Frege: Philosophy of Mathematics

The following essay reconsiders the ontological and logical issues around Frege’s Basic Law (V). If focuses less on Russell’s Paradox, as most treatments of Frege’s Grundgesetze der Arithmetik (GGA)1 do, but rather on the relation between Frege’s Basic Law (V) and Cantor’s Theorem (CT). So for the most part the inconsistency of Naïve Comprehension (in the context of standard Second Order Logic) will not concern us, but rather the ontological issues central to the conflict between (BLV) and (CT). These ontological (...) issues are interesting in their own right. And if and only if in case ontological considerations make a strong case for something like (BLV) we have to trouble us with inconsistency and paraconsistency. These ontological issues also lead to a renewed methodological reflection what to assume or recognize as an axiom. (shrink)

A version of Frege's theorem can be proved in a plural logic with pair abstraction. We talk through this and discuss the philosophical implications of the result.

This essay aims to redress the contention that epistemic possibility cannot be a guide to the principles of modal metaphysics. I introduce a novel epistemic two-dimensional truthmaker semantics. I argue that the interaction between the two-dimensional framework and the mereological parthood relation, which is super-rigid, enables epistemic possibilities and truthmakers with regard to parthood to be a guide to its metaphysical profile. I specify, further, a two-dimensional formula encoding the relation between the epistemic possibility and verification of essential properties obtaining (...) and their metaphysical possibility or verification. I then generalize the approach to haecceitistic properties. I also examine the Julius Caesar problem as a test case. I conclude by addressing objections from the indeterminacy of ontological principles relative to the space of epistemic possibilities, and from the consistency of epistemic modal space. (shrink)

The principal goal of this entry is to present Frege's Theorem (i.e., the proof that the Dedekind-Peano axioms for number theory can be derived in second-order logic supplemented only by Hume's Principle) in the most logically perspicuous manner. We strive to present Frege's Theorem by representing the ideas and claims involved in the proof in clear and well-established modern logical notation. This prepares one to better prepared to understand Frege's own notation and derivations, and read Frege's original work (whether in (...) German or in translation). Moreover, this should prepare the reader to understand a number of scholarly books and articles in the secondary literature on Frege's work. (shrink)

Frege's definition of the real numbers, as envisaged in the second volume of Grundgesetze der Arithmetik, is fatally flawed by the inconsistency of Frege's ill-fated Basic Law V. We restate Frege's definition in a consistent logical framework and investigate whether it can provide a logical foundation of real analysis. Our conclusion will deem it doubtful that such a foundation along the lines of Frege's own indications is possible at all.

Some tools introduced by Linnebo to show that mathematical entities are thin objects can also be applied to non-mathematical entities, which have been thought to be thin as well for a variety of reasons. In this paper, I discuss some difficulties and opportunities concerning the application of abstraction and interpretational modalities to mereological sums. In particular, I show that on one hand some prima facie attractive candidates for the role of an explanatory plural abstraction principle for mereological sums (in terms (...) of pluralities of summed entities) are not really explanatory; on the other hand, singular abstraction principles (in terms of single summed entities) are materially inadequate. Nonetheless, explanatory criteria of identity and conditions of existence for mereological sums are provided by classical extensional mereology independent of abstraction principles. Thus, given classical extensional mereology, the reasons why, according to Linnebo, mathematical abstracted entities are thin also hold for mereological sums. Finally, I contend that interpretational modalities can be used to characterise the process by which a subject adds sums of previously admitted entities to the domain of quantification. (shrink)

One prominent criticism of the abstractionist program is the so-called Bad Company objection. The complaint is that abstraction principles cannot in general be a legitimate way to introduce mathematical theories, since some of them are inconsistent. The most notorious example, of course, is Frege’s Basic Law V. A common response to the objection suggests that an abstraction principle can be used to legitimately introduce a mathematical theory precisely when it is stable: when it can be made true on all sufficiently (...) large domains. In this paper, we raise a worry for this response to the Bad Company objection. We argue, perhaps surprisingly, that it requires very strong assumptions about the range of the second-order quantifiers; assumptions that the abstractionist should reject. (shrink)

In this review I briefly explain the most important points of each chapter of Dummett's book, and critically discuss some of them. Special attention is given to the criticisms of Crispin Wright's interpretation of Frege's Platonism, and also to Dummett's interpretation of the role(s) of the context principle in Frege's thought.

