O artigo levanta questões que interessam ao tema do magistério filosófico: o sentido original do conceito de filosofia, sua realização peculiar no método pedagógico de Sócrates, a ideia da filosofia como saber e como ciência. É a filosofia, para o professor e para o aluno, principalmente busca ou aquisição? O artigo considera, neste contexto, a magnitude do empreendimento filosófico e, por outro lado, as possibilidades da razão humana, que as filosofias mesmas diversamente tematizam.El artículo plantea cuestiones que interesan al magisterio (...) filosófico: el sentido original del concepto de filosofía, su realización peculiar en el método de Sócrates, la idea de la filosofía como saber y como ciencia. Es la filosofía, para el profesor y para el alumno, principalmente búsqueda o adquisición? El artículo considera, en este contexto, la magnitud del emprendimiento filosóficoy, por otro lado, las posibilidades de la razón humana, que las mismas filosofías diversamente tematizan. (shrink)

Roughly, a proof of a theorem, is “pure” if it draws only on what is “close” or “intrinsic” to that theorem. Mathematicians employ a variety of terms to identify pure proofs, saying that a pure proof is one that avoids what is “extrinsic,” “extraneous,” “distant,” “remote,” “alien,” or “foreign” to the problem or theorem under investigation. In the background of these attributions is the view that there is a distance measure (or a variety of such measures) between mathematical statements and (...) proofs. Mathematicians have paid little attention to specifying such distance measures precisely because in practice certain methods of proof have seemed self- evidently impure by design: think for instance of analytic geometry and analytic number theory. By contrast, mathematicians have paid considerable attention to whether such impurities are a good thing or to be avoided, and some have claimed that they are valuable because generally impure proofs are simpler than pure proofs. This article is an investigation of this claim, formulated more precisely by proof- theoretic means. After assembling evidence from proof theory that may be thought to support this claim, we will argue that on the contrary this evidence does not support the claim. (shrink)

Traditional geometry concerns itself with planimetric and stereometric considerations, which are at the root of the division between plane and solid geometry. To raise the issue of the relation between these two areas brings with it a host of different problems that pertain to mathematical practice, epistemology, semantics, ontology, methodology, and logic. In addition, issues of psychology and pedagogy are also important here. To our knowledge there is no single contribution that studies in detail even one of the aforementioned areas.

Many mathematicians have cited depth as an important value in their research. However, there is no single widely accepted account of mathematical depth. This article is an attempt to bridge this gap. The strategy is to begin with a discussion of Szemerédi's theorem, which says that each subset of the natural numbers that is sufficiently dense contains an arithmetical progression of arbitrary length. This theorem has been judged deep by many mathematicians, and so makes for a good case on which (...) to focus in analyzing mathematical depth. After introducing the theorem, four accounts of mathematical depth will be considered. (shrink)

This article will consider imagination in mathematics from a historical point of view, noting the key moments in its conception during the ancient, modern and contemporary eras.

Many mathematicians and philosophers of mathematics believe some proofs contain elements extraneous to what is being proved. In this paper I discuss extraneousness generally, and then consider a specific proposal for measuring extraneousness syntactically. This specific proposal uses Gentzen's cut-elimination theorem. I argue that the proposal fails, and that we should be skeptical about the usefulness of syntactic extraneousness measures.

Schizophrenia is usually described as a fragmentation of subjective experience and the impossibility to engage in meaningful cultural and intersubjective practices. Although the term schizophrenia is less than 100 years old, madness is generally believed to have accompanied mankind through its historical and cultural ontogeny. What does it mean to be “mad”? The failure to adopt social practices or to internalize cultural values of common sense? Despite the vast amount of literature and research, it seems that the study of schizophrenia (...) and of the psychoses is suffering from a generic disintegration. In this introduction, we offer an historical overview of the variety of theories and approaches to schizophrenia. We also provide an overview of how the authors in this volume attempt an integrative account where training, practice, theory and research are considered as parts of a larger whole. This is a varied and pluralistic volume, and it is up to the readers to make use of different chapters according to their own needs. (shrink)

Over the years many mathematicians have voiced a preference for proofs that stay “close” to the statements being proved, avoiding “foreign”, “extraneous”, or “remote” considerations. Such proofs have come to be known as “pure”. Purity issues have arisen repeatedly in the practice of arithmetic; a famous instance is the question of complex-analytic considerations in the proof of the prime number theorem. This article surveys several such issues, and discusses ways in which logical considerations shed light on these issues.

