The Present Is Big With The Future • Comment 1

Posted on June 2, 2017 by Jon Awbrey

Re: Peirce List • John Sowa
Re: Peirce List • Jon Awbrey

Here is a passage from Leibniz where he half decrypts half encrypts the big idea sparking his discovery of the differential calculus.

The Present Is Big With The Future

Now that I have proved sufficiently that everything comes to pass according to determinate reasons, there cannot be any more difficulty over these principles of God’s foreknowledge.  Although these determinations do not compel, they cannot but be certain, and they foreshadow what shall happen.

It is true that God sees all at once the whole sequence of this universe, when he chooses it, and that thus he has no need of the connexion of effects and causes in order to foresee these effects.  But since his wisdom causes him to choose a sequence in perfect connexion, he cannot but see one part of the sequence in the other.

It is one of the rules of my system of general harmony, that the present is big with the future, and that he who sees all sees in that which is that which shall be.

What is more, I have proved conclusively that God sees in each portion of the universe the whole universe, owing to the perfect connexion of things.  He is infinitely more discerning than Pythagoras, who judged the height of Hercules by the size of his footprint.  There must therefore be no doubt that effects follow their causes determinately, in spite of contingency and even of freedom, which nevertheless exist together with certainty or determination.

I have a vague memory of having once looked on the Latin text, where the word big was gravis, meaning pregnant, in the original.  But it was a long time ago, and I’ll need to check that out again sometime.

Incidentally, working out the logical analogue of differential calculus is the object of my efforts on differential logic.  This work led me to develop an extension of Peirce’s alpha graphs that is efficient enough in both conceptual and computational terms to carry the load.  For an introduction, see the following articles.

Reference

  • Gottfried Wilhelm (Freiherr von) Leibniz, Theodicy : Essays on the Goodness of God, the Freedom of Man, and the Origin of Evil, edited with an introduction by Austin Farrer, translated by E.M. Huggard from C.J. Gerhardt’s edition of the Collected Philosophical Works, 1875–1890.  Routledge 1951.  Open Court 1985.  Paragraph 360, page 341.

cc: Peirce List

Posted in C.S. Peirce, Determination, Differential Calculus, Differential Logic, Hologrammautomaton, Infinitesimals, Leibniz, Logic, Logical Graphs, Mathematics, Peirce, Preëstablished Harmony, Propositional Calculus, Visualization | Tagged , , , , , , , , , , , , , | 10 Comments

The Difference That Makes A Difference That Peirce Makes • 8

Posted on May 31, 2017 by Jon Awbrey

Re: Peirce List • James Albrecht

Among the subtle shifts in scientific thinking that occurred in the mid 1800s, George Boole gave us a functional interpretation of logic, associating every propositional expression — at the most basic level of logic we now describe in terms of boolean algebras, boolean functions, propositional calculi, or Peirce’s alpha graphs — with a function from a universe of discourse X to a domain of two values, say \mathbb{B} = \{ 0, 1 \}, normally interpreted as logical values, false and true, respectively.  This may seem like a small change so far as conceptual revolutions go but it made a big difference in the future development, growth, and power of our logical systems.

Among other things, the functional interpretation of logic enables the construction of a bridge from propositional logic, whose subject matter now consists of functions of the form f : X \to \mathbb{B}, to probability theory, that deals with probability distributions or probability densities of the form p : X \to [0, 1], with values in the unit interval [0, 1] of the real number line \mathbb{R}.  This allows us to view propositional logic as a special case within the frame of a more general statistical theory.  This turns out to be a very useful perspective in real-world research when it comes to moving back and forth between qualitative observations and the data given by quantitative measurement.  And it gives us a bridge still further, connecting deductive and inductive reasoning, as Boole well envisioned.

