A Calculus of Number Based on Spatial Forms - University of ...

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PREFACE This work attempts to clarify mathematics, to make complex ideas accessible to those with less mathematical training. In this thesis, I have reduced number and operator representation down to three forms, revealing a simplicity to numbers. I hope this work may aid those who have struggled to understand numbers and computation, as well as elementary algebra, for I surely consider these topics in new light. I am not a mathematician, so writing a thesis about mathematics has been achallenge for me. For years, I have struggled to understand what research in mathematics is all about that I might contribute to its continuing progress. Alas, essential concepts continue to elude me and I remain uncertain about what mathematicians are actually doing. Nonetheless, I believe mywork to have some relevance to them and have tried to describe it in a manner that would avail mathematical researchers to make use <strong>of</strong> it, though my audience also includes the less mathematically inclined. Ihave benetted from many discussions with friends and colleagues during the development <strong>of</strong> this material. Most recognize some insight here, though I am not certain whether they liked the material itself or just the idea <strong>of</strong> trying to make mathematics easier to understand. I thank William Bricken with my heart and soul. William introduced me to boundary mathematics and has served as my mentor throughout this research|his guidance and direction has made it all happen. I thank the rest <strong>of</strong> my thesis committee: Judith Ramey, William Winn, and Steven Tanimoto. Each <strong>of</strong> them played unique roles in the development <strong>of</strong> this material. I was privileged to have had such excellent support for my research. I credit my zest for research toPenelope Sanderson, with whom I studied vii
as an undergraduate at the University <strong>of</strong> Illinois. Penny challenged me with interesting research problems and gave me opportunities to presentmy ndings to receptive audiences. I am forever grateful to her. I thank the students and sta <strong>of</strong> the Human Interface Technology Laboratory and its director, Dr. Thomas A. Furness, for providing a stimulating environment for developing these ideas. I thank Kimberley Osberg, Dav Lion, and Greg Woodward in particular for their personal support. I thank US West for funding part <strong>of</strong> this research and I thank all members <strong>of</strong> the consortium for supporting the lab. Though I wrote this document at the HIT Lab in Seattle, I \polished the turd" at Interval Research inPalo Alto. I thank Dick Shoup, the Natural Computing group, and everyone at Interval for their support <strong>of</strong> my research. I extend my warmest appreciation to Ann Miller, with whom I shared the joy <strong>of</strong> discovering many <strong>of</strong> the ideas herein. Ann taught mehumility and compassion and will always be in my heart. Finally, I thank Laura Grant, the love<strong>of</strong>my life. I could never have nished this thesis without her intellectual and emotional support. Though it took a while, I nally managed to \swallow the toad!" Jerey Mark James Human Interface Technology Laboratory October 14, 1993 viii
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Page 16 and 17: 3 Table 1.1: Simple Calculations in
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Page 32 and 33: 19 connectivities denes the system
Page 34 and 35: Chapter 4 INSTANCE AND ABSTRACT 4.1
Page 36 and 37: 23 The third rule creates new eleme
Page 38 and 39: 25 Distribution: (A[BC]) : =(A[B])(
Page 40 and 41: 27 Multiplication of</stron
Page 42 and 43: 29 Table 4.1: The Two-Boundary <str
Page 44 and 45: 31 transformations. 5.2.1 Elements
Page 46 and 47: 33 Inversion: A = : Equivalent Expr
Page 48 and 49: 35 Boundary integers add by collect
Page 50 and 51: 37 Multiplication requires manageme
Page 52 and 53: 39 Proof. Rational
Page 54 and 55: 41 For example, a half-cardinality
Page 56 and 57: Chapter 6 PHASE 6.1 Introduction Th
Page 58 and 59: 45 Using J, two concise and useful
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47 The ambiguous sign makes the car
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49 Table 6.1: Transcendentals in th
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Chapter 7 FUTURE WORK 7.1 Introduct
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53 ematics problems could be phrase
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55 boundary form, in search <strong
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57 Technology interface sof
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59 such asintegrals and Fourier tra
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61 mulas specify how they transform
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63 dominion A theorem stating that
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BIBLIOGRAPHY [1] B. Banaschewski. O
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67 [28] L. Lamport. LaTeX: A Docume
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69 A.2 Functions The standard algeb
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71 Distribution of
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73 Radians 1+e i =0 ([]) = Euler's
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75 B.2 Square Root The following sq
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77 The a coecient can now be remove
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79 f = (([[(([][]) [(([[b]][]))])]]
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