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

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25 Distribution: (A[BC]) : =(A[B])(A[C]) Equivalent Expressions Template Replacement ([][])([][]) ([][ ]) A = [];B = ;C = (([[]][])) (([[]][])([[]][])) A = [[]];B = ;C = ([][ ]) ([][])([][]) A = [];B = ;C = ([ ][]) A = [];B = ;C = Figure 4.2: Examples <strong>of</strong> Distribution. 4.3 Natural <strong>Number</strong>s The elements <strong>of</strong> the two-boundary calculus map to natural numbers, complete with arithmetic operations <strong>of</strong> addition, multiplication, and exponentiation. Space is treated as addition so that collecting elements adds them. Natural numbers are constructed simply by accumulating the unit formed by the empty instance boundary, : The elements ; ; ;::: form the set <strong>of</strong> natural numbers, N, within the boundary calculus. The successor function, S(A) ! A, inductively builds N from the element : 4.3.1 Addition Elements a; b 2 N add by collection, ab: The set N is closed under addition because collecting elements can only form an element <strong>of</strong> equal or greater cardinality, which still an element <strong>of</strong>N Addition is commutative because spatial collection is unordered. Addition is associative because spatial collection makes no grouping distinctions for multiple additions. The additive identity is the void: an element remains uneected when collected with the void. When adding these natural numbers, no calculation is necessary. They are initially
26 phrased in their canonical form. For example, 3 + 2 rewrites as : 4.3.2 Multiplication Elements a; b 2 N multiply by the form ([a][b]): The set N is closed under the multiplication. Pro<strong>of</strong>. The product <strong>of</strong> a 2 N and initial element, , reduces to a by involution, ([a][])=([a])=a: If the product <strong>of</strong> a and b, ([a][b]), isintheN, the product <strong>of</strong> a and the successor <strong>of</strong> b is also in the set, ([a][b])=([a][b])([a][])=([a][b])a: Because addition is closed, this result is in N. Therefore, by induction all products <strong>of</strong> a; b 2 N are in N. Multiplication is commutative because [a] and [b] are unordered within the outer instance. Associativity <strong>of</strong> binary multiplication is shown by two applications <strong>of</strong> involution: ([([a][b])][c])=([a][b][c])=([a][([b][c])]): The multiplicative identity is the unit, : An element multiplied by the unit reduces to itself by involution, ([a][])=([a])=a: In the boundary calculus, the void represents zero. Accordingly, multiplication by the void is void, ([a][])= : Pro<strong>of</strong>. By distribution, a product with the void is equal to two copies <strong>of</strong> the same, ([a][ ])([a][ ])=([a][ ]): The equation ee = e is true when e is void or when collection <strong>of</strong> e is idempotent. Since collection is assumed additive, ee 6= e unless e is void. Therefore, ([a][ ])= :
Page 1 and 2: A Calculus <strong
Page 3 and 4: University ofWashi
Page 5 and 6: 4.2 The Two-Boundary Calcul
Page 7 and 8: A.2 Functions : : : : : : : : : : :
Page 9 and 10: LIST OF TABLES 1.1 Simple Calculati
Page 11 and 12: as an undergraduate at the Universi
Page 14 and 15: Chapter 1 INTRODUCTION Numb
Page 16 and 17: 3 Table 1.1: Simple Calculations in
Page 18 and 19: 5 contents, they are also drawn as
Page 20 and 21: 7 Table 1.4: Axioms of</str
Page 22 and 23: Chapter 2 PRIOR WORK Boundary mathe
Page 24 and 25: 11 3 2 Given Number</stro
Page 26 and 27: 13 this system uses string rewrite
Page 28 and 29: 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 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 80 and 81: 67 [28] L. Lamport. LaTeX: A Docume
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
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79 f = (([[(([][]) [(([[b]][]))])]]
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