In 1996, W. Veldman and F. Waaldijk present a constructive (intuitionistic) proof for the homogeneity of the ordered structure of the Cauchy real numbers, and so this result holds in any topos with natural number object. However, it is well known that the real numbers objects obtained by the traditional constructions of Cauchy sequences and Dedekind cuts are not necessarily isomorphic in an arbitrary topos with natural numbers object. Consequently, Veldman and Waaldijk's result does not apply to the ordered (...) structure of Dedekind real numbers in toposes. The main result to be proved in the present paper is that the ordered structure of the Dedekind real numbers object is homogeneous, in any topos with natural numbers object. This result is obtained within the framework of local set theory. (shrink)

La critica che Ludwig Wittgenstein muove ai presupposti filosofici del dibattito sui fondamenti della matematica si estende oltre le tematiche specifiche di tale dibattito ed investe un’intera tradizione di pensiero. Dietro alle argomentazioni di Frege e Russell, alle dimostrazioni di Cantor e di Dedekind, al programma di Hilbert e al teorema di Gödel, c’è uno sfondo filosofico che le acute osservazioni del filosofo ci permette di smascherare. Questa ricerca, articolata in quattro studi, intende presentare alcuni momenti fondamentali del dibattito (...) sui fondamenti e la relativa critica di Wittgenstein. Si prenderà in considerazione la definizione di limite di Cantor, la sezione di Dedekind, la definizione logicista di numero naturale ed infine il problema della scoperta in matematica. In conclusione si cercherà di interpretare il cosiddetto anti-platonismo di Wittgenstein. (shrink)

In Zermelo-Fraenkel set theory with the axiom of choice every set has the same cardinal number as some ordinal. Von Rimscha has weakened this condition to “Every set has the same cardinal number as some transitive set”. In set theory without the axiom of choice, we study the deductive strength of this and similar statements introduced by von Rimscha.

avec de nombreux textes inédits Pierre Dugac. APPENDICE XVII Wilhelm WEBER : Brie)'e an Richard Dedekind (Cod. Ms. Richard Dedekind 14, II; Niedersàchsische Staats- und Universitàts- bibliothek Gôttingen) Gôttingen 10. Aug.

By the middle of the nineteenth century, it had become clear to mathematicians that the study of finite field extensions of the rational numbers is indispensable to number theory, even if one’s ultimate goal is to understand properties of diophantine expressions and equations in the ordinary integers. It can happen, however, that the “integers” in such extensions fail to satisfy unique factorization, a property that is central to reasoning about the ordinary integers. In 1844, Ernst Kummer observed that unique factorization (...) fails for the cyclotomic integers with exponent 23, i.e. the ring Z[ζ] of integers of the field Q(ζ), where ζ is a primitive twenty-third root of unity. In 1847, he published his theory of “ideal divisors” for cyclotomic integers with prime exponent. This was to remedy the situation by introducing, for each such ring of integers, an enlarged domain of divisors, and showing that each integer factors uniquely as a product of these. He did not actually construct these integers, but, rather, showed how one could characterize their behavior qua divisibility in terms of ordinary operations on the associated ring of integers. (shrink)

1. If A is strongly amorphous , then its power set P is dually Dedekind infinite, i. e., every function from P onto P is injective. 2. The class of “inexhaustible” sets is not closed under supersets unless AC holds.

© The Author [2017]. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.comRichard Dedekind was a contemporary of Bernhard Riemann, Georg Cantor, and Gottlob Frege, among others. Together, they revolutionized mathematics and logic in the second half of the nineteenth century. Dedekind had an especially strong influence on David Hilbert, Ernst Zermelo, Emmy Noether, and Nicolas Bourbaki, who completed that revolution in the twentieth century. With respect to mainstream mathematics, he is best known for (...) his contributions to algebra and number theory. With respect to logic and the foundations of mathematics, many of his technical results — his conceptualization of the natural and real numbers, his analysis of proofs by mathematical induction and definitions by recursion (extended to the transfinite by Zermelo, John von... (shrink)

