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

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41 For example, a half-cardinality is given by ([a]): Collection <strong>of</strong> two <strong>of</strong> these reduces to a: ([a])([a]) Given ([([a])][]) Cardinality ([a][]) Involution ([a]) Inversion a Involution Applied to addition combinations, fractional cardinality builds fractions. The form ([a]) translates to a=2. In the above pro<strong>of</strong> this fraction is doubled, as a=2+a=2 =2(a=2) = a(2=2) = a: Applied to multiplicative combinations, fractional cardinalities build roots. The form (([[b]])) translates to b 1=2 . In a pro<strong>of</strong> similar to the above but with a=[b], this root is squared to double the cardinality, as b 1=2 b 1=2 =(b 1=2 ) 2 =b 2=2 =b: Since fractional cardinalities form roots, they allow construction <strong>of</strong> the algebraic irrationals. 5.6 Conclusions Though the two-boundary calculus only represents natural numbers, simply adding an inverse boundary extends the boundary numbers through the algebraic irrationals. The three axioms completely dene computation on these numbers and functions. The three-boundary calculus is summarized in Table 5.1. The inverse boundary serves manyinverse functions due to the structures provided by the instance and abstract boundaries. It is dened in the most fundamental way that an inverse can be dened, utilizing the void to cancel inverses to nothing. The spatial formalism makes this inverse denition possible and powerful. Kauman dened an inverse boundary <strong>of</strong> this sort but did not have the other structures to utilize it in a general form. The inverse is powerful but it introduces problems, such as division by zero. Though the two-boundary calculus was solid, the three-boundary calculus has a single exception, preventing rules from applying to all congurations. When performing
42 Table 5.1: The Three-Boundary <strong>Calculus</strong> <strong>Number</strong>s 0 ! 1 ! 2! ,1! ,2! 1=2 ! () 2=3 ! ([]) p 3 ! (([[]])) 3p 2 ! (([[]])) Operators x ! x ,x ! 1=x ! () x + y ! xy x , y ! x x y ! ([x][y]) x=y ! ([x]) x y ! (([[x]][y])) x ,y ! (([[x]][])) yp x ! (([[x]])) Rules ([A]) = : A = : [(A)] (A[BC]) = : (A[B])(A[C]) A = : a computation, constraints on variables must be propagated through to avoid paradoxes.
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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 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 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
Page 92: 79 f = (([[(([][]) [(([[b]][]))])]]
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