Inquiry Into Inquiry

⚠ It’s A Trap ⚠

Jon Awbrey — Sun, 19 May 2013 03:00:15 +0000

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.

Château Descartes

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 on the coordinate planes and ask whether these dyadic relations uniquely determine If so, we say 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)

Triadic Relation Irreducibility : 3

Jon Awbrey — Fri, 17 May 2013 03:00:10 +0000

References

Relation Theory

Sign Relations

Triadic Relations

Relation Composition

Relation Construction

Relation Reduction

Related Readings

grabitational singularity

Jon Awbrey — Thu, 16 May 2013 15:30:20 +0000

the trouble with a bubble
on a pyramid top
is the point
when it
pop
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What part do arguments from authority play in mathematical reasoning?

Jon Awbrey — Tue, 14 May 2013 19:45:07 +0000

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.

Information = Comprehension × Extension : 1

Jon Awbrey — Wed, 08 May 2013 16:48:03 +0000

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

C.S. Peirce • Upon Logical Comprehension and Extension
My Notes • Information = Comprehension × Extension

Rock On

Jon Awbrey — Mon, 06 May 2013 17:20:05 +0000

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.

Finding a Needle in a Cactus Patch

Jon Awbrey — Tue, 30 Apr 2013 14:48:04 +0000

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 —

Minimal Negation Operators

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.

Cactus Calculus

Revolt, Freedom, Passion

Jon Awbrey — Tue, 23 Apr 2013 21:36:53 +0000

Thus I draw from the absurd three consequences, which are my revolt, my freedom, and my passion. By the mere activity of consciousness I transform into a rule of life what was an invitation to death — and I refuse suicide. I know, to be sure, the dull resonance that vibrates throughout these days. Yet I have but a word to say: that it is necessary. When Nietzsche writes: “It clearly seems that the chief thing in heaven and on earth is to obey at length and in a single direction: in the long run there results something for which it is worth the trouble of living on this earth as, for example, virtue, art, music, the dance, reason, the mind — something that transfigures, something delicate, mad, or divine,” he elucidates the rule of a really distinguished code of ethics. But he also points the way of the absurd man. Obeying the flame is both the easiest and the hardest thing to do. However, it is good for man to judge himself occasionally. He is alone in being able to do so.

(64–65)

Absurdum Quid

Jon Awbrey — Mon, 22 Apr 2013 21:12:06 +0000

I am thus justified in saying that the feeling of absurdity does not spring from the mere scrutiny of a fact or an impression, but that it bursts from the comparison between a bare fact and a certain reality, between an action and the world that transcends it. The absurd is essentially a divorce. It lies in neither of the elements compared; it is born of their confrontation.

In this particular case and on the plane of intelligence, I can therefore say that the Absurd is not in man (if such a metaphor could have a meaning) nor in the world, but in their presence together. For the moment it is the only bond uniting them. If I wish to limit myself to facts, I know what man wants, I know what the world offers him, and now I can say that I also know what links them. I have no need to dig deeper. A single certainty is enough for the seeker. He simply has to derive all the consequences from it.

The immediate consequence is also a rule of method. The odd trinity brought to light in this way is certainly not a startling discovery. But it resembles the data of experience in that it is both infinitely simple and infinitely complicated. Its first distinguishing feature in this regard is that it cannot be divided. To destroy one of its terms is to destroy the whole. There can be no absurd outside the human mind. Thus, like everything else, the absurd ends with death. But there can be no absurd outside this world either. And it is by this elementary criterion that I judge the notion of the absurd to be essential and consider that it can stand as the first of my truths.

(30–31)

Infinite Uses → Finite Means

Jon Awbrey — Thu, 18 Apr 2013 03:00:21 +0000

The idea that a language is based on a system of rules determining the interpretation of its infinitely many sentences is by no means novel. Well over a century ago, it was expressed with reasonable clarity by Wilhelm von Humboldt in his famous but rarely studied introduction to general linguistics (Humboldt, 1836). His view that a language “makes infinite use of finite means” and that its grammar must describe the processes that make this possible is, furthermore, an outgrowth of a persistent concern, within rationalistic philosophy of language and mind, with this “creative” aspect of language use.	(v)
⁂
Although it was well understood that linguistic processes are in some sense “creative”, the technical devices for expressing a system of recursive processes were simply not available until much more recently. In fact, a real understanding of how a language can (in Humboldt’s words) “make infinite use of finite means” has developed only within the last thirty years, in the course of studies in the foundations of mathematics.	(8)

Noam Chomsky (1965), Aspects of the Theory of Syntax, MIT Press, Cambridge, MA.

☞ Pigeonhole Principle