Reference and Computation in Intuitionistic Type Theory By J. G. Granstrom¨ Uppsala MMVIII Contents Contents iii List of Figures v List of Tables vii Preface ix Introduction xi Chapter I. Prolegomena 1 § 1. A threefold correspondence 1 § 2. The acts of the mind 5 § 3. The principle of compositionality 7 § 4. Lingua characteristica 10 Chapter II. Truth and Knowledge 13 § 1. The meaning of meaning 13 § 2. A division of being 16 § 3. Mathematical entities 19 § 4. Judgement and assertion 24 § 5. Reasoning and demonstration 27 § 6. The proposition 28 § 7. The laws of logic 39 § 8. Variables and generality 48 § 9. Division of definitions 54 Chapter III. The Notion of Set 57 § 1. A history of set-like notions 57 § 2. Set-theoretical notation 64 § 3. Making universal concepts into objects of thought 64 § 4. Canonical sets and elements 69 § 5. How to define a canonical set 75 § 6. More canonical sets 81 Chapter IV. Reference and Computation 85 § 1. Functions, algorithms, and programs 86 § 2. The concept of function 90 iv § 3. A formalization of computation 95 § 4. Noncanonical sets and elements 101 § 5. Nominal definitions 108 § 6. Functions as objects 110 § 7. Families of sets 114 Chapter V. Assumption and Substitution 121 § 1. The concept of function revisited 121 § 2. Hypothetical assertions 125 § 3. The calculus of substitutions 136 § 4. Sets and elements in hypothetical assertions 148 § 5. Closures and the λ-calculus 152 § 6. The disjoint union of a family of sets 160 § 7. Elimination rules 162 § 8. Propositions as sets 170 Chapter VI. Intuitionism 175 § 1. The intuitionistic interpretation of apagoge 175 § 2. The law of excluded middle 184 § 3. The philosophy of mathematics 192 Summary in Swedish 197 Bibliography 199 Index of Proper Names 211 Index of Subjects 215 List of Figures 1 The relation between object, concept, and expression 3 2 Meaning, referent, and term 14 3 Mediate vs. immediate reference 15 4 Division of modes of being into real and ideal 17 5 The threefold correspondence for universal concepts 18 6 The threefold correspondence for beings of reason 22 7 The disjoint union of a family of sets 116 8 Example of a dependently typed function 118 9 A context, telescopic in the sense of de Bruijn 127 List of Tables 1 Division of logic according to the acts of the mind 6 2 The interpretation of the propositional connectives 31 3 Classification of mathematical categorems 51 4 Four different notions of set 59 5 Classification of assertions and inference rules 77 6 Overview of seven different notions of function 122 7 The Curry-Howard correspondence 171 8 Two types of knowledge 179 Omnia cum veterum sint explorata libellis, Multa loqui breviter sit novitatis opus.† †Preface to Phocas’ grammar, quoted in de Bury, Philobiblon, p. 50. Preface y interest in the foundations of mathematics began while I was writing a dissertation on class number formulae as a student of mathematics at Stockholm University. The M modern treatment of this subject involves a great deal of ring theory wherein the machinery of extensional set theory and the law of excluded middle are fully employed ; furthermore, in modern treatments, many central concepts in this field are indirectly defined. In trying to read such a modern treatment, two foundational problems become very tangible. First, that one can read a proof of a theorem, see that it follows the rules of the game, but still not accept the conclusion.1 Next, that one can read the definition of some characteristic, say of a number field, without getting a hint of how to compute it.2 These problems spurred me on to studying the foundations of math- ematics, albeit on my own and with little success. Disillusioned, I even- tually gave up mathematics and accepted an offer to work as a computer programmer. This brought me for the first time into contact with LISP and functional programming, something which later proved very valu- able. Need I add that my dissertation on class number formulae never got finished ? A couple of years later occurred one of those twists of fate which in retrospect seem providential. While still working as a computer pro- grammer, I decided to attend Prof. Per Martin-L¨of’s course on type theory, given at Stockholm University, 2003. Not only did I attend, but it seems as if I had the necessary background to actually understand what was going on. At this point I started to inquire Prof. Martin-L¨of about every aspect of his type theory and I have not stopped yet. I soon cancelled my job and started to look for a scholarship to write a PhD dissertation with type theory as the subject. It did not take me long to be accepted to FMB, which is an abbreviation in Swedish which translates as Graduate school in Mathematics and Computing. At FMB, I had the fortune of getting Prof. Erik Palmgren as supervisor and, thanks to the generosity of FMB, we managed to get funding for me to also have Prof. Martin-L¨ofas supervisor. 