*Archimedes**’*.

**Mathematical**‘‘‘**Double**’’’__Dialectic__
Dear

**,***Readers*The purpose of the

**blog-entry is to share, with**

__present__**, a**

*you***passage from a discourse by**

*key***o**

__ur__

*co**-*, Karl Seldon, entitled

**founder***‘*.

**Dialectic in the Ancient Occident**’
Without any further ado, here is that passage --

**“**. . . Aristotle assigned the credit for inventing or discovering “dialectic”, or the “method” of “dialectic”, to Zeno of Elea, in Aristotle’s

**.”**

*Sophist******

“We think that Aristotle is referring to Zeno’s explicit
formulation and practice of the method of «

*reductio ad absurdum*» ‘dis-proof’, or of “indirect” refutation, later of widespread use, especially in**mathematics, albeit controversially so.”***modern*
“In this method, one adopts --

**to be true, momentaneously, and ‘‘‘for the sake of argument’’’ -- the very proposition that one wishes to refute, e.g., a proposition held to be true by a philosophical adversary.”***assumes*
“One then deduces, rigorously, from that ostensively
affirmed proposition, another proposition, one that propositionally contradicts
that assumed proposition itself, and/or that contradicts still another
proposition that one’s opponent holds to be true.”

“Since, only a false proposition can formally deduce to
a[nother] false proposition, and/or to a proposition contradictory to what is
given, this contradictory result forces one’s opponent, logically, rationally,
to abandon the proposition formerly held true by same.”

“I.e., this deductive result strikes back at its
starting-point, at the opponent’s proposition, from the affirmation of which
one had begun in this “indirect” argument, thus propositionally-negating that
affirmation with the full force of formal logic.”

“Socrates’ later ‘‘‘dialogic’’’, “dialectical” practice of «

*elenchus*» is a kind extension of ‘‘‘Zeno’s Dialectic’’’.”
“Practicing his method of «

*elenchus*», e.g., as portrayed in Plato’s dialogues, Socrates would induce his dialogue partner to state that interlocutor’s best definition of a key concept or category, e.g., that of “Justice”. Socrates then, in a questions-and-answers exchange, would cross-examine that interlocutor about that stated definition, until, ideally, Socrates had convicted his dialogue partner of the view that the stated definition leads inescapably to absurd, impossible, and/or undesirable consequences, hopefully inducing the partner, thereby, to retract that definition statement, or at least to drastically amend it.”
“Perhaps, rather than as ‘‘‘Zeno’s Dialectic’’’, it would be
better to describe Zeno’s Method of argument as ‘a formal-logical foreshadowing
of dialectic’ as we have come to know it, in the Occident, especially with and
since Hegel’s work.”

“Zeno’s Method does share some key features with that more
developed dialectic.”

“It exhibits a kind of

*‘‘‘*and**self**-**refl**y’’’__e__xivit*‘*, in which the faulty conclusion of its deductive argument reflects back upon, and propositionally negates, its own source in the propositional assumption(s) from which that argument issued.”**self**-**refl**’__u__xivity
“It also features a kind of foreshadowing of the method of

**, central, e.g., to Marx’s critique of political economy. It does so by adopting, as premise, the proposition to be criticized/refuted, and letting that proposition’s own implications refute it.”***immanent critique*
“However, Zeno’s Method features only deductive and absolute
falsification -- i.e., absolute, propositional negation. It is propositional,

**categorial. It is without cumulative, categorial progression. Consequently, it does not comprehend or allow a***not***,***determinate**partial***of the idea(s)-system under its immanent critique. It «***negation***»***aufheben**-***conserves****a single shred of that idea(s)-system and its categories/concepts, while it also does not «***not***»***aufheben**-*any part of that system, nor, as said already, does it only**elevate***partially**/determinately**aufheben**»- ‘‘‘*that system, yielding an improved system.”**negate**’’’
“Nevertheless, if one were to combine, and unify, these Ancient
traditions of “dialectic” -- the Zenoan «

**» and the Socratean «***reductio***», with the Platonian***elenchus**‘*, the “dividing according to kinds” of Plato’s «**-**__ideo__**taxonomics**’*arithmo*__i__**» “as a guide on the voyage of discourse” [**__eide__tiko__i__*Sophist*, 253b-254d], and with the Platonian grasp of human thinking, including of dialectical thinking, as ‘‘‘a dialogue that one carries on with[in] oneself”’ [*Theaetetus*], plus with the ‘‘‘questions-and-answers’’’ rules of ‘dialogue-ic’ disputation already seeded in the Socratean «**» practice, but much further elaborated and codified in post-Aristotelian Ancient “dialectic”, and, further still, in Medieval “dialectic”, plus with the ‘algorithmic-heuristic dialectic’ that has emerged in***elenchus**recent***times, at least since 1995, one might arrive at a universal algorithmic-heuristic method for ‘‘‘systematic dialectics’’’.***modern*
One might arrive at a

