History: 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)

This comprehensive historical account concerns that non-void intersection region between Riemann zeta function and entire function theory, with a view towards possible physical applications.

In the late 1940s and early 1950s Lorenzen developed his operative logic and mathematics, a form of constructive mathematics. Nowadays this is mostly seen as the precursor to the more well-known dialogical logic and one could assumed that the same philosophical motivations were present in both works. However we want to show that this is not always the case. In particular, we claim, that Lorenzen’s well-known rejection of the actual infinite as stated in Lorenzen (1957) was not a major motivation (...) for operative logic and mathematics. In this article, we claim that this is in fact not the case. Rather, we argue for a shift that happened in Lorenzen’s treatment of the infinite from the early to the late 1950s. His early motivation for the development of operativism is concerned with a critique of the Cantorian notion of set and related questions about the notion of countability and uncountability; only later, his motivation switches to focusing on the concept of infinity and the debate about actual and potential infinity. (shrink)

The reception of Newton's Principia in 1687 led to the attempt of many European scholars to ‘mathematicise' their field of expertise. An important example of this ‘mathematicisation' lies in the work of Irish-Scottish philosopher Francis Hutcheson, a key figure in the Scottish Enlightenment. This essay aims to discuss the mathematical aspects of Hutcheson's work and its impact on British thought in the following centuries, providing a case in point for the importance of the interactions between mathematics and philosophy throughout time.

Frege and Peano started in 1896 a debate where they contrasted the respective conceptions on the theory and practice of mathematical definitions. Which was (if any) the influence of the Frege-Peano debate on the conceptions by the two authors on the theme of defining in mathematics and which was the role played by this debate in the broader context of their scientific interaction?

Du Châtelet holds that mathematical representations play an explanatory role in natural science. Moreover, things proceed in nature as they do in geometry. How should we square these assertions with Du Châtelet’s idealism about mathematical objects, on which they are ‘fictions’ dependent on acts of abstraction? The question is especially pressing because some of her important interlocutors (Wolff, Maupertuis, and Voltaire) denied that mathematics informs us about the properties of real things. After situating Du Châtelet in this debate, this chapter (...) argues, first, that her account of the origins of mathematical objects is less subjectivist than it might seem. Mathematical objects are non-arbitrary, public entities. While mathematical objects are partly mind-dependent, so too are material things. Mathematical objects can approximate the material. Second, it is argued that this moderate metaphysical position underlies Du Châtelet’s persistent claims that mathematics is required for certain kinds of general knowledge, including in natural science. The chapter concludes with an illustrative example: an analysis of Du Châtelet’s argument that matter is continuous. A key but overlooked premise in the argument is that mathematical representations and material nature correspond. (shrink)

I begin by outlining Du Châtelet’s ontology of mathematical objects: she is an idealist, and mathematical objects are fictions dependent on acts of abstraction. Next, I consider how this idealism can be reconciled with her endorsement of necessary truths in mathematics, which are grounded in essences that we do not create. Finally, I discuss how mathematics and physics relate within Du Châtelet’s idealism. Because the primary objects of physics are partly grounded in the same kinds of acts as yield mathematical (...) objects, she thinks we are sometimes licensed in drawing conclusions about physical things from mathematical premises. (shrink)

Claudius Ptolemy’s mathematical astronomy originated in Alexandria in Egypt under Roman rule in the second century CE and held for more than a millennium, even beyond the Copernican theories (sixteenth century). To trace the flourishing of such mathematical creativity requires an understanding of Ptolemy’s philosophy of mathematical practice, the ancient commentators of Ptolemaic works, and the historical context of Alexandria in Egypt, a multicultural city which became a cradle of cultures of mathematical practices and blossomed into the Ptolemaic system and (...) its several related outputs between the second and fourth centuries CE. Ptolemy’s mathematical practice will be explored under three lenses: (a) an examination of the mathematical prose of Ptolemy’s Almagest alongside the corresponding passages in its Alexandrian commentaries; (b) the social, material, and semantic layers of Ptolemy’s mathematics; and (c) the conception of true philosophy in Ptolemy’s theory of knowledge. Examining the history and philosophy of Ptolemy’s mathematical practice in Alexandria (1) shows that Ptolemy’s ideals were usually betrayed by the practitioners of his mathematics and (2) proves that its main aspect is constituted by mathematical knowledge arranged in tabular format, that is the so-called astronomical tables, to which Ptolemy’s mathematics owes its successful legacy. (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)

