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Tue Aug 20 10:57:12 PDT 2013


    Recent Changes and thoughts about the MATHS Language

      2013-08-20 Tue Aug 20 10:56 Linked examples of tabular proofs to notes on proofs

      [ logic_25_Proofs.html ]

      2013-06-11 Tue Jun 11 16:33 Add some notes on Topology

      [ math_91_Topology.html ]

      2013-03-25 Mon Mar 25 11:35 Warning about circular semantics

      [ intro_note.html ]

      2013-02-20 Wed Feb 20 07:03 Corrected an example of integration

      The correction is in one of the most incomplete area [ math_94_Calculus.html ] of the MATHS language -- the calculus.

      2013-01-27 Sun Jan 27 11:20 Not a bug but a feature

      Mentally reviewing the script for "Names2dictionary" that I use to index all the pages in a directory I realized that is will not index items that are not unique. So the two deffinitions of "Update" in the page below will not be indexed.

      Note: The MATHS language would give them different follow qualified names because they are defined in different contexts. But the tool is a dumb shell script that can not handle context dependencies...

      I will now fix the page to use different names:-(

      2013-01-27 Sun Jan 27 10:36 Another indexing bug

      Wondering why some defined operations in [ logic_44_n-aryrelations.html ] are not indexed correctly.

      This is the time in the year when I revisit the topic of how to use simple set theory to document operations in a use case...

      2012-12-07 Fri Dec 7 06:42 Working on abstract and universal algebras

      [ math_43_Algebras.html ] Discovered that headings and definitions with greek symbols
      Σ for example were not being indexed correctly. This area is very skimpy.

      2012-11-20 Tue Nov 20 07:07 Improved notes on strings

      [ math_62_Strings.html#EVENT ]

      2012-10-14 Sun Oct 14 12:36 Improving notes on Language Theory

      [ math_63_Languages.html ]

      2012-09-26 Wed Sep 26 09:46 Adding a new algebraic step

      [ logic_25_Proofs.html#algebraic_step ]

      This allows you to combine equations with algebraic operations ... It follows from Euclid's Laws...

      2012-08-28 Tue Aug 28 13:06 Updating theory of Probability

      Please review [ math_81_Probabillity.html ] and contact me with corrections!

      2012-08-14 Tue Aug 14 19:39 Venn diagrams and complements

      [ notn_16_Classification.html ] (Classifications and Venn Diagrams).

      [ logic_30_Sets.html#Conveniences ] (Defining the meaning of a unary complement ~A when A is a subset of elements of type T).

      2012-07-26 Thu Jul 26 12:40 Structured arguments -- to be done

      [ notn_5_Form.html ]

      2012-07-12 Thu Jul 12 08:58 Trying to bake the logic of events

      [ logic_8_Natural_Language.html#Even%20more%20unbaked%20ideas%20on%20time%20and%20events ]

      2012-06-30 Sat Jun 30 07:31 Added Unary set complements

      [ logic_30_Sets.html ]

      2012-06-26 Tue Jun 26 16:11 Added a note on Free Nets and Objects

      [ math_12_Structure.html ]

      2012-06-11 Mon Jun 11 16:53 Tydied up permuation groups a little

      Lots nore work to be done with permutations and cycles:-)

      [ math_34_Groups.html ]

      2012-06-02 Sat Jun 2 15:27 Added some more definitions to Matrices

      [ math_45_Three_Operators.html#Matrices ]

      2012-05-31 Thu May 31 08:41 Added some definitions and results to sets

      After reading the discussion on StackOverflow/Mathematics of the definition of a topological space I added some notes and results concerned with expressions like "|{}" and "&{}".

      [ logic_30_Sets.html ] [ logic_31_Families_of_Sets.html ]

      2012-05-22 Tue May 22 10:27 Created a new page on statistical formula

      [ math_85_Statistics.html ]

      2012-05-19 Sat May 19 08:07 Watched video and editted Groups monoids and semigroups

      I watched this [ abstract-algebra ] which is a video from Harvard University that introduces groups by reviewing matrices and linear algebra... which led me to review my own notes [ math_31_One_Associative_Op.html ] [ math_32_Semigroups.html ] [ math_33_Monoids.html ] [ math_34_Groups.html ] and make some small changes.

