A common analysis of Edmund Husserl’s early works on the philosophy of logic and mathematics presents these writings as the result of a combination of two distinct strands of influence: on the one hand a mathematical influence due to his teachers is Berlin, such as Karl Weierstrass, and on the other hand a philosophical influence due to his later studies in Vienna with Franz Brentano. However, the formative influences on Husserl’s early philosophy cannot be so cleanly (...) separated into a philosophical and a mathematical pathway. Growing evidence indicates that a Brentanist philosophy of mathematics was already in place before Husserl. Rather than an original combination at the confluence of two different streams, his early writings represent an elaboration of topics and problems that were already being discussed in the School of Brentano within a pre-existing framework. The traditional account understandably neglects Brentano’s own work on the philosophy of mathematics and logic, which can be found mostly in his unpublished manuscripts and lectures, and various works by Brentano’s students on the philosophy of mathematics which have only recently emerged from obscurity. Husserl’s early works must be correctly placed in this preceding context in order to be fully understood and correctly assessed. (shrink)

Method may be second only to substance-dualism as the best-known among Descartes's enthusiasms. But knowing that Descartes wants to promote good method is one thing; knowing what exactly he wants to promote is another. Two views seem fairly widespread. The first rests on the claim that Descartes endorses a purely procedural picture of reason, so that right reasoning is a matter of proprieties of operation, rather than respect for its objects. On this view, a method for regulating our reason would (...) offer general rules of procedure, abstracted as much as possible from the content of particular problems. Second is the view that Descartes maintains what we might call an ‘intellectualist’ approach to method, one that restricts right reasoning to operations internal to the mind, and allows the use of external bodily resources only as initial inputs or as helpful props— convenient, but marginal to the procedure and readily eliminated from it. (shrink)

Method may be second only to substance-dualism as the best-known among Descartes's enthusiasms. But knowing that Descartes wants to promote good method is one thing; knowing what exactly he wants to promote is another. Two views seem fairly widespread. The first rests on the claim that Descartes endorses a purely procedural picture of reason, so that right reasoning is a matter of proprieties of operation, rather than respect for its objects. On this view, a method for regulating our reason would (...) offer general rules of procedure, abstracted as much as possible from the content of particular problems. Second is the view that Descartes maintains what we might call an ‘intellectualist’ approach to method, one that restricts right reasoning to operations internal to the mind, and allows the use of external bodily resources only as initial inputs or as helpful props— convenient, but marginal to the procedure and readily eliminated from it. (shrink)

Abstract:Although best known to English-speaking readers as the general editor of the Dictionary of Untranslatables, the work of French philologist and philosopher Barbara Cassin is eclectic, encompassing literary studies, ancient philosophy, rhetoric, translation theory, psychoanalysis, politics, and more. From Presocratic philosophy to more recent reflections on Big Tech and democracy, Cassin's work is rooted in "sophistics," an approach that emphasizes the primacy of language in shaping our interactions with the world. Situating this sophistical approach vis-à-vis classical philology (Bollack) and the (...) philosophical tradition (Heidegger, Derrida), this essay explores the contribution of "sophistical reading" to our understanding of philosophical and literary texts. Showing how Cassin uses rhetorical theory to problematize any simple opposition of the latter, the essay concludes with a critical evaluation of Cassin's reading of Helen, Euripides' most sophistical play. (shrink)

This book offers a detailed account of the importance of psychoanalysis in Derrida's thought. Based on close readings of texts from the whole of his career, including less well-known and previously unpublished material, it sheds new light on the crucial role of psychoanalysis in shaping Derrida's response to a number of key questions. These questions range from the psyche's relationship to technology to the role of fiction and metaphor in scientific discourse, from the relationship between memory and the archive to (...) the status of the political in deconstruction. Focusing on Freud but proposing new readings of texts by Lacan, Torok and Abraham, Laplanche and Pontalis, amongst other seminal figures in contemporary French thought, the book argues that Derrida's writings on psychoanalysis can also provide an important bridge between deconstruction and the recent materialist turn in the humanities. Challenging a still prevalent 'textualist' reading of Derrida's work, it explores the ongoing contribution of deconstruction and psychoanalysis to pressing issues in critical thought today, from the localizing models of the neurosciences and the omnipresence of digital technology to the politics of affect in an age of terror. (shrink)

The emergence of mathematical probability has something to do with dice games: all the early discussions (Cardano, Galileo, Pascal) suggest as much. Although this has long been recognized, the problem is that gambling at dice has been a popular pastime since antiquity. Why, then, did gamblers wait until the sixteenth century ce to calculate the math of dicing? Many theories have been offerred, but there may be a simple solution: early-modern gamblers played different sorts of dice games than (...) in antiquity. While ancients diced at communal risk, early-moderns diced at individualized risk. These individual wager-games, for example, the game “hazard”, incentivized solving questions like “how likely is it to roll a triple-six?”. Before the early modern period there was no gambling game to which working out mathematical probability would have brought an advantage. (shrink)

Religion and science dialogues that orbit around rational method, knowledge, and truth are often, though not always, contentious. In this article, I suggest a different cluster of gravitational points around which religion and science dialogues might usefully travel: philosophical anthropology, ethics, and love. I propose seeing morality as a natural outgrowth of the human desire to establish and maintain social bonds so as not to experience the condition of being alone. Humans, of all animals, need to feel loved—defined as a (...) compassionate present-with in dynamic dyadic relation such that one experiences the sense of mattering—but that need has an equally natural tendency to be met by creating biased us-and-them distinctions. A “critical” natural ethics, then, is one in which we become aware of and work to undermine our tendency to reify in-group distinctions between “us” and “them.” Religious communities that work intentionally on this can be seen, to some extent, as laboratories of love—or as sites for co-creating knowledge in perilous times. (shrink)

