This paper is concerned with the genesis of mathematical knowledge. While some philosophers might argue that mathematics has no real subject matter and thus is not a body of knowledge, I will not try to dissuade them directly. I shall not attempt such a refutation because it seems clear to me that mathematicians do know such things as the Mean Value Theorem, The Fundamental Theorem of Arithmetic, Godel's Theorems, etc. Moreover, this is much more evident to (...) me than any philosophical view of mathematics I know of — including my own. So I am going to take mathematics as my starting point. (shrink)

The purpose of this study is to assess Husserl’s debt to Lotze’s philosophy during the Halle period (1886-1901). I shall first track the sources of Husserl’s knowledge of Lotze’s philosophy during his studies with Brentano in Vienna and then with Stumpf in Halle. I shall then briefly comment on Husserl’s references to Lotze in his early work and research manuscripts for the second volume of his Philosophy of Arithmetic. In the third section, I examine Lotze’s influence on Husserl’s antipsychologistic (...) turn in the mid-1890s. The fourth section is a commentary on Husserl’s manuscript entitled “Microcosmus,” to which he explicitly refers in his Prolegomena, and which he planned to publish as an annex of his Logical Investigations. This work contains a detailed analysis of the third book of Lotze’s 1874 Logic. The last section examines Husserl’s arguments against logical psychologism in his Prolegomena, which I discuss through the lens of Stumpf’s critique of psychologism in his paper “Psychology and theory of knowledge”. I argue that Stumpf’s early works on this topic make it possible to establish a connection between Lotze’s interpretation of Plato’s theory of Ideas and Husserl’s antipsychologism. My hypothesis is that Stumpf’s analyses represent the background of Husserl's critique of logical psychologism in his Logical Investigations. I shall conclude this study by showing that Husserl’s position with respect to Lotze’s philosophy remains basically unchanged after the publication of his Logical Investigations, and that Husserl’s main criticism of Lotze pertains, in the final analysis, to the absence of a theory of intentionality in Lotze’s philosophy. (shrink)

This paper is a nontechnical exposition of the author's view that mathematics is a science of patterns and that mathematical objects are positions in patterns. the new elements in this paper are epistemological, i.e., first steps towards a postulational theory of the genesis of our knowledge of patterns.

Origins of the Copernican Revolution that led to modern science genesis can be explained only by the joint influence of external and internal factors. The author tries to take this influence into account with a help of his own growth of knowledge model according to which the growth of science consists in interaction, interpenetration and unification of various scientific research programmes spreading from different cultural milieux. Copernican Revolution consisted in revealation and elimination of the gap between Ptolemy’s (...) class='Hi'>mathematical astronomy and Aristotelian qualitative physics. But the very realization of the gap between physics and astronomy appeared to be possible because at least at its first stages modern science was a result of Christian Weltanschaugung development with its aspiration for elimination of pagan components. Of all the external factors religion was the strongest one. Key words: scientific revolution, Christian weltanschaugung, modernity, Copernicus, Ptolemy. (shrink)

Predictive policing has become a new panacea for crime prevention. However, we still know too little about the performance of computational methods in the context of predictive policing. The paper provides a detailed analysis of existing approaches to algorithmic crime forecasting. First, it is explained how predictive policing makes use of predictive models to generate crime forecasts. Afterwards, three epistemologies of predictive policing are distinguished: mathematical social science, social physics and machine learning. Finally, it is shown that these epistemologies (...) have significant implications for the constitution of predictive knowledge in terms of its genesis, scope, intelligibility and accessibility. It is the different ways future crimes are rendered knowledgeable in order to act upon them that reaffirm or reconfigure the status of criminological knowledge within the criminal justice system, direct the attention of law enforcement agencies to particular types of crimes and criminals and blank out others, satisfy the claim for the meaningfulness of predictions or break with it and allow professionals to understand the algorithmic systems they shall rely on or turn them into a black box. By distinguishing epistemologies and analysing their implications, this analysis provides insight into the techno-scientific foundations of predictive policing and enables us to critically engage with the socio-technical practices of algorithmic crime forecasting. (shrink)