From the early eighteenth century onward, primacy for the invention of binary numeration and arithmetic was almost universally credited to the German polymath Gottfried Wilhelm Leibniz (1646–1716). Then, in 1922, Frank Vigor Morley (1899–1980) noted that an unpublished manuscript of the English mathematician, astronomer, and alchemist Thomas Harriot (1560–1621) contained the numbers 1 to 8 in binary. Morley’s only comment was that this foray into binary was “certainly prior to the usual dates given for binary numeration”. Almost thirty years later, (...) John William Shirley (1908–1988) published reproductions of two of Harriot’s undated manuscript pages, which, he claimed, showed that Harriot had invented binary numeration “nearly a century before Leibniz’s time”. But while Shirley correctly asserted that Harriot had invented binary numeration, he made no attempt to explain how or when Harriot had done so. Curiously, few since Shirley’s time have attempted to answer these questions, despite their obvious importance. After all, Harriot was, as far as we know, the first to invent binary. Accordingly, answering the how and when questions about Harriot’s invention of binary is the aim of this short paper. (shrink)

Cantor’s abstractionist account of cardinal numbers has been criticized by Frege as a psychological theory of numbers which leads to contradiction. The aim of the paper is to meet these objections by proposing a reassessment of Cantor’s proposal based upon the set theoretic framework of Bourbaki—called BK—which is a First-order set theory extended with Hilbert’s \-operator. Moreover, it is argued that the BK system and the \-operator provide a faithful reconstruction of Cantor’s insights on cardinal numbers. I will introduce first (...) the axiomatic setting of BK and the definition of cardinal numbers by means of the \-operator. Then, after presenting Cantor’s abstractionist theory, I will point out two assumptions concerning the definition of cardinal numbers that are deeply rooted in Cantor’s work. I will claim that these assumptions are supported as well by the BK definition of cardinal numbers, which will be compared to those of Zermelo–von Neumann and Frege–Russell. On the basis of these similarities, I will make use of the BK framework in meeting Frege’s objections to Cantor’s proposal. A key ingredient in the defence of Cantorian abstraction will be played by the role of representative sets, which are arbitrarily denoted by the \-operator in the BK definition of cardinal numbers. (shrink)

[Accepted for publication in Lakatos's Undone Work: The Practical Turn and the Division of Philosophy of Mathematics and Philosophy of Science, special issue of Kriterion: Journal of Philosophy. Edited by S. Nagler, H. Pilin, and D. Sarikaya.] Lakatos’s analysis of progress and degeneration in the Methodology of Scientific Research Programmes is well-known. Less known, however, are his thoughts on degeneration in Proofs and Refutations. I propose and motivate two new criteria for degeneration based on the discussion in Proofs and Refutations (...) – superfluity and authoritarianism. I show how these criteria augment the account in Methodology of Scientific Research Programmes, providing a generalized Lakatosian account of progress and degeneration. I then apply this generalized account to a key transition point in the history of entropy – the transition to an information-theoretic interpretation of entropy – by assessing Jaynes’s 1957 paper on information theory and statistical mechanics. (shrink)

