A Meno Acid

Posted on March 21, 2013 by Jon Awbrey

What answers to the Meno Paradox
Comes in the moment of realizing —
Gathering together the building blocks
Is just the beginning of the building.

Posted in Artificial Intelligence, Cognitive Science, Education, Inquiry, Learning, Meno, Philosophy, Plato, Teaching, Verse | Tagged , , , , , , , , , | Leave a comment

Indicator Functions • 1

Posted on March 13, 2013 by Jon Awbrey

Re: R.J. Lipton and K.W. Regan • Who Invented Boolean Functions?

One of the things it helps to understand about 19th Century mathematicians, and those who built the bridge to the 20th, is that they were capable of high abstraction — in Peirce’s case a cut above what is common today — and yet they remained close enough to the point where abstract forms are teased away from the concrete materials of mathematical inquiry to maintain a sense of connection between the two.  There are few better places to see this connection than in the medium of venn diagrams.  But venn diagrams are such familiar pictures that it’s easy to overlook their subtleties, so it may be useful to spend some time developing the finer points of what they picture.

There are actually several types of boolean functions depicted in the typical venn diagram.  Each has the boolean domain \mathbb{B} = \{ 0, 1 \} or one of its powers \mathbb{B}^k as its functional codomain but its functional domain need not be limited to a finite cardinality.  To sort their variety, consider the array of functional arrows in the following figure.

Indicator Functions

Suppose X is a universe of discourse represented by the rectangular area of a Venn diagram.  Note that the set X itself may have any cardinality.  The most general type of Boolean function is a map f : X \to \mathbb{B}.  This is known as a Boolean-valued function since only its functional values need be in \mathbb{B}.

A function of the type f : X \to \mathbb{B} is called a characteristic function in set theory or an indicator function in probability and statistics since it can be taken to characterize or indicate a particular subset S of X, namely, the fiber or inverse image of the value 1, for which we have the notation and definition f^{-1}(1) = \{ x \in X : f(x) = 1 \}.

The notation f_S is often used for the characteristic function of a subset S of X.  Putting all the pieces together then, we have f_S^{-1}(1) = S \subseteq X.

To be continued …

Posted in Abstraction, Boole, Boolean Functions, C.S. Peirce, Category Theory, Characteristic Functions, Euler, Indicator Functions, John Venn, Logic, Mathematics, Peirce, Propositional Calculus, Set Theory, Venn Diagrams, Visualization | Tagged , , , , , , , , , , , , , , , | Leave a comment

Riffs and Rotes • 2

Posted on March 3, 2013 by Jon Awbrey

Re: Peter Cameron • Addition and Multiplication of Natural Numbers

The interaction between addition and multiplication in the natural numbers has long been an interest of mine, leading to broader questions about the relationship between algebra and combinatorics.  My gropings with those enigmas led me to the structures of Riffs and Rotes, extracting what we might think of as the “purely combinatorial” properties of primes factorizations.  Thinking of the additive structure of the positive integers as embodied in their total linear ordering, the following two questions arise.

  • How much of the natural ordering of the natural numbers is purely combinatorial?
  • What additional axioms on the partial orders of Riffs and Rotes would restore their natural order?

Reference

cc: Category Theory • Cybernetics • Ontolog Forum • Peirce List (1) (2) • SeqFan
cc: FB | Riffs and Rotes • Laws of Form • Structural Modeling • Systems Science

Posted in Arithmetic, Combinatorics, Graph Theory, Group Theory, Logic, Mathematics, Number Theory, Riffs and Rotes | Tagged , , , , , , , | Leave a comment

Oracles

Posted on March 1, 2013 by Jon Awbrey

Computing, in its way, and science, in its broader way,
both involve the relation between what appears limited
and what appears not. Whether you believe in divinity
or not, and whether you believe that humanity contains
a spark of divinity or not, we have to acknowledge that
our powers as oracles are limited and, even if they were
not, problems of relative computability would still arise
in the striving of oracles to communicate with one another.

Posted in Communication, Computability, Computing, Inquiry, Oracles, Relative Computability, Science, Universals | Tagged , , , , , , , | 2 Comments

Quotiens?