      2012-05-15 Tue May 15 13:20 Improved partly baked operations in Nets

      [ math_12_Structure.html ] [ intro_objects.html ]

      2011-12-13 Tue Dec 13 14:26 Working on theoretical models of numbers

      [ logic_6_Numbers..Strings.html#Models of numbers ]

      2011-12-07 Wed Dec 7 09:04 Revised notes on Automata and systems

      [ math_71_Auto...Systems.html ]

      2011-11-09 Wed Nov 9 10:24 Fixed an error and the old notation for citations

      Fixed a stupid error
    1. x^1 = 1 ??? in [ math.syntax.html ] while thinking about changing the syntax of subscripts and citations.

      In the original vision of MATHS (or PINAPL as I called it) I thought that a special syntax would be needed for citations and references:

       	[ Author Year Letter ]
      but the invention of the ".See" directive, the "$" link and the notation
       	(citation): reference.
      the original is starting to look obsolete. It also makes the rendering of subscripts a lot harder. This needs thinking through...

      2011-11-08 Tue Nov 8 08:10 Defined more general set comprension

      Just extended the bare-bones definition
    2. {f(a)|| a:A, W(a)} = {y||for some a:A (y=f(a) and W(a))}, to
    3. {e || x:X, y:Y, ... , W} in [ logic_30_Sets.html ] and corrected a tale of relational operators in [ logic_44_n-aryrelations.html ]

      2011-10-11 Tue Oct 11 11:04 More on uniqueness and definite descriptions

      [ logic_11_Equality_etc.html ]

      2011-10-08 Sat Oct 8 10:47 ORDER and Kuratowski

      Working on links in [ math_21_Order.html ]

      2011-10-02 Sun Oct 2 11:46 Added concept of Finite Subsets of a set

      [ logic_30_Sets.html#Finite_sets ] and improving [ type.html#Finite_sets ]

      2011-07-25 Mon Jul 25 10:03 Fixed typo in theory of Orders

      While thinking about the ontological proof of the existence of God I reread my notes on orders and found and fixed some typographical errors: [ math_21_Order.html ] I also added a couple of humrorous only serious links to [ logic_20_Proofs100.html ] about the length of proofs.

      2011-05-06 Fri May 6 09:43 Added a list of Proofs that are not Proofs

      [ logic_2_Proofs.html ]

      2011-03-23 Wed Mar 23 16:29 Tried to improve notes on types

      [ types.html ]

      2011-02-28 Mon Feb 28 07:46 Reorganising Modal logic and Natural language

      Moving things between [ logic_9_Modalities.html ] and [ logic_8_Natural_Language.html ] plus linking to relations and automata...

      2011-02-27 Sun Feb 27 11:07 Corrected SIMPLE TEMPORAL LOGIC

      [ logic_9_Modalities.html ] -- I've made a subtle change to bring it in line with older temporal logics like LTL and CTL.

      If you have a reference to SIMPLE_TEMPORAL_LOGIC or SIMPLE_DISCRETE_TEMPORAL_LOGIC please check to see if my change has invalidated your work.

      2011-02-25 Fri Feb 25 17:23 Linked LTL to Wikipedia and added some notes

      [ logic_9_Modalities.html#LTL ]

      2011 starts above

      2010-12-15 Wed Dec 15 13:14 Classifications Ontologies and Venn

      Just expanded my notes on documenting classifications and ontologies [ notn_16_Classification.html ] by showing how simple subsets lead to complex partitions of a universe of discourse.

      2010-12-13 Mon Dec 13 14:29 Redid an update that disapeared in the crash of 2008

      I finally had time to remove the deprecated old notation that allowed multiple definitions in a line. I had to rewite one or two of these in about 20 files/pages in this web site.

      I fixed this years ago and then lost the fix after a crash... and lost energy to redo it until now.

      2010-12-08 Wed Dec 8 16:44 Added quotations from Lancelot Hogben

      [ logic_20_Proofs100.html#Proofs come after the thinking and experimentation ] [ logic_20_Proofs100.html#Problems with Word Problems ]

      2010-11-20 Sat Nov 20 11:58 Partly Baked Idea -- OO Operations

      Working on using formal methods and MATHS within Craig Larman's UP to express Domain Model contracts for steps leads me to work on [ math_12_Structure.html#Partly Baked Idea -- Operations on Objects ] it may need major modification and possible removal in time.