This paper aims to clarify Merleau-Ponty’s contribution to an embodied-enactive account of mathematical cognition. I first identify the main points of interest in the current discussions of embodied higher cognition and explain how they relate to Merleau-Ponty and his sources, in particular Husserl’s late works. Subsequently, I explain these convergences in greater detail by more specifically discussing the domains of geometry and algebra and by clarifying the role of gestalt psychology in Merleau-Ponty’s account. Beyond that, I explain how, for Merleau-Ponty, (...) mathematical cognition requires not only the presence and actual manipulation of some concrete perceptible symbols but, more strongly, how it is fundamentally linked to the structural transformation of the concrete configurations of symbolic systems to which these symbols appertain. Furthemore, I fill a gap in the literature by explaining Merleau-Ponty’s claim that these structural transformations are operated through motor intentionality. This makes it possible, in turn, to contrast Merleau-Ponty’s approach to ontologically idealistic and realistic views on mathematical objects. On Merleau-Ponty’s account, mathematical objects are relational entities, that is, gestalts that necessarily imply situated cognizers to whom they afford a specific type of engagement in the world and on whom they depend in their eventual structural transformations. I argue that, by attributing a strongly constitutive role to phenomenal configurations and their motor transformation in mathematical thinking, Merleau-Ponty contributes to clarifying the worldly, historical, and socio-cultural aspects of mathematical truths without compromising what we perceive as their universality, certainty, and necessity. (shrink)

1. INTRODUCTIONIt has been some time now since the philosophical community has learned to appreciate Husserl’s contribution to the philosophies of logic, mathematics, and science in general, despite still some prejudices and misinterpretations in certain academic circles incapable of reading Husserl beyond the incompetent and malicious review which Frege wrote in 1894 of his Philosophie der Arithmetik (PA) [1891/2003], hereafter Hua XII.Husserl’s philosophy of mathematics, in particular, has been the subject of many articles and books and has attracted the attention (...) of many interpreters and philosophers who have found in Husserl fruitful ideas with which to tackle the many philosophical problems elicited by mathematics — its ontology, its epistemology, and its somehow surprising utility, even when not indispensable, in modern science. This original and interesting work by Professor Hartimo is a welcome addition to this already respectable bibliography. The book is essentially exegetical and presents a reading of Husserl’s philosophy, particularly his philosophy of mathematics, as found in some of his most important published works, which were reactions to the development of mathematics in his time, of which he was, as the mathematician he used to be, an attentive and well-informed observer. (shrink)

The Hilbert program was actually a specific approach for proving consistency, a kind of constructive model theory. Quantifiers were supposed to be replaced by ε-terms. εxA(x) was supposed to denote a witness to ∃xA(x), or something arbitrary if there is none. The Hilbertians claimed that in any proof in a number-theoretic system S, each ε-term can be replaced by a numeral, making each line provable and true. This implies that S must not only be consistent, but also 1-consistent. Here we (...) show that if the result is supposed to be provable within S, a statement about all Pi-0-2 statements that subsumes itself within its own scope must be provable, yielding a contradiction. The result resembles Gödel's but arises naturally out of the Hilbert program itself. (shrink)

For several decades, Carnap's philosophy of mathematics used to be either dismissed or ignored. It was perceived as a form of linguistic conventionalism and thus taken to rely on the bankrupt notion of truth by convention. However, recent scholarship has revealed a more subtle picture. It has been forcefully argued that Carnap is not a linguistic conventionalist in any straightforward sense, and that supposedly decisive objections against his position target a straw man. This raises two questions. First, how exactly should (...) we characterise Carnap's actual philosophy of mathematics? Secondly, is his position an attractive alternative to established views? I will tackle these issues by looking at Carnap's response to the incompleteness theorems. Drawing on arguments put forward by Gödel and Beth, I show that some crucial aspects of Carnap's positive account have remained underdeveloped. Suggestions on what a full evaluation of Carnap's position requires are made. (shrink)

We give an overview of Lakatos’ life, his philosophy of mathematics and science, as well as of this issue. Firstly, we briefly delineate Lakatos’ key contributions to philosophy: his anti-formalist philosophy of mathematics, and his methodology of scientific research programmes in the philosophy of science. Secondly, we outline the themes and structure of the masterclass Lakatos’ Undone Work – The Practical Turn and the Division of Philosophy of Mathematics and Philosophy of Science, which gave rise to this special issue. Lastly, (...) we provide a summary of the contributions to this issue. (shrink)