Posted in C.S. Peirce, Complementarity, Inquiry, Laws of Form, Logic, Mathematics, Peirce, Philosophy, Physics, Pragmatism, Quantum Mechanics, Relativity, Science, Scientific Method, Semiotics, Spencer Brown | Tagged , , , , , , , , , , , , , , , | Leave a comment

The Difference That Makes A Difference That Peirce Makes • 7

Posted on May 31, 2017 by Jon Awbrey

Re: Peirce List • GF • GF • GR

In our “Inquiry as Action : Risk of Inquiry” paper, originally presented at a conference on “Hermeneutics and the Human Sciences”, Susan and I sought to trace the interminglings of signs and inquiry and the theories thereof.  We pursued their trajectory through three points of reference:  Aristotle, Peirce, and Dewey.  We noted both convergences and divergences in the views of the assembled authors, and the course of true signs never did run smooth, as everyone knows, or eventually finds out

We characterized Aristotle’s treatment “On Interpretation” (where the implied relationship between a sign and its object is a two-step linkage that pivots on what Peirce would call an interpretant sign) as “in part a reasonable approximation and in part a suggestive metaphor, suitable as a first approach to a complex subject”.  It makes for a good start, but ultimately falls short of grasping the full triadicity of sign relations.

References

  • Awbrey, J.L., and Awbrey, S.M. (1992), “Interpretation as Action : The Risk of Inquiry”, The Eleventh International Human Science Research Conference, Oakland University, Rochester, Michigan.
  • Awbrey, J.L., and Awbrey, S.M. (1995), “Interpretation as Action : The Risk of Inquiry”, Inquiry : Critical Thinking Across the Disciplines 15(1), pp. 40–52.  Archive, Journal, Online.
Posted in C.S. Peirce, Complementarity, Inquiry, Laws of Form, Logic, Mathematics, Peirce, Philosophy, Physics, Pragmatism, Quantum Mechanics, Relativity, Science, Scientific Method, Semiotics, Spencer Brown | Tagged , , , , , , , , , , , , , , , | Leave a comment

The Difference That Makes A Difference That Peirce Makes • 6

Posted on May 30, 2017 by Jon Awbrey

Re: Peirce List • Gary Fuhrman

The uses to which Susan Awbrey and I turned Aristotle’s passage from De Interp can be found in our paper from 1992/1995.

  • Awbrey, J.L., and Awbrey, S.M. (1992), “Interpretation as Action : The Risk of Inquiry”, The Eleventh International Human Science Research Conference, Oakland University, Rochester, Michigan.
  • Awbrey, J.L., and Awbrey, S.M. (1995), “Interpretation as Action : The Risk of Inquiry”, Inquiry : Critical Thinking Across the Disciplines 15(1), pp. 40–52.  Archive, Journal, Online.

To get the ball rolling, or ping-ponging as the case may be, let me refer to a few points from our Inquiry paper that came to mind as I skimmed Gary Fuhrman’s post on “Rhematics” and Gary Richmond’s comment on it.

The main thing that strikes me is a thing that never ceases to surprise me — I see there remains a persistent desire to parse symbols into simpler signs like icons and indices, or to say that genuine triadicity has its genesis in some kind of coitus between degenerate species.  I suppose bi-o-logical metaphors are bound to lead innocents down that path, and I guess we all fall into the sinns of simile from time to time, but due care of our semiotic souls should keep us from turning that error into doctrine, if we wit what’s good for us.

Posted in C.S. Peirce, Complementarity, Inquiry, Laws of Form, Logic, Mathematics, Peirce, Philosophy, Physics, Pragmatism, Quantum Mechanics, Relativity, Science, Scientific Method, Semiotics, Spencer Brown | Tagged , , , , , , , , , , , , , , , | Leave a comment

The Difference That Makes A Difference That Peirce Makes • 5

Posted on May 17, 2017 by Jon Awbrey

Re: Peirce List • Gary Richmond

When I think back to the conceptual changes my first university physics courses put me through, a single unifying theme emerges.  Relativity Theory and Quantum Mechanics had a way of making the observer an active participant in the action observed, having a local habitation, a frame of reference, and a bounded sphere of influence within the universe, no longer an outsider looking in.  As I soon discovered in my wanderings through the libraries and bookstores of my local habitation, this very theme was long ago prefigured in the corpus of C.S. Peirce’s work, most strikingly in his Logic of Relatives and Pragmatic Maxim, taken as a basis for his relational theories of information, inquiry, and signs.

