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.