Frege says, at the end of a discussion of formalism in the Foundations of Arithmetic, that his own foundational program “could be called formal” but is “completely different” from the view he has just criticized. This essay examines Frege’s relationship to Hermann Hankel, his main formalist interlocutor in the Foundations, in order to make sense of these claims. The investigation reveals a surprising result: Frege’s foundational program actually has quite a lot in common with Hankel’s. This undercuts Frege’s claim that (...) his own view is completely different from Hankel’s formalism, and motivates a closer examination of where the differences lie. On the interpretation offered here, Frege shares important parts of the formalist perspective, but differs in recognizing a kind of content for arithmetical terms which can only be made available via proof from prior postulates. (shrink)

The human attempts to access, measure and organize physical phenomena have led to a manifold construction of mathematical and physical spaces. We will survey the evolution of geometries from Euclid to the Algebraic Geometry of the 20th century. The role of Persian/Arabic Algebra in this transition and its Western symbolic development is emphasized. In this relation, we will also discuss changes in the ontological attitudes toward mathematics and its applications. Historically, the encounter of geometric and algebraic perspectives enriched the mathematical (...) practices and their foundations. Yet, the collapse of Euclidean certitudes, of over 2300 years, and the crisis in the mathematical analysis of the 19th century, led to the exclusion of “geometric judgments” from the foundations of Mathematics. After the success and the limits of the logico-formal analysis, it is necessary to broaden our foundational tools and re-examine the interactions with natural sciences. In particular, the way the geometric and algebraic approaches organize knowledge is analyzed as a cross-disciplinary and cross-cultural issue and will be examined in Mathematical Physics and Biology. We finally discuss how the current notions of mathematical (phase) “space” should be revisited for the purposes of life sciences. (shrink)

I investigate the role of geometric intuition in Frege’s early mathematical works and the significance of his view of the role of intuition in geometry to properly understanding the aims of his logicist project. I critically evaluate the interpretations of Mark Wilson, Jamie Tappenden, and Michael Dummett. The final analysis that I provide clarifies the relationship of Frege’s restricted logicist project to dominant trends in German mathematical research, in particular to Weierstrassian arithmetization and to the Riemannian conceptual/geometrical tradition at Göttingen. (...) Concurring with Tappenden, I hold that Frege’s logicism should not be understood as a continuing a project of reductionist arithmetization. However, Frege does not quite take up the Riemannian banner either. His logicism supports a hierarchical understanding of the structure of mathematical knowledge, according to which arithmetic is applicable to geometry but not vice versa because the former is more general, as revealed by the strictly logical nature of its objects in comparison to the intuitional nature of geometric objects. I suggest, in particular, that Frege intended that foundational work would show the use of geometric intuition in complex analysis, a source of error for Riemann that Weierstrass was proud to have uncovered, to be inessential. (shrink)

It is widely believed that some puzzling and provocative remarks that Frege makes in his late writings indicate he rejected independence arguments in geometry, particularly arguments for the independence of the parallels axiom. I show that this is mistaken: Frege distinguished two approaches to independence arguments and his puzzling remarks apply only to one of them. Not only did Frege not reject independence arguments across the board, but also he had an interesting positive proposal about the logical structure of correct (...) independence arguments, deriving from the geometrical principle of duality and the associated idea of substitution invariance. The discussion also serves as a useful focal point for independently interesting details of Frege's mathematical environment. This feeds into a currently active scholarly debate because Frege's supposed attitude to independence arguments has been taken to support a widely accepted thesis (proposed by Ricketts among others) concerning Frege's attitude toward metatheory in general. I show that this thesis gains no support from Frege's puzzling remarks about independence arguments. (shrink)

Richard Tieszen [Tieszen, R. (2005). Philosophy and Phenomenological Research, LXX(1), 153–173.] has argued that the group-theoretical approach to modern geometry can be seen as a realization of Edmund Husserl’s view of eidetic intuition. In support of Tieszen’s claim, the present article discusses Husserl’s approach to geometry in 1886–1902. Husserl’s first detailed discussion of the concept of group and invariants under transformations takes place in his notes on Hilbert’s Memoir Ueber die Grundlagen der Geometrie that Hilbert wrote during the winter 1901–1902. (...) Husserl’s interest in the Memoir is a continuation of his long-standing concern about analytic geometry and in particular Riemann and Helmholtz’s approach to geometry. Husserl favored a non-metrical approach to geometry; thus the topological nature of Hilbert’s Memoir must have been intriguing to him. The task of phenomenology is to describe the givenness of this logos, hence Husserl needed to develop the notion of eidetic intuition. (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)

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.

