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Is Boolean algebra a subject?

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  • Vaughan Pratt
    A while back I asked how people felt these days about whether to allow algebras to be empty (absent any zeroary operations in the signature of course). I ve
    Message 1 of 8 , Mar 14, 2011
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      A while back I asked how people felt these days about whether to allow
      algebras to be empty (absent any zeroary operations in the signature of
      course). I've now come up with an even more trivial question: is
      "Boolean algebra" a subject whose objects form a subtopic, or is it
      equally a subject and an object?

      Mathematically trivial, that is. According to the preface of the
      revision by Givant and Halmos of Halmos's 1963 Lectures on Boolean
      algebras, "Outside the realm of mathematics, Boolean algebra has found
      applications in such diverse areas as anthropology, biology, chemistry,
      ecology, economics, sociology, and especially computer science and
      philosophy. For example, in computer science, Boolean algebra is used in
      electronic circuit design (gating networks), programming languages,
      databases, and complexity theory."

      In those areas "outside the realm of mathematics" it is less clear what
      counts as trivial. Many researchers with IQ's over 150 inquiring about
      Boolean algebras have never taken a course in group theory or been
      exposed to any other kind of "modern" or abstract algebra, so their
      perspective may be very different from yours.

      My question arises from the heated debate at
      http://en.wikipedia.org/wiki/Talk:Boolean_algebra
      as to whether the Wikipedia article "Boolean algebra" should be an
      article or a disambiguation ("dab") page. The argument for the latter
      is perhaps most clearly stated by the following extract from that talk page:

      "A Boolean algebra is not Boolean algebra. I don't mind if Boolean
      algebras are treated somewhere in the mass-noun article, presumably
      rather late in the article. But I think there should be a stand-alone
      article on the object, and I think that topic is important enough to
      share equal billing with the mass-noun sense, even if the latter at some
      point mentions the object. --Trovatore (talk) 09:33, 13 March 2011 (UTC)"

      The counterargument is based on
      http://en.wikipedia.org/wiki/Wikipedia:DAB#Broad_concept_articles
      which says "Where the primary topic of a term is a general topic that
      can be divided into subtopics ... the unqualified title should contain
      an article about the general topic rather than a disambiguation page."

      The trivial question then is, are Boolean algebras a subtopic of the
      topic of Boolean algebra, or is the term "Boolean algebra" a genuinely
      ambiguous phrase ("mass-noun" vs. "count-noun") calling for a
      disambiguation page?

      This is the sort of question that those not yet tenured should not touch
      with a ten-foot pole. At the other end of that spectrum, emeriti like
      myself can rant about it without repercussions, as you will see me doing
      on that talk page in spades. ;)

      Vaughan Pratt
    • Steve Vickers
      Dear Vaughan, If you think about algebra ( Elementary algebra ) v. algebras ( Algebraic structures of various kinds including rings and fields) it s
      Message 2 of 8 , Mar 14, 2011
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        Dear Vaughan,

        If you think about algebra ("Elementary algebra") v. algebras
        ("Algebraic structures" of various kinds including rings and fields)
        it's analogous, so my thought is it should be possible to mimic what's
        done there (only it's simpler because "algebra" is so widely used in
        different senses). You have elementary Boolean algebra, and the
        algebraic structures that embody it. Probably not so hard to have a
        general page, subtopic pages, and - to be on the safe side - a
        disambiguation page. You already have all those for "Algebra".

        At present there's a big problem of duplication.

        All the best,

        Steve.

