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

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61 mulas specify how they transform: some functions have several representations while others are unique and some combinations can be simplied while others cannot. Because the visual form does not convey these relationships, much external knowledge must be brought to the task <strong>of</strong> doing mathematics. In this way, standard notation fails to support learning mathematics. Availability and application <strong>of</strong> rules are not suggested by the forms. Knowledge <strong>of</strong> these rules remains distinct and separate from this mathematical interface. Dissociating knowledge from appearance leaves an impression that the knowledge is arbitrary and without foundation. Only by accepting this knowledge on faith can a student bypass the contradictions <strong>of</strong> form introduced by this dissociation. The failure to instantiate mathematical knowledge in its visual forms further contributes to the misconception that mathematics is a purely cognitive activity. In doing so, mathematical ability is politely restricted to only the \gifted" population, another failure <strong>of</strong> mathematics education. The boundary calculus can impact mathematics education. It provides another reference for the implementation <strong>of</strong> formal mathematical concepts and arguably presents a better interface to mathematical behavior. 8.6 Conclusions The stages <strong>of</strong> mathematical representation can be readdressed under the paradigm <strong>of</strong> boundary mathematics so that the activities <strong>of</strong> mathematics are more clearly delineated. The boundary calculus isolates manipulative constraints, leaving conceptual objects as constructions <strong>of</strong> the boundary forms. The delineation gives those particular constructions meaning and benet that is usually hidden. Standard choices for organizing numbers are actually optimized for certain criteria. Opening up representation with the boundary calculus makes the criteria <strong>of</strong> representation and the choices <strong>of</strong> structure more explicit.
GLOSSARY abstract Atype <strong>of</strong> boundary in the calculus <strong>of</strong> number. Abstract forms the black hole when empty. Written as 2 when empty or[A]around boundary conguration A. anti-logarithm Functional interpretation <strong>of</strong> the instance boundary, e x ! (x): black hole The abstract boundary with no contents. The black hole dominates its context. Written as 2: boundary Drawing <strong>of</strong> a distinction. (); []; and are types <strong>of</strong> boundaries. boundary mathematics A representational and computational paradigm based on boundary forms that transform by match and substitute. cardinality A theorem stating that a repeated form can be rewritten using a single reference. Written as A:::A=([A][:::]): collection Spatial justaposition <strong>of</strong> boundary congurations, as AB. Collections are unordered, ungrouped, and variary. conguration Any algebraic boundary expression, including void. content The space inside <strong>of</strong> a distinction. e.g. (content). context The space outside <strong>of</strong> a distinction. e.g. context(). distinction A cleaving <strong>of</strong> space. Drawn as a boundary, (): distribution An axiom that denes the modier form as distributiveover collection. Written as (A[BC]) : =(A[B])(A[C]):
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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
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 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 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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