El artículo versa sobre la relación entre epistemes, espacio y poder. Cuestiona la presunción de universalidad tal como es pensada por la tradición hegemónica europea -eurocentrismo- pero rechaza cualquier pretensión de universalidad unidireccional venga de donde venga. Propone la idea de múltiples universalidades posibles partiendo del supuesto de la incompletud de toda cultura. El artículo invita a la crítica de la colonialidad del saber y del poder que están imbricados mutuamente, y a la atención que debemos prestar al lugar de (...) enunciación de cada discurso, lugar al mismo tiempo social y geográfico. Parte del pensamiento subalterno y valoriza los espacios de conflicto como epistémicamente densos, aportando además que el debate acerca de los territorios quizás sea la mejor expresión de la naturalización de las relaciones sociales y de poder que el territorio comporta, significando, en verdad, cuán profundas son las transformaciones en curso en los días actuales. (shrink)

During the last few months, several pioneer genome‐wide transcriptomic, proteomic and metabolomic studies have revolutionised the understanding of bacterial biological processes, leading to a picture that resembles eukaryotic complexity. Technological advances such as next‐generation high‐throughput sequencing and high‐density oligonucleotide microarrays have allowed the determination, in several bacteria, of the entire boundaries of all expressed transcripts. Consequently, novel RNA‐mediated regulatory mechanisms have been discovered including multifunctional RNAs. Moreover, resolution of bacterial proteome organisation (interactome) and global protein localisation (localizome) have unveiled an (...) unanticipated complexity that highlights the significance of protein multifunctionality and localisation in the cell. Also, analysis of a complete bacterial metabolic network has again revealed a high fraction of multifunctional enzymes and an unexpectedly high level of metabolic responses and adaptation. Altogether, these novel approaches have permitted the deciphering of the entire physiological landscape of one of the smallest bacteria, Mycoplasma pneumoniae. Here, we summarise and discuss recent findings aimed at defining the blueprint of any prokaryote. (shrink)

Review & Reply. Review of Gonçalves, R.T. (tr.) Lucrécio. Sobre a Natureza das Coisas(2021).

The Turing test has been studied and run as a controlled experiment and found to be underspecified and poorly designed. On the other hand, it has been defended and still attracts interest as a test for true artificial intelligence (AI). Scientists and philosophers regret the test’s current status, acknowledging that the situation is at odds with the intellectual standards of Turing’s works. This article refers to this as the Turing Test Dilemma, following the observation that the test has been under (...) discussion for over seventy years and still is widely seen as either too bad or too good to be a valuable experiment for AI. An argument that solves the dilemma is presented, which relies on reconstructing the Turing test as a thought experiment in the modern scientific tradition. It is argued that Turing’s exposition of the imitation game satisfies Mach’s characterization of the basic method of thought experiments and that Turing’s uses of his test satisfy Popper’s conception of the critical and heuristic uses of thought experiments and Kuhn’s association of thought experiments to conceptual change. It is emphasized how Turing methodically varied the imitation game design to address specific challenges posed to him by other thinkers and how his test illustrates a property of the phenomenon of intelligence and suggests a hypothesis on machine learning. This reconstruction of the Turing test provides a rapprochement to the conflicting views on its value in the literature. (shrink)

Turing’s much debated test has turned 70 and is still fairly controversial. His 1950 paper is seen as a complex and multilayered text, and key questions about it remain largely unanswered. Why did Turing select learning from experience as the best approach to achieve machine intelligence? Why did he spend several years working with chess playing as a task to illustrate and test for machine intelligence only to trade it out for conversational question-answering in 1950? Why did Turing refer to (...) gender imitation in a test for machine intelligence? In this article, I shall address these questions by unveiling social, historical and epistemological roots of the so-called Turing test. I will draw attention to a historical fact that has been only scarcely observed in the secondary literature thus far, namely that Turing’s 1950 test emerged out of a controversy over the cognitive capabilities of digital computers, most notably out of debates with physicist and computer pioneer Douglas Hartree, chemist and philosopher Michael Polanyi, and neurosurgeon Geoffrey Jefferson. Seen in its historical context, Turing’s 1950 paper can be understood as essentially a reply to a series of challenges posed to him by these thinkers arguing against his view that machines can think. Turing did propose gender learning and imitation as one of his various imitation tests for machine intelligence, and I argue here that this was done in response to Jefferson's suggestion that gendered behavior is causally related to the physiology of sex hormones. (shrink)