Axioms for the continuum, or smooth real line R. These include the usual axioms for a commutative ring with unit expressed in terms of two operations + and i , and two distinguished elements 0 ≠ 1. In addition we stipulate that R is a local ring, i.e., the following axiom: ∃y x i y = 1 ∨ ∃y (1 – x) i y = 1. Axioms for the strict order relation < on R. These are: 1. a < b (...) and b < c implies a < c. 2. ¬(a < a) 3. a < b implies a + c < b + c for any c. ≤ 4. a < b and 0 < c implies acbc.. (shrink)

In Hájek et al. (J Symb Logic 65(2):669–682, 2000) the authors introduce the concept of supersound logic, proving that first-order Gödel logic enjoys this property, whilst first-order Łukasiewicz and product logics do not; in Hájek and Shepherdson (Ann Pure Appl Logic 109(1–2):65–69, 2001) this result is improved showing that, among the logics given by continuous t-norms, Gödel logic is the only one that is supersound. In this paper we will generalize the previous results. Two conditions will be presented: the first (...) one implies the supersoundness and the second one non-supersoundness. To develop these results we will use, between the other machineries, the techniques of completions of MTL-chains developed in Labuschagne and van Alten (Proceedings of the ninth international conference on intelligent technologies, 2008) and van Alten (2009). We list some of the main results. The first-order versions of MTL, SMTL, IMTL, WNM, NM, RDP are supersound; the first-order version of an axiomatic extension of BL is supersound if and only it is n-potent (i.e. it proves the formula ${\varphi^{n}\,\to\,\varphi^{n\,{+}\,1}}$ for some ${n\,\in\,\mathbb{N}^+}$ ). Concerning the negative results, we have that the first-order versions of ΠMTL, WCMTL and of each non-n-potent axiomatic extension of BL are not supersound. (shrink)

O objetivo deste trabalho é traçar alguns paralelos históricos que existem entre a oposição Darstellung/Beschreibung tal como ela é usada por Wittgenstein no período intermediário de seu pensamento e a mesma oposição em alguns matemáticos do século XIX, pertencentes a duas escolas distintas que divergiam quanto ao método de produção matemática. Em seguida, retornaremos aos escritos de Wittgenstein, buscando mostrar em que medida esta investigação histórica pode ser útil para iluminar alguns pontos de sua filosofia e levantar alguns problemas que (...) surgiram neste contexto. (shrink)

El objetivo de este artículo es presentar y conectar el panorama intelectual inmediato, especialmente en el ámbito matemático, que Husserl se apropió durante sus primeros años de formación, centrándonos en las investigaciones de B. Riemann, R. Dedekind y G. Cantor. Se busca, en particular, evidenciar cómo Husserl reinterpreta filosóficamente los conceptos matemáticos que estos autores utilizaron. Partimos de la hipótesis de que, en el horizonte intelectual del joven Husserl, existen puntos de convergencia con el surgimiento de la moderna teoría (...) de conjuntos. Reconocer estos antecedentes justifica plenamente el trabajo que aquí presentaré. Comenzaré, por tanto, revisando las obras de Riemann, Dedekind y Cantor, al tiempo que mostraré su influencia en la obra de Husserl, y concluiré con las reflexiones pertinentes. (shrink)

The paper discusses a relatively underexamined element of Wittgenstein’s philosophy of mathematics associated with his critique of set theory. I outline Wittgenstein’s objections to the theories of Dedekind and Cantor, including the confounding of extension and intension, the faulty definition of the infinite set as infinite extension and the critique of Cantor’s diagonal proof. One of Wittgenstein’s major objections to set theory was that the concept of the size of infinite sets, which Cantor expressed by means of symbols אₒ (...) and c, had no application, i.e. that there was no grammatical technique that could show how such expressions were to be used. Notions of set theory are, so to speak, exterior – they find themselves outdoors, outside of what we usually do. They form a discourse that takes us beyond the horizon of everydayness and commonality. They are like an engine idling of the language of mathematics. (shrink)

The computability of reals was introduced by Alan Turing [20] by means of decimal representations. But the equivalent notion can also be introduced accordingly if the binary expansion, Dedekind cut or Cauchy sequence representations are considered instead. In other words, the computability of reals is independent of their representations. However, as it is shown by Specker [19] and Ko [9], the primitive recursiveness and polynomial time computability of the reals do depend on the representation. In this paper, we explore (...) how the weak computability of reals depends on the representation. To this end, we introduce three notions of weak computability in a way similar to the Ershov's hierarchy of Δ02-sets of natural numbers based on the binary expansion, Dedekind cut and Cauchy sequence, respectively. This leads to a series of classes of reals with different levels of computability. We investigate systematically questions as on which level these notions are equivalent. We also compare them with other known classes of reals like c. e. and d-c. e. reals. (shrink)