The ways in which the sciences have been delineated and categorized throughout history provide insights into the formation, stabilization, and establishment of scientific systems of knowledge. The Dresdener school’s approach for explaining and categorizing the genesis of the engineering disciplines is still valid, but needs to be complemented by further-reaching methodological and theoretical reflections. Pierre Bourdieu’s theory of social practice is applied to the question of how individual agents succeed in influencing decisively a discipline’s changing object orientation, institutionalisation (...) and self-reproduction. Through the accumulation of social, cultural and economic capital, they succeed in realising their own organisational ideas and scientific programs. Key concepts for the analysis include the struggle for power and resources, monopolies of interpretation, and the degree of autonomy. A case study from the Aachener Technische Hochschule shows that the consolidation of ferrous metallurgy can be conceived as a symbolical struggle between Fritz Wüst, professor for ferrous metallurgy, and the German Iron and Steel Institute, leading to a construction of a system of differences in which scientists accepted being scientists rather than entrepreneurs, and entrepreneurs accepted becoming entrepreneurs and renounced science. (shrink)

This book argues against the view that mathematical knowledge is a priori,contending that mathematics is an empirical science and develops historically,just as ...

What were the reasons of the Copernican Revolution ? How did modern science (created by a bunch of ambitious intellectuals) manage to force out the old one created by Aristotle and Ptolemy, rooted in millennial traditions and strongly supported by the Church? What deep internal causes and strong social movements took part in the genesis, development and victory of modern science? The author comes to a new picture of Copernican Revolution on the basis of the elaborated model of scientific (...) revolutions that takes into account some recent advances in philosophy, sociology and history of science. The model was initially invented to describe Einstein’s Revolution of the XX century beginning. The model considers the growth of knowledge as interaction, interpenetration and unification of the research programmes, springing out of different cultural traditions. Thus, Copernican Revolution appears as a result of revealation and (partial) resolution of the dualism , of the gap between Ptolemy’s mathematical astronomy and Aristotelian qualitative physics. The works of Copernicus, Galileo, Kepler and Newton were all the stages of mathematics descendance from skies to earth and reciprocal extrapolation of earth physics on skies. The model elaborated enables to reassess the role of some social factors crucial for the scientific revolution. It is argued that initially modern science was a result of the development of Christian Weltanschaugung . Later the main support came from the absolute monarchies. In the long run the creators of modern science appeared to be the “apparatchics” of the “regime of truth” built-in state machine. Natural science became a part of ideological state apparatus providing not only scientific education but the internalization of values crucial for the functioning of state. -/- . (shrink)

This book draws its inspiration from Hilbert, Wittgenstein, Cavaillès and Lakatos and is designed to reconfigure contemporary philosophy of mathematics by making the growth of knowledge rather than its foundations central to the study of mathematical rationality, and by analyzing the notion of growth in historical as well as logical terms. Not a mere compendium of opinions, it is organised in dialogical forms, with each philosophical thesis answered by one or more historical case studies designed to support, complicate (...) or question it. The first part of the book examines the role of scientific theory and empirical fact in the growth of mathematical knowledge. The second examines the role of abstraction, analysis and axiomatization. The third raises the question of whether the growth of mathematical knowledge constitutes progress, and how progress may be understood. Readership: Students and scholars concerned with the history and philosophy of mathematics and the formal sciences. (shrink)

In his book The Value of Science Poincaré criticizes a certain view on the growth of mathematical knowledge: “The advance of science is not comparable to the changes of a city, where old edifices are pitilessly torn down to give place to new ones, but to the continuous evolution of zoological types which develop ceaselessly and end by becoming unrecognizable to the common sight, but where an expert eye finds always traces of the prior work of the centuries (...) past” (Poincaré 1958, p. 14). The view criticized by Poincaré corresponds to Frege’s idea that the development of mathematics can be described as an activity of system building, where each system is supposed to provide a complete representation for a certain mathematical field and must be pitilessly torn down whenever it fails to achieve such an aim. All facts concerning any mathematical field must be fully organized in a given system because “in mathematics we must always strive after a system that is complete in itself” (Frege 1979, p. 279). Frege is aware that systems introduce rigidity and are in conflict with the actual development of mathematics because “in history we have development; a system is static”, but he sticks to the view that “science only comes to fruition in a system” because “only through a system can we achieve complete clarity and order” (Frege 1979, p. 242). He even goes so far as saying that “no science can be so enveloped in obscurity as mathematics, if it fails to construct a system” (Frege 1979, p. 242). By ‘system’ Frege means ‘axiomatic system’. In his view, in mathematics we cannot rest content with the fact that “we are convinced of something, but we must strive to obtain a clear insight into the network of inferences that support our conviction”, that is, to find “what the primitive truths are”, because “only in this way can a system be constructed” (Frege 1979, p. 205). The primitive truths are the principles of the axiomatic system. Frege’s stress on the role of systems also determines his views on the growth of mathematical knowledge.. (shrink)