The two brothers Julius and Adolf Hurwitz were born in the middle of the nineteenth century in a small town near Hanover (not far from Göttingen). Already during their schooldays, the two of them became acquainted with mathematical problems and both started to study mathematics, but while the younger brother Adolf turned out to be extremely successful in his research, the elder brother and his work seem to be almost forgotten. This paper examines the lives and works of (...) the two brothers with particular emphasis on the contributions of Julius Hurwitz, and the subsequent reception of their research. It deals with the development of an arithmetical theory for complex continued fractions by Julius and Adolf Hurwitz around 1890 and its rediscovery in the twentieth century. (shrink)

In 1931 Kurt Gödel published his incompleteness results, and some years later Church and Turing showed that the decision problem for certain systems of symbolic logic has a negative solution. However, already in 1921 the young logician Emil Post worked on similar problems which resulted in what he called an “anticipation” of these results. For several reasons though he did not submit these results to a journal until 1941. This failure ‘to be the first’, did not discourage him: his contributions (...) to mathematical logic and its foundations should not be underestimated. It is the purpose of this article to show that an interest in the early work of Emil Post should be motivated not only by this historical fact, but also by the fact that Post’s approach and method differs substantially from those offered by Gödel, Turing and Church. In this paper it will be shown how this method evolved in his early work and how it finally led him to his results. (shrink)

This book makes available to the English reader nearly all of the shorter philosophical works, published or unpublished, that Husserl produced on the way to the phenomenological breakthrough recorded in his Logical Investigations of 1900-1901. Here one sees Husserl's method emerging step by step, and such crucial substantive conclusions as that concerning the nature of Ideal entities and the status the intentional `relation' and its `objects'. Husserl's literary encounters with many of the leading thinkers of his day illuminates both (...) the context and the content of his thought. Many of the groundbreaking analyses provided in these texts were never again to be given the thorough expositions found in these early writings. Early Writings in the Philosophy of Logic and Mathematics is essential reading for students of Husserl and all those who enquire into the nature of mathematical and logical knowledge. (shrink)

The work of Jacques Derrida is often characterized as anti-scientific, and his philosophy of language taken to mean we are sealed off from empirical reality, confined to our metaphysical prison. This position is reinforced by the fact that his forerunners, Heidegger and Nietzsche, did diminish the importance of the sciences, and argued that we are enclosed within the limits of language. Today, philosophy continues to deconstruct the nature/culture distinction, and challenge the meaning of materialism, but in recent decades has realized (...) that this work requires, in addition to a critique of the modern concept of science, a rehabilitation of the sciences outside their metaphysical definition. The fact that Derrida continues to be understood as an anti-science thinker has led to the exclusion of his work from this project. In this paper, I show that Derrida, while deconstruction the metaphysical concepts of science, nature and empiricism, in fact takes the mathematical sciences as an important force of deconstructing, and develops an interpretation of empiricism that points to a non-metaphysical understanding of it. From this perspective, Derrida's work is useful for thinking through the relation of the human to language and nature in the age of globalization and anthropogenic climate change. (shrink)

In reading the Tractatus, one gets the impression that Wittgenstein, having resolved to his satisfaction the problems about language, logic, science, and mathematics, sets these painstakingly articulated findings in a disproportionately skimpy setting. There is a perfunctory ontology at the beginning, which is highly original as well as austere and perplexing; and at the end he hurries even more than usual through ethics, aesthetics and religion—as if the silence was already coming upon him, prematurely. The Notebooks 1914–1916 help a (...) good deal in understanding this skimpy setting. They give little direct indication of the ontological overture, apart from their frequent reiteration that there must be simples of some sort if the sense of expressions is to be determinate, but they give a fuller treatment to the other topics. This is particularly true of the latter half of 1916, when this parcel of topics seems to have become uppermost in Wittgenstein’s mind. Though it is exceedingly difficult to know what to make of what are in effect discarded notes, some of the entries are just too interesting to ignore. I wish to consider what light they throw on his thought at roughly the Tractatus period, and in particular on the ontology that apparently springs up full grown at the beginning of the Tractatus. (shrink)

This volume will be of particular interest to researchers working in the history, and in the philosophy, of logic and mathematics, and more generally, to ...

Introduction: mathematization and the language of nature -- Realists and nominalists : language and mathematics before the scientific revolution -- Ontology recapitulates epistemology : Gassendi, epicurean atomism, and nominalism -- British empiricism, nominalism, and constructivism -- Three mathematicians : constructivist epistemology and the new mathematical methods -- Conclusion: mathematization and the nature of language.