      2010-11-17 Wed Nov 17 09:46 Working on Z-like specifications etc

      I want to make sure that [ math_14_Dynamics.html ] includes the kind of operation contracts that turn up in Z and Use case realizations...

      2010-11-01 Mon Nov 1 10:53 Refurbishing the MATHS theory of types

      [ types.mth ]

      2010-10-26 Tue Oct 26 08:47 Infix priorities standard

      [ math_11_STANDARD.html#Infix operations ]

      more work to do...

      2010-10-25 Mon Oct 25 10:17 Starting to rework the theory and practice of grammars

      Avoid these for the nonce: notn_12_Expressions.mth math_63_Languages.mth notn_13_Docn_Syntax.mth notn_3_Conveniences.mth, sorry for any inconvenience.... just patching a 11 year old hole.

      2010-09-17 Fri Sep 17 10:45 Publishing my log of changes

      My tools automatically log changes in ".Q.log" and I'm going to publish it as [ log.txt ] in this directory.

      2010-08-11 Wed Aug 11 11:31 Added some theorems on the subset relation

      [ logic_30_Sets.html#subsets ]

      2010-07-28 Wed Jul 28 07:34 Improving notes on grammars

      I've gone back to the original [ ../monograph/ ] to improve [ intro_grammar.html ] and [ math_63_Languages.html ] on Context Free Grammars in particular.

      2010-05-25 Tue May 25 14:30 Corrected and improved notes on distfix or mixfix predicates

      I wanted there to be a way to include natural language into the predicate calculus.... [ logic_10_PC_LPC.html#Roadworks ] any thoughts would be welcome.... this is still an PBI -- Partly Baked Idea.

      2010-05-04 Tue May 4 17:02 Found out more about product topologies

      My notes on topology [ math_91_Topology.html ] are based on some note I wrote 3 or 4 decades ago.... and never used. But I may need them for working on, of all things, cellular automata. SO a quick trip to the Wikipedia.

      2010-03-08 Mon Mar 4 12:30 How Lewis Carrol Solved Problems

      Like Lewis Carrol I control my personal demons when I can't get to sleep by mulling of mathematics. It usually sends me straight to sleep. In this case I went to sleep thinking about reasoning with non-unique definite descriptions. Could you get a paradox by allowing Euclid's "Substitute equals for equals" when one of the terms was not unique:
    4. "The present King of France"
    5. "The pen on my desk"

      And I awoke remembering a little booklet full of mathematical paradoxes from my childhod in Guildford, Surrey, England. It was a Euclidean proof of a ridiculous result that is distroyed when your try to draw the construction. This shows you that one of the points can not exist....

      So I have hardened the rules for equality -- now you should establish uniqueness before doing algebra...

      2010-03-04 Thu Mar 4 07:27 Struggling with non-unique definite descriptions

      2010-03-03 Wed Mar 3 11:54 Improved notes on one and the

      [ logic_11_Equality_etc.html ]

      2010-03-01 Mon Mar 1 12:49 Added the Logic book

      I've added a bibliographic item for "the logic book" to [ logic_20_Proofs100.html ] about proving thinsg.

      2010 starts

      2009-12-18 Fri Dec 18 11:12 Rediscovered definition

      While looking over some documents from the 1990's that have just been regurgitated from the Faculty RPT system I discovered some fun definitions of addition, multiplication, and power in terms of adding and subtracting 1... [ intro_function.html#Arithmetic_operators ] , enjoy!

      2009-12-16 Wed Dec 16 14:53 Toulmin Arguments

      In memory of Dr. Stephen E. Toulmin who died on December 4th I've revised my notes [ notn_5_Form.html#Toulmin Arguments and Rationales ] on them. He sounds like a good man who I wish I had known...