        Vaughan Pratt wrote:
        >
        >
        > A while back I asked how people felt these days about whether to allow
        > algebras to be empty (absent any zeroary operations in the signature of
        > course). I've now come up with an even more trivial question: is
        > "Boolean algebra" a subject whose objects form a subtopic, or is it
        > equally a subject and an object?
        >
        > Mathematically trivial, that is. According to the preface of the
        > revision by Givant and Halmos of Halmos's 1963 Lectures on Boolean
        > algebras, "Outside the realm of mathematics, Boolean algebra has found
        > applications in such diverse areas as anthropology, biology, chemistry,
        > ecology, economics, sociology, and especially computer science and
        > philosophy. For example, in computer science, Boolean algebra is used in
        > electronic circuit design (gating networks), programming languages,
        > databases, and complexity theory."
        >
        > In those areas "outside the realm of mathematics" it is less clear what
        > counts as trivial. Many researchers with IQ's over 150 inquiring about
        > Boolean algebras have never taken a course in group theory or been
        > exposed to any other kind of "modern" or abstract algebra, so their
        > perspective may be very different from yours.
        >
        > My question arises from the heated debate at
        > http://en.wikipedia.org/wiki/Talk:Boolean_algebra
        > as to whether the Wikipedia article "Boolean algebra" should be an
        > article or a disambiguation ("dab") page. The argument for the latter
        > is perhaps most clearly stated by the following extract from that talk page:
        >
        > "A Boolean algebra is not Boolean algebra. I don't mind if Boolean
        > algebras are treated somewhere in the mass-noun article, presumably
        > rather late in the article. But I think there should be a stand-alone
        > article on the object, and I think that topic is important enough to
        > share equal billing with the mass-noun sense, even if the latter at some
        > point mentions the object. --Trovatore (talk) 09:33, 13 March 2011 (UTC)"
        >
        > The counterargument is based on
        > http://en.wikipedia.org/wiki/Wikipedia:DAB#Broad_concept_articles
        > which says "Where the primary topic of a term is a general topic that
        > can be divided into subtopics ... the unqualified title should contain
        > an article about the general topic rather than a disambiguation page."
        >
        > The trivial question then is, are Boolean algebras a subtopic of the
        > topic of Boolean algebra, or is the term "Boolean algebra" a genuinely
        > ambiguous phrase ("mass-noun" vs. "count-noun") calling for a
        > disambiguation page?
        >
        > This is the sort of question that those not yet tenured should not touch
        > with a ten-foot pole. At the other end of that spectrum, emeriti like
        > myself can rant about it without repercussions, as you will see me doing
        > on that talk page in spades. ;)
        >
        > Vaughan Pratt
        >
        >
      • Jens Doll
        Here are a few thoughts about the subject under investigation: set theory ... Boolean algebra is also a pattern of thinking in natural sciences, because of
        Message 3 of 8 , Mar 15, 2011
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          Here are a few thoughts about the subject under investigation:

          set theory
          ----------
          Boolean algebra is also a pattern of thinking in natural sciences,
          because of it's isomorphic relation to set theory. Geo-, bio-, psycho-,
          sociology and many more use it as a classification pattern for
          scientific phenomena. So it is contained as a filter in scientific
          thinking. The majority of theories is base on the law of the excluded
          middle. Also the Wiki people are infected by two valued thinking.

          many valued logics
          -------------------
          If on the other hand we consider logics, Boolean algebra is one of
          many, where the variety goes from two- over three-valued logics in
          Kripke style to continuous causalities in Minkowski spaces. I can
          imagine thinking machines which use different logics.

          circularity
          ---------
          If you consider decomposition of algebraic field structures you have the
          field (+-*/,{0,1}) as a base for countable algebraic fields.

          Jens
        • Steve Vickers
          Even in the context of many-valued logics (I m thinking in particular of the frame-valued logics that arise in toposes of sheaves), Boolean algebras have an
          Message 4 of 8 , Mar 16, 2011
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            Even in the context of many-valued logics (I'm thinking in particular of
            the frame-valued logics that arise in toposes of sheaves), Boolean
            algebras have an important role to play as the algebraic counterpart of
            totally disconnected compact Hausdorff spaces such as Cantor space
            2^omega. (By Stone's representation theorem Boolean algebras are the
            lattices of clopens of such. In the absence of choice this still makes
            good sense if one understands topology in a point-free way.)

            What's more the duality between sets and Stone Boolean algebras
            (Proposition 3.2 in Johnstone's "Stone Spaces") would in principle allow
            us to use the category of Boolean algebras as a foundational substitute
            for that of sets. The duality is not the well-known classical duality of
            sets with CABAs, but starts from the restriction of that to finite
            decidable sets/BAs and then exploits the fact that any set or BA is a
            filtered colimit of finite decidable ones. (In Johnstone's book the
            result is starred to indicate use of choice. Nonetheless, the folklore
            is that it works point-free without choice.)

            Steve.

