Treatise on Intuitionistic Type Theory

Intuitionistic type theory can be described, somewhat boldly, as a partial fulfillment of the dream of a universal language for science. This book expounds several aspects of intuitionistic type theory, such as the notion of set, reference vs. computation, assumption, and substitution. Moreover, the book includes philosophically relevant sections on the principle of composition...

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Intuitionistic type theory can be described, somewhat boldly, as a partial fulfillment of the dream of a universal language for science. This book expounds several aspects of intuitionistic type theory, such as the notion of set, reference vs. computation, assumption, and substitution. Moreover, the book includes philosophically relevant sections on the principle of compositionality, lingua characteristica, epistemology, propositional logic, intuitionism, and the law of excluded middle. Ample historical references are given throughout the book.

Johan G. Granström (1977) holds an Uppsala doctorate in mathematical logic (2009). He had the privilege of having Em.Prof. Per Martin-Löf, the father of dependent types, as doctoral supervisor (2003-2009), along with Prof. Erik Palmgren, a renowned expert in constructive mathematics.

Dr. Granström has been a short-term research fellow at Ludwig-Maximilians-Universität München (...

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Johan G. Granström (1977) holds an Uppsala doctorate in mathematical logic (2009). He had the privilege of having Em.Prof. Per Martin-Löf, the father of dependent types, as doctoral supervisor (2003-2009), along with Prof. Erik Palmgren, a renowned expert in constructive mathematics.

Dr. Granström has been a short-term research fellow at Ludwig-Maximilians-Universität München (2006-2007) and a research associate in formal methods for MDA at King’s College London (2009). Before entering into doctoral studies he was employed in the computer industry as systems developer, consultant, and software architect (1998-2003). He worked as Systems and Solutions Architect at Svea Ekonomi (2009-2011) and is currently employed by Google, Zürich (2011- ).

Contents.- List of Figures.- List of Tables.- Introduction.- Chapter I. Prolegomena. 1. A treefold correspondence.- 2. The acts of the mind.- 3. The principle of compositionality.- 4. Lingua characteristica.- Chapter II. Truth of Knowledge. 1. The meaning of meaning.- 2. A division of being.- 3. Mathematical entities.- 4. Judgement and assertion.- 5. Reasoning and demonstration.- 6. The proposition.- 7. The laws of logic.- 8. Variables and generality.- 9. Division of definitions.- Chapter III. The Notion of Set. 1. A History of set-like notions.- 2. Set-theoretical notation.- 3. Making universal concepts ito objects of thought.- 4. Canonical sets and elements.- 5. How to define a canonical set.- 6. More canonical sets.- Chapter IV. Reference and Computation. 1. Functions, algorithms, and programs.- 2. The concept of function.- 3. A formalization of computation.- 4. Noncanonical sets and elements.- 5. Nominal definitions.- 6. Functions as objects.- 7. Families of sets.- Chapter V. Assumption and Substitution. 1. The concept of function revisited.- 2. Hypothetical assertions.- 3. The calculus of substitutions.- 4. Sets and elements in hypothetical assertions.- 5. Closures and λ -calculas.- 6. The disjoint union of a family of sets.- 7. Elimination rukes.- 8. Propositions as sets.- Chapter VI. Intuitionism. 1. The intuitionistic interpretation of apagoge.- 2. the law of excluded middle.- 3. The philosophy of mathematics.- Bibliography.- Index of Proper Names.- Index of Subjects.-

Contents.- List of Figures.- List of Tables.- Introduction.- Chapter I. Prolegomena. 1. A treefold correspondence.- 2. The acts of the mind.- 3. The principle of compositionality.- 4. Lingua characteristica.- Chapter II. Truth of Knowledge. 1. The meaning of meaning.- 2. A division of being.- 3. Mathematical entities.- 4. Judgement and assertion.- 5. Reasoning and demonstration.- 6. The proposition.- 7. The laws of logic.- 8. Variables and generality.- 9. Division of definitions.- Chapter III. The Notion of Set. 1. A History of set-like notions.- 2. Set-theoretical notation.- 3. Making universal concepts ito objects of thought.- 4. Canonical sets and elements.- 5. How to define a canonical set.- 6. More canonical sets.- Chapter IV. Reference and Computation. 1. Functions, algorithms, and programs.- 2. The concept of function.- 3. A formalization of computation.- 4. Noncanonical sets and elements.- 5. Nominal definitions.- 6. Functions as objects.- 7. Families of sets.- Chapter V. Assumption and Substitution. 1. The concept of function revisited.- 2. Hypothetical assertions.- 3. The calculus of substitutions.- 4. Sets and elements in hypothetical assertions.- 5. Closures and λ -calculas.- 6. The disjoint union of a family of sets.- 7. Elimination rukes.- 8. Propositions as sets.- Chapter VI. Intuitionism. 1. The intuitionistic interpretation of apagoge.- 2. the law of excluded middle.- 3. The philosophy of mathematics.- Bibliography.- Index of Proper Names.- Index of Subjects.-
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