Charles Sanders Peirce, George Spencer Brown, and Me

Re: Laws Of Form DiscussionJames Bowery

James Bowery left a comment on my blog and opened a thread in the Yahoo! group devoted to discussing the mathematics of George Spencer Brown’s Laws of Form.  I’ve been meaning to join that discussion as soon as I could work up the time and concentration to think about it … at long last I think I can do that now.  I’ll use the above heading on this blog to post any bits from my side of the conversation that I think might serve a wider audience.

It’s been a long time since I joined a new discussion group so I thought I’d start by posting a bit of the old-fashioned self-intro.

Posted in Abstraction, Amphecks, Analogy, Animata, Automated Research Tools, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Cybernetics, Deduction, Diagrammatic Reasoning, Differential Logic, Duality, Form, Graph Theory, Iconicity, Information Theory, Inquiry, Laws of Form, Logic, Logical Graphs, Mathematics, Model Theory, Painted Cacti, Peirce, Pragmatic Maxim, Proof Theory, Propositional Calculus, Semiotics, Sign Relational Manifolds, Spencer Brown, Symbolism, Systems Theory, Theorem Proving, Topology, Visualization, Zeroth Order Logic | Tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | Leave a comment

¿Shifting Paradigms? • 5

Re: Peter CameronInfinity and Foundation

We always encounter a multitude of problems whenever we try to rationalize mathematics by reducing it to logic, where logic itself is reduced to a purely deductive style.  A number of thinkers have proposed it is time — well past time — to stop counting so heavily on that idea and to join a Declaration of Independence for Mathematics.

Posted in Algorithms, Boole, C.S. Peirce, Combinatorics, Computation, Foundations of Mathematics, Inquiry, Laws of Form, Leibniz, Logic, Mathematics, Model Theory, Paradigms, Peirce, Proof Theory, Spencer Brown | Tagged , , , , , , , , , , , , , , , | 2 Comments

{ Information = Comprehension × Extension } • Discussion 6

Re: Peirce List Discussion • JAJFSJA

What interests me so much about Peirce’s first legislation of the “laws of information” in his 1865–1866 “Logic of Science” is that the primal twins of Inquiry and Semiotics nestle so closely in their first nest that we can see their kinship far better and more easily than ever we will again.  (I am cautiously optimistic their further development won’t go the same way it did for Rome.)

More than that, whatever disclaimers Peirce may issue about his own originality, I don’t think anyone can fairly encounter his definition of a term’s information as “the measure of its superfluous comprehension” without being downright shocked at its novelty.

Posted in Abduction, C.S. Peirce, Comprehension, Deduction, Extension, Hypothesis, Icon Index Symbol, Induction, Inference, Information, Information = Comprehension × Extension, Information Theory, Inquiry, Intension, Logic, Logic of Science, Peirce, Peirce's Categories, Pragmatism, Science, Scientific Method, Semiotic Information, Semiotics, Sign Relations | Tagged , , , , , , , , , , , , , , , , , , , , , , , | Leave a comment

{ Information = Comprehension × Extension } • Discussion 5

Re: Peirce List Discussion • John Sowa

What you say goes to the heart of a problem I saw in Natural Propositions, whether it was Peirce’s account or Stjernfelt’s analysis I did not have time to decide as the schedule of the slow reading went too fast for me to take it up on the List.  I marked the critical passages and my copy of the book is around here someplace but I am trying to stay focused on the subject matter and the set of problems I introduced under the above subject line.

There are many issues about cross-disciplinary communication, the varieties of quasi-religious belief about the uses of words in the whole proposition/sentence/statement complex, the various uses Peirce and others use across contexts, disciplines, historical time, and even within the same discussion.  But I think it’s best to hold the forte on that for now.

Posted in Abduction, C.S. Peirce, Comprehension, Deduction, Extension, Hypothesis, Icon Index Symbol, Induction, Inference, Information, Information = Comprehension × Extension, Information Theory, Inquiry, Intension, Logic, Logic of Science, Peirce, Peirce's Categories, Pragmatism, Science, Scientific Method, Semiotic Information, Semiotics, Sign Relations | Tagged , , , , , , , , , , , , , , , , , , , , , , , | Leave a comment

{ Information = Comprehension × Extension } • Discussion 4

Re: Peirce List Discussion • Jeffrey Brian Downard

JBD quoting CSP:
I restricted myself to terms, because at the time this chapter was first written (1867), I had not remarked that the whole doctrine of breadth and depth was equally applicable to propositions and to arguments.  The breadth of a proposition is the aggregate of possible states of things in which it is true;  the breadth of an argument is the aggregate of possible cases to which it applies.  The depth of a proposition is the total of fact which it asserts of the state of things to which it is applied;  the depth of an argument is the importance of the conclusions which it draws.  In fact, every proposition and every argument can be regarded as a term. —1893.  (C.S. Peirce, Collected Papers, CP 2.407 n. 1)

A very apt quote.  It reinforces an impression I had formed and tried to express on several occasions under the heading of contemporary category theory and computer science jargon about “polymorphism”.