This article presents evidence supporting the claim that ethical reasoning is a skill that can be taught and assessed. We propose a working definition of ethical reasoning as 1) the ability to identify, analyze, and weigh moral aspects of a particular situation, and 2) to make decisions that are informed and warranted by the moral investigation. The evidence consists of a description of an ethical reasoning education program—Ethical Reasoning in Action —designed to increase ethical reasoning skills in a variety of (...) situations and areas of life. ERiA is housed at a public, major comprehensive U.S. university—James Madison University—and assessment of the program focuses on interventions delivered prior to and during orientation for incoming first-year students. Findings indicate that the interventions measurably enhance the ability of undergraduate students to reason ethically. ERiA’s competency-targeted program and positive student learning outcomes offers a promising model for higher education ethics programs seeking to connect classroom learning in ethics to decision-making in everyday life. (shrink)

Professor Atiyah is one of the greatest living mathematicians and is well known throughout the mathematical world. He is a recipient of the Fields Medal, the mathematical equivalent of the Nobel Prize, and is still at the peak of his career. His huge number of published papers, focusing on the areas of algebraic geometry and topology, have here been collected into six volumes, divided thematically for easy reference by individuals interested in a particular subject. This first volume covers his (...) class='Hi'>early and general papers dating back to his first paper written in 1952 whilst being a second-year undergraduate student. (shrink)

Background While it is generally acknowledged that self-prescribing among physicians poses some risk, research finds such behaviour to be common and in certain cases accepted by the medical community. Largely absent from the literature is knowledge about other activities doctors perform for their own medical care or for the informal treatment of family and friends. This study examined the variety, frequency and association of behaviours doctors report providing informally. Informal care included prescriptions, as well as any other type of personal (...) medical treatment (eg, monitoring chronic or serious conditions). Method A survey was sent to 2500 randomly-selected physicians in Colorado, 600 individuals returned questionnaires with usable data. The authors hypothesised: (1) physicians would prescribe the same types of treatment at home as they prescribed professionally; and (2) physicians who informally prescribed addictive medications would be more likely to engage in other types of informal medical care. Results Physicians who wrote prescriptions for antibiotics, psychotropics and opioids at work were more likely to prescribe these medications at home. Those prescribing addictive drugs outside of the office treated more serious illnesses in emergency situations, more chronic conditions and more major medical/surgical conditions informally than did those not routinely prescribing addictive medications. Physicians reported a variety of informal care behaviour and high frequency of informal care to family and friends. Discussion The frequency and variety of informal care reported in this study strongly argues for profession-wide discussion about ethical and guideline considerations for such behaviour. These areas are discussed in the paper. (shrink)

The paper deals with some major themes in early Cassirer’s philosophy of mathema- tics. It appears, that the basis of his thinking about mathematical objects and mathematical concept formation is his Neo-Kantian idealistic theory of concepts which he developed in opposition to what is called the „traditional theory of concepts“ going back to Aristotle. Cassirer often seeks to confirm his philo- sophical insights concerning mathematics by the interpretations the works of significant mathematicians. Therefore, the second part of (...) the paper deals with Cassirer’s attempt to find such a confirmation in famous Dedekind’s theory of natural numbers. Cassirer’s philosophical attitude to Dedekind’s theory is compared with that of Russell. The author raises the sceptical question of whether Cassirer’s view of mathematics – as developed in his early period – could be a sufficient or at least plausible basis for solving philosophical problems of the foundations of mathematics of that time. (shrink)