For certain classes of Dedekind domains S we want to characterize S-modules U such that Ext=0 for some module SMQ. We shall call these modules M-Whitehead modules. On the one hand we will show that assuming all M-Whitehead modules U are S0-free, i.e. US0 is a free S0-module where S0 is the nucleus of M. On the other hand if there is a ladder system on a stationary subset of ω1 that satisfies 2-uniformization, then there exists a non-S0-free M-Whitehead (...) module. Conversely, we will show that in the special case of Abelian groups the existence of a non-S0-free R-Whitehead group–here R is a rational group–implies that there is a ladder system on a stationary subset of ω1 that satisfies 2-uniformization. (shrink)

In 1891, Hilbert came to the conclusion that geometry is not a science of the physical space but is an abstract science. Hilbert’s new position was repeated in the introduction to Grundlagen der Geometrie in 1899. The aim of this paper is to show the sources of change of Hilbert’s position in 1891. We showed that the di-rect source of the change was having heard a lecture by Wiener on projective geome-try. Wiener presented there a project of geometry as a (...) science of abstract objects i.e. objects as sets of minimal number of – only relational – properties. We showed that this idea was derived by Wiener from Dedekind’s arithmetic of natural numbers, pre-sented in 1888 in Was sind und was sollen die Zahlen? Thus – and this is the essential thesis of our paper – the original source of Hilbert’s idea of geometry as a abstract science is included in Dedekind’s concept of arithmetic. (shrink)

We provide a general criterion for Fraenkel–Mostowski models of $${\textsf{ZFA}}$$ (i.e. Zermelo–Fraenkel set theory weakened to permit the existence of atoms) which implies “every linearly ordered set can be well ordered” ( $${\textsf{LW}}$$ ), and look at six models for $${\textsf{ZFA}}$$ which satisfy this criterion (and thus $${\textsf{LW}}$$ is true in these models) and “every Dedekind finite set is finite” ( $${\textsf{DF}}={\textsf{F}}$$ ) is true, and also consider various forms of choice for well-ordered families of well orderable sets in (...) these models. In Model 1, the axiom of multiple choice for countably infinite families of countably infinite sets ( $${\textsf{MC}}_{\aleph _{0}}^{\aleph _{0}}$$ ) is false. It was the open question of whether or not such a model exists (from Howard and Tachtsis “On metrizability and compactness of certain products without the Axiom of Choice”) that provided the motivation for this paper. In Model 2, which is constructed by first choosing an uncountable regular cardinal in the ground model, a strong form of Dependent choice is true, while the axiom of choice for well-ordered families of finite sets ( $${\textsf{AC}}^{{\textsf{WO}}}_{{\textsf{fin}}}$$ ) is false. Also in this model the axiom of multiple choice for well-ordered families of well orderable sets fails. Model 3 is similar to Model 2 except for the status of $${\textsf{AC}}^{{\textsf{WO}}}_{{\textsf{fin}}}$$ which is unknown. Models 4 and 5 are variations of Model 3. In Model 4 $${\textsf{AC}}_{\textrm{fin}}^{{\textsf{WO}}}$$ is true. The construction of Model 5 begins by choosing a regular successor cardinal in the ground model. Model 6 is the only one in which $$2{\mathfrak {m}} = {\mathfrak {m}}$$ for every infinite cardinal number $${\mathfrak {m}}$$. We show that the union of a well-ordered family of well orderable sets is well orderable in Model 6 and that the axiom of multiple countable choice is false. (shrink)