In Thin Objects: An Abstractionist Account, Linnebo offers what he describes as a “simple and definitive” solution to the bad company problem facing abstractionist accounts of mathematics. “Bad” abstraction principles can be rendered “good” by taking abstraction to have a predicative character. But the resulting predicative axioms are too weak to recover substantial portions of mathematics. Linnebo pursues two quite different strategies to overcome this weakness in the case of set theory and arithmetic. I argue that neither infinitely iterated abstraction (...) nor abstraction on possible specifications fully resolves the bad company problem. (shrink)

The notion of the extension of a concept has been used in logic for a long time. It is usually considered to be closely connected to the intuitive notion of a set and thus seems as though it should be embedded into set theory. However, there are significant differences between this “logical” concept of set and the notion of set (class) as defined via standard axiomatic systems of set theory; it may, therefore, be quite misleading to consider the two concepts (...) as being continuous with each other. When we look at the writings of Gottlob Frege and consider the development of his attitude to extensions, we can see what the differences consist in and which of the two notions is more apt to be used in foundations of logic. Frege himself eventually rejected sets entirely. (shrink)

Mathematical formalism is the the view that numbers are "signs" and that arithmetic is like a game played with such signs. Frege's colleague Thomae defended formalism using an analogy with chess, and Frege's critique of this analogy has had a major influence on discussions in analytic philosophy about signs, rules, meaning, and mathematics. Here I offer a new interpretation of formalism as defended by Thomae and his predecessors, paying close attention to the mathematical details and historical context. I argue that (...) for Thomae, the formal standpoint is an _algebraic perspective_ on a domain of objects, and a "sign" is not a linguistic expression or mark, but a representation of an object within that perspective. Thomae exploits a shift into this perspective to give a purely algebraic construction of the real numbers from the rational numbers. I suggest that Thomae's chess analogy is intended to provide a model for such shifts in perspective. (shrink)

In this article, I try to shed new light on Frege’s envisaged definitional introduction of real and complex numbers in _Die Grundlagen der Arithmetik_ (1884) and the status of cross-sortal identity claims with side glances at _Grundgesetze der Arithmetik_ (vol. I 1893, vol. II 1903). As far as I can see, this topic has not yet been discussed in the context of _Grundlagen_. I show why Frege’s strategy in the case of the projected definitions of real and complex numbers in (...) _Grundlagen_ is modelled on his definitional introduction of cardinal numbers in two steps, tentatively via a contextual definition and finally and definitively via an explicit definition. I argue that the strategy leaves a few important questions open, in particular one relating to the status of the envisioned abstraction principles for the real and complex numbers and another concerning the proper handling of cross-sortal identity claims. (shrink)

The aim of this article is to examine Frege’s view of the context principle in his mature philosophical doctrine. Here, the author argues that the context principle is embodied in the contextual explanation of value-ranges presented in Basic Laws of Arithmetic. The contextual explanation of value-ranges plays essentially the same role as the context principle in The Foundations of Arithmetic. It is supposed to show how a reference to natural numbers is possible. Moreover, the author argues against the view that (...) the context principle should be separated from the recarving thesis. This aspect of Frege’s view leads the author to the rejection of the purely referential view of the context principle, which is fairly widely accepted in the current literature. (shrink)

Review of Vols 1 & 2 of Stephen Houlgate's 'Hegel on Being'.

Kant's claim that arithmetical truths are synthetic is famously contradicted by Frege, who considers them to be analytical. It may seem that this is a mere dispute about linguistic labels, since both Kant and Frege agree that arithmetical truths are a priori and informative, and, therefore, it is only a matter of how one chooses to call them. I argue that the choice between calling arithmetic “synthetic” or “analytic” has a deeper significance. I claim that the dispute is not a (...) merely linguistic one, but one involving the status of arithmetic as a science. Namely, it is a dispute about what truths are appropriate to use in grounding a scientific discipline. I interpret Frege's critique of Kant as being mainly about the lack of distinction between the context of justification and the context of discovery. (shrink)

This paper presents three explanations of why Frege took the universal, rather than the existential, quantifier as primitive in his formalization of logic. The first two explanations provide technical reasons related to how Frege formalizes the logic of truth-functions and the logic of quantification. The third, philosophical explanation locates the reason in Frege's logicist goal of analyzing arithmetical concepts---especially the concepts of 0 and 1---in purely logical terms.

