Mathematical Intuitionism and Intersubjectivity: A Critical Exposition of Arguments for Intuitionism

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Springer Science & Business Media, Jun 30, 1999 - Science - 220 pages
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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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Contents

Introduction
1
2 Intersubjectivity and Conditions of Intersubjectivity
4
3 Mathematicians on Intersubjectivity
12
Brouwers Philosophy
17
1 The Knowing Subject
18
12 Brouwer and the Problem of Other Minds
22
13 The Basic Intuition of Twoity
27
2 Mathematics and Intuition
29
Heytings Argument
103
1 Against Intuitionistic Philosophy for Intuitionistic Psychology
104
2 Intuition as SelfEvidence
108
3 The Neutrality Argument
112
What Do They Prove?
113
32 From Ontological Neutrality to the Repudiation of Bivalent Truth
119
4 The Semantical Argument
126
41 A Note on Heytings Views on Formalization and Logic
137

22 What Is Not Intuitionistically Intuitive?
36
23 Some Objections to Brouwers Concept of Intuition
40
Brouwers Notion of Possibility
44
3 Language Truth and Relationship Between Logic and Mathematics
48
32 Against Hilberts Program
51
33 Brouwers and the AxiomaticDeductive Method
54
34 What Is Logic?
59
35 Mathematics vs Logic
62
36 Against Begriffe
65
37 What Is Truth?
67
38 The Validity of Laws of Logic
69
382 A Reconstruction of Brouwers Argument
71
383 Indeterminacy or Infinity
74
384 Strong Counterexamples to the Generalized Excluded Middle
79
4 Intersubjectivity in Brouwers Conception of Mathematics
83
41 Psychologism Subjectivism Solipsism?
84
the Mentalist Condition
89
43 How Can One Communicate about Mental Constructions?
90
5 Conclusions about Brouwers Philosophy
100
5 Intersubjectivity in Heytings Conception
139
6 Resume of Heytings Arguments
144
Dummetts Case for Intuitionism
147
2 Dummett on Semantic Theories
151
22 Programmatic Interpretation
155
23 Skeletal Semantics
161
3 Dummett on Meaning and its Basis
164
32 Sense Force and Holism
170
33 Sense and Semantic Theory
173
4 The LanguageLearning Argument
176
42 The Ingredient of Meaning That Transcends Use
181
43 Why Intuitionistic Provability? Holism to the Rescue
187
5 Resume of Dummetts Argument
192
Conclusions
194
APPENDIX
197
NOTES
203
BIBLIOGRAPHY
207
INDEX
213
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