In this book, Michael Potter offers a fresh and compelling portrait of the birth and first several decades of analytic philosophy, one of the most important periods in philosophy’s long history. He focuses on the period between the publication of Gottlob Frege’s _Begriffsschrift _in 1879 and Frank Ramsey’s death in 1930. Potter--one of the most influential writers on late 19 th and early 20 th century philosophy--presents a deep but accessible account of the break with Absolute Idealism and Neo-Kantianism, (...) specifically, and more generally with many of the metaphysical preoccupations of philosophy’s preceding history. Potter’s focus is on philosophical logic and philosophy of mathematics, but he also relies heavily on important issues in metaphysics and meta-ethics to complete his story. The book provides an essential starting point for any student or philosopher attempting to understand Frege, Russell, Wittgenstein, and Ramsey as well as their interactions and their intellectual milieux. It will also be of interest to a great many philosophers today who want to illuminate the problems they work on by better knowing their origins. KEY FEATURES: 1. Discusses the interconnections of Frege, Russell and Wittgenstein—founding thinkers in the history of analytic philosophy—and also brings the neglected Frank Ramsey into this conversation, providing a unique focus and depth to an introductory text 2. Increases the general awareness of the importance of the history of analytic philosophy for today’s non-historical debates, giving the book appeal in all areas of analytic philosophy 3. Written by one of the most influential philosophers of logic and writers in the history of analytic philosophy 4. Written for upper-level undergraduates, guaranteeing widespread accessibility 5. Includes coverage of topics and issues neglected in competing publications, including Russell’s _Principles_, solipsism in the _Tractatus_, and the contributions of Frank Ramsey 6. Emphasizes the chronological development of authors’ views so as to provide a better understanding of their motivation. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Abraham, Planter of Mathematics":Histories of Mathematics and Astrology in Early Modern EuropeNicholas PopperFrancis Bacon's 1605 Advancement of Learning proposed to dedicatee James I a massive reorganization of the institutions, goals, and methods of generating and transmitting knowledge. The numerous defects crippling the contemporary educational regime, Bacon claimed, should be addressed by strengthening emphasis on philosophy and natural knowledge. To that end, university positions were to be (...) created devoted to "Artes and Sciences at large," rather than to the professions. High salaries would render lecturers "able and sufficient," undistracted from their task. Most famously, he argued that teaching of the "operatiue studie of many Scyences" should involve sophisticated technical education. The study of natural philosophy demanded not only books, but globes, astrolabes, and other "instrumentals." Most significantly, yielding reliable and meaningful knowledge from experiential gleanings required a rigorous system of deductive reasoning.The legacy of this colossal proposal has earned Bacon honored status as devisor of the scientific method.1 But Bacon's educational reform extended [End Page 87] beyond the methods of producing and transmitting knowledge. To facilitate more efficient "vse and administration" of the knowledge produced by his system, he also demanded a searching examination of the history of learning. This history would provide a mirror enabling his contemporaries to deploy the fruits of his method, by considering how learning in the past had been used successfully or ill-advisedly. Producing this history was the ambition of the Advancement, for he explained: "no man hath propounded to himselfe the generall state of learning to bee described and represented from age to age, as many have done the works of Nature, & the State civile and Ecclesiastical." He continued:And yet I am not ignorant that in divers particular sciences, as of the Iurisconsults, the Mathematicians, the Rhetoricians, the Philosophers, there are set down some smal memorials of the Schooles, Authors, and Bookes: and so likewise some barren relations touching the Invention of Arts, or usages. But a iust story of learning, containing the Antiquities and Originalls of Knowledges, & their Sects; their Inventions, their Traditions; their diverse Administrations and Managings; their Flourishings, their Oppositions, Decayes, Depressions, Oblivions, Removes, with the causes, and occasions of them, and all other events concerning learning, throughout the ages of the world; I may truly affirme to be wanting.2Bacon thus positioned himself not only as a Father of Modern Science, but also a Father of the History of Science.Following Bacon's suggestion, I will examine the "small memorials" of the history of mathematics—and particularly the mathematical art of astrology—in the sixteenth and seventeenth centuries. My conclusion, however, will not bear out Bacon's claims. Despite his frustration, Bacon was only one of many early modern scholars appraising the role of mathematics within history. And he devoted less energy than others to mapping its origins and tracing its transmissions between communities. In fact, Bacon's 1605 proposal for a history of mathematics was already out-of-date. Discussions [End Page 88] of the history of mathematics had been rife on the continent and in England throughout the previous century.3The evidence that Renaissance scholars inherited was sprawling and inconclusive. Several genealogies for mathematics could be found within classical Greek, Latin, and patristic references. One lineage claimed mathematics began in ancient Assyria, where the priestly caste, the Chaldeans, practiced a form of mathematics that seemed a corrupt admixture of philosophy, medicine, and religion, and relied heavily on observation of the heavens. Other scholars traced the origins to Egypt, claiming that the field developed to survey lands frequently flooded by the Nile. The real problem for ancient, late antique, and medieval scholars had not been the origins of mathematics, but what exactly mathematics were. For some, the term strictly referred to the quadrivium: geometry, arithmetic, astronomy, and music. But for others, mathematics' origins amongst the Egyptians or Chaldeans inextricably linked it to forms of astrological divination, augury, and necromancy that were unsavory to both Latin and Christian traditions. Mathematici were included alongside ghastly lists of Magi, Brahmins, Aruspices, Genethliaci, and other diabolic practitioners of idolatrous magic. These might or might not be distinguished from mathematici such as Pythagoras... (shrink)

Georg Agricola’s De Re Metallica (1556) is an impressive work about mining arts and sciences, still used as a reference work to understand the mining world of his time and to analyze the relationships between scholars and practitioners. This essay begins by studying how Agricola presents the underground surveying, also known as geometria subterranea. How does the author’s mathematics relate to contemporary geometry or to actual surveying practices? Second, the essay contrasts his scholarly approach with sixteenth-century administrative and technical (...) documents, as well as mining sermons. Comparing the practical geometry contained in the De Re Metallica with different kinds of sources helps to question the relationships between the arts and natural and exact sciences in the practical settings of early modern Europe. Finally, it argues for a distinction between the inquiry on the historical progress of mathematics in civil life and philosophical questions about the rise of new sciences. (shrink)

While Garfinkel’s early work, captured in Studies in Ethnomethodology, has received a lot of attention and discussion, this has not been the case for his later work since the 1970s. In this paper, we critically examine the aims of Garfinkel’s later ethnomethodological studies of work programme and evaluate key ideas such as the ‘missing what’ in the sociology of work, ‘the unique adequacy requirements of methods’, and the notion of ‘hybrid studies’. We do so through a detailed engagement with (...) a study that has frequently been singled out as exemplary by Garfinkel for his studies of work programme, namely Livingston’s The Ethnomethodological Foundations of Mathematics. We show how Livingston uses the proof of Gödel’s Incompleteness Theorem as a way to exhibit the work involved in understanding mathematical proofs. We then discuss how Livingston uses this example to introduce a distinction between the written ‘proof account’ and the associated ‘lived work’ of working through that proof, which we argue allows Livingston to provide a powerful critique of a formalist understanding of the objectivity of mathematics. We then discuss three aspects that we find problematic in Livingston’s and Garfinkel’s claims about Livingston’s study. Firstly, we question whether written proofs are best conceived of as descriptions or accounts. Secondly, we interrogate whether an ethnomethodological study could teach mathematicians how to make discoveries. Thirdly, we throw doubt on Garfinkel’s claim that Livingston’s results are results in mathematics. We conclude this paper with a discussion of how Livingston’s study Gödel’s proof highlights key tensions in Garfinkel’s later work. Firstly, we argue that there exists an ambiguity in Garfinkel’s treatment of texts as ‘incompetent’. Secondly, we show that Garfinkel’s attempt to extend his idea of classical studies from sociology to other professions and disciplines is problematic. Finally, we question Garfinkel’s proposals to reorient ethnomethodological studies away from sociological audiences and ask whether ethnomethodological studies promise the delivery of a ‘large prize’ or, rather, provide something like ‘helpful therapy’. (shrink)

