## Mathematical Intuitionism and Intersubjectivity: A Critical Exposition of Arguments for IntuitionismIn 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 |

207 | |

213 | |

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Mathematical Intuitionism and Intersubjectivity: A Critical Exposition of ... Tomasz Placek No preview available - 2010 |

### Common terms and phrases

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