University of New Mexico UNM Digital Repository Philosophy ETDs Electronic Theses and Dissertations 6-26-2015 Formalism and the Notion of Truth Joseph Spencer Follow this and additional works at: https://digitalrepository.unm.edu/phil_etds Recommended Citation Spencer, Joseph. "Formalism and the Notion of Truth." (2015). https://digitalrepository.unm.edu/phil_etds/10 This Dissertation is brought to you for free and open access by the Electronic Theses and Dissertations at UNM Digital Repository. It has been accepted for inclusion in Philosophy ETDs by an authorized administrator of UNM Digital Repository. For more information, please contact [email protected]. Joseph M. Spencer Candidate Philosophy Department This dissertation is approved, and it is acceptable in quality and form for publication: Approved by the Dissertation Committee: Paul M. Livingston, Chairperson Iain Thomson Kelly Becker William Martin i FORMALISM AND THE NOTION OF TRUTH by JOSEPH M. SPENCER B.A., Philosophy, Brigham Young University, 2004 M.L.I.S., Library Science, San José State University, 2009 M.A., Philosophy, University of New Mexico, 2012 DISSERTATION Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy Philosophy The University of New Mexico Albuquerque, New Mexico May, 2015 ii FORMALISM AND THE NOTION OF TRUTH by Joseph M. Spencer B.A., Philosophy, Brigham Young University, 2004 M.L.I.S., Library Science, San José State University, 2009 M.A., Philosophy, University of New Mexico, 2012 Ph.D., Philosophy, University of New Mexico, 2015 ABSTRACT The most widely acknowledged conceptions of truth take some kind of relation to be at truth’s core. This dissertation attempts to establish that an adequate conception of this relation begins with an investigation of the entanglement of the formal and the material as set forth in the model theoretical development of set theoretical mathematics. Truth concerns first and most crucially a certain commerce across the border between the formal and the material, between the ideal and the real. The entanglement of the formal and the material must be thought in itself, apart from or prior to any assimilation into philosophical schemas committed to larger metaphysical claims. This is accomplished in model theory. The twentieth century witnessed two attempts at bringing model theoretical mathematics to bear on accounting philosophically for the concept of truth: that of Alfred iii Tarski, and that of Alain Badiou. In order to investigate the relevance of model theory to the task of working out a philosophical conception of truth, this dissertation investigates, through comparative work, these two thinkers. It is necessary to see where their projects converge in important ways, as well as where their projects diverge in equally important ways. What brings their work into close proximity is their shared conviction that truth must be thought in light of model theory. Nonetheless, the two do not agree about exactly how model theory sheds light on truth. Comparative study thus reveals both a shared site for thinking and a struggle over the significance of that site. Agreement between Tarski and Badiou concerns the excess of the purely formal over itself, marked by the generation of an undecidable statement within formal systems of a certain level of complexity. Both thinkers determine that this formal excess touches on the material, and both further determine that the consequent entanglement of the formal and the material provides the basic frame for any philosophical consideration of truth. The point of disagreement is ultimately rooted in a difference of opinion about the adequacy of thinking the concept of truth independently of an account of what philosophers of science call theory change. iv TABLE OF CONTENTS INTRODUCTION.................................................................................................... 1 CHAPTER 1 – PRELIMINARY CONSIDERATIONS ....................................... 7 Thesis and Argument ..................................................................................... 9 Comparative Study ......................................................................................... 12 Badiou on Tarski ............................................................................................ 19 Balibar on Badiou .......................................................................................... 27 CHAPTER 2 – TARSKI’S PROJECT ................................................................... 36 Material Adequacy ......................................................................................... 39 The Definition of Truth .................................................................................. 50 Truth and Satisfaction .................................................................................... 61 CHAPTER 3 – BADIOU’S PROJECT .................................................................. 76 The Question of Being ................................................................................... 79 Voids and Events ............................................................................................ 92 Set Theory and Truth .....................................................................................106 CHAPTER 4 – TRUTH, SET THEORY, AND MODELS ...................................117 Divergence, Convergence ..............................................................................120 Model Theory and Ontology ..........................................................................126 The Debate .....................................................................................................138 Material Adequacy and Model Theory ..........................................................150 CONCLUSION ........................................................................................................154 WORKS CITED.......................................................................................................158 v INTRODUCTION For as long as truth has been investigated philosophically, it has been recognized that there must be some kind of relation at work in truth, whether that relation has been taken to be a matter of correspondence between minds and objects, of coherence among terms or claims made regardless of inaccessible states of affairs, or of fulfillment of intentions and aims in practical encounters with the world. In the course of this dissertation, I attempt to establish that an adequate conception of the relation that lies at the heart of truth begins with an investigation of the entanglement of the formal and the material as this is set forth in the model theoretical development of set theoretical mathematics. Truth, I argue, concerns first and most crucially a certain commerce across the border between the formal and the material, between the ideal and the real. But this means that essential to posing the question of truth is establishing the basic nature of this commerce. The entanglement of the formal and the material must be thought in itself, apart from or prior to any assimilation into philosophical schemas committed to larger metaphysical claims. This is accomplished, on my argument, in model theory. Strange as it might sound to say it, model theory presents a purely formal theory of the entanglement of the formal and the material. For this reason, it establishes what might be called the site of the problem of truth. Significantly, the twentieth century witnessed two attempts at bringing model theoretical mathematics to bear on accounting philosophically for the concept of truth. The first was the well-known work of Alfred Tarski, undertaken over the course of the 1 1930s and 1940s. The second was the less-familiar work of Alain Badiou, launched in the 1960s but brought to its first real flowering only in the 1980s. To investigate the relevance of model theory to the task of working out a philosophical conception of truth, it is necessary to investigate quite closely the work of these two thinkers. As a result, much of the work I undertake in the course of this dissertation is dedicated to close comparative readings of their most important writings on truth. It is necessary to see where their projects converge in important ways, as well as where their projects diverge in equally important ways. What seems rather clearly to bring their work into close proximity is what I take to be their shared conviction that truth must be thought beginning from a chiefly formal investigation of the entanglement of the formal and the material, and that in light of model theory. On my interpretation, in other words, Tarski and Badiou agree on the site of the problem of truth. Nonetheless and unsurprisingly, the two do not agree about exactly how model theory sheds light on truth. As a result, by looking comparatively at these two thinkers’ work, it is possible to present a kind of struggle over the significance of model theoretical considerations for any thinking of truth. The essence of the agreement between Tarski and Badiou is not difficult to summarize. Both thinkers focus on a certain excess of the purely formal over itself, marked by the generation of an undecidable statement within formal systems of a certain level of complexity. Both thinkers then determine that this excess concerns, precisely, the material, and both thinkers further determine that the consequent entanglement of the formal and the material provides the basic frame for any philosophical consideration of