The Language of Mathematics [E-Book] : A Linguistic and Philosophical Investigation / by Mohan Ganesalingam.
Ganesalingam, Mohan.
Berlin, Heidelberg : Springer, 2013
XX, 260 p. 15 illus. digital.
englisch
Printed edition: 9783642370113
9783642370120
10.1007/978-3-642-37012-0
Lecture notes in computer science ; 7805
Full Text
Table of Contents:
  • Introduction.-1.1 Challenges
  • 1.2 Concepts.-1.2.1 Linguistics and Mathematic.-1.2.2 Time
  • 1.2.3 Full Adaptivity
  • .3 Scope
  • 1.4 Structure
  • 1.5 Previous Analyses
  • 1.5.1 Ranta
  • 1.5.2 de Bruijn
  • 1.5.3 Computer Languages
  • 1.5.4 Other Work
  • 2 The Language of Mathematics
  • 2.1 Text and Symbol
  • 2.2 Adaptivity
  • 2.3 Textual Mathematics
  • 2.4 Symbolic Mathematics. -2.4.1 Ranta’s Account and Its Limitations
  • 2.4.2 Surface Phenomena
  • 2.4.3 Grammatical Status
  • 2.4.4 Variables
  • 2.4.5 Presuppositions
  • 2.4.6 Symbolic Constructions
  • 2.5 Rhetorical Structure
  • 2.5.1 Blocks
  • 2.5.2 Variables and Assumptions
  • 2.6 Reanalysis
  • 3 Theoretical Framework
  • 3.1 Syntax
  • 3.2 Types
  • 3.3 Semantics
  • 3.3.1 The Inadequacy of First-Order Logic
  • 3.3.2 Discourse Representation Theory
  • 3.3.3 Semantic Functions
  • 3.3.4 Representing Variables
  • 3.3.5 Localisable Presuppositions
  • 3.3.6 Plurals
  • 3.3.7 Compositionality
  • 3.3.8 Ambiguity and Type
  • 3.4 Adaptivity
  • 3.4.1 Definitions in Mathematics
  • 3.4.2 Real Definitions and Functional Categories
  • 3.5 Rhetorical Structure
  • 3.5.1 Explanation
  • 3.5.2 Blocks
  • 3.5.3 Variables and Assumptions
  • 3.5.4 Related Work: DRT in NaProChe
  • 3.6 Conclusion
  • 4 Ambiguity.-4.1 Ambiguity in Symbolic Mathematics.-4.1.1 Ambiguity in Symbolic Material.-4.1.2 Survey: Ambiguity in Formal Languages.-4.1.3 Failure of Standard Mechanisms
  • 4.1.4 Discussion.-4.1.5 Disambiguation without Type
  • 4.2 Ambiguity in Textual Mathematics.-4.2.1 Survey: Ambiguity in Natural Languages.-4.2.2 Ambiguity in Textual Mathematics
  • 4.2.3 Disambiguation without Type
  • 4.3 Text and Symbol
  • 4.3.1 Dependence of Symbol on Text
  • 4.3.2 Dependence of Text on Symbol
  • 4.3.3 Text and Symbol: Conclusion
  • 4.4 Conclusion
  • 5 Type
  • 5.1 Distinguishing Notions of Type
  • 5.1.1 Types as Formal Tags
  • 5.1.2 Types as Properties
  • 5.2 Notions of Type in Mathematics
  • 5.2.1 Aspect as Formal Tags
  • .2.2 Aspect as Properties
  • 5.3 Type Distinctions in Mathematics
  • 5.3.1 Methodology
  • 5.3.2 Examining the Foundations
  • 5.3.3 Simple Distinctions
  • 5.3.4 Non-extensionality.-5.3.5 Homogeneity and Open Types
  • 5.4 Types in Mathematics
  • 5.4.1 Presenting Type: Syntax and Semantics
  • 5.4.2 Fundamental Type
  • 5.4.3 Relational Type
  • 5.4.4 Inferential Type
  • 5.4.5 Type Inference
  • 5.4.6 Type Parametrism
  • 5.4.7 Subtyping
  • 5.4.8 Type Coercion
  • 5.5 Types and Type Theory
  • 6 TypedParsing
  • 6.1 Type Assignment
  • .1.1 Mechanisms
  • 6.1.2 Example
  • 6.2 Type Requirements
  • 6.3 Parsing
  • 6.3.1 Type
  • 6.3.2 Variables.-6.3.3 Structural Disambiguation
  • 6.3.4 Type Cast Minimisation
  • 6.3.5 Symmetry Breaking
  • 6.4 Example
  • 6.5 Further Work
  • 7 Foundations
  • 7.1 Approach
  • 7.2 False Starts
  • 7.2.1 All Objects as Sets
  • 7.2.2 Hierarchy of Numbers
  • 7.2.3 Summary of Standard Picture
  • 7.2.4 Invisible Embeddings
  • 7.2.5 Introducing Ontogeny
  • 7.2.6 Redefinition
  • 7.2.7 Manual Replacement
  • 7.2.8 Identification and Conservativity
  • 7.2.9 Isomorphisms Are Inadequate
  • 7.3 Central Problems
  • 7.3.1 Ontology and Epistemology
  • 7.3.2 Identification
  • 7.3.3 Ontogeny
  • 7.4 Formalism
  • 7.4.1 Abstraction
  • 7.4.2 Identification
  • 7.5 Application.-7.5.1 Simple Objects.-7.5.2 Natural Numbers
  • 7.5.3 Integers
  • 7.5.4 Other Numbers
  • 7.5.5 Sets and Categories
  • 7.5.6 Numbers and Late Identification
  • 7.6 Further Work
  • 8 Extensions
  • 8.1 Textual Extensions
  • 8.2 Symbolic Extensions
  • 8.3 Covert Arguments
  • Conclusion.