Information Resistance • Ω

The hardest thing to understand about information is people’s resistance to it.

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Signs Of Signs • 4

Re: Michael HarrisLanguage About Language

But then inevitably I find myself wondering whether a proof assistant, or even a formal system, can make the distinction between “technical” and “fundamental” questions. There seems to be no logical distinction. The formalist answer might involve algorithmic complexity, but I don’t think that sheds any useful light on the question. The materialist answer (often? usually?) amounts to just-so stories involving Darwin, and lions on the savannah, and maybe an elephant, or at least a mammoth. I don’t find these very satisfying either and would prefer to find something in between, and I would feel vindicated if it could be proved (in I don’t know what formal system) that the capacity to make such a distinction entails appreciation of music.

Peirce proposed a distinction between corollarial and theorematic reasoning in mathematics that strikes me as similar to the distinction that Michael Harris seeks between technical and fundamental questions.

I can’t say I have a lot of insight into how the line might be drawn, but I recall a number of traditions pointing to the etymology of theorem as having to do with the observation of objects and practices whose depth of detail always escapes full accounting by any number of partial views.

On the subject of music, all I have is this incidental —

Riffs & Rotes

Perhaps it takes a number theorist to appreciate it …

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Signs Of Signs • 3

Re: Michael HarrisLanguage About Language

And if we don’t, who puts us away?

One’s answer, or at least one’s initial response to that question will turn on how one feels about formal realities.  As I understand it, reality is that which persists in thumping us on the head until we get what it’s trying to tell us.  Are there formal realities, forms that drive us in that way?

Discussions like these tend to begin by supposing we can form a distinction between external and internal.  That is a formal hypothesis, not yet born out as a formal reality.  Are there formal realities that drive us to recognize them, to pick them out of a crowd of formal possibilities?

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Signs Of Signs • 2

Re: Michael HarrisLanguage About Language

I compared mathematics to a “consensual hallucination,” like virtual reality, and I continue to believe that the aim is to get (consensually) to the point where that hallucination is a second nature.

I think that’s called coherentism, normally contrasted with or complementary to objectivism.  It’s the philosophy of a gang of co-conspirators who think, “We’ll get off scot-free so long as we all keep our stories straight.”

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Signs Of Signs • 1

Re: Michael HarrisLanguage About Language

There is a language and a corresponding literature that approaches logic and mathematics as related species of communication and information gathering, namely, the pragmatic-semiotic tradition passed on to us through the lifelong efforts of C.S. Peirce.  It is by no means a dead language, but it continues to fly beneath the radar of many trackers in logic and math today.  Still, the resource remains for those who are ready intuit to dip.

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Animated Logical Graphs : 9

Re: Ken ReganThe Shapes of Computations

The insight that it takes to find a succinct axiom set for a theoretical domain falls under the heading of abductive or retroductive reasoning, a knack as yet refractory to computational attack, but once we’ve lucked on a select-enough set of axioms we can develop theorems that afford a more navigable course through the subject.

For example, back on the range of propositional calculus, it takes but a few pivotal theorems and the lever of mathematical induction to derive the Case Analysis-Synthesis Theorem (CAST), which provides a bridge between proof-theoretic methods that demand a modicum of insight and model-theoretic methods that can be run routinely.

Posted in Amphecks, Animata, Automated Research Tools, Boolean Algebra, Boolean Functions, Cactus Graphs, Computational Complexity, Constraint Satisfaction Problems, Diagrammatic Reasoning, Graph Theory, Inquiry Driven Education, Laws of Form, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Model Theory, Painted Cacti, Peirce, Proof Theory, Propositional Calculus, Propositional Equation Reasoning Systems, Spencer Brown, Theorem Proving, Visualization | Tagged , , , , , , , , , , , , , , , , , , , , , , , , | 1 Comment

Animated Logical Graphs : 8

Re: Ken ReganThe Shapes of Computations

The most striking example of a “Primitive Insight Proof” (PIP❢) known to me is the Dawes–Utting proof of the Double Negation Theorem from the CSP–GSB axioms for propositional logic. There is a graphically illustrated discussion at the following location:

I cannot hazard a guess what order of insight it took to find that proof — for me it would have involved a whole lot of random search through the space of possible proofs, and that’s even if I got the notion to look for one in the first place.

There is of course a much deeper order of insight into the mathematical form of logical reasoning that it took C.S. Peirce to arrive at his maximally elegant 4-axiom set.

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