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The Pennsylvania State University the Graduate School The Pennsylvania State University The Graduate School Department of Philosophy THE DISCOVERY OF MATHEMATICAL PROBABILITY THEORY: A CASE STUDY IN THE LOGIC OF MATHEMATICAL INQUIRY A Thesis in Philosophy by Daniel Gerardo Campos © 2005 Daniel Gerardo Campos Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy December 2005 The thesis of Daniel Gerardo Campos was reviewed and approved* by the following: Douglas Anderson Associate Professor of Philosophy Thesis Advisor Co-Chair of Committee Emily Grosholz Professor of Philosophy Co-Chair of Committee Dale Jacquette Professor of Philosophy Catherine Kemp Assistant Professor of Philosophy Michael Rovine Associate Professor of Human Development and Family Studies John Christman Associate Professor of Philosophy Head of the Department of Philosophy *Signatures are on file in the Graduate School iii ABSTRACT What is the logic at work in the inquiring activity of mathematicians? I address this question, which pertains to the philosophical debate over whether, in addition to a logic of justification of mathematical knowledge, there is a logic of inquiry and discovery at work in actual mathematical research. Based on the philosophy of Charles Sanders Peirce (1839-1914), I propose that there is a logic of mathematical inquiry, and I expound its form. I argue that even though there are not rules or algorithms that will lead to breakthrough discoveries and successful inquiry with absolute certainty, Peirce’s philosophy provides a way to describe (i) the conditions for the possibility of mathematical discovery; (ii) the actual method of inquiry in mathematics and its associated heuristic techniques; and (iii) the logical form of reasoning that warrants the application of mathematical theories to the study of actual scientific problems in nature. With regard to (i), I discuss the role of the problem-context of discovery and describe the epistemic conditions necessary for carrying out mathematical reasoning. With respect to (ii), I argue that experimental hypothesis-making in the course of analytical problem- solving, and not deduction from axioms, is the actual method of mathematical research. Regarding (iii), I argue that abduction and analogy warrant the application of mathematical theories to the study of actual scientific problems. The discovery and early development of mathematical probability, culminating with Jacob Bernoulli’s Ars Conjectandi (1713), serves as the historical case study to examine critically the proposed logic of mathematical inquiry. I discuss the practical implications of the proposed logic of inquiry for a philosophy of mathematical education. iv TABLE OF CONTENTS LIST OF FIGURES .....................................................................................................vii ACKNOWLEDGEMENTS.........................................................................................viii Chapter 1 Introduction ................................................................................................1 Chapter 2 The Peircean Framework: Peirce’s Philosophy of Inquiry ........................11 2.1 Peirce’s Trichotomic.......................................................................................11 2.1.1 The Peircean Categories: Firstness, Secondness, and Thirdness .........13 2.1.2 Triadic Relations and the Mathematical Origin of the Categories.......18 2.2 The Triad in Reasoning ..................................................................................24 2.2.1 The Triad in Signs: Icons, Indices, and Symbols .................................26 2.2.2 The Triad in Arguments: Deduction, Induction, and Abduction..........30 2.2.3 On Peirce’s Distinction between Induction and Abduction .................43 2.3 Peirce’s Conception of Mathematics..............................................................57 2.3.1 Hypothetical States of Things ..............................................................58 2.3.1.1 Diagrams and Icons....................................................................65 2.3.1.2 The Applicability of Mathematical Theories .............................69 2.3.2 Necessary Reasoning, Experimentation, and Observation...................74 2.3.3 Pure Mathematical Reasoning versus Poietic Creation........................82 2.4 Mathematics as Creative, Precise, Experimental, Open-Ended Inquiry.........88 Chapter 3 The Context of Mathematical Discovery: The Case of Mathematical Probability Theory................................................................................................95 3.1 The 1654 Pascal-Fermat Correspondence and the Problem-Context of Discovery.......................................................................................................104 3.1.1 The ‘Problem with Dice’......................................................................105 3.1.2 The ‘Problem of Points’ and Expectation ............................................118 3.2 The Creation of a Hypothetical State of Affairs and the Origins of Mathematical Probability Theory..................................................................130 3.3 Peircean Considerations and Implications.....................................................141 Chapter 4 Epistemic Conditions for the Possibility of Mathematical Discovery .......146 4.1 Epistemic Abilities..........................................................................................146 4.1.1 Imagination...........................................................................................148 4.1.2 Concentration .......................................................................................158 4.1.3 Generalization.......................................................................................163 4.2 The Community of Inquirers ..........................................................................171 4.2.1 Systems of Representation ...................................................................174 4.2.2 Existing Mathematical Knowledge ......................................................180 v 4.2.3 Dialogical Criticism..............................................................................187 4.3 Pragmatic Upshot Towards a Logic of Mathematical Inquiry .......................192 Chapter 5 The Method of Mathematical Inquiry and the Heuristics of Discovery ....198 5.1 Heuristic Methods for Creating ‘Framing Hypotheses’ .................................200 5.1.1 Abstraction ...........................................................................................200 5.1.2 Framing Analogy..................................................................................205 5.2 The ‘Analytic Method’ of Mathematics and the Heuristics of ‘Analytical Hypothesis-Making’......................................................................................210 5.2.1 Lessons from Huygens’s General Method of Solution for the Problem of Points...................................................................................211 5.2.1.1 The ‘Analytic Method’ of Mathematics.....................................213 5.2.1.2 Generalization and Particularization as Analytical Heuristics...219 5.2.2 Lesson’s from the Demonstration of Bernoulli’s Theorem..................224 5.2.3 Pragmatic Upshot of the Historical Lessons: Towards a Logic of Mathematical Inquiry .............................................................................238 Chapter 6 The Leibniz-Bernoulli Correspondence: The Abductive Warrant of Bernoulli’s Theorem.............................................................................................242 6.1 The Correspondence .......................................................................................243 6.2 Bernoulli’s Hypothesis as an Inference to the Best Explanation....................259 6.3 Bernoulli’s Hypothesis as a Case of Abduction .............................................274 6.3.1 Formulating and Weighing Plausible Explanatory Hypotheses in Response to Living Doubt......................................................................279 6.3.2 ‘Abductive Insight’ in Bernoulli’s Hypothesis.....................................287 6.3.3 ‘Simplicity’ in Bernoulli’s Hypothesis.................................................292 6.4 Comparing Both Accounts of Bernoulli’s Ampliative Inference...................294 Chapter 7 Abduction as Rational Ampliative Inference: Objections and Replies......300 7.1 The Descriptive Adequacy of the Abductive Model ......................................301 7.2 The Problem of Truthful Hypothesizing.........................................................308 7.2.1 The Abductive Faculty .........................................................................312 7.2.2 The Simplicity of Abductive Hypotheses.............................................320 7.2.3 Bernoulli’s Truthful Hypothesizing .....................................................326 Chapter 8 Explanation and Reality .............................................................................329 8.1 True Probabilities as a priori Real Dispositions.............................................331 8.2 A priori Real Dispositions as Explanatory of a posteriori Statistical Regularities....................................................................................................344 8.2.1 Some Possible Objections ....................................................................354 8.2.1.1 Bernoulli’s ‘Propensity’ Interpretation?.....................................354 8.2.1.2 Bernoulli’s
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