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Tue May 22 10:26:04 PDT 2012

Contents


    How to use the MATHS site

    MATHS is a simple way of recording mathematical and logical systems on a computer. I can generate web pages from it. This site is a kind of hypertext encyclopedia of logic and mathematics. Some I've found to be useful and others are used in the texts and research in computer science and software engineering. Use it as a reference or as the basis for your own work. If you do quote from this site include a link or reference to the source and let me know by Email. If I know you are linking to part of the site I can leave the link in place for you.

    This page has a lot of pointers to ready made mathematical systems. If you are new here focus on the links to files with names that start "intro"( see Introduction below). For more (simpler) samples see: [ Search in index ]

    MATHS

      Background

        History

        I developed this notation to be a usable way of documenting and reasoning about complex discrete systems. It is designed to workable on any ASCII based system and to be translated into different forms for display and browsing. It is a formal language with room for informality. It is designed to allow reusable and distributed formal (and informal) documentation. As a result it is a hypertext system.

        As a test I have transcribed 30 years of hand written and typed notes. These are in files with names that start with 'logic' or 'math'.

        Another test was to document MATHS using MATHS. The files with names that start 'notn' and 'types' contain this information.

        For a "Vision statement" see the MATHS Manifesto: [ 10.manifesto.html ]

        Introduction

        [ intro_README.html ] , . . . [ intro_note.html ] , . . . [ intro_copyright.html ] , and the following introductory topics:

        [ intro_characters.html ] , . . . [ intro_documentation.html ] , . . . [ intro_dynamics.html ] , . . . [ intro_ebnf.html ] , . . . [ intro_function.html ] , . . . [ intro_grammar.html ] , . . . [ intro_logic.html ] and [ logic_history.html ] , . . . [ intro_records.html ] , . . . [ intro_relation.html ] , . . . [ intro_sets.html ] , . . . [ intro_standard.html ] , . . . [ intro_strings.html ] , . . . [ intro_structure.html ]

        How Mathematics differs from pure logic [ math_10_Intro.html ]

        Also see [ Mathematics ] below.

        STANDARD

        The Standard set of assumptions and notations [ intro_standard.html ] and [ math_11_STANDARD.html ]

        [ Logic ] [ intro_logic.html ]

        The Notation Used to Generate these Notes

          Syntax

          In essence a document is divided into a hierarchical outline with sections, subsections, paragraphs and lines. Sections have headlines that can act as identifiers. Paragraphs are eith informal comments, definitions, assumptions or proved results. The system is designed to make the searching and translation of the documents as easy as possible.


          (introductions to MATHS): [ intro_characters.html ] [ intro_documentation.html ] [ intro_ebnf.html ] [ notn_10_Lexicon.html ]


          (overview): of notation [ math.syntax.html ] [ notn_00_README.html ]


          (names and expressions in MATHS): in MATHS [ notn_11_Names.html ] [ notn_12_Expressions.html ]
          (intro): [ intro_documentation.html ]


          (syntax of MATHS): [ notn_13_Docn_Syntax.html ]


          (semantics of MATHS): [ notn_14_Docn_Semantics.html ] [ notn_15_Naming_Documentn.html ]


          (structure): [ notn_2_Structure.html ]
          (UML model): [ maths.mdl ]
          (conveniences): [ notn_3_Conveniences.html ]
          (reuse):
          (extension): [ notn_4_Re_Use_.html ]


          (formatting): [ notn_5_Form.html ]
          (tables): [ notn_9_Tables.html ]


          (algebras): [ notn_6_Algebra.html ] [ notn_7_OO_vs_Algebra.html ]


          (reasoning): [ notn_8_Evidence.html ]

          Notation

          Introduction to the notation: [ intro_characters.html ] [ intro_documentation.html ]

          A description of the notation: A lexicon [ notn_10_Lexicon.html ]

          More details: [ notn_11_Names.html ] [ notn_12_Expressions.html ] [ notn_13_Docn_Syntax.html ] [ intro_ebnf.html ] [ notn_14_Docn_Semantics.html ] [ intro_grammar.html ] [ notn_15_Naming_Documentn.html ]

          Specific Notations: [ intro_documentation.html ] [ notn_2_Structure.html ] [ notn_3_Conveniences.html ] [ notn_4_Re_Use_.html ] [ notn_5_Form.html ] [ notn_8_Evidence.html ]

          Mathematical and Logical Rules: [ intro_standard.html ] [ notn_11_Names.html ] [ notn_12_Expressions.html ] [ intro_dynamics.html ] [ intro_function.html ] [ intro_logic.html ] [ intro_sets.html ] [ intro_relation.html ] [ intro_strings.html ] [ intro_structure.html ] [ notn_6_Algebra.html ] [ notn_7_OO_vs_Algebra.html ]