Discussions of the foundations of mathematics and their history are frequently restricted to logical issues in a narrow sense, or else to traditional problems of analytic philosophy. From Dedekind to Gödel: Essays on the Development of the Foundations of Mathematics illustrates the much greater variety of the actual developments in the foundations during the period covered. The viewpoints that serve this purpose included the foundational ideas of working mathematicians, such as Kronecker, Dedekind, Borel and the early Hilbert, (...) and the development of notions like model and modelling, arbitrary function, completeness, and non-Archimedean structures. The philosophers discussed include not only the household names in logic, but also Husserl, Wittgenstein and Ramsey. Needless to say, such logically-oriented thinkers as Frege, Russell and Gödel are not entirely neglected, either. Audience: Everybody interested in the philosophy and/or history of mathematics will find this book interesting, giving frequently novel insights. (shrink)

Though he is known primarily for his mathematics, John Wallis was also a prominent natural philosopher and experimentalist. Like many experimental philosophers, including his colleagues in the Royal So­ciety, Wallis sought to identify the mathematical laws that govern natural phenomena. However, I argue that Wallis’s particular understanding of the laws of nature was informed by his reading of a thirteenth–century optical treatise by Robert Grosseteste, De lineis, angulis et figuris, which expresses the principle that “Nature doth not work by (...) Election.” Wallis’s use of this principle in his Discourse of Gravity and Gravitation helps to clarify his understanding of natural laws. According to Wallis, since nature cannot choose to act one way or another, natural phenomena are unfailingly regular, and it is this that allows them to be predicted, generalized, and described by mathematical rules. Furthermore, I argue that Wallis’s reading of Grosseteste reveals one way that medieval scholarship contributed to the “mathematization of nature” in the early modern period: historically–minded scholars like Wallis found insightful philosophical principles in medieval sources, and they transformed and redeployed these principles to suit the needs of early modern natural philosophy. (shrink)

This paper explores the work of Nicolas Rashevsky, a Russian émigré theoretical physicist who developed a program in "mathematical biophysics" at the University of Chicago during the 1930s. Stressing the complexity of many biological phenomena, Rashevsky argued that the methods of theoretical physics -- namely mathematics -- were needed to "simplify" complex biological processes such as cell division and nerve conduction. A maverick of sorts, Rashevsky was a conspicuous figure in the biological community during the 1930s and early (...) 1940s: he participated in several Cold Spring Harbor symposia and received several years of funding from the Rockefeller Foundation. However, in contrast to many other physicists who moved into biology, Rashevsky's work was almost entirely theoretical, and he eventually faced resistance to his mathematical methods. Through an examination of the conceptual, institutional, and scientific context of Rashevsky's work, this paper seeks to understand some of the reasons behind this resistance. (shrink)

Analytic philosophy - arguably the most important philosophical movement in the 20th century - has gained a new historical self-consciousness, particularly about it's own origins. The period between 1880 and 1930 saw the most important work of its founding figures (Frege, Russell, Moore, Wittgenstein) take root and flourish. The fifteen previously-unpublished essays in this collection explore different facets of this period, with an emphasis on the vital intellectual relationship between Frege and the early Wittgenstein.

The novel use of symbolism in early modern mathematics poses both philosophical and historical questions. How can we trace its development and transmission through manuscript sources? Is it intrinsically related to the emergence of symbolic algebra? How does symbolism relate to the use of diagrams? What are the consequences of symbolic reasoning on our understanding of nature? Can a symbolic language enable new forms of reasoning? Does a universal symbolic language exist which enable us to express all knowledge? (...) This book brings together a collection of papers that address all these and related questions which were initially posed at a conference held in Ghent (Belgium) in August 2009. Scholars working on philosophy of science, history of philosophy and history of mathematics provide an insight into the role and function of symbolic representations in the development of early modern mathematics. The papers cover the period from early abbaco arithmetic and algebra (14h century) up to Leibniz (early 18th century). (shrink)

In his book on The Mathematics of Great Amateurs Coolidge starts the chapter on Bolzano saying that he included Bolzano because it seemed interesting to him ‘that a man who was a remarkable pulpit orator, only removed from his chair for his political opinions, should have thought so far into the deepest problems of a science which he never taught in a professional capacity’ [Coolidge, 1990, p. 195]. In fact, considering Bolzano's poor health and his enormous productivity in his (...) ‘professional areas’, the results, the breadth and the extent of this ‘great amateur's’ thoughts on mathematics, and its proper scientific presentation are simply astonishing. Even this new selection of more than 650 printed pages of text, well edited and translated by Steve Russ, represents only a part of Bolzano's legacy on mathematical topics: the complete German edition of Bolzano's writings, the Bernard Bolzano-Gesamtausgabe edited by J. Berg, E. Morscher, B. van Rootselaar, and others is projected to include over 120 volumes of which more than 36 volumes will contain his mathematical works. About 25 volumes have been published so far in this mathematical subseries including Bolzano's extensive Miscellanea Mathematica .To most readers of this journal, Bolzano will be well-known for his contributions to logic—his ‘logic of variation’ is viewed as a precursor of the modern semantic conception of logical consequence—and his contributions to the foundations of calculus, especially his ‘purely analytic proof’ of the intermediate-value theorem and the Bolzano-Weierstrass theorem, or his early discovery of a counterintuitive function which is continuous in its domain, but nowhere-differentiable.Despite his lifelong interest in the foundations of mathematics, Bolzano studied theology and taught philosophy of religion in his native city, Prague, from 1805 until his dismissal in 1820. Part of Bolzano's duties …. (shrink)