            Jens Doll wrote:
            > Here are a few thoughts about the subject under investigation:
            >
            > set theory
            > ----------
            > Boolean algebra is also a pattern of thinking in natural sciences,
            > because of it's isomorphic relation to set theory. Geo-, bio-, psycho-,
            > sociology and many more use it as a classification pattern for
            > scientific phenomena. So it is contained as a filter in scientific
            > thinking. The majority of theories is base on the law of the excluded
            > middle. Also the Wiki people are infected by two valued thinking.
            >
            > many valued logics
            > -------------------
            > If on the other hand we consider logics, Boolean algebra is one of
            > many, where the variety goes from two- over three-valued logics in
            > Kripke style to continuous causalities in Minkowski spaces. I can
            > imagine thinking machines which use different logics.
            >
            > circularity
            > ---------
            > If you consider decomposition of algebraic field structures you have the
            > field (+-*/,{0,1}) as a base for countable algebraic fields.
            >
            > Jens
            >
            >
            > ------------------------------------
            >
            > Yahoo! Groups Links
            >
            >
            >
          • Insall, Matt
            A while ago, Vaughan Pratt wrote: ``... ... My question arises from the heated debate at http://en.wikipedia.org/wiki/Talk:Boolean_algebra as to whether the
            Message 5 of 8 , Mar 16, 2011
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              A while ago, Vaughan Pratt wrote:

               

              ``…


              My question arises from the heated debate at
              http://en.wikipedia.org/wiki/Talk:Boolean_algebra
              as to whether the Wikipedia article "Boolean algebra" should be an
              article or a disambiguation ("dab") page. The argument for the latter
              is perhaps most clearly stated by the following extract from that talk page:

              "A Boolean algebra is not Boolean algebra. I don't mind if Boolean
              algebras are treated somewhere in the mass-noun article, presumably
              rather late in the article. But I think there should be a stand-alone
              article on the object, and I think that topic is important enough to
              share equal billing with the mass-noun sense, even if the latter at some
              point mentions the object. --Trovatore (talk) 09:33, 13 March 2011 (UTC)"


              …’’

               

               

              A Boolean algebra is a structure that is an object of study of the theory of Boolean algebras, just as a group is a structure that is the object of study of the theory of groups.  Of course, here I mean the informal concept of ``the theory of …’’, not a formal concept, like ``the first order theory of …’’ or ``the second order theory of …’’.  It is unfortunate that anyone would `argue’ (heatedly) over this however.   It is equally unfortunate, I think, that terminology is bandied about without concern for the grammatical role it plays in  the communication of ideas.  For example, if I teach group theory (aka the theory of groups), I will not write a treatise in which I say something like `` Group is a logical calculus of permutations, developed by …’’ (see the first line of the following Wikipedia page: http://en.wikipedia.org/wiki/Boolean_algebra_(logic)); for in fact,  this would be (a) grammatically disturbing, (b) terribly narrow-minded (in spite of various representation theorems in the theory of groups, in which it is shown that every group is isomorphic to a group of permutations) as a description of group theory, and (c) terribly narrow-minded as a description of the calculus of permutations and the theory of permutations.  When narrow-minded descriptions of  entire theories, or narrow-minded definitions, are promulgated as the ``mainstream’’ or ``encyclopedic’’ notion of a concept, the retreat from abstraction is noted by those who better understand the nature of these things, but the authoritative declaration is taken, by those who do not work in the areas being rendered for the general public, as proclamations from the mouth of the wiki-omniscience, making the task of improving humankind’s collective understanding of useful abstraction more and more burdensome.  The student who reads Wikipedia, for example, may then go to the logic class, and decry the abstract definition of a lattice because it is abstract and therefore not specifically Boolean, in spite of the usefulness of lattices in non-classical logics (like quantum logic).  We see this unfortunate phenomenon frequently, in, for example, linear algebra classes, where students who have been told (or actually indoctrinated without being ``told’’, using a ``concrete’’ approach) by some ``authority’’ (textbook, or instructor of some pre-requisite course) that linear algebra is merely a course in the manipulation of matrices with at most 5 rows and 7 columns, whose entries are all integers.  Another similarly unfortunate phenomenon rears its head in logic classes when we speak of Boolean algebras to students (fortunately this particular misunderstanding is NOT promulgated by Wikipedia) who then wonder why we make such a big deal about a structure with (they, mistakenly, think) only two elements.  (These students have been, as a colleague once put it, ``abused’’ in some previous class, where someone told them that in Boolean Logic there are only two truth values, and then that the numbers zero and one are used for THE elements of THE Boolean Algebra.)  When such ``abuse’’ occurs, later instructors, researchers, writers, etc, find that their task begins with an inefficiency-producing process of ``disabusing’’ the readership of various misconceptions.  Please, let us get Wikipedia and other sociologically authoritative resources to avoid being the initiators of any of the ``abuse’’;  thus, let us avoid, or even eradicate, the apparent use, as definitions, of narrow, or ``concrete’’, versions of topics from the ``main page’’ definitions, especially in mathematics and logic.  (This is not intended, however, as a diatribe against examples.  For in fact, even on the main pages, we should, imho, have more examples, not less.  But since an example is not a definition or a proof or a theorem, let us not even appear to promulgate one such example as if it were the definition, or as if it were a theorem, or as if it were a proof, for any abstract concept.) 