It is more this level of underground conceptual revolution that comes to mind when I think of Peirce’s impact on the development of physical theory, needless to say science in general, more than any particular doctrines about continua, especially since continua posed no novelty to classical mechanics, indeed, if anything, were more catholic within its realm, while quantum mechanics introduced an irreducible aspect of discreteness to physics.

Posted in C.S. Peirce, Complementarity, Inquiry, Laws of Form, Logic, Mathematics, Peirce, Philosophy, Physics, Pragmatism, Quantum Mechanics, Relativity, Science, Scientific Method, Semiotics, Spencer Brown | Tagged , , , , , , , , , , , , , , , | Leave a comment

The Difference That Makes A Difference That Peirce Makes • 4

Posted on May 14, 2017 by Jon Awbrey

Re: Peirce List • Mike Bergman

The mathematical perspectives and theories that made modern physics possible, perhaps even inevitable, were developed by many mathematicians, both abstract and applied, all throughout the 19th Century.  There was a definite sea change in the way scientists began to view the relationship between mathematical models and the physical world, passing from a monolithic concept to variational choices among multiple approaches, models, perspectives, and theories.

Charles Sanders Peirce was an astute observer and active participant in that transformation but it has always been difficult to trace his true impact on its course — so much of what he contributed operated underground, rhizome like, and without recognition.

But I think it’s fair to say that Peirce articulated the springs and catches of the workings of science better than any other reflective practitioner in his or subsequent times.  And I think the full import of his information‑theoretic and pragmatic‑semiotic approaches to scientific inquiry is a task for the future to work out.

Posted in C.S. Peirce, Complementarity, Inquiry, Laws of Form, Logic, Mathematics, Physics, Pragmatism, Quantum Mechanics, Relativity, Scientific Method, Semiotics, Spencer Brown, Visualization | Tagged , , , , , , , , , , , , , | Leave a comment

Icon Index Symbol • 17

Posted on May 9, 2017 by Jon Awbrey

Questions Concerning Certain Faculties Claimed For Signs

Re: Peirce List (1) (2) (3) (4) (5) • Helmut Raulien

Our object being to clarify the relationships among icons, indices, and symbols, I believe the maximum benefit possible at this point is to be gained from studying the simple examples of triadic relations and sign relations discussed in the following places.

Once we get used to dealing with small examples like that we can move on to tackling more complex examples on the order of those we might encounter in realistic applications.

The sort of sign relation we usually encounter in practice will be a subset L of a cartesian product O \times S \times I, where the object, sign, and interpretant domains all have infinitely many members in principle, though of course we tend to get by with finite samples at any given moment and it may even be possible to start small and build capacity over time.

All the objects we need to reference in a given application will go into the object domain O and all the signs and interpretants we need to denote these objects will go into the sign domain S and the interpretant domain I.

It may be useful to note at this point that there are such things as monadic projections:

\begin{array}{lll} \mathrm{proj}_O & : & O \times S \times I \to O \\[4pt] \mathrm{proj}_S & : & O \times S \times I \to S \\[4pt] \mathrm{proj}_I & : & O \times S \times I \to I \end{array}

For example, \mathrm{proj}_O (L) gives the set of all elements in O actually appearing as first correlates in L, sometimes called the O-range.

There are interesting classes of relations taking place internal to the various domains.  For example, syntactic relations or parsing relations operate within the sign domain, relating complex signs to their component signs.