The aim of this paper is twofold: (1) to show the principal aspects of the way in which Newton conceived his mathematical concepts and methods and applied them to rational mechanics in his Principia; (2) to explain how the editors of the Geneva Edition interpreted, clarified, and made accessible to a broader public Newton’s perfect but often elliptic proofs. Following this line of inquiry, we will explain the successes of Newton’s mechanics, but also the problematic aspects of his perfect geometrical (...) methods, more elegant, but less malleable than analytical procedures, of which Newton himself was one of the inventors. Furthermore, we will also consider the way in which Newtonianism was spread before in England and afterwards on continental Europe. In this respect the Geneva Edition plays a fundamental role because of its complete apparatus of notes, and because it appeared only thirteen years after the publication of the third edition of the Principia (1726). Finally, we will also confront some problems connected to the metaphysics of calculus. Therefore, the case of Newton is one of those in which, starting from mathematics applied to physics, it is possible to connect an impressive series of fundamental arguments such as the role of mathematics in science; the comparison between Newton’s geometrical methods and analytical methods; the way in which Newtonianism was spread as well as the philosophical implications of Newton’s mathematical concepts. (shrink)

The Polish philosophy of mathematics in the 19th century had its origins in the Romantic period under the influence of the then-predominant idealist philosophies. The decline of Romantic philosophy precipitated changes in general philosophy, but what is less well known is how it triggered changes in the philosophy of mathematics. In this paper, we discuss how the Polish philosophy of mathematics evolved from the metaphysical approach that had been formed during the Romantic era to the more modern positivistic paradigm. These (...) evolutionary changes are attributed to the philosophers Henryk Struve, Antoni Molicki and Julian Ochorowicz, and mathematicians Karol Hertz and Samuel Dickstein. We also show how implicit ideas (i.e., those not declared openly) from the area between the philosophy of science and general philosophy played a crucial role in the paradigm shift in the Polish philosophy of mathematics. (shrink)

We provide a new interpretation of Zeno’s Paradox of Measure that begins by giving a substantive account, drawn from Aristotle’s text, of the fact that points lack magnitude. The main elements of this account are (1) the Axiom of Archimedes which states that there are no infinitesimal magnitudes, and (2) the principle that all assignments of magnitude, or lack thereof, must be grounded in the magnitude of line segments, the primary objects to which the notion of linear magnitude applies. Armed (...) with this account, we are ineluctably driven to introduce a highly constructive notion of (outer) measure based exclusively on the total magnitude of potentially infinite collections of line segments. The Paradox of Measure then consists in the proof that every finite or potentially infinite collection of points lacks magnitude with respect to this notion of measure. We observe that the Paradox of Measure, thus understood, troubled analysts into the 1880’s, despite their knowledge that the linear continuum is uncountable. The Paradox was ultimately resolved by Borel in his thesis of 1893, as a corollary to his celebrated result that every countable open cover of a closed line segment has a finite sub-cover, a result he later called the “First Fundamental Theorem of Measure Theory.” This achievement of Borel has not been sufficiently appreciated. We conclude with a metamathematical analysis of the resolution of the paradox made possible by recent results in reverse mathematics. (shrink)

