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From the Outside Looking In: Can Mathematical Certainty Be Secured Without Being Mathematically Certain That It Has Been?

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From the Outside Looking In: Can Mathematical Certainty Be Secured Without Being Mathematically Certain That It Has Been? From the Outside Looking In: Can mathematical certainty be secured without being mathematically certain that it has been? Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University Matthew Souba, MSc, MLitt Graduate Program in Philosophy The Ohio State University 2019 Dissertation Committee: Neil Tennant, Advisor Stewart Shapiro Christopher Pincock Copyright by Matthew Souba 2019 Abstract The primary aim of this dissertation is to discuss the epistemological fallout of Gödel’s Incompleteness Theorems on Hilbert’s Program. In particular our focus will be on the philo- sophical upshot of certain proof-theoretic results in the literature. We begin by sketching the historical development up to, and including, Hilbert’s mature program, discussing Hilbert’s views in both their mathematical and their philosophical guises. Gödel’s Incompleteness Theorems are standardly taken as showing that Hilbert’s Pro- gram, as intended, fails. Michael Detlefsen maintains that they do not. Detlefsen’s argu- ments are the focus of chapter 3. The argument from the first incompleteness theorem, as presented by Detlefsen, takes the form of a dilemma to the effect that either the infini- tistic theory is incomplete with respect to a certain subclass of real sentences or it is not a conservative extension over the finitistic theory. He contends that Hilbert need not be committed to either of these horns, and, as such the argument from the first incompleteness theorem does no damage to Hilbert’s program. His argument against the second incomplete- ness theorem as refuting Hilbert’s Program, what he calls the stability problem, concerns the particular formalization of the consistency statement shown unprovable by Gödel’s theorem, and endorses what are called Rosser systems. The success of Detlefsen’s arguments critically depends upon the precise characterization of what exactly Hilbert’s program is. It is our contention that despite Detlefsen’s attempts, both of the arguments (from the first and sec- ond incompleteness theorems) are devastating to Hilbert. The view that Detlefsen puts forth is better understood as a modified version of Hilbert’s general program cast as a particularly ii strict form of instrumentalism. We end by analyzing the coherence of Detlefsen’s proposal, independently of the historical Hilbert. In response to Gödel’s Incompleteness theorems several modified or partial Hilbert’s pro- grams have been pursued. In chapter 3 we consider one such version due to Gentzen that enlarges the methods to be admitted in consistency proofs. By giving up the stress on strictly finitary reasoning and liberalizing what counts as epistemically acceptable, Genzten was able to prove that PA is consistent by appeal to the principle of quantifier-free transfinite induction up to the ordinal 0. Gentzen’s method proceeds by means of ordinal assignments to, and reduction procedures for, possible proofs of contradiction, showing such proofs to be impossible. We first present Gentzen’s method in order to provide a systematic overview of the structure of his proof and its philosophical motivations. We then consider the modern version of the proof as presented by Gaisi Takeuti. Central to Takeuti’s proof is the demon- stration of the claim that whenever a concrete method of constructing decreasing sequences of ordinals is given, any such decreasing sequence must be finite. Takeuti takes the construc- tive demonstration of this result as being of particular philosophical and epistemic value. The central theme that comes out of the philosophical discussion is that such a result can only be understood “from the outside” of the system. Our discussion of Takeuti shows how this theme generalizes, and will be of central importance in the rest of the dissertation. Predicativity is similar in spirit to the work of Gentzen discussed above in the sense of liberalizing what counts as an epistemically safe starting point. The basic idea behind Predicativism is the acceptance of the natural numbers as basic to human understanding and then to see just how much of mathematics can be shown on the basis of this starting point. The discussion of Predicativity is important for two reasons. First, it signals a shift in how one understands foundational work in mathematics. In particular it signals a shift from a focus on security and justification to that of determining the limits of particular philosophical views. It serves as a lynchpin, so to speak, between the early foundational work discussed in the first half of the dissertation and the more contemporary foundational work discussed iii in the second part. To see this clearly we begin chapter 5 by sketching some of the early historical developments of Predicativity, focusing on Russell and Weyl. We then look at more recent technical