“For to think does not mean to be an abstract 'I', but an 'I' which has at the same time the significance of intrinsic being, of having itself for object, or of relating itself to objective being in such a way that its significance is the being-for-self of the consciousness for which it is [an object]. For in thinking, the object does not present itself in picture-thoughts but in Notions, i.e. in a distinct being-in-itself or intrinsic being, consciousness being immediately aware that this is not anything distinct from itself.”
I think the following is true—whether it's Hegel is another question:
Thinking an object means more than representing it for consciousness. For what can be represented to consciousness is not what can be thought. I can represent a circular thing to myself but I cannot represent a circle. The most I can do is represent something as representative of a circle. But even this "as" does not appear within representation. Only by hypothesizing the circular as the equal distance of points from a center on a plane do I form the concept of the circle. This hypothesis can be drawn through a figure, or better, through the tracing of a figure, and continually depends on the repetition of such a figure, in order to persist as the superseding of this figure. The circular is the determinate nothing constituted by the vanishing of a circular figure.
This concept of circularity is only a model: the discovery of the circular as the vanishing of a figure is not yet thinking, but it is like thinking. It is like thinking in that it supersedes what can be represented and is this superseding. Thinking, however, does not inhere in a mathematical approach to things, but in the disclosure of the thinghood of things through work. The proper object of consciousness shows up not in the light of the indiscriminate overturning of representation, but in the light of the good. At first this light shines in the proximate good which makes the object of work show up as something to be developed. Something that has to be worked into shape presents itself to the worker as resistance: it resists its own being—what it is supposed to be. Abstractly, the thing seems just to be what it is. In the light of the good which shines through work, it shows up as the concept of itself, which is to say that it shows up as really being what the working consciousness, referring to the good, has placed upon it to become. What from the outside seems to be a projection of the working consciousness on its object shows up in this light as the true being of the thing in itself.
Interesting, Amos...I would say it kind of sounds like (platonism's) Plato with a Hegelian twist.
ReplyDeleteBut I have one question regarding the 'analogy' you draw to mathematical hypothesis. You write:
"This concept of circularity is only a model: the discovery of the circular as the vanishing of a figure is not yet thinking, but it is like thinking. It is like thinking in that it supersedes what can be represented and is this superseding."
To pick up on that last senstence: according to the Ancients, it does indeed supercede what can be "represented", that is to say, what can be produced by φαντασία in Aristotle, and, in the same spirit, is revealed by the only impurely intelligible in Platonic διανοῖα. BUT it is by no means also, as you claim, this superceding. For the geometrical figure is esteemed as a midway point between τὰ φύσικα and τα νοήματα because it enjoys the repose of what is timeless and without process, that is, what is separate from ὕλη. And this is borne out by our own familiar conception of a circle: we do not think of a process but of a stable entity. The superceding required to "recall" or "abstract unto" the true and actual circle from our geometrical construction is not to be confused with the circle itself.
Just to make the point succinctly: why should one include the process whereby he came to know the mathematicals in his conception of the mathematicals themselves? THis would be a Hegelian move --but he would do it based upon his re-interpretation of time and the time-less Concept it leads to. But when you write that the circular itself is discovered as the VANISHING of a figure, I am tempted to simply ask why? Isn't the circle or even the circular (which is derivative of the circle) arrived at when the geometrical construction has already come under intellectual erasure? I will quote an older post at Seynsgeschichte that argues to this effect in the comment below.
ReplyDelete"Even before Aristotle, didn't Plato, the first to witness 'the universal' as such ---at least the first to do so and "live to tell the tale", didn't he already have something like this in mind when he had inscribed over the very entrance to his acedemy the imperative: ᾿Αγεωμέτρητος μειδεὶς εἰσίτω!, i.e., that no one un-geometrical be admitted, or in other words, that one must know geometry before passing through these gates. Plato wasn't here talking about the mere capacity to practice geometry correctly, he was talking about the knowledge of geometrical things, of the "geometricals". I can draw a dot and call it a point, but I don't speak with geometrical knowledge regarding this point unless I also know that the dot is precisely not the point. The point, unlike the dot, has no shape, has no borders, and has no place, even though it is known as the end of a line. To offer an image of the point is to miss the point. But the genuine geometer knows precisely what he is missing."