I argue for an interpretation of the connection between Descartes’ early mathematics and metaphysics that centers on the standard of geometrical intelligibility that characterizes Descartes’ mathematical work during the period 1619 to 1637. This approach remains sensitive to the innovations of Descartes’ system of geometry and, I claim, sheds important light on the relationship between his landmark Geometry and his first metaphysics of nature, which is presented in Le monde. In particular, I argue that the same standard of (...) clear and distinct motions for construction that allows Descartes to distinguish ‘geometric’ from ‘imaginary’ curves in the domain of mathematics is adopted in Le monde as Descartes details God’s construction of nature. I also show how, on this interpretation, the metaphysics of Le monde can fruitfully be brought to bear on Descartes’ attempted solution to the Pappus problem, which he presents in Book I of the Geometry. My general goal is to show that attention to the standard of intelligibility Descartes invokes in these different areas of inquiry grants us a richer view of the connection between his early mathematics and philosophy than an approach that assumes a common method is what binds his work in these domains together.Keywords: René Descartes; Geometry; Mathematics; Intelligibility; Metaphysics. (shrink)

ArgumentTwo simultaneous episodes in late nineteenth-century mathematical research, one by Karl Hermann Brunn and another by Hermann Minkowski, have been described as the origin of the theory of convex bodies. This article aims to understand and explain how and why the concept of such bodies emerged in these two trajectories of mathematical research; and why Minkowski's – and not Brunn's – strand of thought led to the development of a theory of convexity. Concrete pieces of Brunn's and Minkowski's mathematical work (...) in the two episodes will, from the perspective of the above questions, be presented and analyzed with the use of the methodological framework of epistemic objects, techniques, and configurations as adapted from Hans-Jörg Rheinberger's work on empirical sciences to the historiography of mathematics by Moritz Epple. Based on detailed descriptions and a comparison of the objects and techniques that Brunn and Minkowski studied and used in these pieces it will be concluded that Brunn and Minkowski worked in different epistemic configurations, and it will be argued that this had a significant influence on the mathematics they developed for those bodies, which can provide answers to the two research questions listed above. (shrink)

In spite of Ernst Cassirer’s criticisms of psychologism throughout Substance and Function, in the final chapter he issues a demand for a “psychology of relations” that can do justice to the subjective dimensions of mathematics and natural science. Although these remarks remain somewhat promissory, the fact that this is how Cassirer chooses to conclude Substance and Function recommends it as a topic worthy of serious consideration. In this paper, I argue that in order to work out the details of (...) Cassirer’s psychology of relations in Substance and Function, we need to situate it within two broader frameworks. First, I position Cassirer’s view in relation to the view of psychology and logic endorsed by his Marburg Neo-Kantian predecessors, Hermann Cohen and Paul Natorp. Second, I augment Cassirer’s early account of the psychology of relations in Substance and Function with the more mature view of psychology that he presents in The Philosophy of Symbolic Forms. By placing Cassirer’s account within these contexts, I claim that we gain insight into his psychology of relations, not just as it pertains to mathematics and natural science, but to culture as a whole. Moreover, I maintain that pursuing this strategy helps shed light on one of the most controversial features of his philosophy of mathematics and natural science, viz., his theory of the a priori. (shrink)

At various times, mathematicians have been forced to work with inconsistent mathematical theories. Sometimes the inconsistency of the theory in question was apparent (e.g. the early calculus), while at other times it was not (e.g. pre-paradox na¨ıve set theory). The way mathematicians confronted such difficulties is the subject of a great deal of interesting work in the history of mathematics but, apart from the crisis in set theory, there has been very little philosophical work on the topic of (...) inconsistent mathematics. In this paper I will address a couple of philosophical issues arising from the applications of inconsistent mathematics. The first is the issue of whether finding applications for inconsistent mathematics commits us to the existence of inconsistent objects. I then consider what we can learn about a general philosophical account of the applicability of mathematics from successful applications of inconsistent mathematics. (shrink)

This essay will discuss the hegemonic role that texts have come to play in the historiography of subcontinental mathematical traditions. It will argue that texts need to be studied as records of practices of people's working lives, grounded in social hierarchies. We will take particular mathematical texts to show how different occupational registers have come to shape practices that defy the binaries of concrete and abstract, high and low mathematics or the pure and applied conundrum. Measuring, counting and accounting (...) practices as part of the routine work of practitioners performing their caste occupations then provide us with a spectrum of the computational activities that controlled and regulated the lives of people in the past. In the process the act of computing itself gained certain political values such as cunning and manipulation, identified with professions of village accountant and merchant, for example. Drawn from my earlier work on these records, I discuss the occupational role of the accountant as a political functionary who assessed and authenticated the measurements of land and produce in the village, making values of the labor performed by others, and creating avenues for his own proficiency as a mathematical practitioner. (shrink)

