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

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67 [28] L. Lamport. LaTeX: A Document Preparation System. Addison-Wesley Publishing Company, Menlo Park, CA, 1986. [29] National Council <strong>of</strong> Teachers <strong>of</strong> Mathematics, Inc., Reston, Va. Curriculum and Evaluation Standards for School Mathematics, 1989. [30] R.S. Nickerson. Counting, computing, and the representation <strong>of</strong> numbers. Human Factors, 30(2):181{199, 1988. [31] R.A. Orchard. On the laws <strong>of</strong> form. International Journal <strong>of</strong> General Systems, 2:99{106, 1975. [32] D.G. Schwartz. Isomorphisms <strong>of</strong> Spencer Brown's Laws <strong>of</strong> Form and Varela's calculus for self-reference. International Journal <strong>of</strong> General Systems, 6:239{255, 1981. [33] R.G. Shoup. A complex logic for computation with simple interpretations for physics, 1992. Personal communication. [34] F.J. Varela. A calculus for self-reference. International Journal <strong>of</strong> General Systems, 2:5{24, 1975. [35] F.J. Varela. Principles <strong>of</strong> Biological Autonomy. North Holland, New York, 1979. [36] F.J. Varela and J.A. Goguen. The arithmetic <strong>of</strong> closure. Journal <strong>of</strong> Cybernetics, 8:291{324, 1978.
Appendix A CONVERSION A.1 <strong>Number</strong>s The basic numbers types can be expressed in boundary notation. For each number type below, the general structure <strong>of</strong> that type is given in boundary form followed by some examples. Standard Boundary Zero 0 Natural n ::: 1 2 Integer i n or ,1 ,2 Rational q ([i 1 ]) 2=3 ([]) 1=4 () Algebraic Irrational r (([[q 1 ]][q 2 ])) or r 1 ([r 2 ]([[r 3 ]][r 4 ])) 3p 7 (([[]])) p 2=2 (([[]])) Complex c r 1 ([r 2 ]([J])) i (([J])) 2+4i ([]([J])) Transcendental e () (J[J]([J])) i []
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A Calculus <strong
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University ofWashi
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4.2 The Two-Boundary Calcul
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A.2 Functions : : : : : : : : : : :
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LIST OF TABLES 1.1 Simple Calculati
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as an undergraduate at the Universi
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Chapter 1 INTRODUCTION Numb
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3 Table 1.1: Simple Calculations in
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5 contents, they are also drawn as
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7 Table 1.4: Axioms of</str
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Chapter 2 PRIOR WORK Boundary mathe
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11 3 2 Given Number</stro
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13 this system uses string rewrite
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15 Table 2.3: Denition of</
Page 30 and 31: 17 Figure 3.1: The Void. Figure 3
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
Page 60 and 61: 47 The ambiguous sign makes the car
Page 62 and 63: 49 Table 6.1: Transcendentals in th
Page 64 and 65: Chapter 7 FUTURE WORK 7.1 Introduct
Page 66 and 67: 53 ematics problems could be phrase
Page 68 and 69: 55 boundary form, in search <strong
Page 70 and 71: 57 Technology interface sof
Page 72 and 73: 59 such asintegrals and Fourier tra
Page 74 and 75: 61 mulas specify how they transform
Page 76 and 77: 63 dominion A theorem stating that
Page 78 and 79: BIBLIOGRAPHY [1] B. Banaschewski. O
Page 82 and 83: 69 A.2 Functions The standard algeb
Page 84 and 85: 71 Distribution of
Page 86 and 87: 73 Radians 1+e i =0 ([]) = Euler's
Page 88 and 89: 75 B.2 Square Root The following sq
Page 90 and 91: 77 The a coecient can now be remove
Page 92: 79 f = (([[(([][]) [(([[b]][]))])]]
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