Forest Primeval → Riffs & Rotes

Re: Shifting Paradigms? • (1)(2)(3)(4)(5)(6)

Prompted by the discussion of Catalan numbers on the Foundations Of Math List, I dug up a few pieces of early correspondence and later discussions that bear on the graph theory-number theory correspondences that have occupied me these many years and I began putting reformatted transcripts at the following sites:

These are wiki sites, so please feel free to use the associated discussion pages if you have any interest.

Regards,

Jon

Posted in Algebra, Arithmetic, Combinatorics, Forest Primeval, Graph Theory, Group Theory, Integer Sequences, Lambda Calculus, Logic, Mathematics, Model Theory, Number Theory, Paradigms, Peirce, Programming, Proof Theory, Riffs and Rotes | Tagged , , , , , , , , , , , , , , , , | Leave a comment

¿Shifting Paradigms? • 2

Re: Timothy Chow • Shifting Paradigms?

2014 Jul 31

I can’t remember when I first started playing with Gödel codings of graph-theoretic structures, which arose in logical and computational settings, but I remember being egged on in that direction by Martin Gardner’s 1976 column on Catalan numbers, planted plane trees, polygon dissections, etc.  Codings being injections from a combinatorial species S to integers, either non-negatives \mathbb{N} or positives \mathbb{M}, I was especially interested in codings that were also surjective, thereby revealing something about the target domain of arithmetic.

The most interesting bijection I found was one between positive integers \mathbb{M} and finite partial functions from \mathbb{M} to \mathbb{M}.  All of this comes straight out of the primes factorizations.  That type of bijection may remind some people of Dana Scott’s D_\infty.  Corresponding to the positive integers there arose two species of graphical structures, which I dubbed “riffs” and “rotes”.  See these links for more info:

The On-Line Encyclopedia of Integer Sequences (OEIS)

Jon

An interesting tangent to the main subject, but one that I had some ready thoughts on.

Posted in Algebra, Arithmetic, Combinatorics, Graph Theory, Group Theory, Logic, Mathematics, Model Theory, Number Theory, Paradigms, Peirce, Programming, Proof Theory, Riffs and Rotes | Tagged , , , , , , , , , , , , , | Leave a comment

¿Shifting Paradigms? • 1

Re: Dana Scott • Shifting Paradigms?

2014 Jul 28

This is very interesting to me, but not all my posts make it to the list, so I will spend a few days reflecting on it and post a comment on my blog, linked below. Thanks for the stimulating question.

Jon Awbrey
Inquiry Into Inquiry

I’ve been trying to get back to this for over a week now, but there are times when all you can do is document the process, the flow of thought, no matter how slow it goes. So read my ellipsis … and watch this space  

Posted in Logic, Mathematics, Model Theory, Paradigms, Peirce, Programming, Proof Theory | Tagged , , , , , , | Leave a comment

☯ Quantum Mechanics ☯


Particle ☯ Wave

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Why is there so much falsity in the world?

Because people prefer falsity to truth, illusion to reality.

Being the drift of my reflections on the plays I saw at Stratford this summer —
King Lear, King John, Man of La Mancha, Alice Through the Looking-Glass,
Crazy for You, Hay Fever.

Posted in Drama, Falsity, Illusion, Memoir, Question, Reality, Reflection, Shakespeare, The Big Picture, Theatre, Truth | Tagged , , , , , , , , , , | 9 Comments

Doubt, Uncertainty, Dispersion, Entropy : 2

Re: John Baez • Entropy and Information in Biological Systems

Re: To develop the concept of evolutionary games as “learning” processes in which information is gained over time.

My customary recommendation on this point is to look more deeply into the work of C.S. Peirce on the themes of evolution, inquiry, and their interaction.  Peirce stands out as one of the few pioneers in the study of scientific method who managed to avoid the dead-ends of naive deductivisim and naive inductivisim.  He developed Aristotle’s concept of abductive reasoning in a way that anticipated later insights into the dynamics of paradigm shifts.  A question worth exploring in this connection is whether abductive hypothesis formation is the analogue within scientific method of random mutation.

Posted in Biological Systems, Cybernetics, Dispersion, Doubt, Entropy, Evolution, Information, Information Theory, Inquiry, Inquiry Driven Systems, Inquiry Into Inquiry, Learning Theory, Peirce, Peirce List, Uncertainty | Tagged , , , , , , , , , , , , , , | Leave a comment

Doubt, Uncertainty, Dispersion, Entropy : 1

Re: Stephen Rose • The Second Law of Thermodynamics

Just a note to anchor a series of recurring thoughts that come to mind in relation to a Peirce List discussion of entropy etc., but I won’t have much to say on the bio-chemico-physico-thermo-dynamic side of things, so I’ll spin this off under a separate heading.  My interest in this topic arises mainly from my long-time work on inquiry driven systems (1)(2)(3)(4)(5)(6), where understanding the intertwined measures of uncertainty and information is critical to comprehending the dynamics of inquiry.

In a famous passage, Peirce says that inquiry begins with the “irritation of doubt” and ends when the irritation is soothed.  Here we find the same compound of affective and cognitive ingredients that we find in Aristotle’s original recipe for the sign relation.

When we view inquiry as a process taking place in a system the first thing we have to ask is what are the properties or variables that we need to consider in describing the state of the system at any given time.  Taking a Peircean perspective on a system capable of undergoing anything like an inquiry process, we are led to ask what are the conditions for the possibility of a system having “states of uncertainty” and “states of information” as state variables.

Posted in Dispersion, Doubt, Entropy, Information, Information Theory, Inquiry, Inquiry Driven Systems, Inquiry Into Inquiry, Peirce, Peirce List, Uncertainty | Tagged , , , , , , , , , , | Leave a comment