In this article we present and compare two early attempts to establish psychology as an independent scientific discipline that had considerable influence in central Europe: the theories of Johann Friedrich Herbart (1776—1841) and Franz Brentano (1838—1917). While both of them emphasize that psychology ought to be conceived as an empirical science, their conceptions show revealing differences. Herbart starts with metaphysical principles and aims at mathematizing psychology, whereas Brentano rejects all metaphysics and bases his method on a conception of inner (...) perception (as opposed to inner observation) as a secondary consciousness, by means of which one gets to be aware of all of one’s own conscious phenomena. Brentano’s focus on inner perception brings him to deny the claim that there could be unconscious mental phenomena — a view that stands in sharp contrast to Herbart’s emphasis on unconscious, ‘repressed’ presentations as a core element of his mechanics of mind. Herbart, on the other hand, denies any role for psychological experiments, while Brentano encouraged laboratory work, thus paving the road for the more experimental work of his students like Stumpf and Meinong. By briefly tracing the fate of the schools of Herbart and Brentano, respectively, we aim to illustrate their impact on the development of psychological research, mainly in central Europe. (shrink)

This book advances a reading of Wittgenstein’s Tractatus that moves beyond the main interpretative options of the New Wittgenstein debate. It covers Wittgenstein’s approach to language and logic, as well as other areas unduly neglected in the literature, such as his treatment of metaphysics, the natural sciences and value. Tejedor re-contextualises Wittgenstein’s thinking in these areas, plotting its evolution in his diaries, correspondence and pre- Tractatus texts, and developing a fuller picture of its intellectual background. This broadening of the angle (...) of view is central to the interpretative strategy of her book: only by looking at the Tractatus in this richer light can we address the fundamental questions posed by the New Wittgenstein debate – questions concerning the method of the Tractatus , its approach to nonsense and the continuity in Wittgenstein’s philosophy. Wittgenstein’s early work remains insightful, thought-inspiring and relevant to contemporary philosophy of language and science, metaphysics and ethics. Tejedor’s ground-breaking work ultimately conveys a surprisingly positive message concerning the power for ethical transformation that philosophy can have, when it is understood as an activity aimed at increasing conceptual clarification and awareness. (shrink)

The uncountability of$\mathbb {R}$is one of its most basic properties, known far outside of mathematics. Cantor’s 1874 proof of the uncountability of$\mathbb {R}$even appears in the very first paper on set theory, i.e., a historical milestone. In this paper, we study the uncountability of${\mathbb R}$in Kohlenbach’shigher-orderReverse Mathematics (RM for short), in the guise of the following principle:$$\begin{align*}\mathit{for \ a \ countable \ set } \ A\subset \mathbb{R}, \mathit{\ there \ exists } \ y\in \mathbb{R}\setminus A. \end{align*}$$An important conceptual (...) observation is that the usual definition of countable set—based on injections or bijections to${\mathbb N}$—does not seem suitable for the RM-study of mainstream mathematics; we also propose a suitable (equivalent over strong systems) alternative definition of countable set, namelyunion over${\mathbb N}$of finite sets; the latter is known from the literature and closer to how countable sets occur ‘in the wild’. We identify a considerable number of theorems that are equivalent to the centred theorem based on our alternative definition. Perhaps surprisingly, our equivalent theorems involve most basic properties of the Riemann integral, regulated or bounded variation functions, Blumberg’s theorem, and Volterra’s early work circa 1881. Our equivalences are alsorobust, promoting the uncountability of${\mathbb R}$to the status of ‘big’ system in RM. (shrink)

Early mathematical abilities matter for later formal arithmetical performances, school and professional success. Accordingly, it seems central to accurately assess numerical school readiness at school entrance. This is a prerequisite for identifying school-starters who are at risk to encounter difficulties in mathematics and stimulate their acquisition of mathematical fundamentals as soon as possible. In the present study, we present a new test which allows professionals working with children (e.g., teachers, school psychologists, speech therapists, school doctors) to assess children’s (...) numerical school readiness when they enter formal schooling in a simple, rapid and efficient manner. 346 children were assessed at the entrance of 1st grade (6-to-7-year-olds) with the novel numerical school readiness test (T1). In addition, children’s numerical skill levels were evaluated with classical arithmetical tests at T1 and a year later in 2nd grade (T2, 7-to-8-year-olds). Performance in these classical tests in 1st and 2nd grade systematically related to their early mathematical abilities assessed with the numerical school readiness test. By using the present test to evaluate numerical school readiness it is possible to identify pupils at risk for developing poor mathematical skills right from the start of formal schooling. Such a tool is needed, as a child’s scholastic level in mathematic at school entrance (or school readiness) is known to be critical for his or her future school and professional carrier (Currie & Thomas, 1999; Duncan et al., 2007; Pagani, Fitzpatrick, Archambault, & Janosz, 2010; Rivera-Batiz, 1992; Romano, Babchishin, Pagani, & Kohen, 2010). (shrink)

The form of nominalism known as 'mathematical fictionalism' is examined and found wanting, mainly on grounds that go back to an early antinominalist work of Rudolf Carnap that has unfortunately not been paid sufficient attention by more recent writers.