This paper argues that Frege's notoriously long commitment to Kant's thesis that Euclidean geometry is synthetic _a priori_ is best explained by realizing that Frege uses ‘intuition’ in two senses. Frege sometimes adopts the usage presented in Hermann Helmholtz's sign theory of perception. However, when using ‘intuition’ to denote the source of geometric knowledge, he is appealing to Hermann Cohen's use of Kantian terminology. We will see that Cohen reinterpreted Kantian notions, stripping them of any psychological connotation. Cohen's defense of (...) his modified Kantian thesis on the unique status of the Euclidean axioms presents Frege's own views in a much more favorable light. (shrink)

In 1902 there arrived in Jena a letter from Russell laying out a proof that shattered Frege’s confidence in logicism, which is widely taken to be the doctrine according to which every truth of arithmetic is re-expressible without relevant loss as a provable truth about a purely logical object. Frege was persuaded that Russell had exposed a pathology in logicism, which faced him with the task of examining its symptoms, diagnosing its cause, assessing its seriousness, arriving at a treatment option, (...) and making an estimate of future prospects. The symptom was the contradiction that had crept into naïve set theory in the form of the set that provably is and is not its own member. The diagnosis was that it is caused by Basic Law V of the Grundgesetze (Frege, In: Grundgesetze der Arithmetik: Begrifsschriftlich abgeleitet, Jena: Herman Pohle, 1893/1903. Translated into English as Basic Laws of Arithmetic, also edited by Philip A. Ebert and Marcus Rossberg, with Crispin Wright, Oxford: Oxford University Press, 2013). Triage answers the question, “How bad is it?” The answer was that the contradiction irreparably destroys the logicist project. The treatment option was nil. The disease was untreatable. In due course, the prognosis turned out to be that a scaled-down Fregean logic could have an honourable life as a theory inference for various domains of mathematical discourse, but not for domains containing the logical objects required for logicism. Since there aren’t such objects, there aren’t such domains. On the face of it, Frege’s logicist collapse is astonishing. Why wouldn’t he have repaired the fault in Law V and gone back to the business of bringing logicism to an assured realization? In the course of our reflections, we will have nice occasion to consider the good it might have done Frege to have booked some time with Aristotle had he been able to. By the time we’re finished, we’ll have cause to think that in the end the Russell might well have begged the question against Frege. (shrink)

The issues surrounding the Caesar problem are assumed to be inert as far as ongoing mathematics is concerned. This paper aims to correct this impression by spelling out the ways that, in their historical context, Frege's remarks would have had considerable resonance with work that other mathematicians such as Riemann and Dedekind were doing. The search for presentation‐independent characterizations of objects and global definitions was seen as bound up with fundamental methodological questions in complex analysis and number theory.

This paper is about Poincaré’s view of the foundations of geometry. According to the established view, which has been inherited from the logical positivists, Poincaré, like Hilbert, held that axioms in geometry are schemata that provide implicit definitions of geometric terms, a view he expresses by stating that the axioms of geometry are “definitions in disguise.” I argue that this view does not accord well with Poincaré’s core commitment in the philosophy of geometry: the view that geometry is the study (...) of groups of operations. In place of the established view I offer a revised view, according to which Poincaré held that axioms in geometry are in fact assertions about invariants of groups. Groups, as forms of the understanding, are prior in conception to the objects of geometry and afford the proper definition of those objects, according to Poincaré. Poincaré’s view therefore contrasts sharply with Kant’s foundation of geometry in a unique form of sensibility. According to my interpretation, axioms are not definitions in disguise because they themselves implicitly define their terms, but rather because they disguise the definitions which imply them. (shrink)

Setting out to understand Frege, the scholar confronts a roadblock at the outset: We just have little to go on. Much of the unpublished work and correspondence is lost, probably forever. Even the most basic task of imagining Frege's intellectual life is a challenge. The people he studied with and those he spent daily time with are little known to historians of philosophy and logic. To be sure, this makes it hard to answer broad questions like: 'Who influenced Frege?' But (...) the information vacuum also creates local problems of textual interpretation. Say we encounter a sentence that may be read as alluding to a scientific dispute. Should it be read that way? To answer, we'd need to address prior questions. Is it .. (shrink)