Husserl’s notion of definiteness, i.e., completeness is crucial to understanding Husserl’s view of logic, and consequently several related philosophical views, such as his argument against psychologism, his notion of ideality, and his view of formal ontology. Initially Husserl developed the notion of definiteness to clarify Hermann Hankel’s ‘principle of permanence’. One of the first attempts at formulating definiteness can be found in the Philosophy of Arithmetic, where definiteness serves the purpose of the modern notion of ‘soundness’ and leads Husserl to (...) a ‘computational’ view of logic. Inspired by Gauss and Grassmann Husserl then undertakes a further investigation of theories of manifolds. When Husserl subsequently renounces psychologism and changes his view of logic, his idea of definiteness also develops. The notion of definiteness is discussed most extensively in the pair of lectures Husserl gave in front of the mathematical society in Göttingen (1901). A detailed analysis of the lectures, together with an elaboration of Husserl’s lectures on logic beginning in 1895, shows that Husserl meant by definiteness what is today called ‘categoricity’. In so doing Husserl was not doing anything particularly original; since Dedekind’s ‘Was sind und sollen die Zahlen’ (1888) the notion was ‘in the air’. It also characterizes Hilbert’s (1900) notion of completeness. In the end, Husserl’s view of definiteness is discussed in light of Gödel’s (1931) incompleteness results. (shrink)

In this article we give a general categorical construction via reflection functors for various completions of T0-spaces subordinate to sobrification, with a particular emphasis on what we call the -completion, a type of directed completion introduced by Wyler [O. Wyler, Dedekind complete posets and Scott topologies, in: B. Banaschewski, R.-E. Hoffmann , Continuous Lattices Proceedings, Bremen 1979, in: Lecture Notes in Mathematics, vol. 871, Springer Verlag, 1981, pp. 384–389]. A key result is that all completions of a certain type (...) are universal, hence unique . We give a direct definition of the -completion and develop its theory by introducing a variant of the Scott topology, which we call the d-topology. For partially ordered sets the -completion turns out to be a natural dcpo-completion that generalizes the rounded ideal completion. In the latter part of the paper we consider settings in which the -completion agrees with the sobrification respectively the closed ideal completion. (shrink)

In the setting of ZF, i.e., Zermelo–Fraenkel set theory without the Axiom of Choice , we study partitions of Russell-sets into sets each with exactly n elements , for some integer n. We show that if n is odd, then a Russell-set X has an n -ary partition if and only if |X | is divisible by n. Furthermore, we establish that it is relative consistent with ZF that there exists a Russell-set X such that |X | is not divisible (...) by any finite cardinal n > 1. (shrink)

This paper introduces the notion of cW10-creative set, which strengthens that of semicreative set in a similar way as complete creativity strengthens creativity. Two results are proven, both of which imply that not all semicreative sets are cW10-creative. First, it is shown that semicreative Dedekind cuts cannot be cW10-creative; the existence of semicreative Dedekind cuts was shown by Soare. Secondly, it is shown that if A ⊕ B, the join of A and B, is cW10-creative, then either A (...) or B is cW10-creative, and the same is not true with ‘cW10-creative’ replaced by ‘semicreative’. Moreover, sets A, B which provide a counterexample for can be constructed within any given nonrecursive r.e. T-degrees a, b. MSC: 03D30. (shrink)

In this paper we show that the field of the real numbers is an intentional object in the sense specified by Roman Ingarden in his Das literarische Kunstwer and Der Streit um die Existenz der Welt. An ontological characteristics of a classic example of an intentional object, i.e. a literary character, is developed. There are three principal elements of such an object: the author, the text and the entity in which the literary character forms the content. In the case of (...) the real numbers the triad consists of Richard Dedekind, his work Steitigkeit und irratinale Zahlen, and the intentional object determined by this work. Showing that we are indeed faced with the inten-tional object we analyse three moments: two-sidedness of the formal structure of the intentional object, the moment of its existentional derivation, and schematism of the intentional object. Since there are many constructions of the real numbers known in mathematics, we show what relation obtains between Dedekind's and Cantor's constructions of the real numbers as these are taken as intentional objects. Moreover, we show what relation obtains between Dedekind's construction and the axiomatic characteristics of the real numbers. (shrink)

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)