In this paper I begin by extending two results of Solovay; the first characterizes the possible Turing degrees of models of True Arithmetic (TA), the complete first-order theory of the standard model of PA, while the second characterizes the possible Turing degrees of arbitrary completions of P. I extend these two results to characterize the possible Turing degrees of m-diagrams of models of TA and of arbitrary complete extensions of PA. I next give a construction showing that the conditions Solovay (...) identified for his characterization of degrees of models of arbitrary completions of PA cannot be dropped (I showed that these conditions cannot be simplified in the paper. (shrink)

Une preuve est pure si, en gros, elle ne réfère dans son développement qu’à ce qui est « proche » de, ou « intrinsèque » à l’énoncé à prouver. L’infinité des nombres premiers, un théorème classique de l’arithmétique, est un cas d’étude particulièrement riche pour les recherches philosophiques sur la pureté. Deux preuves différentes de ce résultat sont ici considérées, à savoir la preuve euclidienne classique et une preuve « topologique » plus récente proposée par Furstenberg. D’un point de vue (...) naïf, il semblerait que la première soit pure et la seconde impure. Des objections à cette vue naïve sont ici considérées et réfutées. Concernant la preuve euclidienne, la question relève de la logique, notamment de la définissabilité arithmétique de l’addition en termes de successeur et de divisibilité telle que démontrée par Julia Robinson, tandis qu’en ce qui concerne la preuve topologique, la question relève de la sémantique, notamment pour tout ce qui touche au problème de savoir ce qui est « inclus » dans le contenu d’énoncés particuliers.A proof is pure, roughly, if it draws only on what is « close » or « intrinsic » to the statement being proved. The infinitude of prime numbers, a classical theorem of arithmetic, is a rich case study for philosophical investigation of purity. Two different proofs of this result are considered, namely the classical Euclidean proof and a more recent « topological » proof by Furstenberg. Naively the former would seem to be pure and the latter to be impure. Objections to these naive views are considered and met. In the case of the former the issue rests on logical matters, specifically the arithmetic definability of addition in terms of successor and divisibility shown by Julia Robinson, while in the case of the latter the issue rests on semantic matters, specifically with respect to what is « contained » in the content of particular statements. (shrink)

O estudo do valor é indissociável da análise dos conceitos de bom e Bem, pelo que qualquer base metodológica que lhe sirva de alicerce tem de partir do exame destes. Assumindo os riscos do sem-sentido, recusamos uma redução relativista do bom predicativo a bom atributivo e de bom a bom para, bom do ponto de vista de, bom relativo a, e avançamos para a discussão de algumas tentativas de identificação do bom, nomeadamente com o prazer, o desejo e o escolhido. (...) Seguindo-se o esforço de mostrar como bom é insusceptível de uma remissão para o psicológico ou para o subjectivo, o ensaio procura evidenciar a ideia de Bem como o conceito fundamental e indefinível para a compreensão do bom, a partir da sua estruturação teleológica. Propomo-nos, portanto, estabelecer bom = aquilo que remete para o Bem, sendo o conceito de valor o reflexo da relação mediada com o Bem, indissociável do domínio ético. Por fim, consagra-se a possibilidade de uma nova fundação de um método na teoria do valor baseado nos preceitos alcançados, distinguindo-se o estudo do valor em analítico, substantivo e prático. The study of value is inseparable from the analysis of the concepts of good and the Good, so any methodological basis which serves as groundwork must take from these. Assuming the risks of non-sensicality, we propose to refuse a relativistic reduction of predicative good to attributive good and of good to good for, good from the point of view of and good-relative-to and discuss some attempts to identify good, particularly with pleasure, desire and choice. Following the effort to show how good is not liable to be ascribed to the psychological or subjective, the essay seeks to highlight the idea of the Good as the fundamental concept - therefore undefinable - to understand good as from its teleological structure. We therefore propose to establish good = what refers to the Good, and the concept of value as one which reflects the mediated relationship with the Good, inseparable from the ethical. Finally, we establish the possibility of a new foundation of a method of the theory of value based in the reached percepts, distinguishing between analytical, substantive and practical inquiries on value. (shrink)

