Hegel’s Science of Logic

§ 1713

It is a familiar fact that *arithmetic* and the more general *sciences of discrete magnitude* especially, are called *analytical science* and *analysis*. As a matter of fact, their method of cognition is immanently analytical in the highest degree and we shall briefly consider the basis of this fact. All other analytic cognition starts from a concrete material that in itself possesses a contingent manifoldness; on this material depends all distinction of content and progress to a further content.

§ 1714

The material of arithmetic and algebra, on the other hand, is something that has already been made wholly abstract and indeterminate and purged of all peculiarity of relationship, and to which, therefore, every determination and connection is something external. Such a material is the principle of discrete magnitude, the *one. *This relationless atom can be increased to a *plurality, *and externally determined and unified into a sum; this process of increasing and delimiting is an empty progression and determining that never gets beyond the same principle of the abstract one. How *numbers *are further combined and separated depends solely on the positing activity of the cognising subject. *Magnitude *is in general the category within which these determinations are made; it is the determinateness that has become an *indifferent *determinateness, so that the subject matter has no determinateness that might be immanent in it and therefore a *datum *for cognition. Cognition having first provided itself with a contingent variety of numbers, these now constitute the material for further elaboration and manifold relationships. Such relationships, their discovery and elaboration, do not seem, it is true, to be anything immanent in analytic cognition, but something contingent and given; and these relationships and the operations connected with them, too, are usually presented *successively as different *without any observation of an inner connection. Yet it is easy to discover a guiding principle, and that is the immanent principle of analytic identity, which appears in the diverse as *equality; *progress consists in the reduction of the unequal to an ever greater equality. To give an example in the first elements, addition is the combining of quite contingently unequal numbers, multiplication, on the contrary, the combination *of equal* numbers; these again are followed by the relationship of *the equality of amount and unity, *and the relationship of powers makes its appearance.

§ 1715

Now because the determinateness of the subject matter and of the relationships is a *posited *one, the further operation with them is also wholly analytic, and the science of analysis possesses not so much *theorems as problems. *The analytical theorem contains the problem as already solved for it, and the altogether external difference attaching to the two sides equated by the theorem is so unessential that a theorem of this kind would appear as a trivial identity. Kant, it is true, has declared the proposition 5 + 7 = 12, to be a *synthetic *proposition, because the same thing is presented on one side in the form of a plurality, 5 and 7, and on the other side in the form of a unity, 12. But if the analytic proposition is not to mean the completely abstract identity and tautology 12 = 12 and is to contain any advance at all, it must present a difference of some kind, though a difference not based on any quality, on any determinateness of reflection, and still less of the Notion. 5 + 7 and 12 are out and out the same content; the first side also expresses the *demand that *5 and 7 shall be combined in one expression; that is to say, that just as 5 is the result of a counting up in which the counting was quite arbitrarily broken off and could just as well have been continued, so now, in the same way, the counting is to be continued with the condition that the ones to be added shall be seven. The I 2 is therefore a result of 5 and 7 and of an operation which is already posited and in its nature is an act completely external and devoid of any thought, so that it can be performed even by a machine. Here there is not the slightest trace of a transition to an *other; *it is a mere continuation, that is, *repetition, *of the same operation that produced 5 and 7.

§ 1716

The *proof *of a theorem of this kind — and it would require a proof if it were a synthetic proposition — would consist merely in the operation of counting on from 5 for a further 7 ones and in discerning the agreement of the result of this counting with what is otherwise called 12, and which again is nothing else but just that process of counting up to a defined limit. Instead, therefore, of the form of theorem, the form of *problem *is directly chosen, the *demand *for the operation, that is to say, the expression of only *one* side of the equation that would constitute the theorem and whose other side is now to be found. The problem contains the content and states the specific operation that is to be undertaken with it. The operation is not restricted by any unyielding material endowed with specific relationships, but is an external subjective act, whose determinations are accepted with indifference by the material in which they are posited. The entire difference between the conditions laid down in the problem and the result in the *solution, is* merely that the specific mode of union or separation indicated in the former is *actual *in the latter.