ReplyDeleteMy only point here si that the geometrical point, or better, the circle, is neither the constuction of something circular "the drawn circle" nor is it the concept of the circle: i.e. circleness, nor is it the sufficient conditions which that circle-ness requires, i.e. "the equal distance of points from a center on a plane". The reasons why it is not the drawn circle are obvious (though elaborating upon them gets interesting). The reason why it is not the concept of the circle, i.e. "circle", is because a geometrical is always individuated. It is always *a* circle. To think "circle itself" would actually be to engage in metaphysical noesis and not mathematical dia-noesis. THat is how I would roughly lay out where the proper geometrical is to situated...
ReplyDeleteYou may be right about the mathematical as Plato found it. Shortly after publishing this Hegel post (whose secret identity as a post on Plato you have shrewdly found out), I re-read Copleston's chapter on Plato's theory of knowledge, in which he, too, lays out the mathematical as something individuated, citing Aristotle's remarks in the Metaphysics. If you, Aristotle, and Copleston all read Plato this way, I don't see how I can disagree. But still, I wonder what Plato means when he says that "geometers make the arguments for the sake of the square itself and the diagonal itself," and subsequently referring to this sort of object of argument as "intelligible form." And I wonder further why he should never have given an account of knowledge of circle itself as noetic, and why geometry belongs to the portion of a philosophical education which stops short of dialectic, if knowledge of what makes a circle circular already exceeded hypothesis.
ReplyDeleteAs for the superseding, I am only trying to account for the geometer's dependence on phantastic forms. Is it not by seeing past the phantastic form of a figure, letting it vanish, that I come to be in mind of a geometrical?
I meant also to ask, is it not by continuing to see past the phantastic form that I remain in mind of the geometrical?
ReplyDeleteThe question of the process of coming-to-be-in-mind of a geometrical and of the state of remaining-in-mind of a geometrical is, as I understand it, a separate question from my previous comment's central concern, namely, that the process whereby we attain mindfulness of the geometrical should not be equated --at least not without further ado (cf. Hegel) --with that very mindfulness of the geometrical itself and a fortiori with the geometrical itself pure and simple. That being said, however, I believe one may indicate an answer to the former question by reference to the Divided Line. If one interprets the epistemic ascent which the Line itself geometrically attempts to depict as an ascent whose very upward movement is dependent upon an abandonment of what it moves FROM and thereby transcends, then the dianoetics which eventually lead not merely to hypothesis but to noesis of the things themselves (including, e.g. the FORM of Triangle) do not remain a constant requirement of the one who wishes to engage in noesis, any more than opining is required for episteme. By the same principle, then, we must concede that this "Platonic" Plato does not fail to distinguish between coming-to-be-mindful of a geometrical and of remaining-mindful of a geometrical. This perhaps even more clearly conveyed by the "doctrine" of ἀνάμνησις as it is discovered precisely with regard to geometry in the Meno. Here there is no mention of a need for continual recollection after a geometrical has been recalled --and this is above all because what has been recalled is also the fact that one already knew the geometrical, a priori as it were. The necessity of geometrical knowledge is in no wise contingent upon the process by which it is arrived at, nor is that necessity threatened in the future by the danger of another amnesia. Once you have calculated that 2 plus 2 equals four, you have understood that the result does not depend on future calculation and will remain unaltered.
ReplyDeleteI see your point. It is ridiculous to say that the process of coming to be mindful of a geometrical is identical with being mindful. Perhaps even as ridiculous as the geometers' own way of speaking.
ReplyDeleteWhat made me say this ridiculous thing? Just the way it seems to me: that my knowledge of a geometrical as such is somehow inseparable from the determinate negation of any sensible approximation I can imagine. I have to think "that-but-not-that." (As when I think that the guy trying to hammer out "What a Wonderful World" on the piano is not playing "What a Wonderful World," but also that he is not playing it in a special way, which I would not say, for instance, is the same way that he is not playing "Chopsticks.") The circle is what this circular thing which I can picture is failing to be. If I come up with a way of describing the asymptote or limit of this repeated failure, it may be very helpful, but I have lost touch with reality if I think that this description preserves itself as knowledge. Dianoia's measure for the failure of circular figures to be circles is the hypothetical definition, which is not and cannot be grounded in circle-ness itself except in the light of the good. This definition is not grounded in its beginning but hypothesized on the basis of the determinate negation.
Real knowledge of the circle would have to adjust the hypothetical definition according to the good.