The paper discusses this year’s Nobel Prize in physics for experiments of entanglement “establishing the violation of Bell inequalities and pioneering quantum information science” in a much wider, including philosophical context legitimizing by the authority of the Nobel Prize a new scientific area out of “classical” quantum mechanics relevant to Pauli’s “particle” paradigm of energy conservation and thus to the Standard model obeying it. One justifies the eventual future theory of quantum gravitation as belonging to the newly established quantum information (...) science. Entanglement, involving non-Hermitian operators for its rigorous description, non-unitarity as well as nonlocal and superluminal physical signals “spookily” (by Einstein’s flowery epithet) synchronizing and transferring some nonzero action at a distance, can be considered to be quantum gravity so that its local counterpart to be Einstein’s gravitation according to general relativity therefore pioneering an alternative pathway to quantum gravitation different from the “secondary quantization” of the Standard model. So, the experiments of entanglement once they have been awarded by the Nobel Prize launch particularly the relevant theory of quantum gravitation grounded on “quantum information science” thus granted to be nonclassical quantum mechanics in the shared framework of the generalized quantum mechanics obeying rather quantum-information conservation than only energy conservation. The concept of “dark phase” of the universe naturally linked to the very well confirmed “dark matter” and “dark energy” and opposed to its “light phase” inherent to classical quantum mechanics and the Standard model obeys quantum-information conservation, after which reversible causality or the mutual transformation of energy and information are valid. The mythical Big Bang after which energy conservation holds universally is to be replaced by an omnipresent and omnitemporal medium of decoherence of the dark and nonlocal phase into the light and local phase. The former is only an integral image of the latter and borrowed in fact rather from religion than from science. Physical, methodological and proper philosophical conclusions follow from that paradigm shift heralded by this year’s Nobel Prize in physics. For example, the scientific theory of thinking should originate from the dark phase of the universe, as well: probably only approximately modeled by neural networks physically belonging to the light phase thoroughly. A few crucial philosophical sequences follow from the break of Pauli’s paradigm: (1) the establishment of the “dark” phase of the universe as opposed to its “light” phase, only to which the Cartesian dichotomy of “body” and “mind” is valid; (2) quantum information conservation as relevant to the dark phase, furthermore generalizing energy conservation as to its light phase, productively allowing for physical entities to appear “ex nihilo”, i.e., from the dark phase, in which energy and time are yet inseparable from each other; (3) reversible causality as inherent to the dark phase; (4) the interpretation of gravitation only mathematically: as an interpretation of the incompleteness of finiteness to infinity, for example, following the Gödel dichotomy (“either contradiction or incompleteness”) about the relation of arithmetic to set theory; (5) the restriction of the concept of hierarchy only to the light phase; (6) the commensurability of both physical extremes of a quantum and the universe as a whole in the dark phase obeying quantum information conservation and akin to Nicholas of Cusa’s philosophical and theological worldview. (shrink)

This paper aims to show that Frege’s and Hilbert’s mutual disagreement results from different notions of Anschauung and their relation to axioms. In the first section of the paper, evidence is provided to support that Frege and Hilbert were influenced by the same developments of 19th-century geometry, in particular the work of Gauss, Plücker, and von Staudt. The second section of the paper shows that Frege and Hilbert take different approaches to deal with the problems that the developments in 19th-century (...) geometry posed for the traditional Kantian philosophy of mathematics. Frege maintains that Anschauung is a source of knowledge by which we acknowledge geometrical axioms as true. For Hilbert, in contrast, axioms provide one of several correct “pictures” of reality. Hilbert’s position is thereby deeply influenced by epistemological ideas from Hertz and Helmholtz, and, in turn, influenced the neo-Kantian Cassirer. (shrink)

The essays collected in this volume address such questions from different points of view and will interest students and scholars in several branches of scientific knowledge.

Frege's main contributions to logic and the philosophy of mathematics are, on the one hand, his introduction of modern relational and quantificational logic and, on the other, his analysis of the concept of number. My focus in this paper will be on the latter, although the two are closely related, of course, in ways that will also play a role. More specifically, I will discuss Frege's logicist reconceptualization of the natural numbers with the goal of clarifying two aspects: the motivations (...) for its core ideas; the step-by-step development of these ideas, from Begriffsschrift through Die Grundlagen der Arithmetik and Grundgesetze der Arithmetik to Frege's very last writings, indeed even beyond those, to a number of recent "neo-Fregean" proposals for how to update them. One main development, or break, in Frege's views occurred after he was informed of Russell's antinomy. His attempt to come to terms with this antinomy has found some attention in the literature already. It has seldom been analyzed in connection with earlier changes in his views, however, partly because those changes themselves have been largely ignored. Nor has it been discussed much in connection with Frege's basic motivations, as formed in reaction to earlier positions. Doing both in this paper will not only shed new light on his response to Russell's antinomy, but also on other aspects of his views. In addition, it will provide us with a framework for comparing recent updates of these views, thus for assessing the remaining attraction of Frege's general approach. I will proceed as follows: In the first part of the paper (§1.1 and §1.2), I will consider the relationship of Frege's conception of the natural numbers to earlier conceptions, in particular to what I will call the "pluralities conception", thus bringing into sharper focus his core ideas and their motivations. In the next part (§2.1 and §2.2), I will trace the order in which these ideas come up in Frege's writings, as well as the ways in which his position gets modified along the way, both before and after Russell's antinomy.. (shrink)

