FORMAL PROOFS AND REFUTATIONS A DISSERTATION SUBMITTED TO THE DEPARTMENT OF PHILOSOPHY AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY Jesse Alama June, 2009 i C 2009, Jesse Alama All rights reserved. ii I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. —————————————————— (Grigori Mints) Principal Adviser I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. —————————————————— (Solomon Feferman) I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. —————————————————— (Johan van Benthem) I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. —————————————————— (Jeremy Avigad) Approved for the Stanford University Committee on Graduate Studies. —————————————————— iii iv ABSTRACT Abstract Two questions drive the dissertation: • What can one discover in a formal mathematical theory? • What more do we know of a mathematical theorem when it has been formally proved than that it is provable? These questions spring from the provocative philosophy of mathematics of Imre Lakatos. They are tackled in two ways: by evaluating the philosophical foundations of Lakatos’s work, and by studying contemporary work in formal mathematics, specifically formal proof checking technology. The dissertation has a technical part and a philosophical part. The first part considers some philosophical problems raised (or brought into focus) by formal mathematical proofs. The second, technical part attempts to answer mathematical questions raised in the first part. The bridge between the two is a formal proof of a famous mathematical result known as Euler’s polyhedron formula, whose history Lakatos has reconstructed and which serves as the central example for his philosophy of mathematics. The aim of the dissertation is to explore some of the philosophical problems suggested by such formalization efforts. The argument of the dissertation has three components. In the first component, I ex- plain how Lakatos’s philosophy of mathematics poses a challenge to formal proof checking technology. The second component is to respond to the challenge by formalizing Euler’s polyhedron formula. Finally, the third component evaluates the technical, formal proof response. The dissertation is timely because, owing to developments in logic and computing in the last half-century, the concept of a formal proof, which used to be at best a model of mathemat- ical argumentation, has become more concrete and practical. It has now become possible to actually formalize significant mathematical proofs. These contemporary developments are a source of problems for a philosophy of mathematics that is sensitive to mathematical practice. This movement within the philosophy of mathematics is to no small degree in- spired by Lakatos’s work. The time is ripe for returning to some of the basic philosophical problems that Lakatos and other philosophers pointed to long ago, and to examine new problems that come from the development of what might be called proof technology, tools for helping mathematicians construct and evaluate proofs. v In chapter 1, I lay out some of the main questions and problems about formal proofs and explain how they are related to central issues within mainstream philosophy, particularly epistemology and philosophy of science. The development of formal proof technology is based on classical 19th and 20th century results in mathematical logic but depends crucially on computers. Chapter 1 also surveys the variety of uses of computers in mathematical practice and discusses the variety of philosophical problems they pose. The next step in the discussion of formal proofs will be a critical evaluation of the philosophy of mathematics of Imre Lakatos. His Proofs and Refutations (1963) attacks formalist philosophies of mathematics. Since much proof technology is to some extent based on or requires a certain formalist view of mathematics, the question naturally arises how Lakatos’s philosophy bears on these developments. Chapter 2 addresses these concerns. I focus also on some epistemological problems suggested by formal proofs, such as the question of defining rigor and the problem of whether and how one improves one’s justification for a mathematical claim through formalization of a proof of it. The cornerstone of Lakatos’s Proofs and Refutations is a history of a particular math- ematical theorem known as Euler’s polyhedron formula, which is a certain geometrical- combinatorial claim with a rather colorful history. I have formalized a proof of this math- ematical result; chapter 3 contains a discussion of the proof and its formalization. Thanks to the work carried out in chapter 3, Euler’s polyhedron formula (understood in a certain abstract or combinatorial way that is explained in