Mathematical Intuitionism and Intersubjectivity: A Critical Exposition of Arguments for Intuitionism
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.
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2 Intersubjectivity and Conditions of Intersubjectivity
3 Mathematicians on Intersubjectivity
1 The Knowing Subject
12 Brouwer and the Problem of Other Minds
13 The Basic Intuition of Twoity
2 Mathematics and Intuition
1 Against Intuitionistic Philosophy for Intuitionistic Psychology
2 Intuition as SelfEvidence
3 The Neutrality Argument
What Do They Prove?
32 From Ontological Neutrality to the Repudiation of Bivalent Truth
4 The Semantical Argument
41 A Note on Heytings Views on Formalization and Logic
22 What Is Not Intuitionistically Intuitive?
23 Some Objections to Brouwers Concept of Intuition
Brouwers Notion of Possibility
3 Language Truth and Relationship Between Logic and Mathematics
32 Against Hilberts Program
33 Brouwers and the AxiomaticDeductive Method
34 What Is Logic?
35 Mathematics vs Logic
36 Against Begriffe
37 What Is Truth?
38 The Validity of Laws of Logic
382 A Reconstruction of Brouwers Argument
383 Indeterminacy or Infinity
384 Strong Counterexamples to the Generalized Excluded Middle
4 Intersubjectivity in Brouwers Conception of Mathematics
41 Psychologism Subjectivism Solipsism?
the Mentalist Condition
43 How Can One Communicate about Mental Constructions?
5 Conclusions about Brouwers Philosophy
5 Intersubjectivity in Heytings Conception
6 Resume of Heytings Arguments
Dummetts Case for Intuitionism
2 Dummett on Semantic Theories
22 Programmatic Interpretation
23 Skeletal Semantics
3 Dummett on Meaning and its Basis
32 Sense Force and Holism
33 Sense and Semantic Theory
4 The LanguageLearning Argument
42 The Ingredient of Meaning That Transcends Use
43 Why Intuitionistic Provability? Holism to the Rescue
5 Resume of Dummetts Argument
Other editions - View all
ability account of meaning argue argument ascribe assertion assumed assumption atomic sentences Beth trees bivalent truth Brouwer's conception Brouwer's philosophy causal attention choice sequences claim classical logic classical mathematician classical mathematics concept of truth derive disjunction distinction double negation elimination Dummett elements evidence excluded middle formal given grasp Heyting Heyting's Hilbert's Program holds ibid ideal infinity instance intention intersubjectivity introduced intuition of two-ity intuitionism intuitionistic logic intuitionistic mathematics intuitive mathematics ISBN knowledge language laws of logic linguistic communication logical constants manifestation math mathematical constructions mathematical intuitionism mathematical objects mathematical statements mathematical system meaning theory mental constructions mental objects metalanguage mind Moreover natural numbers Nevertheless node notion obtain person philosophy of mathematics possible postulates predicate principle problem proof question real numbers reasoning relation requirement restrictions rules semantic theory semantic values sensations sense sentence solipsism speaker theorem true truth condition undecidable understanding validity words
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