               

               

               

              Matt Insall, PhD
              Associate Professor of Mathematics
              Department of Mathematics and Statistics, suite 315
              Missouri University of Science and Technology
              (formerly University of Missouri - Rolla)

              400 W. 12th St.

              Rolla MO 65409-0020

               

              (573)341-4901
              insall@...

              http://web.mst.edu/~insall/

               

              Essays: 

              http://web.mst.edu/~insall/Essays/Bernoulli%20Functions%20versus%20modern%20functions/

              http://web.mst.edu/~insall/Essays/Consistency%20in%20Mathematics/

              http://web.mst.edu/~insall/Essays/First%20Lines/

              http://web.mst.edu/~insall/Essays/Set%20Theory/

               

              Effective Jan. 1, 2008, UMR became Missouri University of Science and Technology (Missouri S&T)

               


              PS:  visit lulu.com!

               

              viaR03

               

               

               

              From: univalg@yahoogroups.com [mailto:univalg@yahoogroups.com] On Behalf Of Vaughan Pratt
              Sent: Monday, March 14, 2011 2:55 AM
              To: univalg@yahoogroups.com
              Subject: [univalg] Is Boolean algebra a subject?

               

               

              A while back I asked how people felt these days about whether to allow
              algebras to be empty (absent any zeroary operations in the signature of
              course). I've now come up with an even more trivial question: is
              "Boolean algebra" a subject whose objects form a subtopic, or is it
              equally a subject and an object?

              Mathematically trivial, that is. According to the preface of the
              revision by Givant and Halmos of Halmos's 1963 Lectures on Boolean
              algebras, "Outside the realm of mathematics, Boolean algebra has found
              applications in such diverse areas as anthropology, biology, chemistry,
              ecology, economics, sociology, and especially computer science and
              philosophy. For example, in computer science, Boolean algebra is used in
              electronic circuit design (gating networks), programming languages,
              databases, and complexity theory."

              In those areas "outside the realm of mathematics" it is less clear what
              counts as trivial. Many researchers with IQ's over 150 inquiring about
              Boolean algebras have never taken a course in group theory or been
              exposed to any other kind of "modern" or abstract algebra, so their
              perspective may be very different from yours.

              My question arises from the heated debate at
              http://en.wikipedia.org/wiki/Talk:Boolean_algebra
              as to whether the Wikipedia article "Boolean algebra" should be an
              article or a disambiguation ("dab") page. The argument for the latter
              is perhaps most clearly stated by the following extract from that talk page:

              "A Boolean algebra is not Boolean algebra. I don't mind if Boolean
              algebras are treated somewhere in the mass-noun article, presumably
              rather late in the article. But I think there should be a stand-alone
              article on the object, and I think that topic is important enough to
              share equal billing with the mass-noun sense, even if the latter at some
              point mentions the object. --Trovatore (talk) 09:33, 13 March 2011 (UTC)"

              The counterargument is based on
              http://en.wikipedia.org/wiki/Wikipedia:DAB#Broad_concept_articles
              which says "Where the primary topic of a term is a general topic that
              can be divided into subtopics ... the unqualified title should contain
              an article about the general topic rather than a disambiguation page."

              The trivial question then is, are Boolean algebras a subtopic of the
              topic of Boolean algebra, or is the term "Boolean algebra" a genuinely
              ambiguous phrase ("mass-noun" vs. "count-noun") calling for a
              disambiguation page?

