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Riemann Integration in NQA+: the Role of Geometric Subdivision and Its Consequences for the Continuum

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University of North Dakota UND Scholarly Commons Theses and Dissertations Theses, Dissertations, and Senior Projects January 2021 Riemann Integration In NQA+: The Role Of Geometric Subdivision And Its Consequences For The Continuum Matthew J. Lorenz Follow this and additional works at: https://commons.und.edu/theses Recommended Citation Lorenz, Matthew J., "Riemann Integration In NQA+: The Role Of Geometric Subdivision And Its Consequences For The Continuum" (2021). Theses and Dissertations. 3933. https://commons.und.edu/theses/3933 This Thesis is brought to you for free and open access by the Theses, Dissertations, and Senior Projects at UND Scholarly Commons. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of UND Scholarly Commons. For more information, please contact [email protected]. RIEMANN INTEGRATION IN NQA+ THE ROLE OF GEOMETRIC SUBDIVISION AND ITS CONSEQUENCES FOR THE CONTINUUM by Matthew Jon Lorenz Bachelor of Science, University of North Dakota 2018 A Thesis Submitted to the Graduate Faculty of the University of North Dakota in partial fulfillment of the requirements for the degree of Master of Science Grand Forks, North Dakota May 5, 2021 DocuSign Envelope ID: A5DF4CC1-FF26-4BB6-98DA-FA3FD58BF3ED Matthew Lorenz Name: Degree: Master of Science This document, submitted in partial fulfillment of the requirements for the degree from the University of North Dakota, has been read by the Faculty Advisory Committee under whom the work has been done and is hereby approved. ____________________________________ Bruce Dearden ____________________________________ Jeremiah Bartz ____________________________________ Joel Iiams ____________________________________ ____________________________________ ____________________________________ This document is being submitted by the appointed advisory committee as having met all the requirements of the School of Graduate Studies at the University of North Dakota and is hereby approved. ____________________________________ Chris Nelson Dean of the School of Graduate Studies 5/3/2021____________________________________ Date iii PERMISSION TITLE Riemann integration in NQA+: The role of geometric subdivision and its consequences for the continuum DEPARTMENT Mathematics DEGREE Masters of Science In presenting this thesis in partial fulfillment of the requirements for a graduate degree from the University of North Dakota, I agree that the library of this University shall make it freely available for inspection. I further agree that permission for extensive copying for scholarly purposes may be granted by the professor who supervised my thesis work or, in his absence, by the Chairperson of the department or the dean of the School of Graduate Studies. It is understood that any copying or publication or other use of this thesis or part thereof for financial gain shall not be allowed without my written permission. It is also understood that due recognition shall be given to me and to the University of North Dakota in any scholarly use which may be made of any material in my thesis. Matthew Lorenz May 2021 3 Abstract: In the mid 1990s mathematicians Rolando Chuaqui and Patrick Suppes developed a constructive axiomatic system of nonstandard analysis. This project, now called NQA+ (later developed as Elementary Recursive Nonstandard Analysis by Suppes and Richard Sommer) is notable because it has a finitary consistency proof with which we can develop a large fragment of infinitesimal analysis, which they claim represents the mathematical practice characteristic of physics in a manner that does justice to the geometric intuition that facilitated, for example, 17th century indivisible methods. In this thesis I develop the authors’ formulation of geometric subdivision and of the integral as a nonstandard Riemann sum, and provide examples which clarify my main research questions, which are 1) to which practices does geometric subdivision refer? 2) what are their motivations? 3) how does geometric subdivision provide a foundation for NQA+, and what consequences does it have for the continuum of real numbers? 2 Contents Contents i 1 Introduction 1 1.1 General outline . .3 1.2 Overview of NQA+ and relevant literature . .7 2 Basic Analysis and NQA+ 15 2.1 Introduction . 15 2.2 Basic functions, differentials, and derivatives . 15 2.3 Familiar results: The intermediate value theorem . 18 2.4 Working in NSA . 20 2.4.1 An introduction to transfer . 21 2.4.2 An introduction to standard part . 25 2.5 IVT in NSA . 29 2.6 IVT in NQA+ ............................. 36 2.6.1 Hyperfinite partitions . 36 2.6.2 A first look at functions . 37 2.6.3 The selector . 40 2.6.4 IVT≈ ............................. 