Paul Benacerraf and Hilary Putnam. Introduction. Philosophy of mathematics, Selected readings, edited by Paul Benacerraf and Hilary Putnam, Prentice-Hall, Inc., Engle-wood Cliffs, New Jersey, 1964, pp. 1–27. - Rudolf Carnap. The logicist foundations of mathematics. English translation of 3528 by Erna Putnam and Gerald E. Massey. Philosophy of mathematics, Selected readings, edited by Paul Benacerraf and Hilary Putnam, Prentice-Hall, Inc., Engle-wood Cliffs, New Jersey, pp. 31–41. - Arend Heyting. The intuitionist foundations of mathematics. English translation of 3856 by Erna Putnam and Gerald E. Massey. Philosophy of mathematics, Selected readings, edited by Paul Benacerraf and Hilary Putnam, Prentice-Hall, Inc., Engle-wood Cliffs, New Jersey, pp. 42–49. - Johann von Neumann. The formalist foundations of mathematics. English translation of 2998 by Erna Putnam and Gerald E. Massey. Philosophy of mathematics, Selected readings, edited by Paul Benacerraf and Hilary Putnam, Prentice-Hall, Inc., Engle-wood Cliffs, New Jersey, pp. 50–54. - Arend Heyting. Disputation. A reprint of pages 1-12 (the first chapter) and parts of the bibliography of XXI 367. Philosophy of mathematics, Selected readings, edited by Paul Benacerraf and Hilary Putnam, Prentice-Hall, Inc., Engle-wood Cliffs, New Jersey, pp. 55–65. - L. E. J. Brouwer. Intuitionism and formalism. A reprint of 1557. Philosophy of mathematics, Selected readings, edited by Paul Benacerraf and Hilary Putnam, Prentice-Hall, Inc., Engle-wood Cliffs, New Jersey, pp. 66–77. - L. E. J. Brouwer. Consciousness, philosophy, and mathematics. A reprint of pages 1243-1249 of XIV 132. Philosophy of mathematics, Selected readings, edited by Paul Benacerraf and Hilary Putnam, Prentice-Hall, Inc., Engle-wood Cliffs, New Jersey, pp. 78–84. - Gottlob Frege. The concept of number. English translation of pages 67-104, 115-119, of 495 (1884 edn.) by Michael S. Mahoney. Philosophy of mathematics, Selected readings, edited by Paul Benacerraf and Hilary Putnam, Prentice-Hall, Inc., Engle-wood Cliffs, New Jersey, pp. 85–112. - Bertrand Russell. Selections from Introduction to mathematical philosophy. A reprint of pages 1-19, 194-206, of 11126 (1st edn., 1919). Philosophy of mathematics, Selected readings, edited by Paul Benacerraf and Hilary Putnam, Prentice-Hall, Inc., Engle-wood Cliffs, New Jersey, pp. 113–133. - David Hilbert. On the infinite. English translation of 10813 by Erna Putnam and Gerald E. Massey. Philosophy of mathematics, Selected readings, edited by Paul Benacerraf and Hilary Putnam, Prentice-Hall, Inc., Engle-wood Cliffs, New Jersey, pp. 134–151.

1969 ◽  
Vol 34 (1) ◽  
pp. 107-110
Author(s):  
Alec Fisher
Keyword(s):  
New Jersey ◽  
English Translation ◽  
Philosophy Of Mathematics ◽  
Bertrand Russell ◽  
Hilary Putnam ◽  
Foundations Of Mathematics ◽  
Rudolf Carnap ◽  
Von Neumann ◽  
David Hilbert ◽  
And Mathematics