      2009-11-22 Sun Nov 22 11:32 Givens

      I added the definition of terms like "given" and "goal" to [ math_11_STANDARD.html ] and improved my notes on penguins and rebuttals [ logic_20_Proofs100.html#polya ] and [ logic_20_Proofs100.html#Arguments ]

      2009-11-18 Wed Nov 18 16:03 Changed a symbol for the temporal if-then

      Becuase "~>" was overloaded ofr relations and temporal logic, and this might be confusing.... I've chosen to change "~>" to "~~>" in [ logic_9_Modalities.html#SIMPLE_TEMPORAL_LOGIC ] and I hope this doesn't cause any problems.... Let me know if it does.

      2009-11-16 Mon Nov 16 14:39 Late spring cleaning

      [ log.txt ] for a series of small changes linking different pages. STREAMS->STRINGS->STANDARD.

      2009-11-13 Fri Nov 13 13:05 Structured Automata

      I've recovered some definitions of automata (acceptors, generators, and filters) from an old project: [ math_71_Auto...Systems.html#Structured Automata ]

      2009-11-09 Mon Nov 9 14:56 Started a definition of protocols and messages

      [ math_71_Auto...Systems.html#Protocols ]

      2009-11-04 Wed Nov 4 15:43 Improved notes on handling natural language tenses

      [ logic_8_Natural_Language.html#Tenses, Time, and Temporal modalities ]

      2009-11-02 Mon Nov 2 17:54 do and iterate

      Rediscovered the need for a more obvious, if longer word for repetition: [ logic_41_HomogenRelations.html#iterate ] [ intro_relation.html#do ]

      2009-09-28 Mon Sep 28 13:35 Semantics of subsets

      I've added some notes on modelling subsets of sets -- example tall people vs short people: [ notn_16_Classification.html ]

      2009-09-22 Tue Sep 22 15:37 Defined xor and derived inequality of propositions

      [ logic_10_PC_LPC.html ]

      I hadn't noticed the '<>' was implicitly defined for propositions as the 'xor' or non-equivalent operator.

      Also add a table summarizing the commonest propositional operators.

      2009-08-19 Wed Aug 19 14:39 Corrected an error

      Noticed and corrected an error in [ logic_20_Proofs100.html ] where I had miscalculated some elementary algebra.

      More evidence for the need of a proof checker.

      2009-07-13 Mon Jul 13 15:07 Improved notes on using relational calculus to shorten proofs

      [ intro_relation.html ] [ logic_25_Proofs.html ]

      2009-06-11 Thu Jun 11 15:55 Improved proposed notation for problem solving

      [ math_10_Intro.html ] [ notn_13_Docn_Syntax.html ]

      2009-05-19 Tue May 19 06:33 Improved notes on Dynamics

      [ intro_dynamics.html ] [ math_14_Dynamics.html ] [ math_75_Programs.html ] Added notation ?? (s.l'=n) ::= (s'=s~l|n)

      from [ todo.html ] my "to do" list

      [ notn_11_Expressions.html ] [ notn_12_Expressions.html#Free Variables ]

      2009-04-17 Fri Apr 17 06:49 Small change to Process Algebras

      [ math_73_Process_algebra.html ]

      2009-01-31 Sat Jan 31 09:01 Thank you Alejandro

      Alejandro has found an omission in [ logic_9_Modalities.html#AW ] describing the Computational Tree Logic. I fixed it and left a Footnote.... .

      2009-01-14 Wed Jan 14 20:01 Added references to the Tree of Porphyry

      [ logic_8_Natural_Language.html#Stuff ]

      2009-01-09 Fri Jan 9 10:01 Improved notes on Modal Logic

      [ logic_9_Modalities.html ]

      2009 above here

      2008-12-03 Wed Dec 3 11:12 Transfered my unpublished To Do List for MATHS

      I've taken my personal notes of Problems, Partly Baked Ideas, and To Be Dones from my old Palm Pilot and into [ todo.html ] so I can refer to it from my new iPod....

      2008-11-12 Wed Nov 12 14:11 Dimensioned Numbers

      Tidied up the back references in [ math_49_Dimensioned_numbers.html ] and made some small improvements.

      I also extend my mth2html to recognize and render ".{" and ".}" as well as ".(" and ".)". More in the item below.