Frege's Grundgesetze der Arithmetik offers a conception of cpLogic as the study of functions. Among functions are included those that are concepts, i.e. characteristic functions whose values are the logical objects that are the True/the False. What, in Frege's view, are the objects the True/the False? Frege's stroke functions are themselves concepts. His stipulation introducing his negation stroke mentions that it yields [...]. But curiously no accommodating axiom is given, and there is no such theorem. Why is it that some (...) of Frege's informal stipulations never made appearances as axioms? I offer an explanation that sheds new light on the Grundgesetze. No axioms should over-determination the True as a logical object. Perhaps the True = 0, as would be common in the mathematics of characteristic functions. But the logical objects that are cardinal numbers are value ranges correlated with second-level numerical concepts by a non-homogeneous second-level value-range function [...]. The existence of concepts would be ontologically circular if the True is itself a number. We find this circularity perfectly agreeable to Frege, and suggest that he had accepted that the existence of functions that are concepts in his cpLogic may well be ontologically inseparable from the existence of his value-range function. His cpLogic itself stands or falls with the viability of some value-range function. (shrink)

Number words seemingly function both as adjectives attributing cardinality properties to collections, as in Frege’s ‘Jupiter has four moons’, and as names referring to numbers, as in Frege’s ‘The number of Jupiter’s moons is four’. This leads to what Thomas Hofweber calls Frege’s Other Puzzle: How can number words function as modifiers and as singular terms if neither adjectives nor names can serve multiple semantic functions? Whereas most philosophers deny that one of these uses is genuine, we instead argue that (...) number words, like many related expressions, are polymorphic, having multiple uses whose meanings are systematically related via type shifting. (shrink)

In recent years, several scholars have been investigating Frege’s mathematical background, especially in geometry, in order to put his general views on mathematics and logic into proper perspective. In this article I want to continue this line of research and study Frege’s views on geometry in their own right by focussing on his views on a field which occupied center stage in nineteenth century geometry, namely, projective geometry.

In this paper I argue that the two-dimensional character of Frege’s Begriffsschrift plays an epistemological role in his argument for the analyticity of arithmetic. First, I motivate the claim that its two-dimensional character needs a historical explanation. Then, to set the stage, I discuss Frege’s notion of a Begriffsschrift and Kant’s epistemology of mathematics as synthetic a priori and partly grounded in intuition, canvassing Frege’s sharp disagreement on these points. Finally, I argue that the two-dimensional character of Frege’s notations play (...) the epistemological role of facilitating our grasp of logical truths independently of intuition. The rest of this paper critically evaluates Frege’s view and discusses Macbeth’s account. (shrink)

Há muitos pontos de concordância entre Frege e os neokantianos. Isso vale especialmente para os representantes do neokantismo da teoria do valor ou do Sudoeste alemão na tradição de Hermann Lotze. Não discutiremos aqui todos os aspectos dessa proximidade; de acordo com o tema que propomos, ficaremos restritos à filosofia da matemática. A primeira parte do artigo tratará da relação entre aritmética e geometria, mostrando surpreendentes semelhanças entre Frege e o neokantiano Otto Liebmann. A segunda parte discutirá as diferentes recepções (...) do logicismo de Frege por Jonas Cohn, Paul Natorp, Heinrich Rickert, Ernst Cassirer e Bruno Bauch. (shrink)

This paper describes both an exegetical puzzle that lies at the heart of Frege’s writings—how to reconcile his logicism with his definitions and claims about his definitions—and two interpretations that try to resolve that puzzle, what I call the “explicative interpretation” and the “analysis interpretation.” This paper defends the explicative interpretation primarily by criticizing the most careful and sophisticated defenses of the analysis interpretation, those given my Michael Dummett and Patricia Blanchette. Specifically, I argue that Frege’s text either are inconsistent (...) with the analysis interpretation or do not support it. I also defend the explicative interpretation from the recent charge that it cannot make sense of Frege’s logicism. While I do not provide the explicative interpretation’s full solution to the puzzle, I show that its main competitor is seriously problematic. (shrink)