A. J. Ayer. Editor's introduction. Logical positivism, edited by A. J. Ayer, The Free Press, Glencoe, Illinois, 1959, pp. 3–28; also first paperback edition, The Free Press, New York 1966, pp. 3–28. - Bertrand Russell. Logical atomism. A reprint of XXV 333. Logical positivism, edited by A. J. Ayer, The Free Press, Glencoe, Illinois, 1959, pp. 31–50; also ibid., pp. 31–50. - Moritz Schlick. Positivism and realism. A reprint of XVI 67. Logical positivism, edited by A. J. Ayer, The Free Press, Glencoe, Illinois, 1959, pp. 82–107; also ibid., pp. 82–107. - Carl G. Hempel. The empiricist criterion of meaning. A reprint of XVI 293. Logical positivism, edited by A. J. Ayer, The Free Press, Glencoe, Illinois, 1959, pp. 108–129; also ibid., pp. 108–129. - Rudolf Carnap. The old and the new logic. English translation of 3525 by Isaac Levi. Logical positivism, edited by A. J. Ayer, The Free Press, Glencoe, Illinois, 1959, pp. 133–146; also ibid., pp. 133–146. - Hans Hahn. Logic, mathematics and knowledge of nature. English translation of 4193 by Arthur Pap. Logical positivism, edited by A. J. Ayer, The Free Press, Glencoe, Illinois, 1959, pp. 147–161 ; also ibid., pp. 147–161. - Friedrich Waismann. How I see philosophy. A reprint of XXIII 209. Logical positivism, edited by A. J. Ayer, The Free Press, Glencoe, Illinois, 1959, pp. 345–380; also ibid., pp. 345–380.

1970 ◽  
Vol 35 (2) ◽  
pp. 312-312
Author(s):  
Alonzo Church
Keyword(s):  
New York ◽  
English Translation ◽  
Logical Positivism ◽  
Bertrand Russell ◽  
Rudolf Carnap ◽  
Free Press ◽  
Paperback Edition ◽  
Logical Atomism

German Philosophy

Philosophy ◽  
1956 ◽  
Vol 31 (119) ◽  
pp. 358-361
Author(s):  
F. H. Heinemann
Keyword(s):  
Natural Science ◽  
Philosophy Of Mathematics ◽  
Premature Death ◽  
New Venture ◽  
Foundations Of Mathematics ◽  
German Philosophy ◽  
Hermann Weyl ◽  
And Mathematics

Operative Logic and Mathematics would appear to be a new venture. Only a few weeks before his premature death Hermann Weyl, one of the most original mathematicians of our time, the author of a Philosophy of Mathematics and Natural Science and also of a stimulating book on Symmetry, drew my attention to Paul Lorenzen's Einführung in die operative Logik und Mathematik(Springer, Berlin). This book had given him new hope, since GÖdel had discouraged his endeavour to find the foundations of mathematics. “Perhaps,” he added, “Lorenzen's approach promises a way of arriving at reliable foundations.”

Wittgenstein on Philosophy of Logic and Mathematics

2009 ◽  
Author(s):  
Juliet Floyd
Keyword(s):  
Ludwig Wittgenstein ◽  
Philosophy Of Mathematics ◽  
Vienna Circle ◽  
Short Review ◽  
Significant Part ◽  
Foundations Of Mathematics ◽  
Philosophy Of Logic ◽  
And Mathematics

Ludwig Wittgenstein (1889–1951) wrote as much on the philosophy of mathematics and logic as he did on any other topic, leaving at his death thousands of pages of manuscripts, typescripts, notebooks, and correspondence containing remarks on (among others) Brouwer, Cantor, Dedekind, Frege, Hilbert, Poincaré, Skolem, Ramsey, Russell, Gödel, and Turing. He published in his lifetime only a short review (1913) and the Tractatus Logico-Philosophicus (1921), a work whose impact on subsequent analytic philosophy's preoccupation with characterizing the nature of logic was formative. Wittgenstein's reactions to the empiricistic reception of his early work in the Vienna Circle and in work of Russell and Ramsey led to further efforts to clarify and adapt his perspective, stimulated in significant part by developments in the foundations of mathematics of the 1920s and 1930s; these never issued in a second work, though he drafted and redrafted writings more or less continuously for the rest of his life.