The paper deals with Kuhn’s and Lakatos’s ideas related to the so-called “historical turn” and its application to the philosophy of mathematics. In the first part the meaning of the term “postpositivism” is specified. If we lack such a specification we can hardly discuss the philosophy of science that comes “after postpositivism”. With this end in view, the metaphor of “generations” in the philosophy of science is used. It is proposed that we restrict the use of the term “post-positivism” to (...) two and only two philosophical “generations”: the one to which Kuhn, Lakatos and Feyerabend belong, and the previous “generation” to which Wittgenstein, Polanyi, Popper and Quine belong. From this point of view, Bloor, Latour, Pickering, Daston and Galison belong to the “third generation” which represents the philosophy of science “after post-positivism”. The characteristic feature of post-positivism is the combination of decisive impact of logical positivism and its severe criticism. This combination inevitably makes post-positivism a transitional form in the philosophy of science. In the second part the contribution of the “big four” of post-positivist philosophers to the radical change in the philosophy of mathematics in the second half of the 20th century is analyzed. Primarily, they shifted philosophical interest from the logical analysis of formal systems to the historical dynamics of informal mathematics. They also reconsidered the sharp opposition between mathematics and the physical sciences. However, the transitional character of their philosophy manifests itself both in their treatment of mathematics and their way of understanding history. On the one hand, their “heritage” is ambiguous, on the other hand, it opens new perspectives. Neither Kuhn, nor Lakatos, have eliminated completely the methodological barrier positing the fundamental heterogeneity of mathematics and natural science. Neither Lakatos, nor Kuhn, adhered to the viewpoint of relentless historicism. Nevertheless, it is their work that has made these options open for today’s historians and philosophers of science, even for philosophers of mathematics. (shrink)

The paper examines the main points of Yankov’s hypothesis on the rise of Greek mathematics. The novelty of Yankov’s interpretation is that the rise of mathematics is examined within the context of the rise of ontological theories of the early Greek philosophers, which mark the beginning of rational thinking, as understood in the Western tradition.

Since the beginning of the twentieth century, prominent authors including Jean Piaget have drawn attention to Edmond Goblot’s account of mathematical thought experiments. But his contribution to today’s debate has been neglected so far. The main goal of this article is to reconstruct and discuss Goblot’s account of logical operations (the term he used for thought experiments in mathematics) and its interpretation by Piaget against the theoretical background of two open questions in today’s debate: (1) the relationship between empirical and (...) mathematical thought experiments and (2) the question of whether mathematical thought experiments can play a justificatory function in proofs. The main corollary of this analysis is that Piaget’s interpretation is seriously flawed and insufficiently appreciative of important theses of Goblot’s account. First, Goblot can be easily defended against Piaget’s main criticism, and second, Goblot developed ideas about mathematical thought experiments that still deserve attention. (shrink)

Since the beginning of the twentieth century, prominent authors including Jean Piaget have drawn attention to Edmond Goblot’s account of mathematical thought experiments. But his contribution to today’s debate has been neglected so far. The main goal of this article is to reconstruct and discuss Goblot’s account of logical operations (the term he used for thought experiments in mathematics) and its interpretation by Piaget against the theoretical background of two open questions in today’s debate: (1) the relationship between empirical and (...) mathematical thought experiments and (2) the question of whether mathematical thought experiments can play a justificatory function in proofs. The main corollary of this analysis is that Piaget’s interpretation is seriously flawed and insufficiently appreciative of important theses of Goblot’s account. First, Goblot can be easily defended against Piaget’s main criticism, and second, Goblot developed ideas about mathematical thought experiments that still deserve attention. (shrink)

La presente investigación tiene como propósito general asumir un posicionamiento filosófico en clave histórico-social y de tipo anti-relativista para analizar el desarrollo histórico de la matemática, el cual aplicaremos a un caso en particular: la matemática del antiguo Egipto. Para ello se discutirán y criticarán, en primera instancia, determinadas posiciones filosóficas afines al cuasi-empirismo en matemática que, siendo relativistas, permitirán delinear nuestro propio posicionamiento en contraste: la existencia de una «matemática situada». Esta categoría filosófica tendrá como sustento teórico la noción (...) de conocimiento situado. Además, las implicaciones de esta son, según sostenemos, tanto filosóficas como historiográficas, ya que servirá para analizar las características del corpus de problemas matemáticos egipcios registrados en los diversos papiros matemáticos. En particular, haremos referencia a las expresiones lingüísticas egipcias para denotar las diversas operaciones aritméticas y el carácter algorítmico de los problemas, así como también la cuestión epistemológica de la empiria y cómo la perspectiva situada permite superar la dicotomía entre una matemática pura y otra aplicada, que consideramos no es ubicua para abordar la interpretación de la práctica matemática del antiguo país del Nilo. Palabras clave: Filosofía de la matemática, Anti-relativismo, Matemática situada, Extrañeza del pasado, Matemática egipcia. (shrink)