In set theory without the Axiom of Choice, we investigate the set-theoretic strength of the principle NDS which states that there is no function f on the set ω of natural numbers such that for everyn ∈ ω, f ≺ f, where for sets x and y, x ≺ y means that there is a one-to-one map g : x → y, but no one-to-one map h : y → x. It is a long standing open problem whether NDS implies (...) AC. In this paper, among other results, we show that NDS is a strong axiom by establishing that ACLO ↛ NDS in ZFA set theory. The latter result provides a strongly negative answer to the question of whether “every Dedekind-finite set is finite” implies NDS addressed in G. H. Moore “Zermelo’s Axiom of Choice. Its Origins, Development, and Influence” and in P. Howard–J. E. Rubin “Consequences of the Axiom of Choice”. We also prove that ACWO ↛ NDS in ZF and that “for all infinite cardinals m, m + m = m” ↛ NDS in ZFA. (shrink)

A μ-algebra is a model of a first-order theory that is an extension of the theory of bounded lattices, that comes with pairs of terms where μx.f is axiomatized as the least prefixed point of f, whose axioms are equations or equational implications.Standard μ-algebras are complete meaning that their lattice reduct is a complete lattice. We prove that any nontrivial quasivariety of μ-algebras contains a μ-algebra that has no embedding into a complete μ-algebra.We then focus on modal μ-algebras, i.e. algebraic (...) models of the propositional modal μ-calculus. We prove that free modal μ-algebras satisfy a condition–reminiscent of Whitman’s condition for free lattices–which allows us to prove that modal operators are adjoints on free modal μ-algebras, least prefixed points of Σ1-operations satisfy the constructive relation μx.f=logical and operatorn≥0fn. These properties imply the following statement: the MacNeille–Dedekind completion of a free modal μ-algebra is a complete modal μ-algebra and moreover the canonical embedding preserves all the operations in the class image of the fixed point alternation hierarchy. (shrink)

Reviewing the beginnings and the further development of logicistic idea I am mainly focusing on the tradition of Frege and Russell, i. e. the tradition which was in the closest relationship with the enterprise of modern logic. I am trying to assess its connections to other „logistic“ programs such as Dedekind’s structuralism or Peano‘s axiomatism, to underline its specific features and the share in the success of modern logical semantics, including both its substantial contribution and limitations.

Wholes have parts, and wholes are prior to parts according to Aristotle. Aristotle’s accounts of continuity, in _Phys_. V 3 (plus sections in Metaph. Δ 6 and Ι 1) on the one hand and in _Phys_. VI on the other, are specified in terms of ways in which wholes are related to parts. The synthesis account in Phys. V 3 etc. applies primarily to bodies (in, e.g., anatomy). It indicates a variety of ways in which parts of a body are (...) kept together by a common boundary and are thereby combined into a mostly inhomogeneous, functional whole. Only the analysis account in _Phys_. VI applies primarily to linear continua such as movements, paths of movements, and time. The structure it indicates is only superficially described as indefinite divisibility: what matters is the transfer of potential divisions from path to movement and time (and conversely) which, surprisingly, requires an equivalent to Dedekind’s continuity principle to be tacitly presupposed. – In the present paper, my agenda will focus on the exposition of the relevant theories offered by Aristotle in _Phys_. V 3 and _Phys_. VI 1-2, respectively, with a view to the applications envisaged by Aristotle and to the mathematics involved. (shrink)

While many books have been written about Bertrand Russell's philosophy and some on his logic, I. Grattan-Guinness has written the first comprehensive history of the mathematical background, content, and impact of the mathematical logic and philosophy of mathematics that Russell developed with A. N. Whitehead in their Principia mathematica (1910-1913).? This definitive history of a critical period in mathematics includes detailed accounts of the two principal influences upon Russell around 1900: the set theory of Cantor and the mathematical logic of (...) Peano and his followers. Substantial surveys are provided of many related topics and figures of the late nineteenth century: the foundations of mathematical analysis under Weierstrass; the creation of algebraic logic by De Morgan, Boole, Peirce, Schröder, and Jevons; the contributions of Dedekind and Frege; the phenomenology of Husserl; and the proof theory of Hilbert. The many-sided story of the reception is recorded up to 1940, including the rise of logic in Poland and the impact on Vienna Circle philosophers Carnap and Gödel. A strong American theme runs though the story, beginning with the mathematician E. H. Moore and the philosopher Josiah Royce, and stretching through the emergence of Church and Quine, and the 1930s immigration of Carnap and GödeI. Grattan-Guinness draws on around fifty manuscript collections, including the Russell Archives, as well as many original reviews. The bibliography comprises around 1,900 items, bringing to light a wealth of primary materials. Written for mathematicians, logicians, historians, and philosophers--especially those interested in the historical interaction between these disciplines--this authoritative account tells an important story from its most neglected point of view. Whitehead and Russell hoped to show that (much of) mathematics was expressible within their logic; they failed in various ways, but no definitive alternative position emerged then or since. (shrink)