Even though Frege is a major figure in the history of analytic philosophy, it is not surprising that there are still issues surrounding his views, interpreting them, and labeling them. Frege’s view on numbers is typically termed as ‘Platonistic’ or at least a type of Platonism (Reck 2005). Still, the term ‘Platonism’ has views and assumptions ascribed to it that may be misleading and leads to mischaracterizations of Frege’s outlook on numbers and ideas. So, clarification of the term ‘Platonism’ is (...) required to portray Frege’s views more accurately (Reck 2005). This clarification gives us a better picture of what Frege is interested in and what he does not emphasize. Moreover, in such a clarifying process, we find that Frege draws significant influence from Rudolf Hermann Lotze, who is frequently called a Neo-Kantian (Vagnetti 2018). In Lotze’s major work, Logik, Lotze has a central focus on validity, in its most general form as he used it, that investigates various related topics, i.e., concepts, language, etc. (Vagnetti 2018; Lotze 1888). Furthermore, we observe that Frege’s work is so similar to Lotze, that it seems questionable to call his outlook ‘Platonism’. Therefore, attributing ‘Platonism’ to Frege may be a slight misnomer. This paper’s entirety is mostly a synthesis of a variety of articles related to Frege, Lotze, and their respective outlooks and the original works of Frege and Lotze that I use to support the view that the term ‘Platonism’ is a slight issue when predicated to Frege. As such, I include an overview of Frege’s treatment in other work that highlights the usage of the term ‘Platonism’ and how broad its uses tend to be utilized (Balaguer 2006; Burge 1992). In sum, it is observed that the general label ‘Platonism’ becomes less appropriate when we consider Lotze in the picture and contrast Lotze alongside Mr. Frege. Overall, this paper is just an explanatory one of Mr. Frege, the Platonist, and the issues of applying the term ‘Platonism’ onto him as his views are seemingly more of a segue from Lotze. Keywords: Frege, Lotze, number, objectification, Platonism, validity. (shrink)

The paper gives a historically informed reconstruction of Frege's view of statements of number. The reconstruction supports Frege's claim that a statement can be 'about a concept' although it does not contain a singular term referring to the concept. Hence, Frege's philosophy of number is not subject to the problems Frege sees for singular reference to concepts.

I investigate why Frege rejected the theory of types, as Russell presented it to him in their correspondence. Frege claims that it commits one to violations of the law of excluded middle, but this complaint seems to rest on a dogmatic refusal to take Russell’s proposal seriously on its own terms. What is at stake is not so much the truth of a law of logic, but the structure of the hierarchy of the logical categories, something Frege seems to neglect. (...) To come to a better understanding of Frege’s response, I proceed to investigate his conception of the nature of the logical categories, and how it differs from Russell’s. I argue that, for Frege, our grasp of the logical categories cannot be severed from our grasp of the Begriffsschrift notation itself. Russell, on the other hand, attaches no such importance to notation. From Frege’s point of view, Russell has not succeeded in presenting an alternative conception of the logical hierarchy, since such a conception must go in tandem with the development of a notation. Moreover, Frege has good reasons to think that Russell’s proposal does not admit of a suitable notation. (shrink)

According to the Bad Company objection, the fact that Frege’s infamous Basic Law V instantiates the general definitional pattern of higher-order abstraction principles is a good reason to doubt the soundness of this sort of definitions. In this paper I argue against this objection by showing that the definitional pattern of abstraction principles – as extrapolated from §64 of Frege’s Grundlagen– includes an additional requirement (which I call the specificity condition) that is not satisfied by the Basic Law V while (...) is satisfied by other higher-order abstractions such as Hume’s Principle. I also show that the failure of this additional requirement in the case of Basic Law V is engendered by an essential feature of Frege’s conception of logic and thus that Frege himself should not have regarded the Basic Law V as a definition by abstraction. (shrink)

In his logical foundation of arithmetic, Frege faced the problem that the semantic interpretation of his system does not determine the reference of the abstract terms completely. The contextual definition of number, for instance, does not decide whether the number 5 is identical to Julius Caesar. In a late writing, Quine claimed that the indeterminacy of reference established by Frege’s Caesar problem is a special case of the indeterminacy established by his proxy-function argument. The present paper aims to show that (...) Frege’s Caesar problem does not really support the conclusions that Quine draws from the proxy-function argument. On the contrary, it reveals that Quine’s argument is a non sequitur: it does not establish that there are alternative interpretations of our terms that are equally correct, but only that these terms are ambiguous. The latter kind of referential indeterminacy implies that almost all sentences of our overall theory of the world are either false or neither true nor false, because they contain definite descriptions whose uniqueness presupposition is not fulfilled. The proxy-function argument must therefore be regarded as a reductio ad absurdum of Quine’s behaviorist premise that the reference of terms is determined only by our linguistic behavior. (shrink)

Frege’s famous definition of number famously uses the concept of “extension”. Extensions, in the Fregean framework, are susceptible to bringing many difficulties, and, some say, even paradoxes. Therefore, neo-logicist programs want to avoid the problems and to replace the classical Fregean definition of number with Hume’s Principle. I argue that this move, even if it makes sense from a computational point of view, is at odds with Frege’s larger philosophical project. For Frege, I claim, extensions were an important part of (...) his philosophical program of logic-as-an-universal-language. This is why Frege places his project in line with Leibniz’ philosophical project of finding a lingua characterica universalis. (shrink)