developments of Predicativity, culminating with the limitative result that the bound of Predicativity is Γ0. The second reason why Predicativity is important is due to a relevant feature it shares with Finitism. It provides another lucid example of the internal/external divide that concerned the philosophical portion of the discussion of Takeuti mentioned above. This theme emerges as the central point upon which the philosophical value and significance of the epistemic aspect of proof theory hinges. We end chapter 5 by highlighting this connection to Finitism, setting the stage for some philosophical work to be done in chapter 6. The proof-theoretical reduction of the system WKL0 to PRA is taken as perhaps the clearest example of a partial realization of Hilbert’s Program. The concept of reductive proof theory more generally is taken as being foundationally informative in the sense that certain reductions and conservativity results are taken as revealing just how much mathematics is justified on the basis of other, more elementary frameworks. Implicit in this is the notion of epistemic security. We consider as case studies the reductions of the systems IΣ1 and WKL0 to PRA in order to get clear on exactly what such justification and security amounts to. We argue that to properly understand what is meant by these terms requires a closer look at the radicalization of the axiomatic method and the shift to formalism that underlies Hilbert’s Program. After looking at Hilbert’s famous disagreement with Frege we suggest that central to the notions of justification and security is the meaning of what is expressed in formal languages that is eschewed by the shift to formalism. To appreciate this meaning and hence achieve the level of justification and security that is claimed by proof-theoretical reductions requires one to be on the outside of the formal system looking in. The notion of a foundation for mathematics is vague. One can distinguish between what might be called Hilbertian Foundationalism and Naturalistic Holism. In light of Gödel’s results, Foundationalism has largely gone out of style. The lesson from Gödel is commonly iv taken to be that the goal of the Hilbert Program is simply an outdated and unattainable ideal. In its place has emerged the holistic picture whereby a foundation for mathematics is to be understood as a way to bring the seemingly disparate branches of mathematics together in a unifying way. The sharp contrast between Naturalistic Holism and Hilbertian Foundationalism connects chapter 7 to the rest of the dissertation by illustrating how, to many, the philosophical steam has left the Hilbertian machine. We look at one such brand of Naturalistic Holism due to Penelope Maddy centered on set theory as the foundational arena. Maddy’s aim is to understand the proper grounds for the introduction of sets and set-theoretic axioms, as well as a justification of set-theoretic practice. After considering Maddy’s account of how such an understanding proceeds, we provide critical commentary. Central to the discussion is the notion of mathematical depth. We end with some remarks on mathematical methodology followed by a brief discussion of the a priori and its role in bridging the gap between mathematics and philosophy. v Dedication Dedicated to my wife, Giulia, who has given up so much, and to my son, Jack, for whom I would give up everything. vi Acknowledgments I would like to thank all of my past teachers for their time and dedication to providing me an education. Having now, myself, taught others, I understand the sometimes thanklessness of the job. Special thanks to those who noticed something in me and encouraged me to further study. In particular, I thank Glenn Ross, my logic teacher at Franklin & Marshall College, for first introducing me to the beauty of logic and sparking my interest in epistemology. I would like to thank my committee members for their generosity with their time. Chris Pincock provided numerous insightful comments and thought-provoking questions on early versions of this work that me re-think certain things I had taken for granted. I first met Stewart Shapiro while a student at St. Andrews University in Scotland. His willingness to meet with me and discuss my Master’s thesis, and the level of attention that he showed both it and me, was the primary reason why I chose to attend The Ohio State University. His help in the subsequent years has been invaluable. To my advisor, Neil Tennant, who opened the door to my interest in proof theory, specifically with respect to foundational issues. The time that you have dedicated to me, whether talking through philosophical issues, providing suggestions and encouragement, or imparting general life advice has really meant a lot to me. Your devotion as a scholar, teacher, and mentor is truly inspiring. I would like to thank my family and friends for being an outlet from academia. To my mother-in-law, Gabriella. Without your help these past few weeks watching Jack, I never would have been able to complete the final touches on this dissertation on time.
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