      2008-11-06 Thu Nov 6 10:11 Possible improved rendering for algorithms etc

      I've hacked my personal copy of mth2html to translate ".(" and ".)" directives into Ordered Lists with class="Formula", and added this rule to my standard style sheet
       	OL .Formula{
       		list-style-type : none;
      I think the effect is quite pleasing. Here is the formula [ notn_14_Docn_Semantics.html#Translation into a Z Schema ] rewritten using this convention -- first rendered and second as source
    6. For In:Net, Net_to_Z(In)::=with{signature, axioms:Sets} following (
      1. (signature, axioms) := ({},{});
      2. do (
        1. with{piece:piece_of_documentations} (
          1. piece:?In; (
            1. piece in declaration; signature:|declaration_to_Z(piece)
            2. | piece in wff; axioms:|wff_to_Z(piece)
            3. | piece in definition; signature:|definition_to_Z_declaration(piece); axioms:|definition_to_Z_equality(piece)
            4. | piece in (theorem | comment)
      3. end(In);
      4. output_Z_schema(signature, axioms);
        For In:Net, Net_to_Z(In) ::=with{signature, axioms:Sets} following
       	(signature, axioms) := ({},{});
       .( piece in declaration;  signature:|declaration_to_Z(piece)
       		|  piece in wff;  axioms:|wff_to_Z(piece)
       		|  piece in definition;  signature:|definition_to_Z_declaration(piece); axioms:|definition_to_Z_equality(piece)
       		|  piece in (theorem | comment)
       	output_Z_schema(signature, axioms);

      This idea now goes into a trial use period. Comments would help. I'll either retract the idea or put it into the online m2h tool in a week or two. Watch this blog for the next exciting episode!

      2008-11-03 Mon Nov 3 13:11 Worrying about algorithms and expressions

      I am disatisfied with the expression of algorithms in MATHS. There is a formal semantics in terms of an extension to the calculus of relations. This makes algorithms a special kind of formula. But to make them readable -- structured even -- is another matter -- basically you need to use the .List and .Set directives to show the block structure. An example is [ notn_14_Docn_Semantics.html#Translation into a Z Schema ] from the description of the meaning of a piece of MATHS documentation.

      This morning it occured to me that a natural notation would be to allow directives like: ".(", ".)" and perhaps even ".[", ".{", etc. The sematics is of "(...)". But the rendering should be as an indented subexpression. More on this when I've had a chance to do some more thinking. Any opinions can be sent to me here [ hole.html ] and will get full credit.

      2008-11-01 Sat Nov 1 06:11 Working on Types and Duck Typing

      Several modern programming languages -- Python and Ruby, for example -- use what is called "Duck Typing". This means that behavior of an object defines its type, rather than an explicit definition.

      It turns out, to my surprise, that my MATHS language has a kind of Duck Typing because the type of an object is the universal set in which the object is defined. And the type defined by a Net of variables, axioms, definitions, etc. is the type of the unconstrained objects -- the simple Net associated with the original Net...

      Details and changes in [ notn_14_Docn_Semantics.html ] and [ notn_16_Classification.html ]

      As always -- click here [click here [socket symbol] if you can fill this hole] to submit your thoughts.

      2008-10-24 Fri Oct 24 10:10 Small revisions to the A->B notation

      Not a real change of meaning.... just cleaned up the definitions and noted that
    7. A->B->C

      is not clearly defined and should be avoided.

      2008-10-22 Wed Oct 22 16:10 improved notes on the Calculus

      Still a long way to go in [ math_94_Calculus.html ] before we have a complete page.

      2008-10-07 Tue Oct 7 07:10 Corrected syntax of example database definition

      [ math_13_Data_Bases.html#DataBase ]

      Which in turn lead to an extension of the notation for discriminated types [ types.html#type_disciminated_union ]

      2008-09-29 Mon Sep 29 14:09 Corrected a subtle error in Petri net definition

      I had to shuffle the definitions of Node, Places, and Transitions in [ math_76_Concurency.html#PETRI ] so that it is clear that Places and Transitions are the same type of thing and so validating the definition
    8. Arcs::@(Places><Transitions) | @(Transitions><Places)=given.