Incidentally, I do not see the need for any reinterpretation of time to arrive at these observations.
My previous claim regarding the need for a reinterpretation of time applies only to the assertion that "the discovery of the circular [is] the vanishing of a figure", or to the more general assertion that the discovery of the circle "supersedes what can be represented AND IS THIS SUPERSEDING", or, in its most general formulation,"that the process whereby we attain mindfulness of the geometrical should be equated with that very mindfulness of the geometrical itself and even with the geometrical itself pure and simple." I never intended to leave the impression that I thought such assertions worthy of ridicule, only that they depend on a re-construal of the relation obtaining between motion and rest, and even more fundamentally, potency and act, and therefore they absolutely necessitate re-interpretation of time. (I will cont. in the next comment)
ReplyDeleteNow as regards the substance of what you just commented upon, you write: "my knowledge of a geometrical as such is somehow inseparable from the determinate negation of any sensible approximation I can imagine." I am in agreement with you, but we must press on and ask: just how is it inseparable? The answer that the divided line would seem to give is: as means is inseparable from its end and only in that way. In other words, the negation of which you speak is strictly preparatory and temporal. It is necessary that the negation of the phantasm take place EN ROUTE to the geometrical --but ipso facto NEVER when the geometrical has been encountered. This is why Euclid's proofs are cumulative. But it is also why one need not continually reprove them. If the "recollection" of a true geometrical happens once it has happened enough. And this means that is is not subject to the very time in which it was recalled. Thus when you write: "The circle is what this circular thing which I can picture is failing to be", I must respond that the negative moment of the failure of the circular thing is not the τέλος in terms of which the geometrical is discovered. Rather this negative moment is followed by a positive moment, and the difference between the two moment is that one is temporal and the other is not. Evidence for this claim can be found in the fact that in order for failure of the circular thing to take place it must be set against the expectation or demand of being a true circle. In such a case, a true circle, and even before that, the essence of circle has been seen in advance, i.e. has appeared as εἶδος. This anticipatory look of the circle alone enables the circular thing to fail. The negativity of the failure pre-supPOSES a positivity of the geometrical and should not be confused with it. When this positiviy is no longer presupposed but seen in the light of day, the geometrical is dianoetically apprehended and *A* circle is known. But the cause or source of the failure lies not in a given geometrical, but in the form of the circle as such.
ReplyDeleteThus to recapitulate and address your last point: the failure of the circular is made possible by the individual circle which it fails to be (for it can NEVER fail to be "circle itself", since the latter is an impossible expectation for the circular and is beyond the realm of ὑπόθεσις). The individual geometrical is in turn made possible by "circle itself". Now, within this schema it is perfectly justifiable to recognize the multiplicity that "circle itself" must suffer --being, as it is, only one among other forms of geometricals. And of course, one can then ultimately discover τὸ ἕν that is even ἐπέκεινα τῆς οὐσίας. It may even be said we may fittingly call τὸ ἕν by the name Ἄγαθον since its unity lies in the manner in which it surpasses the one which belongs to the problematic of the one and the many. This would be the Plato that Neoplatonism took over. In this way do I find nothing but an accord struck in your statement that "Dianoia's measure for the failure of circular figures to be circles is the hypothetical definition, which is not and cannot be grounded in circle-ness itself except in the light of the good." But for the reasons I have just given in the past two comments, it seems such a statement can and must be carefully separated and even made exclusive of the statement that "This definition is not grounded in its beginning but hypothesized on the basis of the determinate negation."
ReplyDeleteIt occurs to me that there may be a simpler way of spotting the distinction between that failure of the circular and the individual geometrical circle whose explicit knowledge that failure makes possible, yet whose implicit anticipatory look renders the necessary condition for the possibility of said failure.
ReplyDeleteTake, then, a gander at our geometer. What does he know QUA GEOMETER save what is geometrical? He therefore knows the line the circle the total measure of two right angles etc. ---in short his objects are well confessed by Euclid's elements. These are the explicit acquisitions of his filed of conquest. But what of the failure of the circular? Surely the geometer is not ignorant of it, but just as surely it is no object of geometry. (Where is Euclid's axiom or theorem or postulate or discourse regarding such failure?) The failure is known AS A MEANS to the knowledge of the geometrical. We may, though no long as geometers, make such failure an explicit theme of out investigation --but when we do so we admit it not as an overlooked geometrical.