Frege’s famous definition of number famously uses the concept of “extension”. Extensions, in the Fregean framework, are susceptible to bringing many difficulties, and, some say, even paradoxes. Therefore, neo-logicist programs want to avoid the problems and to replace the classical Fregean definition of number with Hume’s Principle. I argue that this move, even if it makes sense from a computational point of view, is at odds with Frege’s larger philosophical project. For Frege, I claim, extensions were an important part of (...) his philosophical program of logic-as-an-universal-language. This is why Frege places his project in line with Leibniz’ philosophical project of finding a lingua characterica universalis. (shrink)

Frege’s famous definition of number famously uses the concept of “extension”. Extensions, in the Fregean framework, are susceptible to bringing many difficulties, and, some say, even paradoxes. Therefore, neo-logicist programs want to avoid the problems and to replace the classical Fregean definition of number with Hume’s Principle. I argue that this move, even if it makes sense from a computational point of view, is at odds with Frege’s larger philosophical project. For Frege, I claim, extensions were an important part of (...) his philosophical program of logic-as-an-universal-language. This is why Frege places his project in line with Leibniz’ philosophical project of finding a lingua characterica universalis. (shrink)

The present paper deals with the ontological status of numbers and considers Frege´s proposal in Grundlagen upon the background of the Post-Kantian semantic turn in analytical philosophy. Through a more systematic study of his philosophical premises, it comes to unearth a first level paradox that would unset earlier still than it was exposed by Russell. It then studies an alternative path that, departing from Frege’s initial premises, drives to a conception of numbers as synthetic a priori in a more Kantian (...) sense. On this basis, it tentatively explores a possible derivation of basic logical rules on their behalf, suggesting a more rudimentary basis to inferential thinking, which supports reconsidering the difference between logical thinking and AI. Finally, it reflects upon the contributions of this approach to the problem of the a priori. (shrink)

ABSTRACT Although Frege’s theory of real numbers in Grundgesetze der Arithmetik, Vol. II, is incomplete, it is possible to provide a logicist justification for the approach he is taking and to construct a plausible completion of his account by an extrapolation which parallels his theory of cardinal numbers.

In 1907 Einstein had the insight that bodies in free fall do not “feel” their own weight. This has been formalized in what is called “the principle of equivalence.” The principle motivated a critical analysis of the Newtonian and special-relativistic concepts of inertia, and it was indispensable to Einstein’s development of his theory of gravitation. A great deal has been written about the principle. Nearly all of this work has focused on the content of the principle and whether it has (...) any content in Einsteinian gravitation, but more remains to be said about its methodological role in the development of the theory. I argue that the principle should be understood as a kind of foundational principle known as a criterion of identity. This work extends and substantiates a recent account of the notion of a criterion of identity by William Demopoulos. Demopoulos argues that the notion can be employed more widely than in the foundations of arithmetic and that we see this in the development of physical theories, in particular space–time theories. This new account forms the basis of a general framework for applying a number of mathematical theories and for distinguishing between applied mathematical theories that are and are not empirically constrained. (shrink)

One of the more distinctive features of Bob Hale and Crispin Wright’s neologicism about arithmetic is their invocation of Frege’s Constraint – roughly, the requirement that the core empirical applications for a class of numbers be “built directly into” their formal characterization. In particular, they maintain that, if adopted, Frege’s Constraint adjudicates in favor of their preferred foundation – Hume’s Principle – and against alternatives, such as the Dedekind-Peano axioms. In what follows we establish two main claims. First, we show (...) that, if sound, Hale and Wright’s arguments for Frege’s Constraint at most establish a version on which the relevant application of the naturals is transitive counting – roughly, the counting procedure by which numerals are used to answer “how many”-questions. Second, we show that this version of Frege’s Constraint fails to adjudicate in favor of Hume’s Principle. If this is the version of Frege’s Constraint that a foundation for arithmetic must respect, then Hume’s Principle no more – and no less – meets the requirement than the Dedekind-Peano axioms do. (shrink)

This paper aims to answer the question of whether or not Frege's solution limited to value-ranges and truth-values proposed to resolve the "problem of indeterminacy of reference" in section 10 of Grundgesetze is a violation of his principle of complete determination, which states that a predicate must be defined to apply for all objects in general. Closely related to this doubt is the common allegation that Frege was unable to solve a persistent version of the Caesar problem for value-ranges. It (...) is argued that, in Frege’s standards of reducing arithmetic to logic, his solution to the indeterminacy does not give rise to any sort of Caesar problem in the book. (shrink)