      2008-09-25 Thu Sep 25 10:09 Citation in theory and practice

      The bibliography [ ../lab.html ] has evolved a notation for citations that was not a part of my original design for MATHS. Similarly the "hole.php" uses the Source directive in an unexpected way. See [ notn_15_Naming_Documentn.html ] for the kludge. The older BNF syntax may be removed or be replaced in time -- it is now deprecated.

      2008-09-10 Wed Sep 10 11:09 Quotes about math and life

      The following quotes were sent to me by Sonesh Rawat via the Contact/comment button on this part of my website:
      1. He who seeks for methods without having a definite problem in mind seeks for the most part in vain.
      2. One should study mathematics because it is only through mathematics that nature can be conceived in harmonious form.
      3. Just as a mountaineer climbs a mountain- because it is there,so a good mathematics students studies new material because it is there.

      I'd like to cap them with a thought of Sigmund Freud:
    9. Man climbs highest when he doesn't know where he is going.

      By the way just fixed the broken link on Systems English below.

      2008-09-09 Tue Sep 9 17:09 Added notes on Grammars

      [ math_63_Languages.html ]

      2008-08-26 Tue Aug 26 13:08 Improved description of Systems English

      Systems English is a tiny subset of Basic English designed to describe systems: [ logic_8_Natural_Language.html#Systems English ]

      2008-08-11 Mon Aug 11 16:08 What -- no topoi

      Somehow I've never gathered any notes of topos theory and topoi. I'm just moting the following link to a simple motivational introduction:
    10. topos::= See http://math.ucr.edu/home/baez/topos.html at the nearby University of California Riverside.

      When developed the notes would be in [ math_25_Categories.html ] and I will plant a hole there for future expansion.

      2008-08-10 Sun Aug 10 12:08 Corrected error in definition of subspace

      [ math 91 Topology.html#Subspaces and relative topologies ]

      2008-06-27 Fri Jun 27 08:06 An Expression with Ambiguous type

      As a result of thinking about a wikipedia article I tripped over the fact that
    11. /Card(n) -- the sets with cardinallity n, is a typically ambiguous expression. As a result runs the risk of introducing a paradox into the system if not used carefully. So I'm changing the definition from
    12. Card:@T->Nat0= map[X](|X|)


    13. For Type T, Card[T]::@T->Nat0= map[X:@T](|X|).
    14. For Type T, X:@T, Card(X)::Nat0= Card[T](X).
    15. For Type T, A:@T, n:Nat0, A@n = { X:@T || Card[T](X) = n }. (Subsets of a given size).

      Here are the pages where changes were made as a result [ types.html ] [ math_25_Categories.html ] [ logic_30_Sets.html ] [ intro_sets.html ] [ intro_standard.html ] [ notn_12_Expressions.html ]

      I wonder if I need a way to express the idea that some expression is not defined in addition to having a way to define things? [click here [socket symbol] undefinitions if you can fill this hole]

      2008-06-20 Fri Jun 20 14:06 Rebuttals in MATHS

    16. But me no Buts!

      I've decided to introduce a simple way to annotate an argument that attempts to rebut the previous argument. Commonly arguments and proofs in MATHS start ".Let" and end ".Close.Let". A Rebuttal will be an argument that is placed between

      For example

      1. (let)|-It is raining.

      2. (let)|-If it rains then I shall get wet.
      3. (-1, -2)|-I shall get wet.

      (Close Let )

      1. I carry an umbrella.
      2. (-1)|-I keep dry.
        1. In August,
        2. I forgot the umbrella,
        3. (-1)|-I got wet.

        (Close But )

      (Close But )

      2008-06-19 Thu Jun 19 07:06 Working on Toulmin Arguments and Rationales

      Today I'll be working on [ notn_5_Form.html ] to describe informal argumentation.

      This is the result of seeing the same reference in two papers in IEEE Transactions on Software Engineering and in "Wittgenstein's Poker".

      More later...

      2008-06-14 Sat Jun 14 10:06 Added source and some distributions

      [ math_81_Probabillity.html ] (sorry for the mis-spelling in this link).

      2008-06-12 Thu Jun 12 05:06 Clarified definitions of prefic union and intersection in Sets

      This span off from Principia Mathematica Section *42 (and there is more below).