scholarly journals L’appareil analytique et ses modèles

2012 ◽  
Vol 7 (2) ◽  
pp. 23-48 ◽  
Author(s):  
Yvon Gauthier
Keyword(s):  
Charles Sanders Peirce ◽  
Jürgen Habermas ◽  
Rudolf Carnap ◽  
Ian Hacking ◽  
John Von Neumann ◽  
Von Neumann ◽  
David Hilbert ◽  
Jurgen Habermas ◽  
Sciences Sociales

Cet article propose une notion constructiviste de modèle dans la théorie physique applicable à la théorie scientifique en général, c’est-à-dire aussi bien dans les sciences exactes que dans les sciences sociales et humaines. La distinction entre appareil analytique et appareil expérimental par la médiation des modèles permet en effet de généraliser une notion qui est d’abord apparue dans les fondements de la physique chez David Hilbert et John von Neumann. Si l’on consent à inverser les flèches ou homomorphismes qui vont de l’appareil analytique, ensemble des structures logicomathématiques, à l’appareil expérimental, ensemble des données empiriques et des procédures expérimentales, on peut remonter par la modélisation des données jusqu’à l’appareil analytique qui assure la consistance ou cohérence logique de la théorie scientifique, qu’elle relève des sciences exactes ou des sciences sociales. Une telle articulation des savoirs peut apparaître formelle, mais elle a l’avantage de rassembler les entreprises scientifiques dans un schème unificateur qui jette une lumière nouvelle sur le débat majeur en philosophie des sciences contemporaine, la confrontation du réalisme et de l’antiréalisme, qui a des répercussions tant en philosophie de la physique qu’en philosophie du langage et en philosophie de la logique, ou encore en philosophie des sciences sociales, si l’on en croit Jürgen Habermas ou les tenants du contructionnisme appelé jadis constructivisme social ou socioconstructivisme. L’article conclut sur la distinction qu’il faut opérer entre le constructivisme logicomathématique et le contructionnisme, comme le dénomme Ian Hacking, pour bien marquer la distance qui sépare les postures fondationnelles ou les options philosophiques dans ce qu’il faut bien appeler « logique de la science », selon l’expression du grand philosophe pragmatiste Charles Sanders Peirce reprise par des empiristes logiques comme Rudolf Carnap.

scholarly journals Intensionality: From Philosophical Logic to Metamathematics

2020 ◽  
pp. 85-89
Author(s):  
Vitaly V. Tselishchev ◽  
◽  
Keyword(s):  
Mathematical Knowledge ◽  
Research Area ◽  
Mathematical Discourse ◽  
Foundations Of Mathematics ◽  
Philosophical Logic ◽  
David Hilbert ◽  
The Status ◽  
And Mathematics ◽  
Theoretical Apparatus

The article is devoted to the study of the status of intensionality in the exact contexts of logical and mathematical theories. The emergence of intensionality in logical and mathematical discourse leads to significant obstacles in its formalization due to the appearance of indirect contexts, the uncertainty of its indication in the theoretical apparatus, as well as the presence of various kinds of difficult-to-account semantic distinctions. The refusal to consider intensionality in logic is connected with Bertrand Russell’s criticism of Alexius Meinong’s intensionality ontology, and with Willard Van Orman Quine’s criticism of the concept of meaning and quantification of modalities. It is shown that this criticism is based on a preference for the theory of indication over the theory of meaning, in terms of the distinction “Bedeutung” and “Sinn” introduced by Gottlob Frege. The extensionality thesis is explicated; by analogy with it the intensionality thesis is constructed. It is shown that complete parallelism is not possible here, and therefore we should proceed from finding cases of extensionality violation. Since the construction of formal logical systems is to a certain extent connected with the programs of the foundations of mathematics, the complex interweaving of philosophical and purely technical questions makes the question of the role of intensionality in mathematics quite confusing. However, there is one clue here: programs in the foundations of mathematics have given rise to metamathematics, which, although it stands alone, is considered a branch of mathematics. It is not by chance that, judging by the problems arising in connection with intensionality, there is a growing suspicion that intensionality can play a significant role in metamathematics. As for the question of the sense in which metamathematics results can be considered mathematical, in terms of the presence of intensional contexts in both disciplines, it is a matter of taste: for example, the autonomy of mathematical knowledge as a result of the desire of mathematicians to eliminate the influence of philosophy that took place in the case of David Hilbert may be worth considering in the context of mathematics. Thus, the rather vague concept of intensionality receives various explications in different contexts, whether it is philosophical logic or metamathematics. In any case, the detection of context intensionality is always associated with a clear narrowing of the research area. It is obvious that the creation of a more general theory of intensionality is possible within a more general framework, in which logic and mathematics must be combined. In this respect, we can hope for the resumption of a logical project, which would be a purely logical consideration made of the natural and the mathematical.