Wittgenstein's paradoxical theses that unproved propositions are meaningless, proofs form new concepts and rules, and contradictions are of limited concern, led to a variety of interpretations, most of them centered on rule-following skepticism. We argue, with the help of C. S. Peirce's distinction between corollarial and theorematic proofs, that his intuitions are better explained by resistance to what we call conceptual omniscience, treating meaning as fixed content specified in advance. We interpret the distinction in the context of modern epistemic logic (...) and semantic information theory, and show how removing conceptual omniscience helps resolve Wittgenstein's paradoxes and explain the puzzle of deduction, its ability to generate new knowledge and meaning. (shrink)

Es común escuchar que el mundo Occidental debe a los árabes el descubrimiento del álgebra. No obstante, el desarrollo de esta disciplina puede interpretarse como un crisol de distintas tradiciones científicas que fue posible gracias a la clasificación, traducción y crítica tanto de los clásicos como de las obras que los árabes obtuvieron de los pueblos que conquistaron. Entre estos trabajos se encontraba Los Elementos de Euclides. Los Elementos fueron cuidadosamente traducidos durante el califato de Al-Ma’mūn por el matemático Mohammed (...) ibn-Musa Al-Khwārizmī, autor de Al-jabr wa’l muqābalah, quien sentó los fundamentos de la disciplina que más tarde sería conocida como álgebra y quien, en la primera parte de su obra, nos proporciona tres métodos para resolver tres tipos de ecuaciones que llama “ecuaciones combinadas”. A lo largo de este artículo se ofrecen argumentos para sostener que los métodos para resolver estas ecuaciones constituyen una reinterpretación, en el terreno algebraico, de los teoremas 6, 7 y 8 del segundo libro de Los Elementos de Euclides. Mi objetivo es dar respuesta a la siguiente pregunta: ¿Qué lectura de Los elementos Euclides posibilitó la emergencia del álgebra en el mundo árabe? Para responderla necesario explorar el acercamiento que los árabes tuvieron con el Libro II de Los Elementos, con el fin de proponer una interpretación de los posibles factores que los llevaron a formular el álgebra. Esto último, en particular, se encuentra en la primera parte de la obra de Al-Khwārizmī. Palabras clave: Álgebra, Al-Khwārizmī, Euclides, Ecuación, Teorema. (shrink)

The Polish philosophy of mathematics in the 19th century is not a well-researched topic. For this period, only five philosophers are usually mentioned, namely Jan Śniadecki, Józef Maria Hoene-Wroński, Henryk Struve, Samuel Dickstein, and Edward Stamm. This limited and incomplete perspective does not allow us to develop a well-balanced picture of the Polish philosophy of mathematics and gauge its influence on 19th- and 20th-century Polish philosophy in general. To somewhat complete our picture of the history of the Polish philosophy of (...) mathematics in those times, we here present the profiles of some lesser-known Polish Romantic philosophers of the 19th century, namely Karol Libelt, Bronisław Trentowski, and Józef Kremer. We discuss their contributions to the philosophy of mathematics and their metaphysical perspectives, and we also show how their metaphysical ideas have found some continuity in the studies of some Catholic philosophers. (shrink)

The design of good notation is a cause that was dear to Charles Babbage's heart throughout his career. He was convinced of the "immense power of signs" (1864, 364), both to rigorously express complex ideas and to facilitate the discovery of new ones. As a young man, he promoted the Leibnizian notation for the calculus in England, and later he developed a Mechanical Notation for designing his computational engines. In addition, he reflected on the principles that underlie the design of (...) good mathematical notations. In this paper, we discuss these reflections, which can be found somewhat scattered in Babbage's writings, for the first time in a systematic way. Babbage's desiderata for mathematical notations are presented as ten guidelines pertinent to notational design and its application to both individual symbols and complex expressions. To illustrate the applicability of these guidelines in non-mathematical domains, some aspects of his Mechanical Notation are also discussed. (shrink)

