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

More documents

Recommendations

Info

Chapter 4 INSTANCE AND ABSTRACT 4.1 Introduction This chapter introduces boundary numbers by dening a two-boundary calculus. The two-boundary calculus maps to natural numbers, complete with addition, multiplication, and power functions. The calculus also builds a structure without parallel in standard notation: a generalized cardinality form that applies to both addition and multiplication. In this calculus, space is treated as addition. A single distinction represents natural numbers by repeating the distinction for the value. This distinction shall be called instance and drawn as a circular boundary when empty \" or as parentheses otherwise \(A):" An empty instance forms the natural number one; it is the unit for counting. Nestings <strong>of</strong> instance are not natural numbers; these forms will be characterized in Chapter 6. A second distinction builds with the rst to make multiplication and power functions. This distinction shall be called the abstract and drawn as a square boundary when empty \2" or as square brackets otherwise \[A]:" An empty abstract forms a non-real number called a black hole. Nestings <strong>of</strong> abstract produce numerical values beyond the scope <strong>of</strong> this thesis and will not be discussed here. Two axioms dene the relationship between instance and abstract. The involution axiom denes them as functional inverses. When either boundary is immediately nested within the other, where nothing lies between them, both boundaries can be erased. The distribution axiom denes a further relationship between them: when an abstract boundary lies within an instance boundary, the instance boundary and abstract boundary and the contentbetween them can be threaded across the contents <strong>of</strong> the abstract boundary. The visual forms in Section 4.2.2 make these rules visually apparent. The two axioms dene the calculations necessary to reduce multiplication and power expressions to the canonical structure <strong>of</strong> natural numbers. The two boundaries also construct a generalized form <strong>of</strong> cardinality. Cardinality
22 rewrites a repeated reference using a single reference combined with a count <strong>of</strong>the repetition. In standard notation, a repeated sum is rewritten as a product with the count, e.g. x + :::+x = nx, and a repeated product is rewritten as an exponent <strong>of</strong> the count, e.g. x :::x= x n . In the boundary calculus, cardinality takes the same form in addition and in multiplication. Because <strong>of</strong> this dual use, the boundary form <strong>of</strong> cardinality is considered to be a generalized form <strong>of</strong> cardinality. This chapter begins by formally dening the two-boundary calculus. 4.2 The Two-Boundary <strong>Calculus</strong> The two-boundary calculus is introduced by dening the set <strong>of</strong> possible constructions, the elements <strong>of</strong> the calculus, and by dening axioms to state their equivalences. 4.2.1 Elements <strong>of</strong> the <strong>Calculus</strong> Elements <strong>of</strong> the calculus are constructed out <strong>of</strong> the void using two boundary distinctions: instance and abstract. Denition. Let B denote the collection <strong>of</strong> all well-formed boundary expressions composed <strong>of</strong> two boundaries, denoted as(:::) and [:::]: B is dened recursively by three rules: 1. The void belongs to B. 2. If a belongs to B, then (a) and [a] each belong to B. 3. If b 1 ;b 2 ;:::;b n (n 1) belong to B, then the unordered collection b 1 b 2 :::b n belongs to B. The rst rule establishes the void as the foundational element in the recursive construction <strong>of</strong> B. Thus, B contains the void. The second rule creates elements <strong>of</strong> B by nesting other elements inside the two boundaries. The rst two elements introduced by this rule are the boundaries with no content: and 2: Further distinction introduces the following elements into B: ();(2);[];[2];(());((2));([]);([2]);[()];[(2)];[[]];[[2]]:
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 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 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]][]))])]]
show all

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

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?