⚠ It’s A Trap ⚠

Posted on May 18, 2013 by

Re: Kenneth W. Regan

The most common mathematical trap I run across has to do with Triadic Relation Irreducibility, as noted and treated by the polymath C.S. Peirce.

This trap lies in the mistaken belief that every 3-place (triadic or ternary) relation can be analyzed purely in terms of 2-place (dyadic or binary) relations — “purely” here meaning without resorting to any 3-place relations in the process.

A notable thinker who not only fell but led many others into this trap is none other than René Descartes, whose problematic maxim I noted in the following post.

As mathematical traps go, this one is hydra-headed.

I don’t know if it’s possible to put a prior restraint on the varieties of relational reduction that might be considered, but usually we are talking about either one of two types of reducibility.

Compositional Reducibility. All triadic relations are irreducible under relational composition, since the composition of two dyadic relations is a dyadic relation, by the definition of relational composition.

Projective Reducibility. Consider the projections of a triadic relation L \subseteq X \times Y \times Z on the coordinate planes X \times Y, ~ X \times Z, ~ Y \times Z and ask whether these dyadic relations uniquely determine L. If so, we say L is projectively reducible, otherwise it is projectively irreducible.

Et Sic Deinceps …

  • More Information on Relation Reduction • MyWikiBiz • PlanetMath
  • Previous Posts on Triadic Relation Irreducibility • (1) • (2) • (3)
Posted in Error, Fallibility, Logic, Logic of Relatives, Mathematical Traps, Mathematics, Peirce, Philosophy, Pragmatism, Relation Theory, Semiotics, Sign Relations, Triadic Relations, Triadicity | Tagged , , , , , , , , , , , , , | 4 Comments

Triadic Relation Irreducibility : 3

Posted on May 16, 2013 by

References

Relation Theory
Sign Relations
Triadic Relations
Relation Composition
Relation Construction
Relation Reduction
Related Readings
Posted in Logic, Logic of Relatives, Mathematics, Peirce, Philosophy, Pragmatism, Relation Theory, Semiotics, Sign Relations, Triadic Relations, Triadicity | Tagged , , , , , , , , , , | 1 Comment

grabitational singularity

Posted on May 16, 2013 by
the trouble with a bubble
on a pyramid top
is the point
when it
pop
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⚠⚠⚠⚠
⚠⚠⚠⚠⚠
⚠⚠⚠⚠⚠⚠
Posted in Singularity, Verse | Tagged , | Leave a comment

What part do arguments from authority play in mathematical reasoning?

Posted on May 14, 2013 by

In forming your answer you may choose to address any or all of the following aspects of the question:

Descriptive
What part do arguments from authority actually play in mathematical reasoning?
Normative
What part do arguments from authority ideally play in mathematical reasoning?
Regulative
What if any discrepancies exist between the actual and the ideal?
What if anything should be done about the discrepancies that exist?

Recycled from a question I asked on MathOverFlow.

Posted in Inquiry, Logic, Information, Inquiry Driven Systems, Artificial Intelligence, Intelligent Systems, Cybernetics, Optimal Control, Control, Philosophy of Science, Scientific Method, Belief Fixation, Authority, Logic of Science, Optimization, Operations Research, History of Mathematics, History of Science, Control Theory, Information Theory, Mathematical Reasoning, Mathematical Intuition, Intuition | Tagged , , , , , , , , , , , , , , , , , , , , , , | 1 Comment

Information = Comprehension × Extension : 1

Posted on May 8, 2013 by

Re: A Most Perplexing Mystery

The inverse relationship between symmetry and diversity — that we see for example in the lattice-inverting map of a Galois correspondence — is a variation on an old theme in logic called the “inverse proportionality of comprehension and extension”.

C.S. Peirce, in his probings of the “laws of information”, found this principle to be a special case of a more general formula, saying that the reciprocal relation holds only when the quantity of information is constant.

Readings

Posted in Comprehension, Extension, Information, Inquiry, Intension, Logic, Logic of Science, Peirce | Tagged , , , , , , , | 10 Comments

Rock On

Posted on May 6, 2013 by
Elsewhere I have brought out the fact that human will had no other purpose than to maintain awareness. But that could not do without discipline. Of all the schools of patience and lucidity, creation is the most effective. It is also the staggering evidence of man’s sole dignity: the dogged revolt against his condition, perseverance in an effort considered sterile. It calls for a daily effort, self-mastery, a precise estimate of the limits of truth, measure, and strength. It constitutes an ascesis. All that “for nothing”, in order to repeat and mark time. But perhaps the great work of art has less importance in itself than in the ordeal it demands of a man and the opportunity it provides him of overcoming his phantoms and approaching a little closer to his naked reality. (115)
⁂  
All that remains is a fate whose outcome alone is fatal. Outside of that single fatality of death, everything, joy or happiness, is liberty. A world remains of which man is the sole master. What bound him was the illusion of another world. The outcome of his thought, ceasing to be renunciatory, flowers in images. It frolics — in myths, to be sure, but myths with no other depth than that of human suffering and, like it, inexhaustible. Not the divine fable that amuses and blinds, but the terrestrial face, gesture, and drama in which are summed up a difficult wisdom and an ephemeral passion. (117–118)
⁂  
I leave Sisyphus at the foot of the mountain! One always finds one’s burden again. But Sisyphus teaches the higher fidelity that negates the gods and raises rocks. He too concludes that all is well. This universe henceforth without a master seems to him neither sterile nor futile. Each atom of that stone, each mineral flake of that night-filled mountain, in itself forms a world. The struggle itself toward the heights is enough to fill a man’s heart. One must imagine Sisyphus happy. (123)

Albert Camus, The Myth of Sisyphus and Other Essays, Justin O’Brien (trans.), Random House, New York, NY, 1991. Originally published in France as Le Mythe de Sisyphe by Librairie Gallimard, 1942. First published in the United States by Alfred A. Knopf, 1955.

Posted in Absurdity, Albert Camus, Diversity, Existentialism, Freedom, Myth, Oedipus, Passion, Revolt, Sisyphus | Tagged , , , , , , , , , | 11 Comments

Finding a Needle in a Cactus Patch

Posted on April 30, 2013 by

Re: Sex, Lies, And Quantum Computers

Don’t know much about quantum computation, but my ventures in graphical syntaxes for propositional calculus did turn up a logical operator whose evaluation process reminded me a little of the themes involved in the collapse of the wave function.

Here is the essential information —

Boolean formulas constructed from minimal negation operators can be given graph-theoretic representation as “decorated” or “painted” versions of rooted cactus graphs.

Here is a place where you can see some pictures and a description of the Fundamental Evaluation Rule for cactus expressions of propositional formulas.

Posted in Boolean Functions, Cactus Graphs, Graph Theory, Logic, Logical Graphs, Minimal Negation Operators, Peirce, Propositional Calculus | Tagged , , , , , , , | 4 Comments