Mathematical Intuitionism And Intersubjectivity

Author : Tomasz Placek
Publisher : Springer Science & Business Media
Page : 229 pages
File Size : 44,7 Mb
Release : 2013-03-09
Category : Science
ISBN : 9789401593151

Get Book

Mathematical Intuitionism and Intersubjectivity by Tomasz Placek Pdf

In 1907 Luitzen Egbertus Jan Brouwer defended his doctoral dissertation on the foundations of mathematics and with this event the modem version of mathematical intuitionism came into being. Brouwer attacked the main currents of the philosophy of mathematics: the formalists and the Platonists. In tum, both these schools began viewing intuitionism as the most harmful party among all known philosophies of mathematics. That was the origin of the now-90-year-old debate over intuitionism. As both sides have appealed in their arguments to philosophical propositions, the discussions have attracted the attention of philosophers as well. One might ask here what role a philosopher can play in controversies over mathematical intuitionism. Can he reasonably enter into disputes among mathematicians? I believe that these disputes call for intervention by a philo sopher. The three best-known arguments for intuitionism, those of Brouwer, Heyting and Dummett, are based on ontological and epistemological claims, or appeal to theses that properly belong to a theory of meaning. Those lines of argument should be investigated in order to find what their assumptions are, whether intuitionistic consequences really follow from those assumptions, and finally, whether the premises are sound and not absurd. The intention of this book is thus to consider seriously the arguments of mathematicians, even if philosophy was not their main field of interest. There is little sense in disputing whether what mathematicians said about the objectivity and reality of mathematical facts belongs to philosophy, or not.

Author : R.L. Tieszen
Publisher : Springer Science & Business Media
Page : 223 pages
File Size : 49,6 Mb
Release : 2012-12-06
Category : Philosophy
ISBN : 9789400922938

Get Book

Mathematical Intuition by R.L. Tieszen Pdf

"Intuition" has perhaps been the least understood and the most abused term in philosophy. It is often the term used when one has no plausible explanation for the source of a given belief or opinion. According to some sceptics, it is understood only in terms of what it is not, and it is not any of the better understood means for acquiring knowledge. In mathematics the term has also unfortunately been used in this way. Thus, intuition is sometimes portrayed as if it were the Third Eye, something only mathematical "mystics", like Ramanujan, possess. In mathematics the notion has also been used in a host of other senses: by "intuitive" one might mean informal, or non-rigourous, or visual, or holistic, or incomplete, or perhaps even convincing in spite of lack of proof. My aim in this book is to sweep all of this aside, to argue that there is a perfectly coherent, philosophically respectable notion of mathematical intuition according to which intuition is a condition necessary for mathemati cal knowledge. I shall argue that mathematical intuition is not any special or mysterious kind of faculty, and that it is possible to make progress in the philosophical analysis of this notion. This kind of undertaking has a precedent in the philosophy of Kant. While I shall be mostly developing ideas about intuition due to Edmund Husser! there will be a kind of Kantian argument underlying the entire book.

Author : Arend Heyting
Publisher : Elsevier
Page : 159 pages
File Size : 49,9 Mb
Release : 1966
Category : Electronic books
ISBN : 9780444534064

Get Book

Intuitionism by Arend Heyting Pdf

Author : Al'bert Grigor'evi_ Dragalin
Publisher : American Mathematical Soc.
Page : 228 pages
File Size : 48,6 Mb
Release : 1988-12-31
Category : Mathematics
ISBN : 9780821845202

Get Book

Mathematical Intuitionism by Al'bert Grigor'evi_ Dragalin Pdf

In the area of mathematical logic, a great deal of attention is now being devoted to the study of nonclassical logics. This book intends to present the most important methods of proof theory in intuitionistic logic and to acquaint the reader with the principal axiomatic theories based on intuitionistic logic.

Author : Carl J. Posy
Publisher : Cambridge University Press
Page : 75 pages
File Size : 47,5 Mb
Release : 2020-11-12
Category : Science
ISBN : 1108723020

Get Book

Mathematical Intuitionism by Carl J. Posy Pdf

L. E. J. Brouwer, the founder of mathematical intuitionism, believed that mathematics and its objects must be humanly graspable. He initiated a program rebuilding modern mathematics according to that principle. This book introduces the reader to the mathematical core of intuitionism - from elementary number theory through to Brouwer's uniform continuity theorem - and to the two central topics of 'formalized intuitionism': formal intuitionistic logic, and formal systems for intuitionistic analysis. Building on that, the book proposes a systematic, philosophical foundation for intuitionism that weaves together doctrines about human grasp, mathematical objects and mathematical truth.

