists exactly one function ⊕ of the kind required by the contextual definition one must only show: <x, y> ≈ < x′, y′> ∧ <u, v> ≈ <u′, v′> → x/y ⊕ u/v = y′/y′ ⊕ u′/v′.
It is clear that for Frege the self-evidence of a truth cannot serve as a general criterion of analyticity. On the one hand, he grants that there are non-evident sentences which are analytic truths, such as the equation “125664 + 37863 = 163527”, for example, (provided that the logicist programme has been successfully carried out for cardinal arithmetic). On the other hand, Frege acknowledges the existence of self-evident, but non-analytic truths, such as the axioms of Euclidean geometry. He characterizes them as synthetic a priori truths, at least in Frege (1884), but he probably sticked to this view until his last writings on sources of knowledge (Erkenntnisquellen) of mathematics and the mathematical natural sciences and on a new attempt at a foundation of arithmetic (cf. Frege 1969, pp. 286 ff., 298 ff.). For a true statement “a = b” to be analytic in Frege’s sense, the identity of the sense(s) of “a” and “b” is a sufficient condition, but it is not a necessary one.
For a detailed discussion of this issue see Schirn (2014b). To avoid misunderstanding or confusion here, let me mention that in Frege (1884) Frege did not yet construe declarative sentences as truth-value names, as names of the True or the False, and hence as a special kind of (complex) proper names that can appropriately flank “=” on both sides. Yet if HP is considered to be a statement of the form “a ↔ b”, the argument above for its epistemic triviality, if “a” and “b” have the same judgeable content or express the same thought, would equally apply. In Frege’s concept-script, Basic Law V is indeed always presented as an equation of the form “a = b”, where “a” itself is an equation of this form, while “b” is what he calls the generality of an equation or of an equality between function-values (cf., for example, Frege 1967, p. 130, 1893, §20): (\( \mathop \varepsilon \limits^{,} \) f(ε) = \( \mathop \alpha \limits^{,} \) g(α)) = (∀x(f(x) = g(x))). The Roman function-letters “f” and “g” are used here to indicate one-place functions of first level (cf. Frege 1893, §19); they are not variables for monadic first-level functions (cf. Frege 1893, §§19–20). Plainly, by appealing to the unbounded generality of Basic Law V qua logical axiom Frege could have formulated it equally well as a universally quantified sentence: “∀f∀g(\( \mathop \varepsilon \limits^{,} \) f(ε) = \( \mathop \alpha \limits^{,} \) g(α)) = (∀x(f(x) = g(x)))”, where “f” and “g” are here variables for monadic first-level functions. In his concept-script, Frege would have used German letters for the function variables. (For the sake of simplicity, I ignore his distinction between the axiom qua true and unprovable thought and the sentence expressing this thought.) Since HP is the exact structural analogue of Basic Law V, the reader of Grundgesetze may have expected that HP likewise appears there as an equation of the form “a = b”. Yet it does not so appear. Anyway, I do not see any deep reason why in Frege (1893) Frege could not have presented HP as an equation by following his notation for Axiom V. Be this as it may, in Frege (1893), §53, HP is introduced as follows: The cardinal number of a concept is equal to the cardinal number of a second concept, if a relation maps the first into the second, and if the converse of this relation maps the second into the first. If we use “→” for the conditional function (Frege 1893, §12), “∈” for Frege’s membership function, namely the relation of an object falling within the extension of a concept (§34), “〉” for the mapping-into by a relation (§38), “∫” for the converse of a relation (§39), and “Nξ” for Frege’s first-level cardinality function (§40), then we could present HP as follows: v ∈ (u ∈ 〉∫ q) → (u ∈ (v ∈ 〉 q) → Nu = Nv).
For details see Schirn (2014b).
See also Dummett’s comment on CRIT 1 in Dummett (1981, p. 324). It remains unclear why the view involved in this criterion, namely that two analytically equivalent sentences express the same thought, should be irreconcilable with Frege’s ideas about sense as stated in other writings. Dummett owes us a plausible explanation why there should be any serious conflict at all.
“F” is here a schematic letter for a monadic first-level predicate.