David Hilbert’s early foundational views, especially those corresponding to the 1890s, are analysed here. I consider strong evidence for the fact that Hilbert was a logicist at that time, following upon Dedekind’s footsteps in his understanding of pure mathematics. This insight makes it possible to throw new light on the evolution of Hilbert’s foundational ideas, including his early contributions to the foundations of geometry and the real number system. The context of Dedekind-style logicism makes it possible to (...) offer a new analysis of the emergence of Hilbert’s famous ideas on mathematical existence, now seen as a revision of basic principles of the “naive logic” of sets. At the same time, careful scrutiny of his published and unpublished work around the turn of the century uncovers deep differences between his ideas about consistency proofs before and after 1904. Along the way, we cover topics such as the role of sets and of the dichotomic conception of set theory in Hilbert’s early axiomatics, and offer detailed analyses of Hilbert’s paradox and of his completeness axiom (Vollständigkeitsaxiom). (shrink)

The aim of this paper is to provide a comprehensive exposition of the early contributions to the so-called Campbell, Baker, Hausdorff, Dynkin Theorem during the years 1890–1950. Related works by Schur, Poincaré, Pascal, Campbell, Baker, Hausdorff, and Dynkin will be investigated and compared. For a full recovery of the original sources, many mathematical details will also be furnished. In particular, we rediscover and comment on a series of five notable papers by Pascal (Lomb Ist Rend, 1901–1902), which nowadays (...) are almost forgotten. (shrink)

What is the source of logical and mathematical truth? This book revitalizes conventionalism as an answer to this question. Conventionalism takes logical and mathematical truth to have their source in linguistic conventions. This was an extremely popular view in the early 20th century, but it was never worked out in detail and is now almost universally rejected in mainstream philosophical circles. Shadows of Syntax is the first book-length treatment and defense of a combined conventionalist theory of logic and (...) class='Hi'>mathematics. It argues that our conventions, in the form of syntactic rules of language use, are perfectly suited to explain the truth, necessity, and a priority of logical and mathematical claims, as well as our logical and mathematical knowledge. (shrink)

Volume 4 of’ “The Early Works” series covers the period of Dewey’s last year and one-half at the University of Michigan and his first half-year at the University of Chicago. In addition to sixteen articles the present volume contains Dewey’s reviews of six books and three articles, verbatim reports of three oral statements made by Dewey, and a full-length book, The Study of Ethics. Like its predecessors in this series, this volume presents a “clear text,” free of interpretive (...) or reference material. Apparatus, including references, corrections, and emendations, is confined to appendix material. Fredson Bowers, the Consulting Textual Editor, has provided an essay on the textual principles and procedures, and Wayne A. R. Leys, Professor of Philosophy at Southern Illinois University, has written an Introduction discussing the relationship between Dewey’s writings of this period and his later work. That Dewey’s scholarship and writing was at an especially high level during 1893 and 1894 may be considered an index to the significance of this two-year period. (shrink)

This book offers a historical explanation of important philosophical problems in logic and mathematics, which have been neglected by the official history of modern logic. It offers extensive information on Gottlob Frege’s logic, discussing which aspects of his logic can be considered truly innovative in its revolution against the Aristotelian logic. It presents the work of Hilbert and his associates and followers with the aim of understanding the revolutionary change in the axiomatic method. Moreover, it offers useful tools to (...) understand Tarski’s and Gödel’s work, explaining why the problems they discussed are still unsolved. Finally, the book reports on some of the most influential positions in contemporary philosophy of mathematics, i.e., Maddy’s mathematical naturalism and Shapiro’s mathematical structuralism. Last but not least, the book introduces Biancani’s Aristotelian philosophy of mathematics as this is considered important to understand current philosophical issue in the applications of mathematics. One of the main purposes of the book is to stimulate readers to reconsider the Aristotelian position, which disappeared almost completely from the scene in logic and mathematics in the early twentieth century. (shrink)

The work addresses the problem of the relationship between science and philosophy in the work of Hermann Weyl. The author begins by discussing Weylls Gottingen tradition. Contrary to standard accounts of this tradition, Edmund Husserl and Georg Cantor are included. The influence of this tradition on Weyl is then illustrated by an examination of Weyl's early philosophy of mathematics. Here Weyl attempts to use Husserl's early phenomenology to amalgamate the thought of Felix Klein, David Hilbert and Cantor. (...) Weyl's "phenomenological period," in which he wrote The Continuum and Space-Time-Matter, is then discussed in detail. The work reveals that for Weyl, philosophy and science, guided by the same telos to construct an objective picture of the world, are involved in a dialectic in which each serves as a corrective and limitation of the other. (shrink)

This paper aims to bring together the study of normative judgments in mathematics as studied by the philosophy of mathematics and verbal hygiene as studied by sociolinguistics. Verbal hygiene (Cameron 1995) refers to the set of normative ideas that language users have about which linguistic practices should be preferred, and the ways in which they go about encouraging or forcing others to adopt their preference. We introduce the notion of mathematical hygiene, which we define in a parallel way (...) as the set of normative discourses regulating mathematical practices and the ways in which mathematicians promote those practices. To clarify our proposal, we present two case studies from 17th century France. First, we exemplify a case of mathematical hygiene proper: Descartes' algebraic geometry and Newton's subsequent criticism of it, a case of (im)purity in mathematics. Then, we compare Descartes' and Newton's mathematical hygiene with verbal hygiene from this period, as exemplified by the work of the grammarian Claude Favre de Vaugelas (Ayres-Bennett 1987). We argue that these early modern normative discourses on mathematics and language respectively can be seen as emanating from a common socio-political program: the development of a new bourgeois intellectual class. We conclude that the study of mathematical hygiene has the potential to yield new understandings of the social aspects of mathematical practice, and that similarities between mathematical and verbal hygiene at certain time periods, such as 17th century France, open up a new area of inquiry at the borders of linguistics and the philosophy of mathematics. (shrink)