Posted on February 28, 2013 by Jon Awbrey 
How many times do I repeat the same experience?
Before I come to see it as the same experience?
Posted in Algorithms, Anamnesis, Arithmetic, Deja Vu, Education, Epistemology, Eternal Return, Inquiry, Learning, Meno, Music, Pattern Recognition, Plato, Poetry, Recursion, Repetition, Rhythm, Teaching | Tagged , , , , , , , , , , , , , , , , , | 2 Comments

Poems and Programs

Posted on February 27, 2013 by Jon Awbrey
Words that do …

A trendy misunderstanding has reared its head as to what the discipline of computing, indeed the logic of science, are all about.  I blame Penrose, of course, but he is only the most recent promulgator of the recurring misunderstanding.

There are only a countable number of computable functions, so it’s no surprise a natural system picked at random will have non-computable functional features.  Saying not all natural systems are computable is like saying not all poems are sonnets.  Writing a program to model a significant aspect of a natural system is like writing a sonnet to express a significant aspect of human experience.  It’s a voluntary limitation programmers and poets accept for the sake of their elective-effective art.

And both have their uses.

Posted in Aesthetics, Artistic Differences, Computability, Effective Description, Ethics, Existential Choice, Finitude, Form, Imagination, Information, Inquiry, Limitation, Logic of Science, Matter, Mortality, Poetry, Programming, Volition | Tagged , , , , , , , , , , , , , , , , , | 1 Comment

I Wonder, Wonder Who

Posted on February 25, 2013 by Jon Awbrey

Re: R.J. Lipton and K.W. Regan • Who Invented Boolean Functions?

The question recalls recent discussions of discovery and invention in the mathematical field, bringing back to mind questions I’ve wondered about for as long as I can remember.

Speaking as an unreconstructed Platonic realist, I am tempted to say Boolean functions are mathematical objects which cannot be invented, only discovered.  But speaking as a semiotic constructivist I would have to concede we do indeed invent all sorts of syntactic systems for talking and thinking about these mathematical objects.  And some calculi can even be better than others for the purpose of calculation, a fact repaying us to consider the alternatives as they work out in practice.

On the third hand, I have more lately been thinking the concepts of discovery and invention, being human constructs like the proverbial concepts of particles and waves, may not be adequate in the final analysis to articulate the reality of the process at hand.  It may well be all these questions are more like the question —

  • Who discovered Orion in the night sky?
Posted in Anamnesis, Aristotle, Boole, Boolean Functions, C.S. Peirce, Discovery, Invention, Learning, Logic, Mathematics, Meno, Model Theory, Peirce, Plato, Propositional Calculus, Recollection, Semiotics, Socrates, Teaching | Tagged , , , , , , , , , , , , , , , , , , | Leave a comment

Notes On Categories • 1

Posted on February 22, 2013 by Jon Awbrey

Continued from “Notes On Categories” (14 Jul 2003) • Inquiry List • Ontology List

NB.  This page is a work in progress.  I will have to dig up some still older notes from the days of pen and paper before I can remember how I left things last.

Here are some notes on a computational approach to category theory I started working on back in the 1980s, all of which work as yet remains in the “Schubert Category” of unfinished symphonies.

It helps me a little bit to write the names of categories in the plural, so as not to confuse them with individuals.  It also helps if I treat the arrows of Arr(C) as the primary entities in the category C, recovering the objects of Obj(C) as secondary entities by collecting all the entities that appear in s(f) = Source(f) and t(f) = Target(f) as one ranges over all of the arrows f in Arr(C).

The last time that I tried to do “categories by computer”, I was using data structures that had the following shapes:

   Category C o              
             /|\             
            / | \            
          ... | ...          
              |              
      Arrow f o              
             / \             
            s   t            
           /     \           
     s(f) o       o t(f)

A functor, then, is something I picture like this:

                   Functor F o                             
                           . | .                           
                         .   |   .                         
                       .     |     .                       
                     .       |       .                     
        Category C o         o         o Category D = CF   
                   |       ./ \.       |                   
                   |     . /   \ .     |                   
                   |   .  /     \  .   |                   
                   | .   /       \   . |                   
           Arrow f o    o         o    o Arrow fF          
                  / \ .   .     .   . / \                  
                 /  .\      . .      /.  \                 
                s .   t     . .     s   . t                
               /.      \  .     .  /      .\               
              o         o         o         o              
              x         y         xF        yF

This is a rough sketch of the actual data structures that I used to represent a functor F as a “matching” between the parallel items of categories C and D.