*‘self-*dialogic’,*‘self-*elenchustic’, monologic*method of discovery*, one that leverages the ‘intra-duality’ of the thoughts of the ‘monologist’ thinker, driven by a recurring, self-iterating «**», that affirms the***reductio*__in__completeness of every typical step of discovery, thereby motivating each next step, advancing by*self-*questioning and*self-*answering about the fruits of each step, via algorithmically-generated, ‘quasi-Goedelian’ potential counter-examples to any completeness claims of each such step, and, thus, by recurring*self-*reflexive,*immanent*__self__*-*of the fruits of each such step, building, by categorial progression qualitative superposition, an advancing, cumulating,**critique***‘*taxonomically’ and**-**__ideo__*‘*ontologically’ progressing account/explanation of the**-**__ideo__**[sub-]totality or [sub-]universe[-of-discourse] being theorized thereby.**__present__
Reordered systematically, and pruned of any lacunae,
digressions,

*cul de sac*dead-ends, unsubstantiated speculations, and false starts, this inner and ‘outerly’ notated/recorded monologue might be editable into a dialectical-systematic*method of presentation*of a theory comprehending and explaining -- and basing predictions regarding the future of -- the**[sub-]totality thus targeted by the**__present__*method of discovery*.
“Thus, Zeno’s discovery is but a formal-logical shadow of
the more fully-blooming dialectic that

**have seen to have emerged later in Terran human [***we***]history.”***psycho*
“However, in Ancient advanced mathematics, Archimedes of
Syracuse [and of Alexandria], perhaps the greatest recorded mathematical genius
of the Ancient Occident, made masterful use -- and ‘‘‘re-doubled’’’, mathematical
use -- of ‘‘‘Zeno’s Dialectic’’’.” He
did so, e.g., in his proofs of his famous formulas for the volumes of various
geometrically-idealized solid, 3-D figures, wherein he also, in several ways, including
via his use of the “method of exhaustion”, anticipated especially the integral
aspect of

**“calculus”.”***modern*
“He did so via a method known as

*“double*«*reductio ad absurdum*»”.
“Per this method, he would first assume, e.g., that the
volume,

**V**, of a certain idealized closed solid shape, was strictly greater than the value generated, for the volume of that figure, by a formula,**F**.”
“He would assume that

**V > F**. He would then deduce, from that assumption, to an “absurdity”.
Next, he would assume the opposite proposition, namely that

**V < F**, and, again, deduce to an “absurdity” also from that, contrary, assumption.”
“Two, opposing «

*reductio ad absurdum*» arguments, hence a*“*«**double***reductio ad absurdum*»”, and a*‘*Zenoan dialectic’.”**double**
“Then, to the extent that any value can only be greater
than, or less than, or equal to any other value, the only remaining option had
thus been shown to be

**V = F**.”
“Thus, by the foregoing, twin refutations, the formula

**F**was the only possibility left standing to model the volume of that figure. That formula was thereby proven:**V = F**must be the true proposition.”
“Note also that, in many ways -- mainly in ways outside the
purview of the present discussion -- Newton’s work, in his

**, which, in effect, founded, from out of Medieval “Natural Philosophy”,**__Principia__**physics, and, indeed,***modern**modern***as a whole [with help from many others of merit], was but a direct extension of Archimedes’ work and method, including Newton’s method of “fluxions” and “fluents” [i.e., of differential and integral calculus, independently discovered also by Leibniz]. ...***science***”*******[see, for example, Richard Robinson,

**[2nd edition], Oxford University Press, 1953, pp. 91-92.].**

__Plato’s Earlier Dialectic__
FYI: Much of the work
of Karl Seldon, and of his collaborators, including work by “yours truly”, is
available, for

**free-of-charge download, via --***your*

*and*
Regards,

Miguel Detonacciones,

Member,

**Foundation****[**__Encyclopedia Dialectica__**F**.**.**__E__**.],**__D__
Officer,

**F**.**.**__E__**. Office of Public Liaison**__D__