Dans cet article, je considère la pratique et la conception de la ri­gueur chez Richard Dedekind qui se dégagent de l’étude d’une sélection de ses travaux les plus importants. Une analyse des mentions multiples de réquisits de rigueur dans les textes de Dedekind amène à constater qu’il lie très étroi­tement la rigueur à la généralité. La première partie de l’article donne à voir les liens serrés tissés par Dedekind entre généralité et rigueur, dans sa théorie des fonctions algébriques co-écrite avec (...) H. Weber, ainsi que dans ses travaux fondationnels et dans ses travaux de théorie des nombres. Dans la seconde partie, j’examine les critères de rigueur qui apparaissent dans la pratique ma­thématique de Dedekind. Je discute l’idéal logique de rigueur dans l’essai de Dedekind sur les entiers naturels, étudié par M. Detlefsen sous l’appellation « Dedekind’s principle » ; puis je m’intéresse à la stratégie de Dedekind pour arithmétiser les mathématiques afin de mettre en évidence qu’il ne s’agit pas d’une approche guidée par un principe purement logique. Ainsi, l’idéal logique de rigueur apparaît comme intimement lié à la pratique d’une autre norme épistémique : la généralité, en lien avec la quête, par Dedekind, de définitions et preuves générales. Dans la dernière partie, j’analyse la demande de généra­lité et la pluralité de conceptions de la généralité que recouvre cette demande, et termine en mettant en avant la relation des définitions aux preuves et de quelle manière une définition générale se pose en condition de rigueur dans les mathématiques dedekindiennes. (shrink)

This article is about Avicenna’s account of syllogisms comprising opposite premises. We examine the applications and the truth conditions of these syllogisms. Finally, we discuss the relation between these syllogisms and the principle of non-contradiction.

Analytic philosophy has been the most influential philosophical movement in 20th century philosophy. It has surely contributed like no other movement to the elucidation and demarcation of philosophical problems. Nonetheless, the empiricist and sometimes even nominalist convictions of orthodox analytic philosophers have served them to inadequately render even philosophers they consider their own and to propound very questionable conceptions.

Logical universalism, a label that has been pinned on to Frege, involves the conflation of two features commonly ascribed to logic: universality and radicality. Logical universality consists in logic being about absolutely everything. Logical radicality, on the other hand, corresponds to there being the one and the same logic that any reasoning must comply with. The first part of this paper quickly remarks that Frege’s conception of logic makes logical universality prevail and does not preclude the admission of different contexts (...) of discourse. The paper then aims to make it clear how Frege’s universalism can make sense of contextuality. Drawing on a suggestion made by Frege in his discussion of Hilbert, it shows that a properly Fregean notion of model can be devised. Taking up a suggestion from Wilfrid Hodges and William Demopoulos that the non-logical constants of a formal language can be compared to indexicals, this paper shows,paceHodges and Demopoulos, that such an understanding of non-logical constants is not beyond Frege’s horizon. A formal framework, based on the modern tool of fibrations, is set out to explain and justify this point. This framework allows one to compare Frege and Tarski, by formalizing Frege’s suggestion and by presenting Tarski’s semantics in a generalized setting. (shrink)

Frege's docent's dissertation Rechnungsmethoden, die sich auf eine Erweiterung des Grössenbegriffes gründen(1874) contains indications of a bold attempt to extend arithmetic. According to it, arithmetic means the science of magnitude, and magnitude must be understood structurally without intuitive support. The main thing is insight into the formal structure of the operation of ?addition?. It turns out that a general ?magnitude domain? coincides with a (commutative) group. This is an interesting connection with simultaneous developments in abstract algebra. As his main application, (...) Frege studies iterations of functions. He does not yet pose the question of existence proofs. Measurement of magnitudes is also connected to numbers, but the discussion is here ambiguous in a way which calls for the systematic account of numbers in Grundgesetze. (shrink)