We propose that, for the purpose of studying theoretical properties of the knowledge of an agent with Artificial General Intelligence (that is, the knowledge of an AGI), a pragmatic way to define such an agent’s knowledge (restricted to the language of Epistemic Arithmetic, or EA) is as follows. We declare an AGI to know an EA-statement φ if and only if that AGI would include φ in the resulting enumeration if that AGI were commanded: “Enumerate all the (...) EA-sentences which you know.” This definition is non-circular because an AGI, being capable of practical English communication, is capable of understanding the everyday English word “know” independently of how any philosopher formally defines knowledge; we elaborate further on the non-circularity of this circular-looking definition. This elegantly solves the problem that different AGIs may have different internal knowledge definitions and yet we want to study knowledge of AGIs in general, without having to study different AGIs separately just because they have separate internal knowledge definitions. Finally, we suggest how this definition of AGI knowledge can be used as a bridge which could allow the AGI research community to import certain abstract results about mechanical knowing agents from mathematical logic. (shrink)

Most scholars think of David Hilbert's program as the most demanding and ideologically motivated attempt to provide a foundation for mathematics, and because they see technical obstacles in the way of realizing the program's goals, they regard it as a failure. Against this view, Curtis Franks argues that Hilbert's deepest and most central insight was that mathematical techniques and practices do not need grounding in any philosophical principles. He weaves together an original historical account, philosophical analysis, and his own (...) development of the meta-mathematics of weak systems of arithmetic to show that the true philosophical significance of Hilbert's program is that it makes the autonomy of mathematics evident. The result is a vision of the early history of modern logic that highlights the rich interaction between its conceptual problems and technical development. (shrink)

If, as I grant, mathematical objects are abstract entities existing outside of space and time, and if the idea of supernaturally grasping abstract entities is scientifically unacceptable, then we need to explain how we can attain mathematical knowledge using our ordinary faculties. I try to meet this challenge through a postulational account of the genesis of our mathematical knowledge, according to which our ancestors introduced mathematical objects by first positing geometric ideals and then (...) postulating abstract mathematical entities. Since positing involves simply introducing a discourse about objects and affirming their existence, positing mathematical objects involves nothing more serious than writing fiction. For this reason, postulational approaches seem better suited for conventionalists; so in the second part of this chapter, I explain how positing in mathematics is different from positing in fiction, and how we can gain knowledge from the former. Finally, I try to make sense of the idea that mathematical postulates are about an independent mathematical reality and that we can refer to that reality through them, by giving an immanent and disquotational account of reference and contrasting it with a transcendent/causal account. (shrink)

Suggests that the recent emphasis on Benacerraf's access problem locates the peculiarity of mathematical knowledge in the wrong place. Instead we should focus on the sense in which mathematical concepts are or might be "armchair concepts" – concepts about which non-trivial knowledge is obtainable a priori.

This paper attempts to show that mathematical knowledge does not grow by a simple process of accumulation and that it is possible to provide a quasi-empirical (in Lakatos's sense) account of mathematical theories. Arguments supporting the first thesis are based on the study of the changes occurred within Eudidean geometry from the time of Euclid to that of Hilbert; whereas those in favour of the second arise from reflections on the criteria for refutation of mathematical theories.

The article is devoted to identifying the value of the phenomenon of aesthetic value and beauty of mathematical knowledge and the beauty of mathematical theory of teaching mathematics. The aesthetic potential of mathematical knowledge allows the use of theater technology in the educational process with the active dialogic interaction between teacher and students. The criteria of beauty in mathematical theories are distinguished: the realization of beauty as the unity of the whole, and in the (...) disclosure of the complex through the elementary; methodological interpretation of the beauty in the community of mathematical structures and optimal information content of the meta-language of mathematics; the practical embodiment of beauty in the formalization of the infinite through the finite. The beauty of mathematics is the force that permeates all the ‘layers of knowledge‘ not along, and across, although the effectiveness of mathematical activity due to aesthetic laws, which do not always lend themselves to unambiguous interpretation. In the article it is stated that, depending on the educational goals of communicative impact on the audience, in fact, ‘mathematical lectures theatricality‘ can have different characteristics, the most important of which are teachers artistry and artistic director’s work of a teacher. This cultural phenomenon that includes the theatrical talent, helps create an atmosphere of cooperation needed in varying degrees of activity for pedagogical interaction. The author believes that such approach, developed on the basis of the Stanislavsky system, allows university professors of mathematics significantly improve mathematical lectures. (shrink)