But that’s a topic a little ways down the road …

cc: Peirce List (1) (2) (3) (4) (5)

Posted in Abduction, Algorithms, Animata, Artificial Intelligence, Automated Research Tools, C.S. Peirce, Cognition, Computation, Data Structures, Deduction, Icon Index Symbol, Induction, Inquiry, Inquiry Driven Systems, Inquiry Into Inquiry, Interpretive Frameworks, Knowledge Representation, Logic, Logic of Relatives, Logic of Science, Logical Graphs, Objective Frameworks, Peirce, Relation Theory, Semiotics, Sign Relations, Triadic Relations, Visualization | Tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , | 16 Comments

Icon Index Symbol • 16

Posted on May 6, 2017 by Jon Awbrey

Questions Concerning Certain Faculties Claimed For Signs

Re: Peirce List (1) (2) (3) (4) (5) • Jon Alan Schmidt • Helmut Raulien

Having lost my train of thought due to a week on the road, I would like to go back and pick up the thread at the following exchange.

JAS:
To be honest, given that the Sign relation is genuinely triadic, I have never fully understood why Peirce initially classified Signs on the basis of one correlate and two dyadic relations.
HR:
I have a guess about that:  I remember from a thread with Jon Awbrey about relation reduction something like the following:  A triadic relation is called irreducible, because it cannot compositionally be reduced to three dyadic relations.  Compositional reduction is the real kind of reduction.  But there is another kind of reduction, called projective (or projectional?) reduction, which is a kind of consolation prize for people who want to reduce.  It is possible for some triadic relations.

The course of discussion after that point left many of the original questions about icons, indices, and symbols unanswered, so I’d like to make another try at addressing them.  The relevant facts about triadic relations and relational reduction can be found at the following locations.

I introduced two examples of triadic relations from mathematics, two examples of sign relations from semiotics, and used them to illustrate the issue of projective reducibility, in another way of putting it, whether the structure of a triadic relation can be reconstructed from the structures of three dyadic relations derived or “projected” from it.

  • In the geometric picture of triadic relations, a dyadic projection is the shadow a 3-dimensional body casts on one of the three coordinate planes.
  • In terms of relational data tables, a dyadic projection is the result of deleting one of the three columns of the table and merging any duplicate rows.
  • In Peircean terms, a projection is a type of “prescision” operation, abstracting a portion of the structure from the original relation and ignoring the rest.

The question then is whether we keep or lose information in passing from a triadic relation to the collection of its dyadic projections.  If there is no loss of information then the triadic relation is said to be reducible to and reconstructible from its dyadic projections.  Otherwise it is said to be irreducible and irreconstructible in the same vein.

cc: Peirce List (1) (2) (3) (4) (5)

Posted in Abduction, Algorithms, Animata, Artificial Intelligence, Automated Research Tools, C.S. Peirce, Cognition, Computation, Data Structures, Deduction, Icon Index Symbol, Induction, Inquiry, Inquiry Driven Systems, Inquiry Into Inquiry, Interpretive Frameworks, Knowledge Representation, Logic, Logic of Relatives, Logic of Science, Logical Graphs, Objective Frameworks, Peirce, Relation Theory, Semiotics, Sign Relations, Triadic Relations, Visualization | Tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , | 16 Comments

Icon Index Symbol • 15

Posted on May 1, 2017 by Jon Awbrey

Questions Concerning Certain Faculties Claimed For Signs

I put down the cup and turn to my mind.  It is up to my mind to find the truth.  But how?  What grave uncertainty, whenever the mind feels overtaken by itself;  when it, the seeker, is also the obscure country where it must seek and where all its baggage will be nothing to it.  Seek?  Not only that:  create.  It is face to face with something that does not yet exist and that only it can accomplish, and bring into its light.

— Proust • In Search of Lost Time

Re: Peirce List (1) (2) • Tom Gollier • Jerry Rhee • John Sowa

That passage from Proust epitomizes for me one of the most distinctive features of the inquiry process, the fact that its object is a state of information that the inquisiturus, the agent of inquiry, may never have known before.