From ancient times to 19th century geometryGeometry symbolized the essence of mathematical thinking and method, but modern philosophy of mathematics seems to have marginalized the philosophical status of geometryGeometry. The roots of this transformation will be sought in the ascendance of logical foundations in place of intuitive primacy as the cornerstone of mathematical certainty in the late 19th century. Nevertheless, geometry and geometrical thinking, in multiple manifestations, have continued to occupy a central place in the practice of mathematics proper. We (...) argue that this, together with advances in the neuroscience of mathematical processes, calls for an expansion of the present limited remit of the philosophy of mathematics. (shrink)

This paper argues that Noether's axiomatic method in algebra cannot be assimilated to Weyl's late view on axiomatics, for his acquiescence to a phenomenological epistemology of correctness led Weyl to resist Noether's principle of detachment.

There is a tension in Emilie Du Châtelet’s thought on mathematics. The objects of mathematics are ideal or fictional entities; nevertheless, mathematics is presented as indispensable for an account of the physical world. After outlining Du Châtelet’s position, and showing how she departs from Christian Wolff’s pessimism about Newtonian mathematical physics, I show that the tension in her position is only apparent. Du Châtelet has a worked-out defense of the explanatory and epistemic need for mathematical objects, consistent with their metaphysical (...) nonfundamentality. I conclude by sketching how Du Châtelet’s conception of mathematical indispensability differs interestingly from many contemporary approaches. (shrink)

We compare the medieval projects of commentaries and disputations with the modern projects of formal ontology and of mathematics.

The article reconstructs Jean Cavaillès’s polemical engagement with Edmund Husserl’s phenomenological philosophy of mathematics. I argue that Cavaillès’s encounter with Husserl clarifies the scope and ambition of Cavaillès’s philosophy of the concept by identifying three interrelated epistemological problems in Husserl’s phenomenological method: (1) Cavaillès claims that Husserl denies a proper content to mathematics by reducing mathematics to logic. (2) This reduction obliges Husserl, in turn, to mischaracterize the significance of the history of mathematics for the philosophy of mathematics. (3) Finally, (...) Husserl’s phenomenology distorts the nature of mathematical experience. Accordingly, Cavaillès’s philosophy of mathematics is premised on a threefold affirmation designed to overcome these inadequacies. Cavaillès claims that mathematics is an autonomous field of conceptual production that cannot be reduced to logical, psychological, or phenomenological descriptions of conceptual genesis. Two important corollaries follow: the history of mathematics must guide the philosophy of mathematics, and mathematical experience itself must be described according to the mechanisms of a novel theory of mathematical abstraction. In what follows, I demonstrate that Cavaillès’s encounter with Husserl does not merely describe a polemical engagement with a rival philosophical position; it allows us to reconstruct Cavaillès’s own highly original contributions to the philosophy of mathematics. (shrink)

In Bertrand Russell's 1903 Principles of Mathematics, he offers an apparently devastating criticism of the neo-Kantian Hermann Cohen's Principle of the Infinitesimal Method and its History (PIM). Russell's criticism is motivated by his concern that Cohen's account of the foundations of calculus saddles mathematics with the paradoxes of the infinitesimal and continuum, and thus threatens the very idea of mathematical truth. This paper defends Cohen against that objection of Russell's, and argues that properly understood, Cohen's views of limits and infinitesimals (...) do not entail the paradoxes of the infinitesimal and continuum. Essential to that defense is an interpretation, developed in the paper, of Cohen's positions in the PIM as deeply rationalist. The interest in developing this interpretation is not just that it reveals how Cohen's views in the PIM avoid the paradoxes of the infinitesimal and continuum. It also reveals some of what is at stake, both historically and philosophically, in Russell's criticism of Cohen. (shrink)