There is never anything simple about the development of Peirce’s views over time so I think the whole issue of information “deserves further research”, as they say.

Posted in Abduction, C.S. Peirce, Comprehension, Deduction, Extension, Hypothesis, Icon Index Symbol, Induction, Inference, Information, Information = Comprehension × Extension, Information Theory, Inquiry, Intension, Logic, Logic of Science, Peirce, Peirce's Categories, Pragmatism, Science, Scientific Method, Semiotic Information, Semiotics, Sign Relations | Tagged , , , , , , , , , , , , , , , , , , , , , , , | Leave a comment

{ Information = Comprehension × Extension } • Discussion 3

Re: Peirce List Discussion • John Sowa

I gave Frederik Stjernfelt’s Natural Propositions a careful reading back when the Peirce List took it up.  Here is an archived topic thread that contains the author’s lead-off plus a sample of ensuing discussion:

I find a few remnants of my own comments and reflections here:

I have in mind getting back to the issues raised by that reading someday but it would take me too far afield from my current focus to do that now.

The short shrift for now is that Peirce is not talking about propositions in the sense of “double signs, informational signs, quasi-propositions, or Dicisigns” at this juncture but rather the simpler sorts of propositions that fall under the heading of the Propositional Calculus as currently understood, adequately and most felicitously dealt with of course by means of Peirce’s own Alpha Graphs.

The concept of information that comes up in this context is rather distinct.  To my way of thinking the earlier notion of information, however roughly cut, is superior in its underlying principles, being more realistic compared to the residual nominalism of the later concept, at least, as interpreted by others.

Posted in Abduction, C.S. Peirce, Comprehension, Deduction, Extension, Hypothesis, Icon Index Symbol, Induction, Inference, Information, Information = Comprehension × Extension, Information Theory, Inquiry, Intension, Logic, Logic of Science, Peirce, Peirce's Categories, Pragmatism, Science, Scientific Method, Semiotic Information, Semiotics, Sign Relations | Tagged , , , , , , , , , , , , , , , , , , , , , , , | Leave a comment

{ Information = Comprehension × Extension } • Discussion 2

Re: Peirce List Discussion • John Sowa

JFS:
A more fundamental term is proposition, which is informally defined as the “meaning” of a sentence.  That meaning is usually analyzed as comprehension (also known as intension) and extension.

The easier-on-the-eyes blog copy of my first Discussion post, from which point it is easier to follow the links to the first six Selections from Peirce, is here:

The word proposition occurs only twice in the first six Selections, once in Selection 2 and once in Selection 4, so maybe it’s worth our pausing to see how Peirce uses the word in this place and time:

The third and last kind of representations are symbols or general representations.  They connote attributes and so connote them as to determine what they denote.  To this class belong all words and all conceptions.  Most combinations of words are also symbols.  A proposition, an argument, even a whole book may be, and should be, a single symbol.  (Peirce 1866, p. 468)

Accordingly, if we are engaged in symbolizing and we come to such a proposition as “Neat, swine, sheep, and deer are herbivorous”, we know firstly that the disjunctive term may be replaced by a true symbol.  (Peirce 1866, p. 469)

For now I’ll just add those two observations to the hopper, and we can take up the issue of propositions in more detail as it arises in the relevant context.

It is good that John Sowa read us the “Freedom Of Interpretation Act” right at the start, as it will serve us in good stead down the road, but again I’ll have to leave its consequences until a few folks have had a chance to delve further into Peirce’s text, at which point I think its significance will become clear.

Posted in Abduction, C.S. Peirce, Comprehension, Deduction, Extension, Hypothesis, Icon Index Symbol, Induction, Inference, Information, Information = Comprehension × Extension, Information Theory, Inquiry, Intension, Logic, Logic of Science, Peirce, Peirce's Categories, Pragmatism, Science, Scientific Method, Semiotic Information, Semiotics, Sign Relations | Tagged , , , , , , , , , , , , , , , , , , , , , , , | Leave a comment