Author : Mark van Atten,Pascal Boldini,Michel Bourdeau,Gerhard Heinzmann
Publisher : Springer Science & Business Media
Page : 432 pages
File Size : 52,8 Mb
Release : 2008-11-09
Category : Science
ISBN : 9783764386535

Get Book

One Hundred Years of Intuitionism (1907-2007) by Mark van Atten,Pascal Boldini,Michel Bourdeau,Gerhard Heinzmann Pdf

Intuitionism is one of the main foundations for mathematics proposed in the twentieth century and its views on logic have also notably become important with the development of theoretical computer science. This book reviews and completes the historical account of intuitionism. It also presents recent philosophical work on intuitionism and gives examples of new technical advances and applications. It brings together 21 contributions from today's leading authors on intuitionism.

Author : Stewart Shapiro
Publisher : Oxford University Press
Page : 856 pages
File Size : 43,7 Mb
Release : 2005-02-10
Category : Mathematics
ISBN : 9780198033998

Get Book

The Oxford Handbook of Philosophy of Mathematics and Logic by Stewart Shapiro Pdf

Mathematics and logic have been central topics of concern since the dawn of philosophy. Since logic is the study of correct reasoning, it is a fundamental branch of epistemology and a priority in any philosophical system. Philosophers have focused on mathematics as a case study for general philosophical issues and for its role in overall knowledge- gathering. Today, philosophy of mathematics and logic remain central disciplines in contemporary philosophy, as evidenced by the regular appearance of articles on these topics in the best mainstream philosophical journals; in fact, the last decade has seen an explosion of scholarly work in these areas. This volume covers these disciplines in a comprehensive and accessible manner, giving the reader an overview of the major problems, positions, and battle lines. The 26 contributed chapters are by established experts in the field, and their articles contain both exposition and criticism as well as substantial development of their own positions. The essays, which are substantially self-contained, serve both to introduce the reader to the subject and to engage in it at its frontiers. Certain major positions are represented by two chapters--one supportive and one critical. The Oxford Handbook of Philosophy of Math and Logic is a ground-breaking reference like no other in its field. It is a central resource to those wishing to learn about the philosophy of mathematics and the philosophy of logic, or some aspect thereof, and to those who actively engage in the discipline, from advanced undergraduates to professional philosophers, mathematicians, and historians.

Author : Bharath Sriraman
Publisher : Springer Nature
Page : 3221 pages
File Size : 49,5 Mb
Release : 2024-06-09
Category : Electronic
ISBN : 9783031408465

Get Book

Handbook of the History and Philosophy of Mathematical Practice by Bharath Sriraman Pdf

Author : Burt Hopkins,John Drummond
Publisher : Routledge
Page : 346 pages
File Size : 46,9 Mb
Release : 2015-03-24
Category : Philosophy
ISBN : 9781317401278

Get Book

The New Yearbook for Phenomenology and Phenomenological Philosophy by Burt Hopkins,John Drummond Pdf

The New Yearbook for Phenomenology and Phenomenological Philosophy provides an annual international forum for phenomenological research in the spirit of Husserl's groundbreaking work and the extension of this work by such figures as Scheler, Heidegger, Sartre, Levinas, Merleau-Ponty and Gadamer.

Author : Stefania Centrone
Publisher : Springer
Page : 526 pages
File Size : 43,6 Mb
Release : 2017-09-07
Category : Philosophy
ISBN : 9789402411324

Get Book

Essays on Husserl's Logic and Philosophy of Mathematics by Stefania Centrone Pdf

Essays on Husserl’s Logic and Philosophy of Mathematics sets out to fill up a lacuna in the present research on Husserl by presenting a precise account of Husserl’s work in the field of logic, of the philosophy of logic and of the philosophy of mathematics. The aim is to provide an in-depth reconstruction and analysis of the discussion between Husserl and his most important interlocutors, and to clarify pivotal ideas of Husserl’s by considering their reception and elaboration by some of his disciples and followers, such as Oskar Becker and Jacob Klein, as well as their influence on some of the most significant logicians and mathematicians of the past century, such as Luitzen E. J. Brouwer, Rudolf Carnap, Kurt Gödel and Hermann Weyl. Most of the papers consider Husserl and another scholar – e.g. Leibniz, Kant, Bolzano, Brentano, Cantor, Frege – and trace out and contextualize lines of influence, points of contact, and points of disagreement. Each essay is written by an expert of the field, and the volume includes contributions both from the analytical tradition and from the phenomenological one.