Arnošt Kolman (1892–1979) was a Czech mathematician, philosopher and Communist official. In this paper, we would like to look at Kolman’s arguments against logical positivism which revolve around the notion of the fetishization of mathematics. Kolman derives his notion of fetishism from Marx’s conception of commodity fetishism. Kolman is aiming to show the fact that an entity (system, structure, logical construction) acquires besides its real existence another formal existence. Fetishism means the fantastic detachment of the physical characteristics of real things (...) or phenomena from these things. We identify Kolman’s two main arguments against logical positivism. In the first argument, Kolman applied Lenin’s arguments against Mach’s empiricism-criticism onto Russell’s neutral monism, i.e. mathematical fetishism is internally related to political conservativism. Kolman’s second main argument is that logical and mathematical fetishes are epistemologically deprived of any historical and dynamic dimension. In the final parts of our paper we place Kolman’s thinking into the context of his time, and furthermore we identify some tenets of mathematical fetishism appearing in Alain Badiou’s mathematical ontology today. (shrink)

In Résolution des quatre principaux problèmes d’architecture (1673) then in Cours d’architecture (1683), the architect–mathematician Nicolas-François Blondel addresses one of the most famous architectural problems of all times, that of the reduction in columns (entasis). The interest of the text lies in the variety of subjects that are linked to this issue. (1) The text is a response to the challenge launched by Curabelle in 1664 under the name Étrenne à tous les architectes; (2) Blondel mathematicizes the problem in the (...) “style of the Ancients”; (3) The problem is reformulated and solved through the continuous drawing of the curve; (4) Blondel refutes the uniqueness of the curve by enumerating a variety of solutions (conchoid, spiral, parabola, ellipse, circle, hyperbola). This exuberance responds to an intention that does not coincide with the state of the art of mathematics at the end of the seventeenth century, nor with the taste for geometry of the Ancients, nor with any pedagogical project. This feature is explained by Blondel’s plan to found architecture on scientific bases. The reasons for his failure are analysed. (shrink)

This edited volume explores the previously underacknowledged 'pre-history' of mathematical structuralism, showing that structuralism has deep roots in the history of modern mathematics. The contributors explore this history along two distinct but interconnected dimensions. First, they reconsider the methodological contributions of major figures in the history of mathematics. Second, they re-examine a range of philosophical reflections from mathematically-inclinded philosophers like Russell, Carnap, and Quine, whose work led to profound conclusions about logical, epistemological, and metaphysic.

Departing from and closing with reflections on issues regarding teaching practices of philosophy of mathematics, I propose a comparison between the main features of the Leibnizian notion of symbolic knowledge and some passages from the Tractatus on arithmetic. I argue that this reading allows (i) to shed a new light on the specificities of the Tractarian definition of number, compared to those of Frege and Russell; (ii) to highlight the understanding of the nature of mathematical knowledge as symbolic or formal (...) knowledge that Wittgenstein mobilizes in his book; (iii) to offer reasons for the claim that Wittgenstein can be considered the philosopher of mathematical practice avant la lettre. The paper ends with an overview, a return to the initial reflection on the connections between research and teaching, and a defense of the reading key used here in terms of its potential for the research in philosophy of mathematics. (shrink)

ABSTRACT: In searching for the origin of Europe and the cultural region/continent that we call “Europe”, at first glance we have to consider at least a double view: on the one hand the geographical understanding which indicates a region or a continent; on the other a certain form of identity and culture described and defined as European. Rodolphe Gasché taking hint from Husserl’s passage ‘Europe is not to be construed simply as a geographical and political entity’ states that a rigorous (...) engagement with what we understand by “Europe” requires that we acknowledge it as involving ‘something else as well’. With regard to the many bequests of Europe, founded in ancient Greece, in this essay I will attempt to elucidate some essential features of its cultural identity such as science and philosophy, and reflect upon several specific aspects: on the origin of Europe, on its roots and heritage, on the concept of culture, and especially on the foundation of sciences (Geometry), which contains a large part of European spirit and civilization. In particular I will address some European historical moments referring to Husserl, Heidegger, the concept of Thaumazein… In the second part of the essay, I shift to Ancient Greece to access the value of the Esprit de géométrie as defined in Proclus on the Commentary on the First Book of Euclid’s. (shrink)