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.

The 2 real determinations of the positive and negative are:

1. The base is a merely different, immediate existence.

Its 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.

2. 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 unity forming their common base.

It is indifferent to the opposition itself and serves here, without any further notion, as a dead base.

⁹²¹ 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 2 different a’s. 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.

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.

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.