We extend an independence result proved in our earlier paper "Solovay's Theorem Cannot Be Simplified" (Annals of Pure and Applied Logic 112 (2001)). Our method uses the Barwise.

El artículo procura analizar el contexto en que surge el reciente debate acerca de agua identificando los principales protagonistas que vienen conformando ese campo que es, al mismo tiempo, teórico, técnico y político. Demuestra como los organismos multilaterales y sus intelectuales (con el discurso de la escasez) vienen impulsando políticas que buscan transferir el papel del Estado en beneficio de los agentes empresariales estimulando procesos de privatización de agua y de su mercantilización. El artículo presenta una serie de casos en (...) que este proceso de privatización aumenta los conflictos en torno del agua. (shrink)

Although values have increasingly received attention in psychiatric literature over the last three decades, their role has been only partially acknowledged in psychiatric classification endeavors. The review p..

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)

A variety of projects in proof theory of relevance to the philosophy of mathematics are surveyed, including Gödel's incompleteness theorems, conservation results, independence results, ordinal analysis, predicativity, reverse mathematics, speed-up results, and provability logics.

O Principado priva o senado de muitos dos seus poderes tradicionais. No entanto, durante a crise de 69 d. C., são vários os candidatos ao lugar de príncipe e os generais que afirmam nos seus discursos poder contar com o apoio do senado. Este artigo desina-se a averiguar o exato valor atribuído a este apoio num contexto em que o senado revela uma generalizada incapacidade de ação política.

Sound and listening are intrinsically linked to how we experience and engage with places and communities. This guide invites landscape architects and urban designers to become soundscape architects and offers practical advice on sound and listening applicable to each stage of a design project: from reading the environment to intervening on it. This book foregrounds listening as an affective mediator between subjects and multispecies environments, and a vehicle to think and conceptualise environmental design beyond prevailing visual and human-centred modes. The (...) guide expands landscape architects' and urban designers' tools and skills to assess existing soundscapes, predict how those soundscapes will be altered through their designs, consider sound as a creative and active part of the design process, and envisage how users might perceive and be affected by those soundscapes as they evolve in time. The volume sits in the interface of research and practice, and incorporates theoretical, methodological and creative contributions from acoustic ecology, ecoacoustics, bioacoustics and sound art. Each of the design stages is illustrated through project examples that demonstrate the many advantages of incorporating attentive listening and sound into Landscape Architecture and Urban Design Practice. The book shows how incorporating listening and sounding as part of the design process promotes slow and subtle ways of practice, adds social and ecological value through the reduction of noise pollution and by monitoring the health of habitats, and enables the design of soundscapes that complement the character and design intent of a scheme and elicit joy and wonder. The book will be of interest to landscape architects, together with other design professionals such as urban designers, architects, geographers and engineers, who play a primary role in the composition of the soundscape. (shrink)

Introduction Context: In Portugal assisted death was approved in February 2020 by the Parliament, although the law is not yet in force. Objectives To find out what doctors think about those practices. Methods A link to a questionnaire was sent by email three times, at intervals of one week, to the doctors registered in the Northern Section of the Portuguese Medical Association, before the Parliamentary approval. Results The questionnaire was returned by 1148 physicians. A minority of doctors would practice a (...) form of assisted death under the present law or if it was legalized, but a higher percentage think that euthanasia should be legalized, and more would like to have that option if they themselves were in a terminal phase of a disease. Religion has a strong influence on the attitudes of doctors, so too does their contact with patients in a terminal phase, as doctors who deal with more patients with far advanced diseases are more likely to be unfavorable to assisted death. On the other hand, younger doctors are more in favor of these practices. Conclusion The small percentage of questionnaires sent back is a weakness in this study and casts doubts on the generalizability of the conclusions. However, this is, so far, the best approximation to the opinions of Portuguese doctors on assisted death. (shrink)