The conclusion that Frege draws from his abortive attempt to define 0, 1 and n + 1 as objects by means of (I)–(III) appears to be this: correct definitions of the individual natural numbers as objects presuppose a prior unobjectionable definition of the cardinality operator “N x φ(x)”. As a matter of fact, after having explicitly defined the cardinal number that belongs to the concept F as the extension of the concept equinumerous with the concept F in Frege (1884, §68), Frege goes on to define “0”, “1”, and “∞1” as well as the predicate “n is a cardinal number” in terms of “N x φ(x)”; “∞1” is his symbol for the smallest infinite number and, hence, corresponds to “ℵ0”. To be sure, before proposing and analyzing the inductive definition of the natural numbers, Frege knew of course that it could not be accepted as a definition of the natural numbers as objects. I do not think that it would have been a serious loss for chapter IV of Grundlagen, entitled ‘The concept of cardinal number’, had Frege decided to cancel Sect. 55, and with it, the problematic Sects. 56–57 as well. By contrast, as we have seen in Sects. 2 and 5, setting up and discussing the tentative contextual definition of the cardinality operator is crucial for Frege’s logicist strategy in Grundlagen, although the definition is also rejected.
Needless to say, in my role as a critical interpreter of Frege’s work I do not consider it my task to attempt to reconcile what appears to be irreconcilable.
He also speaks somewhat inaccurately of an individual number that forms a part of the predicate “the number 4 belongs to”.
When in his ‘Aufzeichnungen für Ludwig Darmstaedter’ Frege asks: “Could the numerals help to form signs for these second-level concepts [that is, for \( {N}_{{x}}^{0} \) φ(x), \( {N}_{{x}}^{1} \) φ(x),…, \( {N}_{{x}}^{\text{n}} \) φ(x)] and yet not be signs in their own right?” (Frege 1969, p. 277), he must, of course, be aware that as constituents figuring in predicates like “\( {N}_{{x}}^{0} \) φ(x)”, etc. numerals are not signs in their own right. This was already clear in Grundlagen. The question is, rather, whether numerals and other numerical expressions ought to be recognized as proper names of objects concerning their occurrence in sentences like those I quoted above. Frege’s position in his ‘Aufzeichnungen’ is that the second-level concepts \( {N}_{{x}}^{0} \) φ(x), \( {N}_{{x}}^{1} \) φ(x), etc. do not, after all, constitute the numbers of arithmetic; “we do not have objects, but concepts. How can we get from these concepts to the numbers of arithmetic in a way that cannot be faulted?” (Frege 1969, p. 277). The ensuing question “Or are there simply no numbers in arithmetic?” I take to be merely rhetorical. In the course of dealing with Russell’s paradox, Frege discusses the possibility of regarding class names and numerals as pseudo proper names, i.e., as parts of signs which had a reference only as wholes (Frege 1903, p. 255). He rejects this route as impassable.
See Frege (1884, §65, 1969, p. 131). Leibniz gave his famous definition of identity in his fragment ‘Non inelegans specimen demonstrandi in abstractis’ (see Leibniz 1875–1890, p. 288; cf. also p. 236 and Leibniz 1903, pp. 362 f.). The definition is framed in terms of the mutual substitutivity of A and B salva veritate, and in its complete version is as follows:
“Eadem sunt quorum unum potest substitui alteri salva veritate. Si sint A et B, et A ingrediatur aliquam propositionem veram, et ibi in aliquo loco ipsius A pro ipso substiuendo B fiat novo propositio seque itidem vera, idque semper succedat in quancunque tali propositione, A et B dicuntur esse eadem; et contra, si eadem sint A et B, procedet substitutio quan dixi.”
Leibniz does not distinguish here between A qua object and “A” qua sign that refers to A. Trivially, two signs “A” and “B” cannot be said to be identical if a certain condition obtains. So I take it that Leibniz intends to define the identity of A and B qua object(s). However, strictly speaking, it is not the object B that can be substituted for the object A in a true sentence or proposition, but it is the sign “B” that can be substituted for “A”. In other places, Leibniz speaks also of identical or coincident terms, but what he has in mind is clearly their equivalence: “Termini aequivalentes sunt, quibus res significantur eadem, ut triangulum et trilaterum” (Leibniz 1903, p. 240; see also p. 363 where he explains the “coincidence” of statements in terms of their mutual substitutivity salva veritate). Yet I do not think that in his definition of identity “eadem sunt” is meant in the sense of “termini aequivalentes sunt”, although according to this reading the talk of substitutivity would be obviously correct. Be this as it may, in Frege (1884), §65, Frege not only adopts Leibniz’s explanation or definition of identity as its own, but he also chooses Leibniz’s way of speaking when he says that in order to justify his attempted definition of the direction of a line, he must show that one can substitute the direction of b everywhere for the direction of a (see in this respect also §107). One can easily show that the mutual substitutivity of two singular terms “a” and “b” in true (extensional) sentences salva veritate by means of which Leibniz defines the identity of a and b, amounts to the coincidence of the complete concepts of a and b. According to Leibniz, the complete concept of an object a is equivalent to the conjunction of all concepts under which a falls (or to the conjunction of all the predicates that truly apply to a). The equivalence of “f(a)” and “f(b)” in the definiens is supposed to mean: if any substitution instance for “f” is a “conjunctive constituent” of the complete concept of a, then it is also part of the complete concept of b, and vice versa. Since “f” is a variable, the complete concepts of a and b must “coincide”, if a = b. In his review of Husserl’s Philosophie der Arithmetik, exactly 10 years after the publication of Grundlagen, Frege points out that he construes Leibniz’s explanation of identity not as a definition. (By contrast, in Frege 1884 he presumably considered it to be a definition). He writes (Frege 1967, p. 184): “Since every definition is an equation, identity itself cannot be defined. Leibniz’s explanation could be called a principle that brings out the nature of the relation of identity, and as such it is of fundamental importance.” Note that in the logical system of Grundgesetze identity is introduced as a primitive function and hence is indefinable in this system.