      I added the definition of prefix union and intersection to [ logic_30_Sets.html ] and explicitly linked [ logic_31_Families_of_Sets.html ] back to logic_30.

    17. 8:35 Added the (missing) syntax for prefixed union and intersection operators. Also added some quick examples of the syntax.

    18. 5:13 pm Added a nonstandard translation of part of PM's Chapter 42: [ logic_30_Sets.html#Chapter_42 ] (enter here at your own risk).

      2008-06-11 Wed Jun 11 15:06 Fixed diagram of constructive dilemma

      [ constructive_dilemma in logic_25_Proofs ]

      2008-06-11 Wed Jun 11 05:06 Stuggling with an ambiguity

      Until last night I've never worried about sets of sets of sets. But they are used in Principia Mathematica's section *42 -- definitely not the answer to life's persistent questions.

      This lead me to discover an ambiguity in expressions like

    19. +((1,2), (3,4)).

      Plus is a SERIAL operator so it means

    20. (1,2) + (3,4).

      And + is overloaded to combine maps and lists so that its value is

    21. (1+3, 2+4) = (4,6).

      It might be thought that because + is overloaded as a prefix function it should be applied (by a second overloading) to the elements of the list

    22. +((1,2), (3,4)) <> ( +(1,2), +(3,4)) = (3, 7).

      I've added notes to [ notn_12_Expressions.html ] that explain that the simplest parsings are used whenever possible.

      2008-06-07 Sat Jun 7 07:06 Tidied up overloading of infix operators

      Read chapter 38 of Principia Mathematica last night. Made me want to review the equivalent definitions in MATHS. As a result I've tidied up [ math_11_STANDARD.html#Infix operations ] which make the following equations correct:
    23. {1,2,3} * 3 = {3,6,9},
    24. {1,2}*{3,5} = {3,5,6,20},
    25. 4*(1,2,3) = {4,8,20),
    26. (3,4,5)*(1,2,3) = (3,8,15).

      2008-06-05 Thu Jun 5 07:06 Improved notes on Proofs

      I've expanded on the reasons for proving something [ logic_20_Proofs100.html#But Why Prove anything Anyway? ] and what can be concluded after deriving some results from some assumptions in [ logic_25_Proofs.html#Block Structure ] to clarify that you must include all the assumptions and any of the results in the conclusion drawn from an argument.

    27. 11:06 Added a note on tables and relational data bases. [ notn_9_Tables.html ]

      2008-05-23 Fri May 23 16:05 Initiating a web log of changes and thoughts

      I'm planning to record most changes to the files in this directory here.

      Here is a quick list of recently changed pages in this part of my site.
      May 5 10:29math_92_Metric_Spaces.html Added Notes on Hilbert spaces used in Quantum Theory
      May 7 17:58math_81_Probabillity.html Added notes on Expected values, means, variances etc.
      May 10 09:02logic_27_Tableaux.html Added a thought about using a table to express a set of cases
      May 13 07:23intro_records.html Added notes on optional and multiple parts in a record
      May 19 14:32logic_30_Sets.html Added to notes on the Axiom of Choice
      May 21 20:10logic_25_Proofs.html Added an example of using table to document proof by cases.
      May 23 06:49math_41_Two_Operators.html Added definition of a dioid

      (Close Table)

      More 2008 changes below here

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    28. MATHS::=./index.html

    . . . . . . . . . ( end of section Recent Changes and thoughts about the MATHS Language) <<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

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


  2. 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).
  3. given::reason="I've been told that...", used to describe a problem.
  4. given::variable="I'll be given a value or object like this...", used to describe a problem.
  5. goal::theorem="The result I'm trying to prove right now".
  6. goal::variable="The value or object I'm trying to find or construct".
  7. 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.
  8. hyp::reason="I assumed this in my last Let/Case/Po/...".
  9. QED::conclusion="Quite Easily Done" or "Quod Erat Demonstrandum", indicates that you have proved what you wanted to prove.
  10. QEF::conclusion="Quite Easily Faked", -- indicate that you have proved that the object you constructed fitted the goal you were given.
  11. RAA::conclusion="Reducto Ad Absurdum". This allows you to discard the last assumption (let) that you introduced.

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