A view of the sources of mathematical knowledge is sketched which emphasizes the close connections between mathematical and empirical knowledge. A platonistic interpretation of mathematical discourse is adopted throughout. Two skeptical views are discussed and rejected. One of these, due to Maturana, is supposed to be based on biological considerations. The other, due to Dummett, is derived from a Wittgensteinian position in the philosophy of language. The paper ends with an elaboration of Gödel's analogy between (...) the mathematician and the physicist. (shrink)

This paper focuses on Kojève’s account of history and philosophy of science. Kojève’s understanding of science can be characterized as internalism, which is evident in his holistic view of philosophy, theology, quantum physics, and the history of classical Newtonian mechanics. It precipitates the facilitation of a further inquiry into the Christian genesis, secular evolution, and subsequent de-Christianization of scientific thought. The paper includes a critical scrutiny of Kojève’s philosophical tenets, followed by a comparative analysis of the views of Hegel, (...) Koyré, and Kojève. The primary objective of this research is to juxtapose Kojève’s doctrines with Hegel’s contemplations on the history and philosophy of science. In addition to identifying affinities, notably the emphasis on the Christian concept of God’s Incarnation for the advancement of science, I draw the distinctions between the positions of Hegel, Kojève, and Koyré, specifically concerning the valuation of mathematical knowledge. (shrink)

The text of Martin Heidegger’s 1927–28 university lecture course on Emmanuel Kant’s Critique of Pure Reason presents a close interpretive reading of the first two parts of this masterpiece of modern philosophy. In this course, Heidegger continues the task he enunciated in Being and Time as the problem of dismatling the history of ontology, using temporality as a clue. Within this context the relation between philosophy, ontology, and fundamental ontology is shown to be rooted in the genesis of the (...) modern mathematical sciences. Heidegger demonstrates that objectification of beings as beings is inseparable from knowledge a priori, the central problem of Kant’s Critique. He concludes that objectification rests on the productive power of imagination, a process that involves temporality, which is the basic constitution of humans as beings. (shrink)

The logician Kurt Godel in 1951 established a disjunctive thesis about the scope and limits of mathematical knowledge: either the mathematical mind is equivalent to a Turing machine (i.e., a computer), or there are absolutely undecidable mathematical problems. In the second half of the twentieth century, attempts have been made to arrive at a stronger conclusion. In particular, arguments have been produced by the philosopher J.R. Lucas and by the physicist and mathematician Roger Penrose that intend (...) to show that the mathematical mind is more powerful than any computer. These arguments, and counterarguments to them, have not convinced the logical and philosophical community. The reason for this is an insufficiency if rigour in the debate. The contributions in this volume move the debate forward by formulating rigorous frameworks and formally spelling out and evaluating arguments that bear on Godel's disjunction in these frameworks. The contributions in this volume have been written by world leading experts in the field. (shrink)

The philosophy of mathematical practice sometimes investigates the social constitution of mathematics but does not always make explicit the philosophical-normative framework that guides the discussion. This chapter investigates some recent proposals in the philosophy of mathematical practice that compare social facts and mathematical objects, discussing similarities and differences. An attempt will be made to identify, through a comparison with three different perspectives in social ontology, the kind of objectivity attributed to mathematical knowledge, the type of (...) representational or non-representational semantics adopted, and the justificatory or coordinative role entrusted to axiomatics. After a brief introduction to key issues in social ontology, Sect. 3 of the chapter offers a survey of contributions by Feferman, Ferreirós, and Cole, highlighting a difference between approaches based on rules, practices, and intentional states, respectively. Section 4.1 also discusses results by Carter, Collin, Giardino, and Pantsar, focusing on differences between the objectivity of knowledge and objectivity of objects and showing that many authors combine a realist, structuralist, and pragmatist perspective, as well as the idea that objectivity comes in degrees. Section 4.2 focuses on the kind of semantics that is adopted in socially oriented approaches. A representational semantics is preferred by authors grounding their views on mental states, whereas a non-representational semantics best fits with views based on practices. Section 5 considers how axiomatics can be understood in a social perspective. Axiomatics generally plays a justificatory role also in theories that aim to explain the social constitution of mathematical knowledge. Yet, if one shifts from approaches based on mental states to approaches anchored on rules or habits, axiomatics can be viewed as an institution based on obligations, functions, coordination problems, and agent’s actions and roles, thus playing an organizational and coordination role. (shrink)

