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Foundations of mathematics
About: Foundations of mathematics is a research topic. Over the lifetime, 1067 publications have been published within this topic receiving 25772 citations.
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TL;DR: A reprint of Frank P. Ramsey's seminal paper "Truth and Probability" written in 1926 and first published posthumous in the 1931 The Foundations of Mathematics and other Logical Essays, ed. R.B. Braithwaite, London: Routledge & Kegan Paul Ltd as discussed by the authors.
Abstract: This chapter is a reprint of Frank P. Ramsey’s seminal paper “Truth and Probability” written in 1926 and first published posthumous in the 1931 The Foundations of Mathematics and other Logical Essays, ed. R.B. Braithwaite, London: Routledge & Kegan Paul Ltd.
1,708 citations
Book•
01 Jan 1903
TL;DR: The first comprehensive treatise on the logical foundations of mathematics written in English was the Principia Mathematica as mentioned in this paper, which was published in 1903 and was the basis for the work of Frege.
Abstract: Published in 1903, this book was the first comprehensive treatise on the logical foundations of mathematics written in English. It sets forth, as far as possible without mathematical and logical symbolism, the grounds in favour of the view that mathematics and logic are identical. It proposes simply that what is commonly called mathematics are merely later deductions from logical premises. It provided the thesis for which Principia Mathematica provided the detailed proof, and introduced the work of Frege to a wider audience. In addition to the new introduction by John Slater, this edition contains Russell's introduction to the 1937 edition in which he defends his position against his formalist and intuitionist critics.
1,626 citations
Book•
14 Jun 2021
TL;DR: The fundamental texts of the great classical period in modern logic, some of them never before available in English translation, are here gathered together for the first time as discussed by the authors, and they are presented with a note that sets it in perspective, explains its importance, and points out difficulties in interpretation.
Abstract: The fundamental texts of the great classical period in modern logic, some of them never before available in English translation, are here gathered together for the first time. Modern logic, heralded by Leibniz, may be said to have been initiated by Boole, De Morgan, and Jevons, but it was the publication in 1879 of Gottlob Frege's "Begriffsschrift" that opened a great epoch in the history of logic by presenting, in full-fledged form, the propositional calculus and quantification theory. Frege's book, translated in its entirety, begins the present volume. The emergence of two new fields, set theory and foundations of mathematics, on the borders of logic, mathematics, and philosophy, is depicted by the texts that follow. Peano and Dedekind illustrate the trend that led to "Principia Mathematica." Burali-Forti, Cantor, Russell, Richard, and Konig mark the appearance of the modern paradoxes. Hilbert, Russell, and Zermelo show various ways of overcoming these paradoxes and initiate, respectively, proof theory, the theory of types, and axiomatic set theory. Skolem generalizes Lowenheim's theorem, and heand Fraenkel amend Zermelo's axiomatization of set theory, while von Neumann offers a somewhat different system. The controversy between Hubert and Brouwer during the twenties is presented in papers of theirs and in others by Weyl, Bernays, Ackermann, and Kolmogorov. The volume concludes with papers by Herbrand and by Godel, including the latter's famous incompleteness paper. Of the forty-five contributions here collected all but five are presented "in extenso." Those not originally written in English have been translated with exemplary care and exactness; the translators are themselves mathematical logicians as well as skilled interpreters of sometimes obscure texts. Each paper is introduced by a note that sets it in perspective, explains its importance, and points out difficulties in interpretation. Editorial comments and footnotes are interpolated where needed, and an extensive bibliography is included.
1,027 citations
Book•
01 Jan 1956
TL;DR: Wittgenstein's work remains, undeniably, now, that off one of those few few philosophers who will be read by all future generations as discussed by the authors, regardless of whether it is true or not.
Abstract: Wittgenstein's work remains, undeniably, now, that off one of those few philosophers who will be read by all future generations.
906 citations
Book•
01 Jan 2005
TL;DR: In this paper, the authors discuss the role of conceptual metaphor in the development of number processing in pre-school children and discuss the importance of conceptual metaphor for number representation in mathematics.
Abstract: Part 1: Cognitive Representations for Number and Mathematics. M. Fayol, X. Seron, About Numerical Representations: Insights from Neuropsychological, Experimental and Developmental Studies. M. Brysbaert, Number Recognition in Different Formats. W. Fias, M.H. Fischer, Spatial Representation of Numbers. J. Tzelgov, D. Ganor-Stern, Automaticity in Processing Ordinal Information. M. Zorzi, I. Stoianov, C. Umilta, Computational Modeling of Numerical Cognition. E.M. Brannon, What Animals Know about Numbers. R. Nunez, G. Lakoff, The Cognitive Foundations of Mathematics: The Role of Conceptual Metaphor. Part 2: Learning and Development of Numerical Skills. S. Cordes, R. Gelman, The Young Numerical Mind: When Does It Count? J. Bisanz, J.L. Sherman, C. Rasmussen, E. Ho, Development of Arithmetic Skills and Knowledge in Pre-school Children. K.F. Miller, M. Kelly, X. Zhou, Learning Mathematics in China and the United States: Cross-cultural Insights into the Nature and Course of Pre-school Mathematical Development. M.-P. Noel, L. Rousselle, C. Mussolin, Magnitude Representation in Children: Its Development and Dysfunction. R.S. Siegler, J.L. Booth, Development of Numerical Estimation: A Review. K.C. Fuson, D. Abrahamson, Understanding Ratio and Proportion as an Example of the Apprehending Zone and Conceptual-phase Problem-solving Models. T. Ben-Zeev, S. Duncan, C. Forbes, Stereotypes and Math Performance. Part 3: Learning and Performance Disabilities in Math and Number Processing. D.C. Geary, M.K. Hoard, Learning Disabilities in Arithmetic and Mathematics: Theoretical and Empirical Perspectives. M.M.M. Mazzocco, M. McCloskey, Math Performance in Girls with Turner or Fragile X Syndrome. M.A. Barnes, B. Smith-Chant, S.H. Landry, Number Processing in Neurodevelopmental Disorders: Spina Bifida Myelomeningocele. M.H. Ashcraft, K.S. Ridley, Math Anxiety and its Cognitive Consequences: A Tutorial Review. Part 4: Calculation and Cognition. N.J. Zbrodoff, G,D. Logan, What Everyone Finds: The Problem Size Effect. J.I.D. Campbell, L.J. Epp, Architectures for Arithmetic. J.-A. LeFevre, D. DeStefano, B. Coleman, T. Shanahan, Mathematical Cognition and Working Memory. J.A. Dixon, Mathematical Problem Solving: The Roles of Exemplar, Schema, and Relational Representations. S. Duverne, P. Lemaire, Aging and Mental Arithmetic. M. Pesenti, Calculation Abilities in Expert Calculators. Part 5: Neuropsychology of Number Processing and Calculation. S. Dehaene, M. Piazza, P. Pinel, L. Cohen, Three Parietal Circuits for Number Processing. B. Butterworth, Developmental Dyscalculia. A. Lochy, F. Domahs, M. Delazer, Rehabilitation of Acquired Calculation and Number Processing Disorders.
691 citations