In Frege (1884, §64), Frege uses the term “zerspalten”.
In Schirn (1996), I have argued that there is ample evidence that in the logical system of Grundgesetze the priority thesis holds only in a restricted sense. It applies neither to the primitive concept- and relation-expressions and their logically simple senses, nor to the first complex concept-expressions of first level (that are formed from the primitive names) and their complex senses. In Grundgesetze, the priority thesis applies, in effect, to all predicates which in the last stage of their entire “constructional history” are obtained by means of gap formation (to be explained below). The constructional chain of a predicate may of course involve an iterated application of a gap formation rule. The priority thesis applies, in particular, to all simple names of complex concepts. A good example would be the formation of “ξ = ξ” by starting with the primitive name “ξ = ζ”. According to Frege’s criterion of the simplicity of an expression (cf. Frege 1903, p. 79), “ξ = ξ” is simple, because it is not put together from signs each of which has a reference (sense) on its own and contributes to determining the reference (sense) of the whole. Yet according to Frege’s use of the term “logically simple”, “ξ = ξ” is not logically simple because it is not one of the primitive, logically simple function-names of the concept-script. While “ξ = ζ” expresses a simple sense, “ξ = ξ” does not. Although Frege does not explain under what conditions the sense of an expression should be regarded as simple, the simplicity of the sense certainly implies that the latter is not further analyzable into simpler or more elementary senses. Every function-name of the system of Grundgesetze that is introduced by means of a definition is simple, but it is not logically simple. It refers to a logically complex function and expresses a complex sense. Admittedly, it remains unclear how Frege could explain the complexity of the sense of a predicate like “ξ = ξ”. Perhaps he could say that the complexity derives from the fact that the sense is not immediately given by means of an elucidation (as is the case for every primitive function-name with the notable exception of the name of the course-of-values function), but is obtained by running through the stepwise construction of “ξ = ξ”, which consists in a successive application of the rule of insertion (= first and second steps of construction) and the subsequent application of the first gap formation rule to “Δ = Δ”. “Δ” is here a schematic letter for a referential proper name. It does not occur in the formal proofs of the basic laws of arithmetic. As to the capital Greek letters “Γ” and “Δ”, Frege (1893, p. 35) (footnote) says that he uses them as names in such a way as if they were to refer to something.
I presume that in Grundlagen Frege would have described second-order abstraction in a way similar to his characterization of first-order abstraction.
Argument-places of the first kind are suitable for the insertion of proper names; argument-places of the second kind are suitable for the insertion of monadic first-level function-names; and argument-places of the third kind are suitable for the insertion of dyadic first-level function-names; cf. Frege (1893, §23).
But see also Dummett (1981, pp. 333 f).
Dummett (1981, p. 271) admits at least that Frege himself never introduced any distinct terminology to differentiate the two types of analysis. Dummett uses the word “analysis” for the process of analyzing a sentence or the thought expressed by it into its ultimate constituents, and reserves the word “decomposition” for the process which I call gap formation. He obviously regards the process of building up a sentence and the corresponding thought from its constituents as the inverse of the process of analysis. But this view cannot be sustained with respect to the formal language of Grundgesetze; see in this respect Schirn (1983, p. 248).
In Schirn (1985), I introduced the term “truth-value name”, which Frege does not use. By a truth-value name I understand a function-value name that has the structure of a sentence, and hence does not only refer to one of the two truth-values, but also expresses a thought. By contrast, the course-of-values name “\( \mathop{\varepsilon}\limits^{,} \)(ε = (ε = ε))” or the definite description “\\( \mathop{\varepsilon}\limits^{,} \)(—ε)” do not express a thought, although, due to Frege’s stipulations in Grundgesetze both names refer to the True. They are names of a truth-value, but they are not truth-value names qua sentences. Both names express a complex, non-propositional sense.