              This is the sort of question that those not yet tenured should not touch
              with a ten-foot pole. At the other end of that spectrum, emeriti like
              myself can rant about it without repercussions, as you will see me doing
              on that talk page in spades. ;)

              Vaughan Pratt

            • Vaughan Pratt
              Matt, was your reply to my question a yes or a no ? Reading between the lines, it sounded as though you were arguing for referring to the subject as the
              Message 6 of 8 , Mar 17, 2011
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                Matt, was your reply to my question a "yes" or a "no"?

                Reading between the lines, it sounded as though you were arguing for
                referring to the subject as "the theory of Boolean algebras" and
                insisting that "Boolean algebra" refer only to the object (as in, "a
                Boolean algebra," "the Boolean algebra B," "Boolean algebras," etc.)

                If so then there should be no need for a disambiguation page, and the
                article titled "Boolean algebra" could start out something like "In
                mathematics, a Boolean algebra is a complemented distributive lattice,"
                or "is any model of the identically true equations in the language AND,
                OR, NOT standardly interpreted over 0 and 1," or "is a Boolean ring."

                Your analogy with group theory would then show that there is no need to
                waste time talking about Venn diagrams, truth tables, schematics for
                digital circuits, Post's operation basis theorem, equivalence of Boolean
                rings and complemented distributive lattices, complexity of
                satisfiability, etc. and the second sentence of the article could go on
                to say that Boolean algebra is primarily about proper subalgebras of
                CABAs (since CABAs themselves are trivial).

                But in that case "algebra" should likewise denote "algebraic structure"
                and the subject should be called "theory of algebraic structures"
                instead of "algebra." And there should be subjects called "theory of
                relation algebras," "theory of commutative algebras," etc. instead of
                subjects called "relation algebra" and "commutative algebra."

                Or did I misunderstand your analogy with groups?

                Also it would help if you could be more specific about the "abuse" being
                initiated by Wikipedia so it can be fixed. Would it be ok if an
                encyclopedia article on Boolean algebras began with "Boolean algebra is
                the algebra of two-valued logic with only sentential connectives"?

                Vaughan Pratt
              • A. Mani
                ... Connections between logic (in the sequent calculus or axiomatic sense) and algebra carry plenty of meta-level assumptions. It is simply not proper to force
                Message 7 of 8 , Mar 18, 2011
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                  Vaughan Pratt wrote:

                  >
                  > Also it would help if you could be more specific about the "abuse" being
                  > initiated by Wikipedia so it can be fixed. Would it be ok if an
                  > encyclopedia article on Boolean algebras began with "Boolean algebra is
                  > the algebra of two-valued logic with only sentential connectives"?


                  Connections between logic (in the sequent calculus or axiomatic sense) and
                  algebra carry plenty of meta-level assumptions. It is simply not proper to
                  force such a view. Intuitionists for example can argue that the connection is
                  flawed.

                  Logic can be understood in purely algebraic terms as well. The distinction
                  between semantics and proof theory is not very deep.

                  An encylopedic work should concentrate on calling a spade a spade.
                  A good way in the present context would be to start from the complemented
                  distributive lattice definition and then proceed to descriptions based on
                  other signatures.

                  Disambiguations for different application areas like logic makes sense, but
                  the algebraic perspective should be the basic one.

                  Boolean Algebras - An Algebraic Perspective
                  Boolean Algebras in Formal Logic
                  Boolean Algebras for Computation


                  Best

                  A. Mani





                  --
                  A. Mani
                  ASL, CLC, AMS, CMS
                  http://www.logicamani.co.cc
                • Fred Linton
                  I fear I d have to answer the question Is Boolean algebra a subject? the same way I d have to answer the similar questions, Is High School Algebra a subject?
                  Message 8 of 8 , Mar 18, 2011
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                    I fear I'd have to answer the question

                    "Is Boolean algebra a subject?"

                    the same way I'd have to answer the similar questions,

                    Is High School Algebra a subject?
                    Is Modern Algebra a subject?
                    Is Commutative Algebra a subject?
                    Is Linear Algebra a subject?
                    Is Multilinear Algebra a subject?
                    Is Abstract Algebra a subject?
                    Is Tensor Algebra a subject?
                    Is Universal Algebra a subject?
                    Is Algebra (plain and simple) a subject?

                    -- namely, with a resounding "Yes." :-) .

                    Cheers, -- Fred [Linton, Wes U Math/CS Emeritus]
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