On knowledge and practices: a manifesto -- The web of practices -- Agents and frameworks -- Complementarity in mathematics -- Ancient Greek mathematics: a role for diagrams -- Advanced math: the hypothetical conception -- Arithmetic certainty -- Mathematics developed: the case of the reals -- Objectivity in mathematical knowledge -- The problem of conceptual understanding.

This thesis is about the nature of proofs in mathematics as it is practiced, contrasting the informal proofs found in practice with formal proofs in formal systems. In the first chapter I present a new argument against the Formalist-Reductionist view that informal proofs are justified as rigorous and correct by corresponding to formal counterparts. The second chapter builds on this to reject arguments from Gödel's paradox and incompleteness theorems to the claim that mathematics is inherently inconsistent, basing my objections on (...) the complexities of the process of formalisation. Chapter 3 looks into the relationship between proofs and the development of the mathematical concepts that feature in them. I deploy Waismann's notion of open texture in the case of mathematical concepts, and discuss both Lakatos and Kneebone's dialectical philosophies of mathematics. I then argue that we can apply work from conceptual engineering to the relationship between formal and informal mathematics. The fourth chapter argues for the importance of mathematical knowledge-how and emphasises the primary role of the activity of proving in securing mathematical knowledge. In the final chapter I develop an account of mathematical knowledge based on virtue epistemology, which I argue provides a better view of proofs and mathematical rigour. (shrink)

Mathematicians do not claim to know a proposition unless they think they possess a proof of it. For all their confidence in the truth of a proposition with weighty non-deductive support, they maintain that, strictly speaking, the proposition remains unknown until such time as someone has proved it. This article challenges this conception of knowledge, which is quasi-universal within mathematics. We present four arguments to the effect that non-deductive evidence can yield knowledge of a mathematical proposition. We (...) also show that some of what mathematicians take to be deductive knowledge is in fact non-deductive. 1 Introduction2 Why It Might Matter3 Two Further Examples and Preliminaries4 An Exclusive Epistemic Virtue of Proof?5 Analyses of Knowledge6 The Inductive Basis of Deduction7 Physical to Mathematical Linkages8 Conclusion. (shrink)

The article deals with historical dynamics of implicit and intuitive elements of mathematical knowledge. The author describes historical dynamics of implicit and intuitive elements and discloses a historical and evolutionary mechanism of building up mathematical knowledge. Each requirement to increase the level of theoretical rigor in mathematics is historically realized as a three-stage process. The first stage considers some general conditions of valid mathematical knowledge recognized by the mathematical community. The second one reveals (...) the level of theoretical rigor increasing, while the third one is characterized by explication of the hidden lemmas. A detailed discussion of historical substantiation of the basic algebra theorem is conducted according to the proposed technique. (shrink)

In Systematicity, Paul Hoyningen-Huene answers the question "What is science?" by proposing that scientific knowledge is primarily distinguished from other forms of knowledge, especially everyday knowledge, by being more systematic. "Science" is here understood in the broadest possible sense, encompassing not only the natural sciences but also mathematics, the social sciences, and the humanities. The author develops his thesis in nine dimensions in which it is claimed that science is more systematic than other forms of knowledge: (...) regarding descriptions, explanations, predictions, the defense of knowledge claims, critical discourse, epistemic connectedness, an ideal of completeness, knowledge generation, and the representation of knowledge. He compares his view with positions on the question held by philosophers from Aristotle to Nicholas Rescher. The book concludes with an exploration of some consequences of Hoyningen-Huene's view concerning the genesis and dynamics of science, the relationship of science and common sense, normative implications of the thesis, and the demarcation criterion between science and pseudo-science. (shrink)