El artículo busca explicar las razones de la conversión al discurso de la sustentabilidad, por parte del sector de los grandes productores de petróleo y el de los latifundios empresariales de monocultivos, que hegemonizan el agro negocio. Para ello, se analiza la forma como históricamente se ha constituido el bloque de poder hegemónico conformado como un complejo de poder técnico-científico-industrial-financiero-militar-mediático, del cual también participan los dueños de los agronegocios -el viejo bloque de poder modernizado. Se concluye que se avecina una (...) verdadera tragedia con la expropiación generalizada que advendrá de la expansión del monocultivo latifundista moderno-colonial de los agro negociantes que quieren someter la producción agrícola a la producción de combustibles de biomasa. Se señala la enorme importancia de comprender adecuadamente el carácter mundial del capitalismo y del papel del desarrollo de las fuerzas productivas que el capital impone al mundo y la necesidad de ejercer el derecho a la diferencia anclada en la defensa del territorio. (shrink)

In a recent paper, Trevor Hedberg and Jordan Huzarevich assessed a number of objections against practical apatheism, concluding that they are either unsuccessful or at least very controversial. The aim of this discussion note is to contribute to their discussion by presenting and analysing a short argument for a reason in favor of apatheism; one that appeals to a particular universalist formulation of the thesis of theodical individualism. After briefly introducing a distinction between a particular and a universal version of (...) apatheism, I present the argument, assess its merits in light of some objections to it, I conclude that at least one reason in favor of a particular practical apatheistic position is warranted by it. Finally, I suggest some alternative reasons that could be advanced in favor of apatheism, and I distinguish between strong and weak forms of apatheism. (shrink)

Arguments on determinism and fate must be placed in human being's industry to over come life uncertainties. For a long time was regarded suitable to postulate an ontologi cal necessity to progress towards a more and more sure knowledge. Nowadays it is tended to believe that determinism is unacceptable either theoretical or practically. Nevertheless it does not mean that the notion of fate has to be legitimazed in all levels.

