Chapter 17 B. Quantity (Die QuantitaBt) Ⅱ. Quantitative (Quantum)

§101 Quantity is inherently exclusive, and the quantity with this exclusiveness is quantification, or quantity with a certain limit. Note: Quantitative is the fixed presence in quantity, pure quantity is equivalent to being, and the degree to be discussed below is equivalent to being-for-itself.The detailed steps of the progression from the scalar to the quantitative are based on the fact that in the scalar the distinction between continuity and separability is at first only latent, whereas in the quantitative the distinction between the two becomes apparent. established.Now, therefore, quantity generally appears as differentiated or limited.But in doing so, quantification is simultaneously split into quantities or specific quantities of an indeterminate number of units.Each definite quantity, by virtue of its distinction from other definite quantities, forms a unit, but on the other hand the unit of this definite quantity is still many.Quantities are then defined as numbers.

§102 In numbers, quantification achieves its development and perfect determinacy.Number contains "one" as its element, and thus contains within itself two qualitative moments: It is a number in its discrete moments, and a unit in its continuous moments. (Explanation) In arithmetic various calculation methods are often cited as accidental ways of dealing with numbers.If these calculation methods are also necessary and have an intelligible meaning, they must be based on a principle, and this principle can only be found in the determination contained in the concept of number itself.Let me try to reveal this principle a little bit: the definition of the concept of number is number and unit, and number itself is the unity of number and unit.But units, when applied to empirical numbers, refer only to the equality of these numbers.Therefore, the principles of various calculation methods must put the number on the proportional relationship between the unit and the number, and find out the equality of the two.

Most ones or numbers themselves are independent of each other, so the units derived from numbers generally appear as an external makeshift.So counting (Rechnen) is really counting (ZaBhle).The difference between various calculation methods lies in the nature of the numbers to be summed up. The principle that determines the nature of numbers is the regulation of units and numbers. Counting is the original method of forming a general number, which is to put together any number of "ones".But as a calculation method, it is to add up those things that are already numbers and are no longer simply "one".

First, the numbers are immediate, and initially completely indeterminate, and therefore generally unequal.The summing up or counting of these numbers is addition. Second, another definition of counting: the numbers are generally equal, so they form a unit, and we get the present number of these units; The calculation of this kind of number is multiplication. In the process of multiplication, no matter how the provisions of number and unit are distributed between two numbers or two factors, no matter which number is used as the number or which number is used as the unit, The result is the same.

Finally, the third determination of counting is the equality of numbers and units.The sum of the numbers determined in this way is multiplication by itself, firstly to the second power. (Finding the higher power of a number is the continuous self-multiplication of this number. This kind of self-multiplication has a formula and can be repeated to an indefinite number of times.) In this third rule, since the unique The complete equality of the existing distinctions, that is, of the distinctions of numbers and units, is therefore nothing but these three methods of calculation.Corresponding to the sum of numbers, according to the same determination of numbers, we get the decomposition of numbers.Therefore, in addition to the three methods mentioned above, which can also be called positive calculation methods, there are three negative calculation methods.

Note: Numbers are generally well-defined quantifications, so we can use this quantification not only to define so-called discrete quantities, but also to determine so-called continuous quantities.Therefore even geometry, when it wants to point out specific figures of space and their proportional relations, has to resort to numbers.