ABSTRACT This paper reveals David Hilbert’s position in the philosophy of mathematics, circa 1900, to be a form of non-eliminative structuralism, predating his formalism. I argue that Hilbert withstands the pressing objections put to him by Frege in the course of the Frege-Hilbert controversy in virtue of this early structuralist approach. To demonstrate that this historical position deserves contemporary attention I show that Hilbertian structuralism avoids a recent wave of objections against non-eliminative structuralists to the effect that they cannot distinguish (...) between structurally identical but importantly distinct mathematical objects, such as the complex roots of $-1$. (shrink)

A speculative investigation of how Frege's logical views change between Begriffsschrift and Grundgesetze and how this might have affected the formal development of logicism.

An overview of what Frege accomplishes in Part II of Grundgesetze, which contains proofs of axioms for arithmetic and several additional results concerning the finite, the infinite, and the relationship between these notions. One might think of this paper as an extremely compressed form of Part II of my book Reading Frege's Grundgesetze.

It would seem that some statements like ‘There are exactly four moons of Jupiter’ and ‘The number of moons of Jupiter is four’ have the same truth-conditions and yet differ in ontological commitment. One strategy to resolve this paradoxical phenomenon is to insist that the statements have not only the same truth-conditions but also the same ontological commitments; the other strategy is to reject the presumption that they have the same truth-conditions. This paper critically examines some popular versions of these (...) two strategies, and defends a new solution according to which the statements have the same ontological commitments and yet differ in truth-conditions. (shrink)

Recent discussions on Fregean and neo-Fregean foundations for arithmetic and real analysis pay much attention to what is called either ‘Application Constraint’ ($AC$) or ‘Frege Constraint’ ($FC$), the requirement that a mathematical theory be so outlined that it immediately allows explaining for its applicability. We distinguish between two constraints, which we, respectively, denote by the latter of these two names, by showing how$AC$generalizes Frege’s views while$FC$comes closer to his original conceptions. Different authors diverge on the interpretation of$FC$and on whether it (...) applies to definitions of both natural and real numbers. Our aim is to trace the origins of$FC$and to explore how different understandings of it can be faithful to Frege’s views about such definitions and to his foundational program. After rehearsing the essential elements of the relevant debate (§1), we appropriately distinguish$AC$from$FC$(§2). We discuss six rationales which may motivate the adoption of different instances of$AC$and$FC$(§3). We turn to the possible interpretations of$FC$(§4), and advance a Semantic$FC$(§4.1), arguing that while it suits Frege’s definition of natural numbers (4.1.1), it cannot reasonably be imposed on definitions of real numbers (§4.1.2), for reasons only partly similar to those offered by Crispin Wright (§4.1.3). We then rehearse a recent exchange between Bob Hale and Vadim Batitzky to shed light on Frege’s conception of real numbers and magnitudes (§4.2). We argue that an Architectonic version of$FC$is indeed faithful to Frege’s definition of real numbers, and compatible with his views on natural ones. Finally, we consider how attributing different instances of$FC$to Frege and appreciating the role of the Architectonic$FC$can provide a more perspicuous understanding of his foundational program, by questioning common pictures of his logicism (§5). (shrink)