scholarly journals Introducing formalism in economics: The growth model of John von Neumann

2010 ◽  
Vol 57 (2) ◽  
pp. 153-172 ◽  
Author(s):  
Sandye Gloria-Palermo
Keyword(s):  
Growth Model ◽  
John Von Neumann ◽  
Von Neumann ◽  
David Hilbert ◽  
The 1950S ◽  
Pragmatic Turn ◽  
The Impact

The objective is to interpret John von Neumann's growth model as a decisive step of the forthcoming formalist revolution of the 1950s in economics. This model gave rise to an impressive variety of comments about its classical or neoclassical underpinnings. We go beyond this traditional criterion and interpret rather this model as the manifestation of von Neumann's involvement in the formalist programme of mathematician David Hilbert. We discuss the impact of Kurt G?del's discoveries on this programme. We show that the growth model reflects the pragmatic turn of the formalist programme after G?del and proposes the extension of modern axiomatisation to economics.

scholarly journals Hilary Putnam on the philosophy of logic and mathematics

2018 ◽  
Vol 33 (2) ◽  
pp. 183
Author(s):  
José Miguel Sagüillo Fernández-Vega
Keyword(s):  
Final Section ◽  
Mathematical Practice ◽  
Logical Truth ◽  
Hilary Putnam ◽  
Philosophy Of Logic ◽  
Logical Validity ◽  
And Mathematics

I discuss Putnam’s conception of logical truth as grounded in his picture of mathematical practice and ontology. i begin by comparing Putnam’s 1971 Philosophy of Logic with Quine’s homonymous book. Next, Putnam’s changing views on modality are surveyed, moving from the modal pre-formal to the de-modalized formal characterization of logical validity. Section three suggests a complementary view of Platonism and modalism underlying different stages of a dynamic mathematical practice. The final section argues for the pervasive platonistic conception of the working mathematician.

scholarly journals PERKEMBANGAN MATEMATIKA DALAM FILSAFAT DAN ALIRAN FORMALISME YANG TERKANDUNG DALAM FILSAFAT MATEMATIKA

Sepren ◽  
2021 ◽  
Vol 2 (2) ◽  
pp. 17-22
Author(s):  
Robin Tarigan
Keyword(s):  
Learning Outcomes ◽  
Historical Perspective ◽  
Philosophy Of Mathematics ◽  
Long Time ◽  
And Mathematics ◽  
The Relationship

Philosophy of mathematics does not add a number of new mathematical theorems or theories, so a philosophy of mathematics is not mathematics. The philosophy of mathematics is an area of ​​reflection about mathematics. After studying for a long time, one needs to reflect on learning outcomes by reflecting on the philosophy of mathematics. Mathematics and philosophy are closely related, compared to other sciences. The reason is that philosophy is the base for studying science and mathematics is the mother of all sciences. There are also those who think that philosophy and mathematics are the mother of all existing knowledge. From a historical perspective, the relationship between philosophy and mathematics underwent a very striking development. This article discusses the development of mathematics in philosophy and the flow of formalism contained in the philosophy of mathematics in particular

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