NB.  I have reverted to the convention I was accustomed to use at the time, where all operators are applied on the right of their arguments.

What the picture says is that the functor F : C → CF takes each arrow f in C to an arrow fF in CF, and each object x in C to an object xF in CF, in such a manner that (fs)F = (fF)s and (ft)F = (fF)t.  To be a functor, F must satisfy the following two systems of equations:

(1x)F   =   1(xF),   for all x in Obj(C).

(f ∘ g)F   =   fF ∘ gF,   for all composable f, g in Arr(C).

That was just how I kept track of things on the computer.

It is, of course, more usual to draw a functor square in the following manner, where we get one such picture for each object x and arrow f in C.

            F            
     x o-------->o xF    
       |         |       
       |         |       
     f |         | fF    
       |         |       
       v         v       
     y o-------->o yF    
            F
Posted in Abstraction, Category Theory, Computing, Graph Theory, Logic, Mathematics, Relation Theory, Type Theory | Tagged , , , , , , , | 8 Comments

Theme One • A Program Of Inquiry 4

Posted on February 14, 2013 by Jon Awbrey

Re: Next Polymath Project • What, When, Where?

Here’s a bit of data on the Theme One Program I worked on all through the 1980s.  My aim was to develop fundamental algorithms and data structures to support an integrated learning and reasoning interface.  I wrote up a pilot version of the program well enough to get a Master’s degree out of it but still haven’t gotten around to writing up the complete documentation.  Below is a link to what I’ve put on the web so far.

I see this project as being related to Tim Gowers’ Proposal on the Mathematics of the Origin of Life, since one of the reasons for doing this work was to get information about the threshold of systems-theoretic complexity necessary to support a capacity for inquiry.

By the end of the 80s, and especially as I returned to grad school in systems engineering during the 90s, my work on this program gradually morphed into a broader study of what I began calling “Inquiry Driven Systems”.  The conceptual background for this study is outlined in the following proposal.

More discussion to follow as I get time …

Inquiry Project Papers

Posted in Artificial Intelligence, C.S. Peirce, Cognition, Computation, Constraint Satisfaction Problems, Cybernetics, Formal Languages, Inquiry, Inquiry Driven Systems, Intelligent Systems, Learning Theory, Logic, Peirce, Semiotics | Tagged , , , , , , , , , , , , , | 8 Comments

Château Descartes

Posted on February 14, 2013 by Jon Awbrey
But if we are to select those dimensions which will be of the greatest assistance to our imagination, we should never attend to more than one or two of them as depicted in our imagination, even though we are well aware that there is an indefinite number involved in the problem at issue.  It is part of the method to distinguish as many dimensions as possible, so that, while attending to as few as possible at the same time, we nevertheless proceed to take in all of them one by one.  (Descartes, CSM, 63).

The final point we should bear in mind is that among the dimensions of a continuous magnitude none is more distinctly conceived than length and breadth, and if we are to compare two different things with each other, we should not attend at the same time to more than these two dimensions in any given figure.  For when we have more than two different things to compare, our method demands that we survey them one by one and concentrate on no more than two of them at once.  (Descartes, CSM, 65).

Reference

  • René Descartes, “Regulae ad Directionem Ingenii”, or “Rules for the Direction of the Mind”, pp. 9–78 in John Cottingham, Robert Stoothoff, and Dugald Murdoch (eds., trans., 1985), The Philosophical Writings of Descartes, Volume 1, Cambridge University Press, Cambridge, UK.
Posted in Analytic Geometry, Cartesian Coordinate System, Cartesian Philosophy, Cartesian Product, Descartes, Dualism, Dyadicism, Inquiry, Logic, Mathematics, Philosophy, Reductionism, Relation Theory | Tagged , , , , , , , , , , , , | 3 Comments