In this paper, I critically discuss Frege’s philosophy of geometry with special emphasis on his position in The Foundations of Arithmetic of 1884. In Sect. 2, I argue that that what Frege calls faculty of intuition in his dissertation is probably meant to refer to a capacity of visualizing geometrical configurations structurally in a way which is essentially the same for most Western educated human beings. I further suggest that according to his Habilitationsschrift it is through spatial intuition that we (...) come to know the axioms of Euclidean geometry. In Sect. 3, I argue that Frege seems right in claiming in The Foundations, §14 that the synthetic nature of the Euclidean axioms follows from the fact that they are independent of one another and of the primitive laws of logic. If the former were dependent on the latter, they would be analytic in Frege’s sense of analyticity. But then they would not be independent of one another and due to their mutual provability would lose their status as axioms of Euclidean geometry, since according to Frege an axiom of a theory T is per definitionen unprovable in T. I further argue that only by invoking pure spatial intuition can Frege “explain” the epistemological status of the axioms of Euclidean geometry completely: their synthetic a priori nature. Finally, I argue that his view about independence in The Foundations, §14 seems to clash with his conception of independence in his mature period. In Sect. 4, I scrutinize Frege’s somewhat vague, but unduly neglected remarks in The Foundations, §26 on space, spatial intuition and the axioms of Euclidean geometry. I argue that for the sake of coherence Frege should have said unambiguously that space is objective, that it is independedent not only of our spatial intuition, but of our mental life altogether including our judgements about space, instead of encouraging the possible conjecture that in his view it contains an objective and a subjective component. I further argue that for Frege the objectivity of both space and the axioms of Euclidean geometry manifests itself in our universal and compulsory acknowledgement of the Euclidean axioms as true. I conclude Sect. 4 by arguing that there is a conflict between the subjectivity of our spatial intuitions as stressed in The Foundations, §26 and Frege’s thesis in his dissertation that the axioms of Euclidean geometry derive their validity from the nature of our faculty of intuition. To resolve this conflict, I propose that in the light of his avowed realism in The Foundations Frege could have replaced his early thesis by saying that although we come to know the axioms of Euclidean geometry through spatial intuition and are justified in acknowledging them as true on the basis of geometrical intuition, their truth is independent not only of the nature of our faculty of intuition and singular acts of intuition, but of our mental processes and activities in general, including the inner mental act of judging. In Sect. 5, I argue that Frege most likely did not adopt Kant’s method of acquiring geometrical knowledge via the ostensive construction of concepts in spatial intuition. In contrast to Kant, Frege holds that the axioms of three-dimensional Euclidean geometry express state of affairs about space obtaining independently of our spatial intuition. In Sect. 6, I conclude with a summarized assessment of Frege’s philosophy of geometry. (shrink)

Foundational projects disagree on whether pure and applied mathematics should be explained together. Proponents of unified accounts like neologicists defend Frege’s Constraint (FC), a principle demanding that an explanation of applicability be provided by mathematical definitions. I reconsider the philosophical import of FC, arguing that usual conceptions are biased by ontological assumptions. I explore more reasonable weaker variants — Moderate and Modest FC — arguing against common opinion that ante rem structuralism (and other) views can meet them. I dispel doubts (...) that such constraints are ‘toothless’, showing they both assuage Frege’s original concerns and accommodate neo-logicist intents by dismissing ‘arrogant’ definitions. (shrink)

Frege writes in Numbers and Arithmetic about kindergarten-numbers and “an a priori mode of cognition” that they may have “a geometrical source.” This resembles recent findings on arithmetical cognition. In my paper, I explore this resemblance between Gottlob Frege’s later position concerning the geometrical source of arithmetical knowledge, and some current positions in the literature dedicated to arithmetical cognition, especially that of Stanislas Dehaene. In my analysis, I shall try to mainly see to what extent logicism is compatible with intuitionism.

This article offers an interpretation of a controversial aspect of Frege’s The Foundations of Arithmetic, the so-called Julius Caesar problem. Frege raises the Caesar problem against proposed purely logical definitions for ‘0’, ‘successor’, and ‘number’, and also against a proposed definition for ‘direction’ as applied to lines in geometry. Dummett and other interpreters have seen in Frege’s criticism a demanding requirement on such definitions, often put by saying that such definitions must provide a criterion of identity of a certain kind. (...) These interpretations are criticized and an alternative interpretation is defended. The Caesar problem is that the proposed definitions fail to well-define ‘number’ and ‘direction’. That is, the proposed definitions, even when taken together with the extra-definitional facts, fail to fix unique semantic extensions for ‘number’ and ‘direction’, and thereby fail to fix unique truth-values for sentences like ‘Caesar is a number’ and ‘England is a direction’. A minor modification of the criticized definitions well-defines ‘0’, ‘successor’ and ‘number’, thereby avoiding the Caesar problem as Frege understands it, but without providing any criterion of number identity in the usual sense. (shrink)

This paper deals with Frege’s early critique of formalism in the philosophy of mathematics. Frege opposes meaningful arithmetic, according to which arithmetical formulas express a sense and arithmetical rules are grounded in the reference of the signs, to formal arithmetic, exemplified in particular by J. Thomae, whose “formal standpoint”, according to Frege, is that arithmetic should be understood as a manipulation of meaningless figures. However, Frege’s discussion of Thomae’s analogy between arithmetic and chess shows that Frege does not understand his (...) main point, which is that we must distinguish conceptually between statements about chess figures, and the chess pieces they represent. Indeed, Thomae’s fruitful use of this comparison undermines the ontological conception of arithmetic represented by Frege. The chess pieces do not go proxy for anything, but playing chess is not for that reason just manipulation of physical chess figures. This is also Wittgenstein’s point when he criticizes Frege for not seeing the “justified side of formalism”. Thomae’s formalism can be clarified by using the distinction between sign and symbol. Symbols are never meaningless or empty forms or signs, since they have a role or function in a symbolism, which one may call their meaning. The discussion of the chess analogy shows that Frege fails to see this operative aspect of the symbolism. We can conclude that the “formal standpoint” that Frege criticizes is his fabrication, rather than anything we can attribute to Thomae. (shrink)