Upshot: In view of Kenny’s clinical insights, Hug’s notes on the intricacies of rational vs. a-rational “knowing” in the design sciences, and Chronaki & Kynigos’s notice of mathematics teachers’ meta-communication on experiences of change, this response reframes the heuristic power of bisociation and suspension of disbelief in the light of Kelly’s notion of “as-if-ism” (constructive alternativism. Doing as-if and playing what-if, I reiterate, are critical to mitigating intra-and inter-personal relations, or meta-communicating. Their epistemic status within the radical constructivist framework is (...) cast in the context of mutually enriching conversational techniques, or language-games, inspired by Maturana’s concepts of “objectivity in parenthesis” and the multiverse. (shrink)

The responses to the questions of why? when?, how?, where?, and in what ways? business ethics should be taught in the BusinessEthics classroom inundate the scholarly literature. Yet, to date, despite some very interesting ideas, with respect to the answers givento the above question, not only has nothing even close to consensus been reached, but this particular area of pedagogy is instagnation—authors still challenge both the very idea of teaching business ethics as well as the practical value of such courses (...) for ourstudents once they graduate to the corporate world.In this paper I will suggest that the reason for this lack of pedagogical progress is that there has been a serious oversight regarding the most important teaching question of all: Who? I will show that the pedagogical issue of whom should be teaching Business Ethics has been largely ignored, skirted or answered incorrectly. I will then boldly argue that the only necessary condition for successful courses inBusiness Ethics is that they be taught by experts in ethics, i.e., Ph.D.s in philosophy. (shrink)

Encore un livre sur saint Thomas, dira-t-on. Que pourrait ajouter celui-ci à tout ce qui existe déjà? Nous voudrions d'abord dans cet ouvrage répondre simplement mais en vérité, et en lisant les textes eux-mêmes de saint Thomas et d'Aristote, à certaines objections ou lieux communs entendus ou lus plus ou moins régulièrement, qui empêchent de commencer à vouloir connaître la pensée de frère Thomas. Il s'agit de mettre en évidence la grande richesse, pour l'intelligence et le coeur humains, de sa (...) pensée philosophique, de ses analyses, de ses manières de poser les questions et d'interroger le réel. Un préambule pour une introduction à la philosophie de Thomas, telle est essentiellement l'intention de ces quatre études et non pas une introduction à la philosophie thomasienne purement et simplement. Elles cherchent ainsi à définir l'esprit d'un apprentissage philosophique qui a aussi en lui tout ce qui convient pour aider à un apprentissage théologique et spirituel, mystique même. (shrink)

