an unbridgeable qualitative opposition: quantities so enormously different from one another that every rational relation, every comparison, between them ceases, that they become quantitatively incommensurable. Ordinary incommensurability, for instance of the circle and the straight line, is also a dialectical qualitative difference; but here it is the difference in quantity of similar magnitudes that increases the difference of quality to the point of incommensurability.
Number. The individual number becomes endowed with quality already in the numerical system itself, and the quality depends on the system used. 9 is not only 1 added together 9 times, but also the basis for 90, 99, 900,000, etc. All numerical laws depend upon and are determined by the system adopted. In dyadic and triadic systems 2 multiplied by 2 does not equal 4, but=100 or=11. In all systems with an odd basic number, the difference between odd and even numbers falls to the ground, e.g., in the system based on 5, 5=10, 10=20, 15=30. Likewise in the same system the sums of digits 3n of products of 3 or 9 (6=11, 9=14). Hence the basic number determines not only its own quality but also that of all the other numbers.
With powers of numbers, the matter goes still further: any number can be conceived as the power of any other number-there are as many logarithmic systems as there are whole and fractional numbers.
One. Nothing looks simpler than quantitative unity, and nothing is more manifold than it, as soon as we investigate it in connection with the corresponding plurality and according to its various modes of origin from plurality. First of all, one is the basic number of the whole positive and negative system of numbers, all other numbers arising by the successive addition of one to itself.
One is the expression of all positive, negative, and fractional powers of one: 12, √1, 1-2 are all equal to one.
It is the content of all fractions in which the numerator and denominator prove to be equal. It is the expression of every number that is raised to the power of zero, and therewith the sole number the logarithm of which is the same in all systems, viz.,=0. Thus one is the boundary that divides all possible systems of logarithms into two parts: if the base is greater than one, then the logarithms of all numbers more than one are positive, and of all numbers less than one negative; if it is smaller than one, the reverse is the case.
Hence, if every number contains unity in itself in as much as it is compounded entirely of ones added together, unity likewise contains all other numbers in itself. This is not only a possibility, in as much as we can construct any number solely of ones, but also a reality, in as much as one is a definite power of every other number. But the very same mathematicians who, without turning a hair, interpolate into their calculations, wherever it suits them, x0=1, or a fraction whose numerator and denominator are equal and which therefore likewise represents one, who therefore apply mathematically the plurality contained in unity, turn up their noses and grimace if they are told in general terms that unity and plurality are inseparable, mutually penetrating concepts and that plurality is not less contained in unity than unity is in plurality. How much this is the case we see as soon as we forsake the field of pure numbers. Already in the measurement of lines, surfaces, and the volumes of bodies it becomes apparent that we can take any desired magnitude of the appropriate order as unity, and the same thing holds for measurement of time, weight, motion, etc. For the measurement of cells even millimetres and milligrams are too large, for the measurement of stellar distances or the velocity of light even the kilometre is uncomfortably small, just as the kilogram for planetary or, even more so, solar masses, Here is seen very clearly what diversity and multiplicity is contained in the concept of unity, at first sight so simple.
Zero, because it is the negation of any definite quantity, is not therefore devoid of content. On the contrary, z