Remark: Opposite Magnitudes of Arithmetic

§ 919

Here is where we must take a look at the notion of the positive and negative as it is employed in arithmetic.

There it is assumed as known; but because it is not grasped in its determinate difference, it does not avoid insoluble difficulties and complications.

We have just found the two real determinations of the positive and negative — apart from the simple notion of their opposition — namely, first, that the base is a merely different, immediate existence whose simple reflection-into-self is distinguished from its positedness, from the opposition itself.

This opposition, therefore, is not regarded as having any truth in and for itself, and though it does belong to the different sides, so that each is simply an opposite, yet, on the other hand, each side exists indifferently on its own, and it does not matter which of the two opposites is regarded as positive or negative.

But secondly, the positive is the positive in and for itself, the negative in and for itself the negative, so that the different sides are not mutually indifferent but this their determination is true in and for itself. These two forms of the positive and negative occur in the very first applications of them in arithmetic.

§ 920

In the first instance, +a and -a are simply opposite magnitudes; a is the implicit [ansichseiende] unity forming their common base; it is indifferent to the opposition itself and serves here, without any further notion, as a dead base. True, -a is defined as the negative, and +a as the positive, but the one is just as much an opposite as the other.

§ 921

Further, a is not merely the simple unity forming the base but, as +a and -a, it is the reflection of these opposites into themselves; there are present two different a’s, and it is a matter of indifference which of them one chooses to define as the positive or negative; both have a separate existence and are positive.

§ 922

According to the first aspect, +y - y = 0; or in -8 + 3, the 3 positive units are negative in the 8. The opposites are cancelled in their combination.

An hour’s journey to the east and the same distance travelled back to the west, cancels the first journey; an amount of liabilities reduces the assets by a similar amount, and an amount of assets reduces the liabilities by the same amount.

At the same time, the hour’s journey to the east is not in itself the positive direction, nor is the journey west the negative direction; on the contrary, these directions are indifferent to this determinateness of the opposition; it is a third point of view outside them that makes one positive and the other negative.

Thus the liabilities, too, are not in and for themselves the negative; they are the negative only in relation to the debtor; for the creditor, they are his positive asset; they are an amount of money or something of a certain value, and this is a liability or an asset according to an external point of view.

§ 923

The opposites certainly cancel one another in their relation, so that the result is zero; but there is also present in them their identical relation, which is indifferent to the opposition itself; in this manner they constitute a one. Just as we have pointed out that the sum of money is only one sum, that the a in +a and -a is only one a, and that the distance covered is only one distance, not two, one going east and the other going west. Similarly, an ordinate y is the same on which ever side of the axis it is taken; so far, +y - y = y; it is only the one ordinate and it has only one determination and law.

§ 924

But again, the opposites are not only a single indifferent term, but two such. For as opposites, they are also reflected into themselves and thus exist as distinct terms.

§ 925

Thus in -8 + 3 there are altogether eleven units present; +y and -y are ordinates on opposite sides of the axis, where each is an existence indifferent to this limit and to their opposition; thus +y - y = 2y. — Also the distance travelled cast and west is the sum of a twofold effort or the sum of two periods of time. Similarly, in economics, a quantum of money or of wealth is not only this one quantum as a means of subsistence but is double: it is a means of subsistence both for the creditor and for the debtor. The wealth of the state is computed not merely as the total of ready money plus the value of property movable and immovable, present in the state; still less is it reckoned as the sum remaining after deduction of liabilities from assets. For capital, even if its respective determinations of assets and liabilities nullified each other, remains first, positive capital, as +a - a = a; and secondly, since it is a liability in a great number of ways, being lent and re-lent, this makes it a very much multiplied capital.

§ 926

But not only are opposite magnitudes, on the one hand, merely opposite as such, and on the other hand, real or indifferent: for although quantum itself is being with an indifferent limit, yet the intrinsically positive and the intrinsically negative also occur in it. For example, a, when it bears no sign, is meant to be taken as positive if it has to be defined. If it were intended to become merely an opposite as such, it could equally well be taken as -a . But the positive sign is given to it immediately, because the positive on its own has the peculiar meaning of the immediate, as self-identical, in contrast to opposition.

§ 927

Further, when positive and negative magnitudes are added and subtracted, they are counted as positive or negative on their own account and not as becoming positive or negative in an external manner merely through the relation of addition and subtraction.

§ 928

In 8 - (-3) the first minus means opposite to 8, but the second minus (-3), counts as opposite in itself, apart from this relation.

This becomes more evident in multiplication and division. Here the positive must essentially be taken as the not-opposite, and the negative, on the other hand, as the opposite, not both determinations equally as only opposites in general. The textbooks stop short at the notion of opposite magnitudes as such in the proofs of the behaviour of the signs in these two species of calculation; these proofs are therefore incomplete and entangled in contradiction. But in multiplication and division plus and minus receive the more determinate meaning of the positive and negative in themselves, because the relation of the factors-they are related to one another as unit and amount-is not a mere relation of increasing and decreasing as in the case of addition and subtraction, but is a qualitative relation, with the result that plus and minus, too, are endowed with the qualitative meaning of the positive and negative. Without this determination and merely from the notion of opposite magnitudes, the false conclusion can easily be drawn that if -a times +a = -a2, conversely +a times -a = +a2. Since one factor is amount and the other unity, and the one which comes first usually means that it takes precedence, the difference between the two expressions -a times +a and +a times -a is that in the former +a is the unit and -a the amount, and in the latter the reverse is the case. Now in the first case it is usually said that if I am to take +a -a times, then I take +a not merely a times but also in the opposite manner, +a times -a; therefore since it is plus, I have to take it negatively, and the product is -a2. But if, in the second case, -a is to be taken +a times, then -a likewise is not to be taken -a times but in the opposite determination, namely +a times. Therefore it follows from the reasoning in the first case, that the product must be +a2. And similarly in the case of division.

