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

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Chapter 2 PRIOR WORK Boundary mathematics was rst explored as a paradigm <strong>of</strong> mathematical representation in the context <strong>of</strong> propositional logic. It has been applied extensively to logic [11, 3, 31, 1, 19, 32] as well as to other systems <strong>of</strong> mathematics, including imaginary logic [25, 34, 35, 33], algebra [13, 27], self-reference and recursion [36, 20, 16, 25, 21], control and deduction in computer science [4, 7, 8, 6, 14], and numbers [5, 10, 24]. The applications <strong>of</strong> boundary mathematics to numbers are <strong>of</strong>ten called boundary numbers. G. Spencer-Brown, Louis Kauman, and William Bricken have each created systems <strong>of</strong> boundary numbers [5, 10, 24, 23, 22]. Their systems share the basic characteristics associated with boundary mathematics but dier in detail. In all <strong>of</strong> them, boundaries serve as the objects <strong>of</strong> the system and as operators on those objects. <strong>Number</strong>s are created out <strong>of</strong> boundaries and boundaries build into arithmetic operations. This common basis <strong>of</strong> form allows calculation directly on the forms, erasing and rearranging to transform an expression. Each <strong>of</strong> these systems are briey described using their creator's original notation. 2.1 Spencer-Brown <strong>Number</strong>s The boundary paradigm began with Spencer-Brown's Laws <strong>of</strong> Form in which he reduced logic to a single distinction [11]. He shows how two fundamental choices <strong>of</strong> reduction, calling and crossing, produce boolean arithmetic. In his work, Spencer- Brown draws a distinction as a mark, , shown below in these rules. Calling Crossing : = : = His number system attempts to adapt calling and crossing to the number domain. He variously treats space as addition and as multiplication but, for simplicity, only the former system will be considered [10, 5].
10 Table 2.1: Denition <strong>of</strong> Spencer-Brown <strong>Number</strong>s. <strong>Number</strong>s 0 ! 1 ! 2 ! 3 ! Operators a + b ! ab ab! a b a b ! b :a Rules at bt ::: : = : = a b ::: t a = a :a = With space as addition, the natural numbers are easily formed. The void acts as zero and counting proceeds by accumulating marks: , , , ::: and so on. <strong>Number</strong>s are added by spatial collection whereas multiplication and power operations are composed using marks. Spencer-Brown's numbers are summarized in Table 2.1. He gives two axioms for calculating on these forms, universe and transfer. With these axioms he derives many theorems, including reexion and null power. Universe and null power are his numerical versions <strong>of</strong> calling and crossing. Universe Transfer at bt ::: Reexion a = a Null Power :a = : = : = a b ::: t In addition to the mark, he uses a vaguely dened colon to disambiguate conicting results whichwould equate 0 1 ! : and 2 ! . This colon does not completely resolve the problems it was introduced for. These numbers add by spatial collection. For example, 3 + 2 = 5. 3+2 Given + <strong>Number</strong> Rewrite Addition Rewrite 5 <strong>Number</strong> Rewrite <strong>Number</strong>s multiply by the form ab ! a b . The form reduces by transfer, which distributes the whole <strong>of</strong> one number throughout the units <strong>of</strong> the other. For example, 3 2=6.
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Page 9 and 10: LIST OF TABLES 1.1 Simple Calculati
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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
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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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