I have thought of inquiry and intelligence in terms of cybernetic or system-theoretic processes ever since my first encounters with the works of Arbib, Ashby, Bateson, McCulloch, Wiener, Young, and others during my undergrad years.  In the early 90s I returned to grad school in a Systems Engineering program with the idea that I might be able to string together many loose threads of unfinished business that continued to tug at my brain.  Here’s a bit I wrote at the outset of that project, that comes to mind in this context:

Prospects for Inquiry Driven Systems • Architecture of Inquiry

It is important to remember that knowledge is a different sort of goal from the run-of-the-mill setpoints a system might have.  The typical goal is a state that a system has actually experienced many times before, like normal body temperature for a human being.  But a particular state of knowledge an intelligent system moves toward may be a state it has never been through before.  The fundamental equivocation on this point expressed in Plato’s Meno, whether learning is functionally equivalent to remembering, was discussed above.  In spite of this quibble, it still seems necessary to regard states of knowledge as a distinctive class.  The reasons for this may lie in the fact that a useful definition of inquiry for human beings necessarily involves a whole community of inquiry.

Resources

cc: Peirce List (1) (2)

Posted in Abduction, Algorithms, Automated Research Tools, C.S. Peirce, Cognition, Computation, Data Structures, Deduction, Icon Index Symbol, Induction, Inquiry Driven Systems, Inquiry Into Inquiry, Interpretive Frameworks, Knowledge Representation, Logic, Logic of Science, Logical Graphs, Objective Frameworks, Relation Theory, Semiotics, Sign Relations, Triadic Relations, Visualization | Tagged , , , , , , , , , , , , , , , , , , , , , , | 16 Comments

Icon Index Symbol • 14

Posted on April 30, 2017 by Jon Awbrey

Questions Concerning Certain Faculties Claimed For Signs

Re: Peirce List (1) (2) (3) • Helmut Raulien
Re: Icon Index Symbol • (10) (11) (12) (13)

Let me sum up the main points of the above exchange before moving on.

Mathematics is useful in our present endeavor because it treats relations in general.  In addition — and multiplication, too — mathematics is chock full of well‑studied examples of triadic relations.  When it comes to the job of analyzing sign relations and teasing out their structures we can save ourselves a lot of trouble and trial and error by examining this record of prior art and adapting its methods to handle sign relations.

On the other hand, there are hints in Peirce’s work of how triadic relations extend across a threshold of complexity in such a way that relations of all higher adicities can be analyzed in terms of 1‑adic, 2‑adic, and 3‑adic relations.  This is where the analogy between Peirce’s category theory and mathematical category theory both forms and breaks.  The utility of categories in mathematical theory derives from the ubiquity of functions in mathematical practice, and functions are dyadic relations.  Still, triadic relations pervade the background of mathematical category theory, being visible in the triadic composition relation and in the concept of “natural transformations”, the clarification of which is one of the original motivations for the subject.  Bringing the triadic roots of category theory into higher relief is one of my motives for bringing about an encounter with Peirce’s categories, an effort to which I have given not a few years of thought.

That brings us to the case of sign relations proper.  These types of triadic relations form our first stepping stones and also our first stumbling blocks in the inquiry into inquiry and I gave some indications why that might be true.  I don’t know if I can do better than that at this time, but I’ll think on it further after that all‑essential secondness of caffeination.

Resources

cc: Peirce List (1) (2) (3)

Posted in Abduction, Algorithms, Automated Research Tools, C.S. Peirce, Cognition, Computation, Data Structures, Deduction, Icon Index Symbol, Induction, Inquiry Driven Systems, Inquiry Into Inquiry, Interpretive Frameworks, Knowledge Representation, Logic, Logic of Science, Logical Graphs, Objective Frameworks, Relation Theory, Semiotics, Sign Relations, Triadic Relations, Visualization | Tagged , , , , , , , , , , , , , , , , , , , , , , | 16 Comments