NYC LOCAL: Tuesday 13 March 2012 Lisp NYC: Jay Sulzberger on John Bell's Two Theorems and the Second Half of George Boole's Laws of Thought

[secretary]

Lisp NYC will meet at 7:00 pm on Tuesday 13 March 2012 at Google
NYC, in the Port of Authority Weapons Building on the Island of
the Manahattoes, which building, for this meeting only, may be
entered at 76 Ninth Avenue, between 15th and 16th Street.

Please RSVP by going to

  http://www.lispnyc.org


Suppose we have a classical circuit of classical probabilistic
gates, with underlying graph like so:

  http://www.panix.com/~jays/EPR-Diagram1.png

We wish to know the possible input output behaviors of such a
circuit.  We will answer this question by using:

1. John S. Bell's First Theorem, which is less well known than
   his Second Theorem

2. George Boole's General Method in Probabilities

3. Results on acyclic database schemata by
   David Maier and others


ad 1: John Bell's Two Theorems are presented in the book "Speakable and
      Unspeakable in Quantum Mechanics", in the chapter "On the
      Einstein-Podolsky-Rosen paradox".  The First Theorem is implicit
      in the discussion on page 15 of the book.  A few pages later, the
      Second Theorem is explicitly laid out.  Here is a BibTeX entry
      for "Speakable and Unspeakable in Quantum Mechanics", taken from
      Google's catalogue:
      
      @book{bell2004speakable,
        title={Speakable and unspeakable in quantum mechanics: collected papers 
on quantum philosophy},
        author={Bell, J.S.},
        isbn={9780521523387},
        lccn={2004557644},
        series={Collected papers on quantum philosophy},
        url={http://books.google.com/books?id=FGnnHxh2YtQC},
        year={2004},
        publisher={Cambridge University Press}
      }


ad 2: A copy of George Boole's book "The Laws of Thought" is at

      
http://ia700208.us.archive.org/3/items/aninvestigationo15114gut/15114-pdf.pdf

      On page vii of this edition is a note:

        In Prop. II., p. 261, by the "absolute probabilities" of the
        events x,y,z.. is meant simply what the probabilities of those
        events ought to be, in order that, regarding them as independent,
        and their probabilities as our only data, the calculated
        probabilities of the same events under the condition V should be
        p,g,r.. The statement of the appended problem of the urn must be
        modified in a similar way. The true solution of that problem, as
        actually stated, is p' = cp, q' = cq, in which c is the arbitrary
        probability of the condition that the ball drawn shall be either
        white, or of marble, or both at once. -See p. 270, CASE II.*

        Accordingly, since by the logical reduction the solution of all
        questions in the theory of probabilities is brought to a form in
        which, from the probabil- ities of simple events, s, t, &c. under
        a given condition, V, it is required to determine the probability
        of some combination. A, of those events under the same condition,
        the principle of the demonstration in Prop. IV. is really the
        following:- "The probability of such combination A under the
        condition V must be calculated as if the events s, t, &c. were
        independent, and possessed of such probabilities as would cause
        the derived probabilities of the said events under the same
        condition V to be such as are assigned to them in the data."
        This principle I regard as axiomatic. At the same time it admits
        of indefinite verification, as well directly as through the
        results of the method of which it forms the basis. I think it
        right to add, that it was in the above form that the principle
        first presented itself to my mind, and that it is thus that I
        have always understood it, the error in the particular problem
        referred to having arisen from inadvertence in the choice of a
        material illustration.

      The note, though short, states the main theorem of
      Boole's General Method in Probabilities.
  

ad 3: David Maier's book The Theory of Relational Databases is at

      http://web.cecs.pdx.edu/~maier/TheoryBook/TRD.html

      Chapter 13 is on acyclic database schemata.  Theorem 13.2
      has been extended so that it answers, for some lambda
      expressions, the following question:

      Given a lambda expression l, we have two cases:

       Case 1: The set of functions computed by the lambda expression,
       when we fill in values for the free variables, becomes greater if
       we are in THE QUANTUM WORLD, than if we are in THE CLASSICAL
       WORLD.

       Case 2: The set of functions computed by the lambda expression,
       when we fill in values for the free variables, is just the same,
       whether we are in THE QUANTUM WORLD or whether we are in THE
       CLASSICAL WORLD.

      Question: Given l, are we in Case 1, or are we in Case 2?


Backround material:

 Judea Pearl's "Causality, Second Edition" presents a careful and
 persuasive explication of the hypotheses of John Bell's First
 Theorem, and includes a long series of clarifications and
 applications of the general theory underlying Bell's First
 Theorem.  Here is a BibTeX entry for "Causality, Second Edition",
 taken from Google's catalogue:

   @book{pearl2009causality,
     title={Causality: models, reasoning, and inference},
     author={Pearl, J.},
     isbn={9780521773621},
     url={http://books.google.com/books?id=6WfQjwEACAAJ},
     year={2009},
     publisher={Cambridge Univ. Press}
   }


 David Kaiser's book "How the Hippies Saved Physics" is a
 wonderful history of the reception of John Bell's Theorem:
  
   http://www.hippiessavedphysics.com/


 In the Nineties of the last century, I and Ed Green and others
 posted, in the Usenet group sci.physics, on Bell's Two Theorems.
 Here are two posts by me:

   
http://groups.google.com/group/sci.physics/msg/9f5bd3d2497039fc?hl=en&dmode=source

   
http://groups.google.com/group/sci.physics/msg/f7a1584f54ee3af4?hl=en&dmode=source

 Google presents, in a not quite convenient form, more of the
 thread, but not all of it, starting at:

   
http://groups.google.com/group/sci.physics/browse_thread/thread/7974a151ee69fdc3/f7a1584f54ee3af4?hl=en&#f7a1584f54ee3af4


Jay Sulzberger <secretary@lxny.org>
Corresponding Secretary LXNY
LXNY is New York's Free Computing Organization.
http://www.lxny.org