Cross-paradigm communication, like cross-disciplinary and cross-cultural communication, can be difficult. Sometimes people do not even recognize the existence of other paradigms, disciplines, cultures, long before it comes to the question of their value. Readers of Peirce know he often uses important words in more primordial senses than later came into fashion. Other times his usage embodies a distinct analysis of the concept in question. More than once I’ve found myself remarking how Peirce “anticipated” some strikingly “modern” idea in logic, mathematics, or science, only to find its roots lay deep in the history of thought. Whether he anticipates a future sense or preserves an ancient sense is not always easy to answer.

]]>Prompted by a recent discussion of prime numbers and complex dynamics on one of the Santa Fe Institute’s FaceBook pages, I posted a link to an old project of mine, going back to a time when I was first learning programming in college and working as an orderly in a hospital x-ray department. Something about the collision of those influences in the medium of my gray matter led me to see curious connections among self-documenting programs, self-indexing data structures, and molecular tagging. Shortly afterwards a couple of *Mathematical Games* columns by Martin Gardner started me thinking about Gödel numbers and links among graph theory, logic, and number theory.

At any rate, here’s a report on what came of that —

]]>cc: Systems Science • Structural Modeling • Ontolog Forum • Laws of Form • Cybernetics

]]>Re: Animated Logical Graphs • 21

- AM:
- Each step on its own, as far as I can follow them, makes sense. You are, if I understand it correctly, trying to figure out something fundamental, the rock bottom reality. When can we expect that results of such a research to become “applicable to more than one of the traditional departments of knowledge”? What kinds of tragedy, disaster, misunderstanding, mismanagement, or failure would/will be preventable by your approach?

The larger questions asked above — interdisciplinary inquiry, the interest in integration, the synthesis of ideas across isolated silos of specialization, and what it might mean for the future — are issues Susan Awbrey and I addressed from a pragmatic semiotic perspective:

- Awbrey, S.M., and Awbrey, J.L. (2001), “Conceptual Barriers to Creating Integrative Universities”,
*Organization : The Interdisciplinary Journal of Organization, Theory, and Society*8(2), Sage Publications, London, UK, 269–284. Abstract. Online. - Awbrey, S.M., and Awbrey, J.L. (1999), “Organizations of Learning or Learning Organizations : The Challenge of Creating Integrative Universities for the Next Century”,
*Second International Conference of the Journal ‘Organization’, Re-Organizing Knowledge, Trans-Forming Institutions : Knowing, Knowledge, and the University in the 21st Century*, University of Massachusetts, Amherst, MA. Online.

From that vantage point, what I’m about here is just a subgoal of a subgoal, panning what bits of elemental substrate can be found ever nearer that elusive “rock bottom reality”.

cc: Structural Modeling • Ontolog Forum • Laws of Form • Cybernetics

]]>In that vein, here’s a Rosetta Stone to give us a grounding in the relationship between boolean functions and our two readings of logical graphs.

cc: Systems Science • Structural Modeling • Ontolog Forum • Laws of Form • Cybernetics

]]>I took a look at Avi’s paper “On the Nature of the Theory of Computation” (OtNotToC). There is naturally a good dose of TOC but little on the type of World-Objective Contact (WOC) it takes to connect with empirical science. Just on that sample it reminds me of projects like Wolfram’s “New Kind Of Science”. They all do a good job of convincing us to use computational media as virtual laboratories for conducting experimental mathematics, but they leave us hanging when it comes to analyzing the relation between what goes on inside the box of computation and the natural world outside the box.

cc: Systems Science • Structural Modeling • Ontolog Forum • Laws of Form • Cybernetics

]]>Re: Animated Logical Graphs • 21

I invoked the general concepts of equivalence and distinction at this point in order to keep the wider backdrop of ideas in mind but since we’ve been focusing on boolean functions to coordinate the semantics of propositional calculi we can get a sense of the links between operations and relations by looking at their relationship in a boolean frame of reference.

Let and a positive integer. Then is the set of -tuples of elements of

- A
*-variable boolean function*is a mapping - A
*-place boolean relation*is a subset of

The correspondence between boolean functions and boolean relations may be articulated as follows:

- Any -place relation as a subset of has a corresponding
*indicator function*(or*characteristic function*) defined by the rule that if is in and if is not in - Any -variable function is the indicator function of a -place relation consisting of all the in where The set is called the
*fiber of*or the*pre-image of*in and is commonly notated as

cc: Systems Science • Structural Modeling • Ontolog Forum • Laws of Form • Cybernetics

]]>Projects giving a central place to computation in scientific inquiry go back to Hobbes and Leibniz, at least, and then came Babbage and Peirce. One of the first issues determining their subsequent development is the degree to which one identifies computation and deduction. The next question concerns how many types of reasoning one counts as contributing to the logic of empirical science:

- Is deduction alone sufficient?
- Are deduction and induction irreducible to each other and sufficient in tandem?
- Are there three irreducible types of inference: abduction, deduction, induction?

cc: Systems Science • Structural Modeling • Ontolog Forum • Laws of Form • Cybernetics

]]>I will have to focus on other business for a couple of weeks — so just by way of reminding myself what we were talking about at this juncture where logical graphs and differential logic intersect, here’s my comment on R.J. Lipton and K.W. Regan’s blog post about Discrepancy Games and Sensitivity.

Just by way of a general observation, concepts like discrepancy, influence, sensitivity, etc. are differential in character, so I tend to think the proper grounds for approaching them more systematically will come from developing the logical analogue of differential geometry.

I took a few steps in this direction some years ago in connection with an effort to understand a certain class of intelligent systems as dynamical systems. There’s a motley assortment of links here:

cc: Systems Science • Structural Modeling • Ontolog Forum • Laws of Form • Cybernetics

]]>cc: Systems Science • Structural Modeling • Ontolog Forum • Laws of Form • Cybernetics

]]>