When in Frege (1884, §64) Frege says that the judgement “The straight line a is parallel to the straight line b” can be construed as an equation, he chooses an infelicitous way of phrasing. This judgement qua sentence (note his use of quotation marks) can never be construed as an equation in a strict sense, although it can be transformed into an equation. The ensuing statement “We replace the symbol || by the more general symbol =, by distributing the particular content of the former symbol to a and to b” is rather metaphorical and far from being clear. The linguistic operation of replacing a symbol by a more general symbol has nothing to do with abstraction. And what is it to mean precisely that the particular content of “=” is distributed between a and b? By the way, replacing the symbol “||” by “=” in “a || b” would be permissible at least from a syntactic point of view. The matter stands differently in the case of HP. The sign for the second level relation of equinumerosity cannot literally be replaced by “=”.
According to Frege, identity is a first-level relation.
Parsons (1997, p. 270) seems to accept Wright's idea that Fregean abstraction effects a reconceptualization for the case of first-order principles. In those cases, he says, “it seems that what we are doing is simply individuating the objects we have in a coarser way, one might say carving up the domain, or a part of it, a little differently.” This is very vague and thus far from being clearer than Wright's characterization. Fine (1998, p. 532) introduces the term “definition by reconceptualization" and says (but does not further explain) that it rests on the idea that new senses may emerge from a reanalysis of a given sense. He further claims that the idea derives from Frege (1884, §§63–64). However, as I have argued, Frege’s attempted contextual definitions via abstraction cannot sensibly be described in such a way that new senses emerge from a reanalysis of a given sense.
It is perfectly possible that Frege planned to compose even a fourth Grundgesetze volume, devoted to the theory of complex numbers.
References
Benacerraf P (1965) What numbers could not be. Philos Rev 74:47–73
Boolos G (1987) The consistency of Frege’s foundations of arithmetic. In: Thomson JJ (ed) On being and saying. Essays for Richard Cartwright. The MIT Press, Cambridge, pp 3–20
Burali-Forti C (1897) A question of transfinite numbers. In: Rediconti of the 28 March meeting of the Circolo matematico di Palermo. English translation in van Heijenooort J (ed) From Frege to Gödel. A soure book in mathematical logic, 1879–1931. Harvard University Press, Cambridge 1967, pp 104-112
Currie G (1986) Continuity and change in Frege’s philosophy of mathematics. In: Haaparanta L, Hintikka J (eds) Frege synthesized. Kluwer Academic Publishers, Dordrecht
Dummett M (1973) Frege. Philosophy of language. Duckworth, London
Dummett M (1981) The interpretation of Frege’s philosophy. Duckworth, London
Dummett M (1991) Frege. Philosophy of mathematics. Duckworth, London
Dummett M (1998) Neo-Fregeans. In bad company. In: Schirn M (ed) The philosophy of mathematics today. Oxford University Press, Oxford, pp 369–387
Fine K (1998) The limits of abstraction. In: Schirn M (ed) The philosophy of mathematics today. Oxford University Press, Oxford, pp 503–629
Frege G (1879) Begriffsschrift. Eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. L. Nebert, Halle. a. S.
