Archimedes’ Mathematical ‘‘‘Double Dialectic’’’.
The purpose of the present blog-entry is to share, with you, a key passage from a discourse by our co-founder, Karl Seldon, entitled ‘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 modern mathematics, albeit controversially so.”
“In this method, one adopts -- assumes 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.”
“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 ‘‘‘self-reflexivity’’’ and ‘self-refluxivity’, 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.”
“It also features a kind of foreshadowing of the method of immanent critique, 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.”
“However, Zeno’s Method features only deductive and absolute falsification -- i.e., absolute, propositional negation. It is propositional, not categorial. It is without cumulative, categorial progression. Consequently, it does not comprehend or allow a determinate, partial negation of the idea(s)-system under its immanent critique. It «aufheben»-conserves not a single shred of that idea(s)-system and its categories/concepts, while it also does not «aufheben»-elevate any part of that system, nor, as said already, does it only partially/determinately aufheben»- ‘‘‘negate’’’ that system, yielding an improved system.”
“Nevertheless, if one were to combine, and unify, these Ancient traditions of “dialectic” -- the Zenoan «reductio» and the Socratean «elenchus», with the Platonian ‘ideo-taxonomics’, the “dividing according to kinds” of Plato’s «arithmoi eidetikoi» “as a guide on the voyage of discourse” [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 «elenchus» 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 recent modern times, at least since 1995, one might arrive at a universal algorithmic-heuristic method for ‘‘‘systematic dialectics’’’.
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 «reductio», that affirms the incompleteness 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-critique of the fruits of each such step, building, by categorial progression qualitative superposition, an advancing, cumulating, ‘ideo-taxonomically’ and ‘ideo-ontologically’ progressing account/explanation of the present [sub-]totality or [sub-]universe[-of-discourse] being theorized thereby.
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 present [sub-]totality thus targeted by the method of discovery.
“Thus, Zeno’s discovery is but a formal-logical shadow of the more fully-blooming dialectic that we have seen to have emerged later in Terran human [psycho]history.”
“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 modern “calculus”.”
“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 ‘double Zenoan dialectic’.”
“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 Principia, which, in effect, founded, from out of Medieval “Natural Philosophy”, modern physics, and, indeed, modern science 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]. ...”
*[see, for example, Richard Robinson, Plato’s Earlier Dialectic [2nd edition], Oxford University Press, 1953, pp. 91-92.].
FYI: Much of the work of Karl Seldon, and of his collaborators, including work by “yours truly”, is available, for your free-of-charge download, via --
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