In Systematicity, Paul Hoyningen-Huene answers the question "What is science?" by proposing that scientific knowledge is primarily distinguished from other forms of knowledge, especially everyday knowledge, by being more systematic. "Science" is here understood in the broadest possible sense, encompassing not only the natural sciences but also mathematics, the social sciences, and the humanities. The author develops his thesis in nine dimensions in which it is claimed that science is more systematic than other forms of knowledge: (...) regarding descriptions, explanations, predictions, the defense of knowledge claims, critical discourse, epistemic connectedness, an ideal of completeness, knowledge generation, and the representation of knowledge. He compares his view with positions on the question held by philosophers from Aristotle to Nicholas Rescher. The book concludes with an exploration of some consequences of Hoyningen-Huene's view concerning the genesis and dynamics of science, the relationship of science and common sense, normative implications of the thesis, and the demarcation criterion between science and pseudo-science. (shrink)

The nature of mathematical knowledge can be understood only by locating the knowing mathematician in an epistemic community. This claim is defended by extending Kripke's version of the Private Language Argument to include informal rules and using Godelian results to argue that such rules rules necessary in mathematics. A committed formalist might evade Kripke's original argument by positing internal mechanisms that determine rule -governed behavior. However, in the presence of informal rules, the formalist position collapses into the extreme (...) skepticism that the Private Language Argument works against. The existence of a community of rule followers provides the only viable alternative to such skepticism. (shrink)

The ArgumentRecent studies in the philosophy of mathematics have increasingly stressed the social and historical dimensions of mathematical practice. Although this new emphasis has fathered interesting new perspectives, it has also blurred the distinction between mathematics and other scientific fields. This distinction can be clarified by examining the special interaction of thebodyandimagesof mathematics.Mathematics has an objective, ever-expanding hard core, the growth of which is conditioned by socially and historically determined images of mathematics. Mathematics also has reflexive capacities unlike those (...) of any other exact science. In no other exact science can the standard methodological framework usedwithinthe discipline also be used to study the nature of the discipline itself.Although it has always been present in mathematical research, reflexive thinking has become increasingly central to mathematics over the past century. Many of the images of the discipline have been dictated by the increase in reflexive thinking which has also determined a great portion of the contemporary philosophy and historiography of mathematics. (shrink)

Mathematics for Hume is the exemplary field of demonstrative knowledge. Ideally, this knowledge is a priori as it arises only from the comparison of ideas without any further empirical input; it is certain because demonstration consist of steps that are intuitively evident and infallible; and it is also necessary because the possibility of its falsity is inconceivable as it would imply a contradiction. But this is only the ideal, because demonstrative sciences are human enterprises and as such they (...) are just as fallible as their human practitioners. According to the reading suggested here, Hume develops a radical sceptical challenge for mathematics, and thereby he undermines the knowledge claims associated with demonstrative reasoning. But Hume does not stop there: he also offers resources for a sceptical solution to this challenge, one that appeals crucially to social practices, and sketches the social genealogy of a community-wide mathematical certainty. While explaining this process, he relies on the conceptual resources of his faculty psychology that helps him to distinguish between the metaphysics and practices of mathematical knowledge. His account explains why we have reasons to be dubious about our reasoning capacities, and also how human nature and sociability offers some remedy from these epistemic adversities. (shrink)

Much attention was devoted, at the Twenty-fifth Congress of the CPSU, to the ideological struggle now in progress and to the need for prompt and decisive refutation of bourgeois ideology. The problem of the genesis of science is now central to fierce ideological debates.

This book provides a survey of the major issues in the philosophy of mathematics, such as ontological questions regarding the nature of mathematical objects, epistemic questions about the acquisition of mathematical knowledge, and the intriguing riddle of the applicability of mathematics to the physical world. Some of these issues go back to the nascent years of mathematics itself, others are just beginning to draw the attention of scholars. In addressing these questions, some of the papers in this (...) volume wrestle with them directly, while others use the writings of philosophers such as Hume and Wittgenstein to approach their problems by way of interpretation and critique. The contributors include prominent philosophers of science and mathematics as well as promising younger scholars. The volume seeks to share the concerns of philosophers of mathematics with a wider audience and will be of interest to historians, mathematicians and philosophers alike. (shrink)

History and Philosophy of Logic, Volume 32, Issue 2, Page 177-183, May 2011.