Or, to say the same thing in different words, what is analysis? The most simple concept of analysis is this: the extraction of a predicate which is already contained in the subject. "The sky is blue" is analytic, because the experience of seeing the sky already contains the experience of seeing the colour blue.
Of course, this notion is too vague and, more problematically, untrue. This account of analysis fails to tell us exactly what the analytic operation itself is doing. That is to say, it fails to tell us how it is possible that we "extract" a predicate, and how it is possible that we see a predicate as "already contained" in a subject.
These are the issues which Hegel tackles in the Science of Logic. Hegel's account is (very roughly) as follows. Analysis requires that the thing to be analyzed be already "given." The givennness of the thing is, however, none other than a mere collection of predicates, what Hegel calls an "accidental manifold." (This manifold is not simply "given," as if it arises out of nowhere - more on this later.) There are no relations established between these predicates. Thus, at this point, no distinction is made between cause/effect, essence/accident, etc. To introduce these distinctions into this manifold is already a synthetic act, that is to say, an act which changes the way in which the manifold appears to us. Now analysis claims that a predicate is contained in a subject. However, what is thought of as "subject," e.g. the sky, is only an element in the manifold which is external to the other element, e.g. "blue." To "extract" the blue is already to shift one's focus from the sky as such to a blue which is necessarily not identical with the sky. Therefore, the act of extraction is in fact a synthetic act. Ergo, the simple concept of analysis is untrue.
What, then, is analysis, or what Hegel calls "analytic cognition?" Hegel's answer is ingenious. Analysis is, to put it in a short formula, the repetition of a rule onto an accidental manifold. In other words, analysis takes a collection of predicates, seeks a rule which is already placing these predicates in relation to each other, re-applies this rule, and sees how things change. Here, change is involved, but nothing is added to the "given" manifold.
The example Hegel gives is arithmetic. By the way, this is the madness and shrewdness of Hegel, to refute another philosophical theory while at the same time explicating his own view. The theory he is targeting is Kant's claim that mathematics is synthetic a priori. For Kant, the operation "5+7=12" is synthetic in the sense that 12 is not immediately "given" in the concepts "5" "+" and "7." Later proponents of this view, such as Wittgenstein, Kripke, and Karatani, further point out that we can introduce different rules to arithmetic which will alter the result of this addition, and since the rules are thus not explicitly given in the equation itself, it must be added from the outside, and this addition is precisely the act of synthesis. Against this view, however, Hegel points out that this account - that math is synthetic by virtue of the implicit rules required to solve an equation - does not in the least refute the possibility that the same equation is analytic. To return to the example, "5+7=12" is based on the rule that the "+" is the combination of quantities, and that in this case the decimal system must be followed (as opposed to, say, the binary numeral system.) These rules are, however, already given with the equation. If we take the manifold "5+7" with these rules, then the other side of the equation just has to be "12" - or, for that matter, "1+2+3+6" etc. There is no need to add anything to this manifold in order to produce the desired result. The fact that there are twelve units in total in the initial manifold is a fact and phenomenon clear as daylight. To express it as "12" or as "5+7" is no different from this point of view.
Hegel then proceeds to more precisely describe the difference between analysis and synthesis in mathematics. All analytic cognition in mathematics is merely the act of solving a problem. Hegel explicitly notes here that machines are capable of analytic cognition, since there is absolutely no need to add something external to what is already given in the manifold. On the other hand, synthetic cognition is a proof which culminates in a theorem. Based on this distinction, it is clear that "5+7" is analytic; for it is not a theorem which requires proof, but rather is merely a problem to be solved. A proof requires the invention of rules. The plausibility of the rule, however, can be checked analytically by comparing it to existing rules and see if it is consistent with the rest. Thus, the latter task is merely a problem.
I find the above account of analytic cognition both convincing and elegant. It nicely explains why machines are capable of calculation but incapable of mathematics proper. It further explains why humans are able to make judgments against situations which have no ready-made solution, while pre-programmed objects simply fail to respond in an appropriate way and thereby also fail to survive.
The account which I summarized here is of course too simplistic and is not doing justice to Hegel's original exposition. In particular, I have left unexplained how the "givenness" of the manifold is in fact a product of a preceding logical operation. It is only by forgetting this condition that a thing appears as "given," but this does not mean that it is actually so. But to go into the original constitution of the given requires another long explanation, which cannot be given here for the time being.