chapter 3) is shown to be a first- order consequence a certain first-order theory of sets. Because of the peculiarities of the particular proof technology with which the formal proof was carried out, the theory of sets that is used is much stronger than what is intuitively required for Euler’s theorem. A natural proof-theoretic question thereby arises: can one do better? Are the strong assumptions really necessary? In chapter 4, I identify a weaker theory in which to carry out a formal proof Euler’s formula. Also discussed are some formal problems about expressibility problems for combinatorial polyhedra, and related issues. In chapter 5, I return to some of the issues that Lakatos raised in connection with formal proofs in light of the formal work that is carried out in chapter 3. This work provides some resources for taking on the two questions that were initially asked. I show that Lakatos’s philosophy, its strong reservations against ‘formalism’ notwithstanding, applies quite naturally to formal mathematics. vi Acknowledgements First of all, my advisor, Grisha Mints, has been supportive and caring throughout my graduate education. I am thankful for my relationship with him, and for the opportunities he gave me throughout my time at Stanford. My dissertation committee has also been quite helpful. I thank Sol Feferman, Johan van Benthem, and Jeremy Avigad for their support in helping me to grow from a student to a scholar. I would also like to thank those who contributed by giving advice, criticism, and sup- port. My family has been especially important in providing a steady base for me. In addition, the following people have all played their roles, small and large (in no particular order): Tomohiro Hoshi, Tobey Scharding, Paolo Mancosu, Marc Pauly, Krista Lawlor, Ulrich Kohlenbach, Branko Grünbaum, Lauren Hartzell, Valeria de Paiva, Darko Sarenac, Quayshawn Spencer, Alan Woods, Herman Geuvers, Audrey Yap, Heidi Dolamore, Jip Veldman, Boris Moroz, Fernando Ferreira, Henk Barendregt, Alexei Angelides, Bas Spit- ters, Patrick Girard, Elizabeth Coppock, Freek Wiedijk, Maria Taylor, David Fernandez, Stephen Simpson, Marvin Greenberg, Lanier Anderson, Michael Friedman, Robert Solovay, Tyler Greene, Peter Koepke, Conor Mayo-Wilson, Michael Beeson, Henry Towsner, Alistair Isaac, Mark Crimmins, Eric Pacuit. I am grateful for all of you. vii Contents Abstract . v Acknowledgements . vii 1 Introduction . 0 2 Formal Proofs in Mathematics . 4 2.1 Introduction . 4 2.2 Formal Proof Technology: Three Strands . 4 2.3 Early Growth of Formal Proof Technology . 6 2.4 Contemporary Developments in Formal Proof Technology . 8 2.4.1 The QED Project . 9 2.5 Digression: Computers in Mathematics . 10 2.6 Formal Proof Technology: A Philosophical Error? . 15 3 A Lakatosian Challenge . 18 3.1 Introduction . 18 3.1.1 Digression: the problem of interpreting Proofs and Refutations . 19 3.2 Main Features of Lakatos’s Philosophy of Mathematics . 21 3.2.1 The method of proofs and refutations . 22 3.2.2 Lakatos on proof (continued) . 24 3.2.3 Digression: Lakatos and Pólya . 27 3.3 Summary of Lakatos scholarship . 28 3.4 Some Basic Philosophical Issues in Lakatos’s Work . 33 3.5 Lakatos and Mathematical Skepticism . 35 3.5.1 Fallibilism in mathematics . 40 3.6 A Lakatosian Challenge . 41 4 A Formal Proof of Euler’s Polyhedron Formula . 44 4.1 Introduction . 44 4.2 A Brief History of Euler’s Polyhedron Formula . 44 4.3 Poincaré’s Proof of Euler’s Polyhedron Formula . 47 4.4 The Formalization . 50 4.4.1 Main formalizations . 50 4.4.1.1 The rank+nullity theorem . 51 4.4.1.2 The vector space of subsets of a set based on symmetric difference . 53 4.4.1.3 Polyhedra . 54 4.5 Discussion . 55 4.5.1 Definition of polyhedron and being a homology sphere . 55 4.5.1.1 Algebraic topological definition of polyhedron . 56 4.5.1.2 Simple connectedness and homology spheres . 60 viii 4.5.2 A proof-theoretic question . 62 4.5.3 Streamlining the formalization . 64 4.6 Conclusion and Further Work . 65 5 Metamathematical Problems about Polyhedra . 68 5.1 Introduction . 68 5.2 Expressibility Problems for Combinatorial Polyhedra . 68 5.2.1 Being a homology sphere . 69 5.2.2 Eulerianness . 71 5.2.2.1 Extending the result: euler characteristic and general-dimensional polyhedra . 73 5.2.2.2 Monadic second-order logic . 77 5.2.2.3 Expressibility using an equicardinality generalized quantifier . 80 5.2.2.4 Expressibility in dyadic existential second-order logic . 82 5.2.3 Convexity ......................................................... 83 5.3 Formal Theories of Polyhedra . 84 5.3.1 Steinitz-Rademacher polyhedral complexes . 84 5.3.1.1 Digression: expressibility of eulerianness in the class of polyhedral complexes .