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Foundations of Geometry California State University, San Bernardino CSUSB ScholarWorks Theses Digitization Project John M. Pfau Library 2008 Foundations of geometry Lawrence Michael Clarke Follow this and additional works at: https://scholarworks.lib.csusb.edu/etd-project Part of the Geometry and Topology Commons Recommended Citation Clarke, Lawrence Michael, "Foundations of geometry" (2008). Theses Digitization Project. 3419. https://scholarworks.lib.csusb.edu/etd-project/3419 This Thesis is brought to you for free and open access by the John M. Pfau Library at CSUSB ScholarWorks. It has been accepted for inclusion in Theses Digitization Project by an authorized administrator of CSUSB ScholarWorks. For more information, please contact [email protected]. Foundations of Geometry A Thesis Presented to the Faculty of California State University, San Bernardino In Partial Fulfillment of the Requirements for the Degree Master of Arts in Mathematics by Lawrence Michael Clarke March 2008 Foundations of Geometry A Thesis Presented to the Faculty of California State University, San Bernardino by Lawrence Michael Clarke March 2008 Approved by: 3)?/08 Murran, Committee Chair Date _ ommi^yee Member Susan Addington, Committee Member 1 Peter Williams, Chair, Department of Mathematics Department of Mathematics iii Abstract In this paper, a brief introduction to the history, and development, of Euclidean Geometry will be followed by a biographical background of David Hilbert, highlighting significant events in his educational and professional life. In an attempt to add rigor to the presentation of Geometry, Hilbert defined concepts and presented five groups of axioms that were mutually independent yet compatible, including introducing axioms of congruence in order to present displacement. He originally presented this new material through a series of lectures during 1898-1899. Using this new presentation of geometry, he demonstrated an algebra of segments in accordance with the laws of arithmetic and showed that this was all possible without the axiom of continuity. In addition, Hilbert created new geometries in order to study Desargues’s Theorem as it relates to these axioms. iv Acknowledgements First, and foremost, I would like to thank God for the many people He put in my path to help me throughout the Master’s program, from classmates, professors and staff at the university to friends and associates that encouraged me along the way. Secondly, I would like to thank my wife and children for the incredibly patient support and understanding that they provided, knowing that the work that I was doing towards my Master’s degree was going to impinge on my commitments to them for a finite period of time. Finally, I would Eke to thank my committee members, each of which were very responsive and helpful throughout the lengthy process of this thesis paper, from the research and writing all the way through to the culmination of the final project. V Table of Contents Abstract iii Acknowledgements iv List of Figures vi 1 Classical Geometry: Euclid 1 2 David Hilbert’s Beginnings 4 3 Introduction to The Foundations of Geometry 9 4 The Five Groups of Axioms 10 5 Compatibility and Mutual Independence of the Axioms 15 6 Independence of the Axioms of Congruence and Continuity 19 7 The Theory of Proportion 23 8 An Algebra of Segments based upon Pascals Theorem 30 9 The Theory of Plane Areas 36 10 Desargues’s Theorem 41 11 Hilbert’s Conclusion 57 Bibliography 58 vi List of Figures 4.1 Axiom II-5: Intersection Between Line and Sides of Triangle ........ 11 5.1 Illustration of the Meaning of Axiom II-5....................................................... 16 5.2 Axiom III: Statement of Parallels....................................................................... 17 6.1 Axiom IV-3: Segments can be added................................................................. 20 7.1 Pascal’s Theory, of Proportion.............................................................................. 25 7.2 Cosine: Hilbert style.............................................................................................. 25 7.3 Angles are interchangeable.................................................................................... 26 7.4 Additional Labeling of Pascal’s Figure.............................................................. 27 7.5 Demonstrating Pascal’s Theorem...................... 28 7.6 Demonstrating Pascal’s Theorem II.................................................................... 29 8.1 Multiplication Defined Geometrically................................................................. 30 8.2 Commutative Law of Multiplication................................................................. 31 8.3 Associative Law of Multiplication....................................................................... 32 8.4 The First Distributive Law................................................................................. 32 8.5 Similar Triangles.................................................................................................... 33 8.6 General Case.......................................................................................................... 34 8.7 Co-ordinate System .............................................................................................. 35 9.1 Triangles of Equal Content ................................................................................. 37 9.2 Triangle with Equal Area to Parallelogram .................................................... 37 9.3 Equal Area versus Equal Content....................................................................... 38 9.4 Area of Triangle .................................................................................................... 39 9.5 Equal Area means Equal Content....................................................................... 40 10.1 Hilbert’s Geometry................................................................................................. 42 10.2 Redefining a Line.................................................................................................... 42 10.3 Congruence of Angles .......................................................................................... 43 10.4 Failure of Desargues’s Theorem.......................................................................... 43 10.5 Geometry Independent of Axioms of Congruence.......................................... 44 vii 10.6 Product of Segments............................................................................................. 45 10.7 Commutative Law of Addition.......................................................................... 46 10.8 Associative Law of Addition................................................................................. 47 10.9 Associative Law of Multiplication....................................................................... 47 lO.lODistributive Laws.................................................................................................... 49 10.11 Commutative Property of Addition.................................................................... 50 10.12The Second Statement of Distributive Property............................................. 51 10.13Straight Lines in Geometry................................................................................. 52 10.14Parallel Lines Create Equal Proportions.......................................................... 53 10.15Axiom II-5 Revisited.............................................................................................. 56 10.16Solutions to Linear Equation.............................................................................. 56 1 Chapter 1 Classical Geometry: Euclid Geometry is an ancient subject that has been around over 4000 years. Its roots are found in one form or another in nearly every human culture. The subject as we know it began in Mesopotamia, Egypt, India, and China. The Egyptians were most likely motivated by the annual flooding of the Nile, and its effect on farming. In their culture, Geometry was all about application, and often the numbers were arrived at through experimentation, observation, and trial and error. It appears that the work in the other aforementioned cultures actually was more advanced than the Egyptians. However, the ancient Greeks (800 B.C. to 150 B.C.) gave credit for the development of Geometry to the Egyptians that pre-dated them. The Greeks brought in a level of abstraction, logical deduction, and proof that had never existed before. This seems to have begun with Thales of Miletus around 600 B.C. and culminated in the works of Euclid 300 years later. Although it was widely known that Euclid had drawn from all his predecessors, nothing has been found from the earlier sources to support the level of rigor in the Math that was presented by Euclid around 300 B.C. He taught the essential features of Math in a much more general sense than had been done before. He laid the axiomatic foundation of theory, setting the standard that was used for over two thousand years. It was from his presentation of Geometry that the world learned how abstraction works. He defined terms, classified objects, and enforced a deductive presentation of theory. He presented material in such a way that a person with minimal mathematical training could benefit. And yet, he was also able to convey topics that are visible only to ’’the eyes of the mind,” as Plato would say. He did this in his presentation of irrational 2 and incommensurable magnitudes. His
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