1Cf. the quotation from Skolem on p. 193. 2Cf. Ch. IV. x Now more than five years ago, in July 2003, I began my PhD studies in Uppsala. The first two years were mainly spent reading courses and getting better acquainted with my subject matter ; the last three years have been spent writing this thesis. During this time, my understanding of type theory has developed, and this explains the distinct characters that the different parts of this thesis have. I soon discovered that a lot in type type theory happens, as it were, under the hood : the more philosophical parts of this thesis is the result of my efforts to find out what that was. Apart from this excursion into philosophy, I have more or less remained faithful to the original plan of writing about matters pertaining to computation and eager evaluation in type theory. In addition to excellent supervision, another benefit of being in Up- psala is the weekly Stockholm-Uppsala logic seminar with its many prominent guests. One especially distinguished and frequent guest at the Stockholm-Uppsala logic seminar has been Prof. Helmut Schwicht- enberg who spent the 2005-2006 winter term in Uppsala. Thanks to Prof. Schwichtenberg I could spend the 2006-2007 winter term at LMU, Munich, under the auspices of the MATHLOGAPS program. This ex- perience has certainly broadened my view on logic. Acknowledgments. I would like thank the following individuals and organizations who in various ways have contributed to the completion of this thesis : my parents and my brother for their continual support ; my supervisors Prof. Martin-L¨ofand Prof. Palmgren for trying to infuse in me high academic standards—if I fail to live up to them, the fault is entirely my own ; Prof. Martin-L¨ofagain, this time for discovering my subject matter ; FMB for financial support ; the Department of Math- ematics at Uppsala University for providing me with a working envi- ronment ; Prof. Schwichtenberg, Dr. Peter Schuster, LMU, Munich, and MATHLOGAPS, for making my stay in Munich possible and enjoyable ; Mr. Bo Hagerf, Rev. H˚akan Lindstr¨om,Prof. Viggo Stoltenberg-Hansen, and Mr. Isidor Johan Kullbom, who have read and commented on early drafts of this thesis ; and all my friends and colleagues with whom I have discussed matters related to type theory. Johan Georg Granstr¨om Uppsala, Sweden October 26th, 2008 Introduction he prerequisites for understanding this thesis have been kept at a minimum. All that is required is some background in mathematics and the ability to follow a rigorous proof. Back- T ground in philosophy and computer science is helpful but not necessary. This thesis is about intuitionistic type theory, but no back- ground in logic is required, even though some familiarity with natural deduction is an advantage. In fact, profound knowledge of extensional set theory can be an impediment, rather than a help, since it is difficult to forget what one already knows. The easiest way to learn intuition- istic type theory is to disregard any preconceptions about logic and set theory and start afresh with the definitions and axioms of intuitionis- tic type theory. Only after having understood the whole system and its methodology, one should make a comparison with what one knew before. The kind of type theory presented in this thesis has been variously called intuitionistic type theory, constructive type theory, dependent type theory, and, after its first expounder, Martin-L¨of’s type theory. As often is the case when one subject has many different names, there are different nuances to them—so also in this case. I have chosen the name intuitionistic type theory because it was the first name applied to the subject and because the type theory in question is the natural adaption of earlier type theories, e.g., the ramified and simple type theories, to intuitionistic principles. In this thesis I will present intuitionistic type theory together with my own contributions to it, resulting in a version of intuitionistic type theory which is essentially backwards compatible with Martin-L¨of’s ver- sion.3 My technical contributions are the following : (1) coinductive definitions of sets are treated on a par with inductive definitions (cf. Def. 4 on p. 70) ; 3Martin-L¨of’s 1984 book Intuitionistic Type Theory contains the essential ideas of intuitionistic type theory, but several important contributions to the system have been presented only in the form of lectures. At present, the most authoritative presentation of intuitionistic type theory is Nordstr¨om, Petersson and Smith, Programming in Martin-L¨of’s Type Theory.