        . . . . . . . . . ( end of section The Notation Used to Generate these Notes) <<Contents | End>>

      . . . . . . . . . ( end of section MATHS) <<Contents | End>>

      Mathematics

        Theories that may be useful rather than traditional

          Using Numbers to represent physical dimensions

        1. (dimensions): [ math_49_Dimensioned_numbers.html ]

          A mathematical theory of record structures and data bases

        2. (data structure): [ intro_structure.html ] [ math_12_Structure.html ]
        3. (data bases): [ math_13_Data_Bases.html ]

          Models of dynamic systems

          [ intro_dynamics.html ]
        4. (dynamical logic): [ math_14_Dynamics.html ]
        5. (dynamic systems): [ math_71_Auto...Systems.html ] [ math_76_Concurency.html ]
        6. (unary algebra): [ math_15_Unary_Algebra.html ]
        7. (enumerations): [ math_77_Enumerations.html ]

        . . . . . . . . . ( end of section Theories that may be useful rather than traditional) <<Contents | End>>

        Traditional Discrete Mathematics

          Posets

          A model of collections of things that can often be put in order.
        1. (order):
        2. (posets): [ math_21_Order.html ]

        3. (cpos): [ math_24_Domains.html ]

          Graphs, Flow diagrams, Categories

        4. (digraphs): [ math_22_graphs.html ]
        5. (flow diagrams): [ math_23_Flow_Diagrams.html ]
        6. (category theory): [ math_25_Categories.html ]

        . . . . . . . . . ( end of section Traditional Discrete Mathematics) <<Contents | End>>

        Algebras

          Notational Abuses

          [ notn_6_Algebra.html ]

          Algebras with one operator

          Common ideas: [ math_31_One_Associative_Op.html ]

          Semigroups/Monoids/Groups

        1. (semigroups): [ math_32_Semigroups.html ]
        2. (monoids): [ math_33_Monoids.html ]
        3. (groups): [ math_34_Groups.html ]

          Algebras with two operators

            Rings
          1. (Rings): [ math_41_Two_Operators.html ]
            Numbers
          2. (number theory): [ math_42_Numbers.html ]
            General
            [ math_43_Algebras.html ]

          . . . . . . . . . ( end of section Algebras with two operators) <<Contents | End>>

          Algebras with More than two operators

        4. (calculus): [ math_44_Formal_Calculus.html ]
        5. (high school algebra): [ math_45_Three_Operators.html ]

        . . . . . . . . . ( end of section Algebras) <<Contents | End>>

        Theories

          A theory of Objects

        1. (genesys): [ math_5_Object_Theory.html ]

          Strings, Grammars, Languages, MetaLanguages, Superstrings

          [ intro_strings.html ] [ intro_grammar.html ]

        2. (strings in theory): [ math_61_String_Theories.html ]
        3. (strings): [ math_62_Strings.html ]
        4. (grammar theory): [ math_63_Languages.html ]
        5. (metalinguistics): [ math_64_Meta_Macros.html ]

          [ math_65_Meta_Linguistics.html ]

        6. (superstrings): [ math_66_SuperStrings.html ]

          Systems, Automata, Games, Programs

        7. (automata):
        8. (systems theory): [ math_71_Auto...Systems.html ]
        9. (system algebra): [ math_72_Systems_Algebra.html ]
        10. (process algebra): [ math_73_Process_algebra.html ]
        11. (games): [ math_74_Games.html ]
        12. (programs): [ math_75_Programs.html ]
        13. (concurrent systems): [ math_76_Concurency.html ]

        . . . . . . . . . ( end of section Theories) <<Contents | End>>

        Probability, Multisets, Bags, Fuzzyiness, Spectra

        1. (probability): [ math_81_Probabillity.html ]
        2. (bags): [ math_82_MultiSets_and_Bags.html ]
        3. (fuzzy sets): [ math_83_Fuzzy_Sets.html ]
        4. (spectra): [ math_84_Spectra.html ]
        5. (statistics): [ math_85_Statistics.html ]

        . . . . . . . . . ( end of section Probability, Multisets, Bags, Fuzzyiness, Spectra) <<Contents | End>>

        Topology

          The theories of space

        1. (set theoretic topology): [ math_91_Topology.html ]
        2. (metric spaces): [ math_92_Metric_Spaces.html ]

          Special Spaces - Graphic, Metric, etc

          [ math_93_Graphics.html ]
        3. (ndimensional calculus): [ math_94_Calculus.html ]
        4. (function spaces): [ math_95_Function_Spaces.html ]