Author : Mark van Atten
Publisher : Springer
Page : 328 pages
File Size : 44,8 Mb
Release : 2014-11-21
Category : Philosophy
ISBN : 9783319100319

Get Book

Essays on Gödel’s Reception of Leibniz, Husserl, and Brouwer by Mark van Atten Pdf

This volume tackles Gödel's two-stage project of first using Husserl's transcendental phenomenology to reconstruct and develop Leibniz' monadology, and then founding classical mathematics on the metaphysics thus obtained. The author analyses the historical and systematic aspects of that project, and then evaluates it, with an emphasis on the second stage. The book is organised around Gödel's use of Leibniz, Husserl and Brouwer. Far from considering past philosophers irrelevant to actual systematic concerns, Gödel embraced the use of historical authors to frame his own philosophical perspective. The philosophies of Leibniz and Husserl define his project, while Brouwer's intuitionism is its principal foil: the close affinities between phenomenology and intuitionism set the bar for Gödel's attempt to go far beyond intuitionism. The four central essays are `Monads and sets', `On the philosophical development of Kurt Gödel', `Gödel and intuitionism', and `Construction and constitution in mathematics'. The first analyses and criticises Gödel's attempt to justify, by an argument from analogy with the monadology, the reflection principle in set theory. It also provides further support for Gödel's idea that the monadology needs to be reconstructed phenomenologically, by showing that the unsupplemented monadology is not able to found mathematics directly. The second studies Gödel's reading of Husserl, its relation to Leibniz' monadology, and its influence on his publishe d writings. The third discusses how on various occasions Brouwer's intuitionism actually inspired Gödel's work, in particular the Dialectica Interpretation. The fourth addresses the question whether classical mathematics admits of the phenomenological foundation that Gödel envisaged, and concludes that it does not. The remaining essays provide further context. The essays collected here were written and published over the last decade. Notes have been added to record further thoughts, changes of mind, connections between the essays, and updates of references.

Author : Markus Sebastiaan Paul Rogier van Atten
Publisher : Cengage Learning
Page : 108 pages
File Size : 49,9 Mb
Release : 2004
Category : Intersubjectivity
ISBN : UCSC:32106016001452

Get Book

On Brouwer by Markus Sebastiaan Paul Rogier van Atten Pdf

This book offers a concise, yet comprehensive, introduction to this philosopher's most important ideas.

Author : Luitzen Egbertus Jan Brouwer
Publisher : Cambridge University Press
Page : 130 pages
File Size : 53,9 Mb
Release : 1981
Category : Mathematics
ISBN : 0521177367

Get Book

Brouwer's Cambridge Lectures on Intuitionism by Luitzen Egbertus Jan Brouwer Pdf

Luitzen Egburtus Jan Brouwer founded a school of thought whose aim was to include mathematics within the framework of intuitionistic philosophy; mathematics was to be regarded as an essentially free development of the human mind. What emerged diverged considerably at some points from tradition, but intuitionism has survived well the struggle between contending schools in the foundations of mathematics and exact philosophy. Originally published in 1981, this monograph contains a series of lectures dealing with most of the fundamental topics such as choice sequences, the continuum, the fan theorem, order and well-order. Brouwer's own powerful style is evident throughout the work.

Author : Dale Jacquette
Publisher : John Wiley & Sons
Page : 832 pages
File Size : 47,8 Mb
Release : 2008-04-15
Category : Philosophy
ISBN : 9781405149945

Get Book

A Companion to Philosophical Logic by Dale Jacquette Pdf

This collection of newly comissioned essays by international contributors offers a representative overview of the most important developments in contemporary philosophical logic. Presents controversies in philosophical implications and applications of formal symbolic logic. Surveys major trends and offers original insights.

Author : Richard L. Tieszen
Publisher : Cambridge University Press
Page : 369 pages
File Size : 48,5 Mb
Release : 2005-06-06
Category : Mathematics
ISBN : 9780521837828

Get Book

Phenomenology, Logic, and the Philosophy of Mathematics by Richard L. Tieszen Pdf

In this 2005 book, logic, mathematical knowledge and objects are explored alongside reason and intuition in the exact sciences.