The project of this Précis de philosophie de la logique et des mathématiques (vol. 1 under the direction of F. Poggiolesi and P. Wagner, vol. 2 under the direction of A. Arana and M. Panza) aims to offer a rich, systematic and clear introduction to the main contemporary debates in the philosophy of mathematics and logic. The two volumes bring together the contributions of thirty researchers (twelve for the philosophy of logic and eighteen for the philosophy of mathematics), specialists (...) in the history or philosophy of logic or mathematics. Each volume consists of ten chapters; each chapter is about forty pages, is independent from the others and deals with a particular philosophical question about logic or mathematics. The objective is to offer to the French-speaking reader a reference book. On many of the issues addressed, this Précis offers the first clear and thorough synthesis that is available in French. -/- This book is intended for third year / master's students, but also for teachers of philosophy in secondary or higher education, for logicians and mathematicians willing to take a philosophical look at the fundamental concepts of their discipline, and for any reader interested in the basic issues discussed in the philosophy of logic and mathematics. Each chapter is self-contained, although some basic training in logic (or in mathematics, as the case may be) is required. The book should be looked at as a comprehensive framework for contemporary discussions in the philosophy of logic and mathematics, while also giving some guidance on the disciplinary content required by these discussions. -/- The first three chapters of the present volume (on the philosophy of mathematics) are devoted to the history of the philosophy of mathematics: from antiquity to the modern era, and from this period to the foundational crisis of the nineteenth century, to the twentieth century. Next, four chapters deal with the crucial questions for the philosophy of mathematics of the twentieth century: the opposition and/or comparison of set theory and category theory as foundational frameworks for mathematics; mathematical constructivism; the analysis of calculability; and Benaceraff's problem. The two following chapters are focused on the philosophy of mathematical practice, by treating the notion of ideals of proof, in particular explanation and purity, and the notion of informal proof and the use of visual artifacts in mathematical argumentation. Finally the last chapter deals with the applicability of mathematics, including the role of probability. -/- In Chapter 1, Sébastien Maronne and David Rabouin present the history of the philosophy of mathematics from antiquity to the modern era. The goal is to show that the philosophy of mathematics has a history as ancient as mathematics and philosophy itself, a history largely continuous with the latter, often by staying out of sync with respect to the former. In Chapter 2, Sébastien Gandon discusses the question of the foundations of mathematics from Kant to the end of the nineteenth century. By tying this question to Kant, he is able to show that the discussion of the foundations of mathematics that occupied a large part of these two centuries has much older roots. Accent is placed on the differences between the framework proposed by Kant for making sense of classical mathematics and the use made by Frege and then Russell of logical tools for giving a different reading of arithmetic and real analysis. The link between between these different conceptions and the evolution of mathematics itself is also underlined. -/- Chapter 3, written by Hourya Benis-Sinaceur and Mirna Džamonja, deals with the evolution of mathematics in the twentieth century, notably the structuralist turn of Dedekind, E. Noether, and Bourbaki, and certain more recent developments. They devote themselves, among other things, to explaining how these developments can be understood as responses to and extensions of questions posed during the foundational debate of the immediately preceding era. -/- In Chapter 4, Jean-Pierre Marquis and Jean-Jacques Szczeciniarz discuss the different foundational options coming from set theory and category theory. They discuss, among other things, Lawvere's program to replace ZF with an axiomatization of the category of sets, the latter judged better adapted to the needs of contemporary mathematics. Different reactions, as much philosophical as technical, raised by this program are also taken into account. -/- Constructive mathematics are the object of Chapter 5, in which Gerhard Heinzmann and Mark van Atten examine in a critical and comparative manner a variety of constructivisms, including intuitionism, constructive type theory, predicativism, and finitism. -/- In Chapter 6, Guido Gherardi and Maël Pegny present computability theory, stressing its philosophical consequences. They discuss the basic concepts of this theory, for the computability of both the natural numbers and the real numbers. They then tackle the theory of computational complexity. -/- In Chapter 7, Andrea Sereni and Fabrice Pataut present Benacerraf's problem, opposing every possible account of mathematical knowledge to the availability of a theory of truth founded on an abstract ontology of mathematical objects. They give an overview of the original problem and its reformulation by Hartry Field, and reconstruct the debate that this problem has raised, by meeting at once the major opposing positions today of the analytic affiliation of the philosophy of mathematics, as much those on the platonist side as on that of nominalism. -/- In Chapter 8, Valeria Giardino and Yacin Hamami treat certain crucial aspects of the philosophy of mathematical practice. They deal in particular with the notion of an informal proof and on the role of artifacts in mathematical reasoning. They consider how such proofs, making use of these means, can contribute to the understanding of theorems and mathematical theories, thanks also to the role of visualisation and the aid of computer tools. -/- Why do mathematics often give several proofs of the same theorem? This is the question that Andrew Arana raises in Chapter 9, introducing the notion of an epistemic ideal and discuss two such ideals, the explanatoriness and purity of proof. -/- Chapter 10 is devoted to the applicability of mathematics in the study of empirical phenomena. After summarizing the history of this question, going back to Plato and Aristotle and passing by Mill and Kant, Daniele Molinini and Marco Panza reconstruct the contemporary debate, in relation, among other things, to questions raised in the philosophy of science. -/- Two appendices complete the volume. In the first Frédéric Patras treats the French tradition in philosophy of mathematics of the 20th century. In the second, Maria Carla Galavotti discusses the notion of probability and its use in science. (shrink)

Why do mathematics often give several proofs of the same theorem? This is the question raised in this article, introducing the notion of an epistemic ideal and discussing two such ideals, the explanatoriness and purity of proof.

Throughout history, mathematicians have expressed preference for solutions to problems that avoid introducing concepts that are in one sense or another “foreign” or “alien” to the problem under investigation. This preference for “purity” (which German writers commonly referred to as “methoden Reinheit”) has taken various forms. It has also been persistent. This notwithstanding, it has not been analyzed at even a basic philosophical level. In this paper we give a basic analysis of one conception of purity—what we call topical purity—and (...) discuss its epistemological significance. (shrink)