The Kyoto Protocol on global warming has provoked great controversy in part because it calls for heavier burdens on wealthy countries than on developing countries in the effort to control climate change. The U.S. Senate voted unanimously to oppose any agreement that does not require emissions reductions in low-income countries. The ethics of this position are examined in this paper which shows that there are good moral reasons for supporting the provisions of the Kyoto Protocol. Such a conclusion follows easily (...) from considerations of distributive justice but can also be supported by more narrowly self-interested arguments. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Augustine's Intellectual Conversion: The Journey from Platonism to ChristianityTravis E. AblesBrian Dobell. Augustine's Intellectual Conversion: The Journey from Platonism to Christianity. Cambridge-New York: Cambridge University Press, 2010. Pp. xvii + 250. Cloth, $82.00.The question of Augustine's Platonism is famously vexed. Since Peter Brown, the standard reading holds that Augustine did not move beyond the Neoplatonism of his early dialogues until he studied the writings of the apostle Paul (...) in the mid-390s CE, after which his distinctive theology begins to emerge. In this book, Brian Dobell's basic claim is that, whatever the validity of the "volitional conversion" of book 8, Augustine is not truly converted to orthodox Christianity until he discards the Photinian (adoptionist) Christology he admits to in Confessions [hereafter conf.] 7.19.25, and discards the Platonic ascent as a means to contemplative union with God, both of which take place with the reading of Paul. But the book is equally motivated by a concern to defend the historicity of conf.; Dobell claims it is perfectly reliable, once we understand the latter half of book 7 not as a few months in 386, but as a compressed narrative spanning Augustine's entire development until his true conversion to Christianity around 395. [End Page 137]Dobell develops this argument by isolating Augustine's twofold way of salvation in the writings of the late 380s. First is the "way of authority," in which Augustine's putative Photinian Christology portrays Christ as a kind of "poor man's Plato, sent to provide a path to salvation for the uneducated masses" (96). The human Jesus is "assumed" by God and is as "a man of outstanding wisdom" (73) but is not the incarnate Son. Dobell analyzes the homo assumptus/susceptus language of the early writings, arguing that Augustine does not truly hold an orthodox Christology until 396, concordant with the development of his theology of operative grace. The second, much longer section of the book concerns the "way of reason," the intellectualist path of salvation expressed in Augustine's early obsession with the liberal disciplines. Dobell dissects the Platonic ascents of conf. 7, adducing verbal and conceptual parallels from texts ca. 387–390 (De immortalitate animae, De ordine, De libero arbitrio, De vera religione) showing how Augustine's métaphysique des degrés d'être, a concept Dobell borrows from Olivier Du Roy, provides both a means of articulating these ascents and an answer to the problem of evil. But by De doctrina christiana, Augustine has decisively rejected the Platonic ascent and the liberal disciplines, and it simply remains for his Christology to come in line for the "intellectual conversion" to be complete.Dobell's book is a significant addition to the current debate. His knowledge of the first decade or so of Augustine's literary output is impressive, and his familiarity with the secondary literature is considerable, though not encyclopedic—he misses the work of J. Patout Burns, which complicates the Pauline thesis through consideration of the anti-Donatist literature. Dobell offers magisterial and invaluable discussions of famously fraught topics such as Augustine's indecisiveness about the preexistence of the soul, or the volitional solution to the problem of evil. There are some problems with Dobell's argument, though. While his main thesis that conf. 7.9.13ff. is a kind of précis of almost a decade of intellectual development deserves serious consideration, and is an elegant solution to debates about the historicity of Augustine's narrative in conf., the supporting arguments Dobell employs are less persuasive. A major portion of his book revolves around the contention that Augustine's early, heretical Photinian Christology extends much longer than modern scholarship has realized, that Augustine rectifies this Christology in conjunction with a Pauline soteriology of grace, and that all of this has to happen before Augustine's conversion is complete. If we suspend the question of normativity inherent in deciding Augustine's "orthodoxy" (particularly prior to Chalcedon), Dobell still does not succeed in showing that the early "pedagogical" homo assumptus Christology is, in fact, adoptionist; indeed, his evidence is often rather weak (for a tacit admission, see 73). On the flip... (shrink)

'Jousse - il a trouvé un filon, comme on dit', confie Bergson dans un entretien avec Lydie Adolphe. La relation que Marcel Jousse, chercheur, professeur, homme d'Église, entretient tout au long de ses quelque quarante ans d'enseignement avec le philosophe Henri Bergson est pour le moins paradoxale. Entre admiration et rejet, Jousse revendique 'sa' lecture de Bergson. Libre et inspirante, souvent critique, voire frôlant la caricature, cette lecture tonique agit en retour sur le lecteur, qu'il soit connaisseur de l'œuvre de (...) Bergson ou pas. Selon Jousse, le geste corporel-manuel, le geste global, est au cœur de la pensée du philosophe. Bergson joue sa pensée avec ses mains, avec tout son corps avant de la jeter dans ses livres. La connaissance des lois anthropologiques du Style oral formulées par Jousse permet de mieux saisir la particularité du style de Bergson, proche de celui de Péguy, et qui s'apparente au style oral des rythmo-mimeurs. Jousse, Bergson : deux noms qui ont marqué leur époque. Le lien fort qui unit les deux penseurs est au cœur de cette enquête - que l'on pourrait aussi écrire 'en quête' - à la fois philosophique et historique. Ce livre s'inscrit dans la longue liste des 'lecteurs de Bergson' qu'ils soient ou non bergsoniens."--Page 4 of cover. (shrink)