§ 929

This is a necessary conclusion in so far as plus and minus are taken only as simply opposite magnitudes: in the first case the minus is credited with the power of altering the plus; but in the second case the plus was not supposed to have the same power over the minus, notwithstanding that it is no less an opposite determination of magnitude than the latter. In point of fact, the plus does not possess this power, for it is to be taken here as qualitatively determined against the minus, the factors having a qualitative relationship to one another. Consequently, the negative here is the intrinsically opposite as such, but the positive is an indeterminate, indifferent sign in general; it is, of course, also the negative, but the negative of the other, not in its own self the negative. A determination as negation, therefore, is introduced solely by the negative, not by the positive.

§ 930

And so -a times -a is also +a2, because the negative a is to be taken not merely in the opposite manner (in that case it would have to be taken as multiplied by -a), but because it is to be taken negatively. But the negation of negation is the positive.

C CONTRADICTION - next section

A IDENTITY B DIFFERENCE (a) Absolute Difference (b) Diversity

Remark: The Law of Diversity

§ 902

Diversity, like identity, is expressed in its own law. And both these laws are held apart as indifferently different, so that each is valid on its own without respect to the other.

§ 903

All things are different, or: there are no two things like each other. This proposition is, in fact, opposed to the law of identity, for it declares: A is distinctive, therefore A is also not A; or: A is unlike something else, so that it is not simply A but rather a specific A. A’s place in the law of identity can be taken by any other substrate, but A as distinctive [als Ungleiches] can no longer be exchanged with any other. True, it is supposed to be distinctive, not from itself, but only from another; but this distinctiveness is its own determination. As self-identical A, it is indeterminate; but as determinate it is the opposite of this; it no longer has only self-identity, but also a negation and therefore a difference of itself from itself within it.

§ 904

That everything is different from everything else is a very superfluous proposition, for things in the plural immediately involve manyness and wholly indeterminate diversity. But the proposition that no two things are completely like each other, expresses more, namely, determinate difference. Two things are not merely two — numerical manyness is only one-and-the-sameness — but they are different through a determination. Ordinary thinking is struck by the proposition that no two things are like each other — as in the story of how Leibniz propounded it at court and caused the ladies to look at the leaves of trees to see whether they could find two alike. Happy times for metaphysics when it was the occupation of courtiers and the testing of its propositions called for no more exertion than to compare leaves! The reason why this proposition is striking lies in what has been said, that two, or numerical manyness, does not contain any determinate difference and that diversity as such, in its abstraction, is at first indifferent to likeness and unlikeness. Ordinary thinking, even when it goes on to a determination of diversity, takes these moments themselves to be mutually indifferent, so that one without the other, the mere likeness of things without unlikeness, suffices to determine whether the things are different even when they are only a numerical many, not unlike, but simply different without further qualification. The law of diversity, on the other hand, asserts that things are different from one another through unlikeness, that the determination of unlikeness belongs to them just as much as that of likeness, for determinate difference is constituted only by both together.

§ 905

Now this proposition that unlikeness must be predicated of all things, surely stands in need of proof; it cannot be set up as an immediate proposition, for even in the ordinary mode of cognition a proof is demanded of the combination of different determinations in a synthetic proposition, or else the indication of a third term in which they are mediated. This proof would have to exhibit the passage of identity into difference, and then the passage of this into determinate difference, into unlikeness. But as a rule this is not done. We have found that diversity or external difference is, in truth, reflected into itself, is difference in its own self, that the indifferent subsistence of the diverse is a mere positedness and therefore not an external, indifferent difference, but a single relation of the two moments.

§ 906

This involves the dissolution and nullity of the law of diversity. Two things are not perfectly alike; so they are at once alike and unlike; alike, simply because they are things, or just two, without further qualification — for each is a thing and a one, no less than the other — but they are unlike ex hypothesi. We are therefore presented with this determination, that both moments, likeness and unlikeness, are different in one and the same thing, or that the difference, while falling asunder, is at the same time one and the same relation. This has therefore passed over into opposition.

§ 907

The togetherness of both predicates is held asunder by the ‘in so far’, namely, when it is said that two things are alike in so far as they are not unlike, or on the one side or in one respect are alike, but on another side or in another respect are unalike. The effect of this is to remove the unity of likeness and unlikeness from the thing, and to adhere to what would be the thing’s own reflection and the merely implicit reflection of likeness and unlikeness, as a reflection external to the thing. But it is this reflection that, in one and the same activity, distinguishes the two sides of likeness and unlikeness, hence contains both in one activity, lets the one show, be reflected, in the other. But the usual tenderness for things, whose only care is that they do not contradict themselves, forgets here as elsewhere that in this way the contradiction is not resolved but merely shifted elsewhere, into subjective or external reflection generally, and this reflection in fact contains in one unity as sublated and mutually referred, the two moments which are enunciated by this removal and displacement as a mere positedness. ®

(c) Opposition

C CONTRADICTION

Remark 1: Unity of Positive and Negative

Remark 2: The Law of the Excluded Middle

Remark 3: The Law of Contradiction