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In the late Platonic dialogue Parmenides, which places the youthful Socrates against one of the elders of Greek Philosophy - Parmenides; Parmenides says that the way to Philosophy is by the dialectic:

Socrates: But, then, what is to become of philosophy? Whither shall we turn, if the ideas are unknown?

to which Parmenides replies:

Yes, said Parmenides; and I think that this arises, Socrates, out of your attempting to define the beautiful, the just, the good, and the ideas generally, without sufficient previous training.

and

I noticed your deficiency, when I heard you talking here with your friend Aristoteles, the day before yesterday. The impulse that carries you towards philosophy is assuredly noble and divine; but there is an art which is called by the vulgar idle talking, and which is often imagined to be useless; in that you must train and exercise yourself, now that you are young, or truth will elude your grasp.

Socrates asks the obvious question:

And what is the nature of this exercise, Parmenides, which you would recommend?

to which he replies:

That which you heard Zeno practising...but I think that you should go a step further, and consider not only the consequences which flow from a given hypothesis, but also the consequences which flow from denying the hypothesis; and that will be still better training for you.

Hegel of course famously developed the notion of:thesis/anti-thesis and sublation to a synthesis; which one can, it appears, discern this in Parmenides; or a hint of it.

Is this correct? Does anyone else, before Hegel, press this notion of the dialectic through in this manner?

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Hegel described himself in the case of dialectic (as in many others) as standing on the shoulders of Kant. Kant's antinomies of reason were examples of dialectic, which Hegel took to a much greater scale.

In modern times it was, more than any other, Kant who resuscitated the name of Dialectic, and restored it to its post of honour. He did it, as we have seen, by working out the Antinomies of the reason. The problem of these Antinomies is no mere subjective piece of work oscillating between one set of grounds and another; it really serves to show that every abstract proposition of understanding, taken precisely as it is given, naturally veers round to its opposite. (Shorter Logic § 81)

Hegel took Plato (not Parmenides) to be the ancient inventor of dialectic. Plato in turn is described as following on Socrates, who is said to be the inventor of "subjective" dialectic, a method or technique. Plato is deemed the father of "objective" dialectic, as an inherent aspect of philosophy.

Among the ancients Plato is termed the inventor of Dialectic; and his right to the name rests on the fact that the Platonic philosophy first gave the free scientific, and thus at the same time the objective, form to Dialectic. Socrates, as we should expect from the general character of his philosophizing, has the dialectical element in a predominantly subjective shape, that of Irony. (ibid.)

Hegel mentions Plato's Parmenides as an example of Platonic dialectic.

If, for instance, the Sophists claimed to be teachers, Socrates by a series of questions forced the Sophist Protagoras to confess that all learning is only recollection. In his more strictly scientific dialogues, Plato employs the dialectical method to show the finitude of all hard and fast terms of understanding. Thus in the Parmenides he deduces the many from the one. In this grand style did Plato treat Dialectic. (ibid.)

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I don't think so. Dialectic as recommended by the Platonic Parmenides is a method of argumentation or rational inquiry. Hegelian dialectic, on the other hand, is a metaphysical principle, one of the most basic properties of the world which every process can be reduced to. Hölderlin is supposed to have anticipated something like this. See Dieter Henrich's "Hegel und Hölderlin" in his "Hegel im Kontext".

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Hegel continues the "development of the philosophical spirit" carried by his predecessors Kant, Descartes etc back to Aristotle and Plato. The philosophical "dialogue" leads back to the Pre-Socratics with Parmenides and Heraclitus among others. (Parmenides opposes "είναι"- (eternal being) to Heraclitus "γίγνεσθαι" (ever changing and becoming and renewing). The Socratic dialogue is not only a way of argument and a method to seek the truth but an imitation by humans of the relation and the dialectic nature between the actual elements of the world.

https://en.wikipedia.org/wiki/Georg_Wilhelm_Friedrich_Hegel#Heraclitus

https://en.wikipedia.org/wiki/Pre-Socratic_philosophy#/media/File:Presocratic_graph.svg

https://en.wikipedia.org/wiki/Dialectic

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