Frege G (1884) Die Grundlagen der Arithmetik. Eine logisch mathematische Untersuchung über den Begriff der Zahl. W. Koebner, Breslau
Frege G (1893) Grundgesetze der Arithmetik. Begriffsschriftlich abgeleitet, vol I. H. Pohle, Jena
Frege G (1903) Grundgesetze der Arithmetik. Begriffsschriftlich abgeleitet, vol II. H. Pohle, Jena
Frege G (1967) Kleine Schriften, ed. I Angelelli. Georg Olms, Hildesheim
Frege G (1969) Nachgelassene Schriften, eds. H. Hermes, F. Kambartel, F. Kaulbach. Felix Meiner, Hamburg
Frege G (1976) Wissenschaftlicher Briefwechsel, eds. G. Gabriel, H. Hermes, F. Kambartel, C. Thiel, A. Veraart. Felix Meiner, Hamburg
Hale B (1987) Abstract objects. Basil Blackwell, Oxford
Hale B (1994) Singular terms (2). In: McGuinness B, Oliveri G (eds) The philosophy of Michael Dummett. Kluwer Academic Publishers, Dordrecht, pp 17–44
Hale B (1996) Singular terms (1). In: Schirn M (ed) Frege: importance and legacy. Walter de Gruyter, Berlin, pp 438–457
Hale B (1997) Grundlagen §64. In: Proceedings of the Aristotelian Society 97:243–261. Reprint in Hale and Wright (2001) with a new postscript added, pp 91–116
Hale B (2000) Reals by abstraction. Philosophia Mathematica 8:100–123. Reprint in Hale and Wright (2001), pp 399–420
Hale B, Wright C (2001) The reason’s proper study. Essays towards a neo-Fregean philosophy of mathematics. Clarendon Press, Oxford
Heck RG (1999) Grundgesetze der Arithmetik I §10. Philos Math 7:258–292
Heck RG (2005) Julius Caesar and basic law V. Dialectica 59:161–178
Hodes H (1984) Logicism and the ontological commitments of arithmetic. J Philos 81:123–149
Leibniz GW (1875–1890) In: Gerhardt C (ed) Die philosophischen Schriften von G.W. Leibniz. Berlin. Reprint Georg Olms, Hildesheim 1960–1961
Leibniz GW (1903) In: Couturat L (ed) Opuscules et fragments inédits de Leibniz. Paris. Reprint Georg Olms, Hildesheim 1966
Linnebo O (ed) (2009) Synthese 170, 321–482. Special issue on the bad company objection
Parsons C (1997) Wright on abstraction and set theory. In: Heck RG (ed) Language, thought, and logic. Essays in honour of Michael Dummett. Clarendon Press, Oxford, pp 263–272
Ruffino M (1998) Logicism: Fregean and neo-Fregean. Manuscrito 21:149–188
Ruffino M (2003) Why Frege would not be a neo-Fregean. Mind 112:51–78
Russell B (1903) The principles of mathematics. Cambridge University Press, Cambridge
Schirn M (1983) Review of Dummett 1981. Erkenntnis 20:243–251
Schirn M (1985) Semantische Vollständigkeit, Wertverlaufsnamen und Freges Kontextprinzip. Grazer Philos Studien 23:79–104
Schirn M (1996) On Frege’s introduction of cardinal numbers as logical objects. In: Schirn M (ed) Frege: importance and legacy. Walter de Gruyter, Berlin, pp 114–173
Schirn M (2013) Frege’s approach to the foundations of analysis (1874–1903). Hist Philos Log 34:266–292
Schirn M (2014a) Frege’s theory of real numbers in consideration of the theories of Cantor, Russell and others. Forthcoming in Link G (ed) Formalism and beyond, Walter de Gruyter, Berlin
Schirn M (2014b) Second-order abstraction before and after Russell’s paradox. Forthcoming in Ebert P, Rossberg M (eds) Essays on Frege’s basic laws of arithmetic. Oxford University Press, Oxford
Schirn M (2014c) Logical objects by abstraction, their criteria of identity and Julius Caesar. Forthcoming
Schirn M (2014d) Reals by abstraction and reals as relations of relations: a note on Frege. Forthcoming
Schirn M (2014e) Die Dichotomie analytisch/synthetisch bei Frege unter Berücksichtigung von Leibniz und Kant. Forthcoming
Simons P (1987) Frege’s theory of real numbers. Hist Philos Log 8:25–44
von Kutschera F (1966) Frege’s Begründung der analysis. Archiv für mathematische Logik und Grundlagenforschung 9:102–111. Reprint in Schirn M (ed) Studien zu Frege—Studies on Frege, vol I.: Logik und Philosophie der Mathematik—Logic and Philosophy of Mathematics. Frommann-Holzboog, Stuttgart–Bad Cannstatt 1976, 301–312
von Kutschera F (1967) Elementare Logik. Springer, Wien
Wright C (1983) Frege’s conception of numbers as objects. Aberdeen University Press, Aberdeen
Wright C (1997) On the philosophical significance of Frege’s theorem. In: Heck RG (ed) Language, thought, and logic. Essays in honour of Michael Dummett. Clarendon Press, Oxford, pp 201–244
Wright C (1998a) On the harmless impredicativity of N (“Hume’s Principle”). In: Schirn M (ed) The philosophy of mathematics today. Oxford University Press, Oxford, pp 339–368
Wright C (1998b) Response to Dummett. In: Schirn M (ed) The philosophy of mathematics today. Oxford University Press, Oxford, pp 369–387
Wright C (1999) Is Hume’s principle analytic? Notre Dame J Formal Log 40:6–30
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Schirn, M. Frege’s Logicism and the Neo-Fregean Project. Axiomathes 24, 207–243 (2014). https://doi.org/10.1007/s10516-013-9222-7
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DOI: https://doi.org/10.1007/s10516-013-9222-7