44 2.7 More familiar results for continuous functions . 46 3 Re-examining 17th Century Geometry 49 3.1 The early calculus . 50 3.2 Cavalieri and geometric intuition . 52 3.2.1 Extending the Greek theory of magnitudes: Infinitesimals and geometric intuition . 54 3.2.2 All the lines, omnes-concepts, and agnosticism toward the composition of continua . 62 3.3 Arithmetization, Intuition, and the Continuum . 71 3.3.1 Boyer’s Interpretation of Cavalieri . 71 3.3.2 17th Century Roots of Arithmetization . 75 3.3.3 Continuity and the Limits of Empirical Verifiability . 79 3.4 Conclusion: Magnitudes and Meaning . 83 4 Hyperfinite approximation and other underlying theories 91 4.1 Halos, liftings, and approximation . 93 4.2 Finite models, physical meaning, and Ω-invariance . 96 4.3 Philosophy and hyperfinite methodologies . 102 4.3.1 Hyperfinite sets, Zeno’s paradoxes, and supertasks . 105 4.3.2 Restricted conceptions of truth and the utility of ideal objects 107 4.3.3 Applications to physics and physical sciences . 111 i ii CONTENTS 5 Riemann Integrals in NQA+ and ERNAA 119 5.1 Nonstandard Riemann summation . 120 5.1.1 Definite integrals as hyperfinite sums . 125 5.1.2 Theorems for hyperfinite sums . 129 5.1.3 Theorems for definite integrals . 133 5.2 Stratification and Riemann Integration in ERNAA .......... 138 5.3 Example from classical physics . 148 6 Conclusion 155 A NQA+ (1993/95) 157 A.1 Logical Axioms and Rules . 157 A.2 Restricted Field Axioms . 158 A.3 Axioms for Natural Numbers and Open Induction . 162 A.4 Axioms for Infinitesimals . 165 Bibliography 167 1 | Introduction In the early 1990’s, Rolando Chuaqui (1935-1994) and Patrick Suppes (1922- 2014) initiated a project to develop a constructive approach to nonstandard analysis. The resulting formal system has since been named NQA+, and was developed by Suppes and Richard Sommer into a more general system called Elementary Recur- sive Nonstandard Analysis (ERNA). It will be seen that ERNA and NQA+ repre- sent a nexus in the history of modern infinitesimals, one of several theories fusing finitary mathematics and logical-axiomatic systems with the motivating physical in- tuition that supported the use of infinitesimal quantities in the 17th and even earlier centuries. What distinguishes ERNA from other subpractices of Nonstandard Analysis and other weak theories, among other things, are some of the bold statements made by the authors concerning the role of the continuum in mathematical practice; for example, that ERNA’s motivating project is to “provide a foundation that is close to the mathematical practice characteristic of theoretical physics,” [83, p. 2] and that it “better matches certain geometric intuitions about the number line,” [84, p. 4] as compared to classical analysis. Amid these statements is the surprising acknowledgment that the type of infinitesimal partition practiced in ERNA has a direct relation to older practices of geometric subdivision; interestingly, it is claimed that the methods in this weakened system of nonstandard analysis have a link to the geometry of, for example, Bonaventura Cavalieri (1598-1647) [89, p. 19]. The goal of this thesis is to fully develop the above statements made by the authors within its primary context of nonstandard Riemann Integration in NQA+. ERNA and NQA+ are a profound restatement of the rules of arithmetic and calculus with a basic set of terms which include terms acting as infinite and in- finitesimal numbers. However, their logic and construction are explicitly finitary and their infinitesimal and infinite terms are usually only symbolically so; all terms 1 2 CHAPTER 1. INTRODUCTION present can finally be put in isomorphism with a finite subset of standard rational numbers, so that we can explicitly calculate our most basic infinitesimal and in- finite terms during the process of solving, for example, a differential equation or, fundamentally, an integral. The explicitly finite constructions possible within the mathematics of ERNA and NQA+ provide the possibility of modeling “a signifi- cant part of the mathematics used in the sciences,” [83] where approximations by rational numbers within a specified region of error are the norm. The result just mentioned, connecting the terms of ERNA to subsets of standard rational numbers, is an isomorphism theorem for ERNA. This surprising result “as- serts that the substructure determined by the interpretation of a finite set of terms in a ‘reasonably sound’ infinite model of nonstandard analysis is isomorphic to finite substructures of the standard rational numbers. This yeields a partial isomorphism from the physically continuous to the physically discrete.” This result . supports the strong physical intuition that no experiment can suc- cessfully distinguish between any physical quantities, even space and time, being continuous or discrete
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