Resumo: o propósito desse artigo é circunscrever e discutir a autocrítica, feita por Wittgenstein no período de 1933-39, a uma das teses mais fundamentais do Tractatus (TLP): ”Há uma e apenas uma análise completa da proposição” (3.25). Chamaremos esse procedimento peculiar de a “Grande análise”. Os argumentos de Wittgenstein contra a sustentabilidade da sua antiga tese podem ser encontrados em algumas passagens do livro Investigações Filosóficas (IF), bem como em passagens do Grande Datiloscrito (BT). Conforme será argumentado nesse artigo, essa (...) autocrítica é uma consequência direta da preocupação de Wittgenstein, no período intermediário, com duas importantes propostas feitas por Frege nos §§18-24 do Os Fundamentos da Aritmética (FA) em relação o seu entendimento da noção de “atribuição numérica”. Nosso objetivo principal será mostrar de que maneira o movimento de autocrítica feito por Wittgenstein em relação a sua antiga concepção de “Grande Análise” teria sido influenciado por suas preocupações, no período intermediário de sua obra, com essas duas propostas feita por Frege no FA a respeito da noção de “atribuição numérica”. A fim de completar tal tarefa, será apresentada inicialmente de modo breve uma versão de como seria a “Grande análise” no TLP. Em seguida, selecionaremos e discutiremos algumas passagens do IF nas quais Wittgenstein recusa a completude e unicidade envolvida em sua prévia concepção tratariana de “análise”. Procuraremos destacar com clareza uma noção que estaria envolvida de um modo essencial na autocrítica de Wittgenstein: a noção de “ver como/ver aspectos”. Como será argumentado, essa noção, assim como a autocrítica feita por Wittgenstein, foram ambas diretamente influenciadas pela noção fregiana de “atribuição numérica”. Para completar essa segunda tarefa, teremos de revisitar e discutir algumas noções pertencentes ao período inicial da filosofia de Wittgenstein -- como a de “propriedades internas e externas” e a distinção entre “dizer e mostrar” -- ambas noções discutidas no TLP. No decorrer de nossa discussão, apresentaremos uma crítica feita por Wittgenstein a Russell, mas também a Frege, sobre a concepção de generalidade, ou mais especificamente, sobre o uso do quantificador existencial, proposta por ambos. Abstract: The purpose of this article is to discuss Wittgenstein 1933-39 self-criticism of one of the most fundamental tractarian theses: There is one and only one complete analysis of the proposition (3.25). Wittgenstein's arguments against his previous thesis can be found in some passages of the Philosophical Investigation (PI) as well as in the Big Typescript (BT). As we shall argue, Wittgenstein's self-critique is directly related to two important ideas proposed by Frege in §§18-24 of The Foundations Arithmetic (FA) with respect to his construal of “number attributions”. Our main goal in this paper will therefore be to show how Wittgenstein's self-criticism of his old conception of the tractarian “analysis” would have been influenced by his concerns in the middle period of his work with these two proposals made by Frege in FA regarding the notion of “numerical attribution”. Our first task will then be to connect Wittgenstein's self-criticisms to his former conception of “analysis” in the Tractatus Logico-Philosophicus (TLP). To accomplish this first goal, we will have to briefly present a version of how such analysis would look like in TLP. Our next task will be to pinpoint some passages of the PI in which Wittgenstein rejects the completeness and uniqueness involved in his previous construal of the analytical process. We will try to circumscribe clearly a crucial notion involved in Wittgenstein's self-critique: the idea of “seeing as/seeing aspects”. Our claim is that this notion, as well as Wittgenstein's self-criticism, were both directly influenced by Frege's construal of the “number attribution” presented in FA. To accomplish this second task, we will have to revisit and discuss some further concepts from an early period of Wittgenstein's philosophy, like “internal and internal properties” and the “saying/showing distinction” in TLP. On our way towards accomplishing this goal, we will present a curious complaint by Wittgenstein regarding Russell's, as well as Frege's conception of generality, or more specifically, on the usage of the existential quantifier suggested by both these authors. (shrink)