§ 1717

It is, therefore, an utterly superfluous bit of scaffolding to apply to these cases the form of geometrical method, which is relevant to synthetic propositions and to add to the *solution *of the problem a *proof *as well. The proof can express nothing but the tautology that the solution is correct because the operation set in the problem has been performed. If the problem is to add several numbers, then the solution is to add them; the proof shows that the solution is correct because the problem was to add, and addition has been carried out. If the problem contains more complex expressions and operations, say for instance, to multiply decimal numbers, and the solution indicates merely the mechanical procedure, a proof does indeed become necessary; but this proof can be nothing else but the analysis of those expressions and of the operation from which the solution proceeds of itself. By this separation of the *solution *as a mechanical procedure, and of the *proof *as a reference back to the nature of the subject matter to be treated, we lose what is precisely the advantage of the analytical problem, namely that *the construction *can be immediately deduced from the problem and can therefore be exhibited as *intelligible *in and for itself; put the other way, the construction is expressly given a defect peculiar to the synthetic method. In the higher analysis, where with the relationship of powers, we are dealing especially with relationships of discrete magnitude that are qualitative and dependent on Notion determinatenesses, the problems and theorems do of course contain synthetic expressions; there other expressions and relationships must be taken as intermediate terms besides those *immediately specified *by the problem or theorem. And, we may add, even these auxiliary terms must be of a kind to be grounded in the consideration and development of some side of the problem or theorem; the synthetic appearance comes solely from the fact that the problem or theorem does not itself already name this side. The problem, for example, of finding the sum of the powers of the roots of an equation is solved by the examination and subsequent connection of the functions which the coefficients of the equation are of the roots. The determination employed in the solution, namely, the functions of the coefficients and their connection, is not already expressed in the problem-for the rest, the development itself is wholly analytical. The same is true of the solution of the equation *x*^{(m-1)} - 1 = 0 with the help of the sine, and also of the immanent algebraic solution, discovered, as is well known, by Gauss, which takes into consideration the *residuum *of *x*^{(m-1)} - 1 divided by *m*, and the so-called primitive roots — one of the most important extensions of analysis in modern times. These solutions are synthetic because the terms employed to help, the sine or the consideration of the residua, are not terms of the problem itself.

§ 1718

The nature of the analysis that considers the so-called infinitesimal differences of variable magnitudes, the analysis of the differential and integral calculus, has been treated in greater detail in the *first* part of this *logic*. It was there shown that there is here an underlying qualitative determination of magnitude which can be grasped only by means of the Notion. The transition to it from magnitude as such is no longer analytic; and therefore mathematics to this day has never succeeded in justifying by its own means, that is, mathematically, the operations that rest on that transition, because the transition is not of a mathematical nature. Leibnitz, who is given the credit of having reduced calculation with infinitesimal differences to a *calculus*, has, as was mentioned in the same place, made the transition in the most inadequate manner possible, a manner that is as completely unphilosophical as it is unmathematical; but once the transition is presupposed — and in the present state of the science it is no more than a presupposition — the further course is certainly only a series of ordinary operations.

§ 1719

It has been remarked that analysis becomes synthetic when it comes to deal with *determinations* that are no longer *posited* by the problems themselves. But the general transition from analytic to synthetic cognition lies in the necesary transition from the form of immediacy to mediation, from abstract identity to difference. Analytic cognition in its activity does not in general go beyond determinations that are self-related; but by virtue of their *determinateness* they are also essentially of such a nature that they *relate themselves to an other*. It has already been remarked that even when analytic cognition goes on to deal with relationships that are not externally give material but thought determinations, it still remains analytic, since for it even these relationships are *given* ones. But becase abstract identity, which alone analytic cognition knows as its own, is essentially the *identity of distinct terms*, identity in this form too must belong to cognition and become for the subjective Notion also the *connection* that is posited by it and identical with it.

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