        . . . . . . . . . ( end of section Topology) <<Contents | End>>

      Principles

        Logic


          (logic101): [ intro_logic.html ] [ logic_0_Intro.html ]
        1. (predicates): [ logic_10_PC_LPC.html ]
        2. (equality): [ logic_11_Equality_etc.html ]
        3. (proofs): [ logic_2_Proofs.html ] [ logic_20_Proofs100.html ] [ logic_25_Proofs.html ] [ logic_27_Tableaux.html ]
        4. (modes): [ logic_9_Modalities.html ]

        . . . . . . . . . ( end of section Logic) <<Contents | End>>

        Sets

          [ intro_sets.html ]
        1. (sets): [ logic_30_Sets.html ]
        2. (families of sets): [ logic_31_Families_of_Sets.html ]
        3. (set theory): [ logic_32_Set_Theory.html ]

          Also see [ fuzzy sets ] [ bags ]

        . . . . . . . . . ( end of section Sets) <<Contents | End>>

        Relations

          [ intro_relation.html ] [ logic_40_Relations.html ]

          [ logic_41_HomogenRelations.html ]

          [ logic_42_Properties_of_Relation.html ]

          [ logic_44_n-aryrelations.html ]

        . . . . . . . . . ( end of section Relations) <<Contents | End>>

        Functions

          [ intro_function.html ]

          [ logic_5_Maps.html ]

        . . . . . . . . . ( end of section Functions) <<Contents | End>>

        Types

          [ intro_records.html ]

          [ types.html ]

          [ math_77_Enumerations.html ]

        . . . . . . . . . ( end of section Types) <<Contents | End>>

      . . . . . . . . . ( end of section Principles) <<Contents | End>>

      More Advanced topics in Logic

        [ intro_standard.html ] [ math_11_STANDARD.html ]

        [ intro_strings.html ] [ logic_6_Numbers..Strings.html ]

        [ logic_7_Semantics.html ]

        [ logic_8_Natural_Language.html ]

      . . . . . . . . . ( end of section More Advanced topics in Logic) <<Contents | End>>

    Logic and Discrete Math On the Internet

      Local Math Page

      [ contents.cgi ]

      Description Logics

      [ index.html ]

    . . . . . . . . . ( end of section Logic and Discrete Math On the Internet) <<Contents | End>>

Notes on MATHS Notation

Special characters are defined in [ intro_characters.html ] that also outlines the syntax of expressions and a document.

Proofs follow a natural deduction style that start with assumptions ("Let") and continue to a consequence ("Close Let") and then discard the assumptions and deduce a conclusion. Look here [ Block Structure in logic_25_Proofs ] for more on the structure and rules.

The notation also allows you to create a new network of variables and constraints. A "Net" has a number of variables (including none) and a number of properties (including none) that connect variables. You can give them a name and then reuse them. The schema, formal system, or an elementary piece of documentation starts with "Net" and finishes "End of Net". For more, see [ notn_13_Docn_Syntax.html ] for these ways of defining and reusing pieces of logic and algebra in your documents. A quick example: a circle might be described by Net{radius:Positive Real, center:Point, area:=π*radius^2, ...}.

For a complete listing of pages in this part of my site by topic see [ home.html ]

Notes on the Underlying Logic of MATHS

The notation used here is a formal language with syntax and a semantics described using traditional formal logic [ logic_0_Intro.html ] plus sets, functions, relations, and other mathematical extensions.

For a more rigorous description of the standard notations see

  • STANDARD::= See http://www.csci.csusb.edu/dick/maths/math_11_STANDARD.html

    Glossary

  • above::reason="I'm too lazy to work out which of the above statements I need here", often the last 3 or 4 statements. The previous and previous but one statments are shown as (-1) and (-2).
  • given::reason="I've been told that...", used to describe a problem.
  • given::variable="I'll be given a value or object like this...", used to describe a problem.
  • goal::theorem="The result I'm trying to prove right now".
  • goal::variable="The value or object I'm trying to find or construct".
  • let::reason="For the sake of argument let...", introduces a temporary hypothesis that survives until the end of the surrounding "Let...Close.Let" block or Case.
  • hyp::reason="I assumed this in my last Let/Case/Po/...".
  • QED::conclusion="Quite Easily Done" or "Quod Erat Demonstrandum", indicates that you have proved what you wanted to prove.
  • QEF::conclusion="Quite Easily Faked", -- indicate that you have proved that the object you constructed fitted the goal you were given.
  • RAA::conclusion="Reducto Ad Absurdum". This allows you to discard the last assumption (let) that you introduced.

    End