Читать книгу Elements of Physiophilosophy - Lorenz Oken - Страница 5

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NOTHING.

31. The highest mathematical idea, or the fundamental principle of all mathematics is the zero = 0.

The whole science of mathematics depends upon zero. Zero alone determines the value in mathematics.

32. Zero is in itself nothing. Mathematics is based upon nothing, and, consequently, arises out of nothing.

33. Out of nothing, therefore, it is possible for something to arise, for mathematics, consisting of propositions, is a something in relation to 0. Mathematics itself were nothing if it had none other than its highest principle zero. In order, therefore, that mathematics may become a real science, it must, in addition to its highest principle, subdivide into a number of details, namely, first of all into numbers, and, finally, into propositions. What is tenable in regard to mathematics must be equally so of all the sciences; they must all resemble mathematics.

34. The first act towards realization or the becoming something, is an origination of Many. All reality can, accordingly, manifest itself only in multiplicity.

That which belongs to the Many is a Definite; this again is a Limited; the Limited is a Finite. The Finite only is real.

The question now arises, how it happens that mathematics becomes a multiplicity, or, what is the same thing, a reality, a something.

35. The reality of mathematics consists in the universality of its quantities; viz. numbers or figures. Every number, and every thing which belongs to mathematics, can be derived from no other source than zero.

Mathematical multiplicity, or its reality, must have proceeded, therefore, out of zero.

36. Zero, however, contains no number and no figure really in itself; it contains, forsooth, neither 1 nor 2, neither a point nor a line within itself. The Singulars or details cannot, therefore, reside in a real, but only an ideal manner in zero; or, in other words, not actually, but only potentially. The conditions here are the same as with all mathematical ideas. We may conceive, e. g., an idea or definition of a triangle in so general a sense that it shall comprehend all triangles, without, however, a definite triangle being actually intended, or without even a triangle actually existing. In order that the idea of the triangle be realized, it must become a definite, in other words, an obtuse or an acute triangle. In short, the idea of the triangle must multiply itself, be self-evolved, or else it is as naught in reference to mathematics, or only a geometrical zero.

The individual objects or figures of mathematics thus attain existence, so far only as the idea comprising them emerges out of itself and assumes an individual character.

It is clear that all individual triangles taken together closely resemble the ideal triangle, or, to express the same in more general terms, that the Real is equivalent to the Ideal, that the former is but the latter which has become dissevered and finite, and that the aggregate of every Finite is equivalent to the Ideal. This will probably be rendered still more distinct by the example of ice and water. The crystals of ice are nothing else than water bounded by definite lines. So, also, are the Real and Ideal no more different from each other than ice and water; both of these, as is well known, are essentially one and the same, and yet are different, the diversity consisting only in the form. It will be shown in the sequel that everything which appears to be essentially different from another, is so only in the form.

The Real and Ideal are one and the same, only under two kinds of form. The latter is the same under an indefinite, eternal, single form; but the Real is also the same, yet under the form of quantity, and, as will be shown, of multiplicity. An infinity resides in both; in the Real an endlessness of individual forms; in the Ideal but one endless form; in the latter case an eternity, in the former an infinity. The quantity and multiplicity of the whole of mathematics is contained in the same manner in the 0, that the quantity and multiplicity of the triangles are in the ideal or primary triangle. Mathematics is a system of nullities or nothings, and this admits of being easily proved.

37. Zero is indeed the universality of mathematics, this, however, is not real, but only ideal. Every number issues out of zero, like the multiplicity of the real triangles out of the primary triangle. This progression of numbers out of zero takes place through a process of becoming determinate and limited; just as the real triangles are only definitions of the absolute triangle. The process of becoming determined is identical with becoming a Finite; becoming real is called becoming finite. Mathematical singulars or numbers can, therefore, be nothing else than zero disintegrated, or rendered real by determination.

What zero is in infinite intensity, that are numbers in endless extensity. Zero is of two forms: under the ideal it is mere intensity; under the real mere extensity, or a series of numbers. The latter is only expanded intensity; the former, extensity concentrated on the point; both are, consequently, one and the same in toto. Numbers are identical with zero; they are zero in a state of extension, while zero is equivalent to numbers in a state of intensity. The sense in which numbers are said to come out of zero is, therefore, very clear; they have not issued forth from zero as if they had previously resided individually therein, but the zero has emerged out of itself, has itself become apparent, and then was it a finite zero, a number. So, also, does the idea of a circle become a real circle, not from the latter emerging from the former, but from this itself becoming manifest. The individual circle is a manifestation or phenomena of the spiritual circle.

38. All realization, therefore, is not the origin of a something that has not previously been; it is only a manifestation, a process of extension taking place in the idea.

Thus the Real does not arise out of the Ideal, but is the Ideal itself in a condition of definition and limitation, as are, e. g. the actual triangle or the actual circle. If, then, the Ideal and Real be one, everything is necessarily identical, and this identity dominates not merely between the Ideal and Real in a general sense, but between all individual members of the Real.

39. The identity of every Different, or of all things among themselves and with the highest unity, is the essence of things. The limitation or definition of the Ideal is their form. Limitation is the Impartient of form.

40. Limitation is originally only a quantitative relation, e. g. the size of the angle in a triangle; later on it becomes also a relation of direction or of position.

In both cases the limitation is only an ideal relation. Realization also takes place, therefore, only in an ideal manner; and the Real is therefore ideal, not simply as it regards its form, but also its essence. Every Plural resembles itself and the highest principle in essence; or, in other words, all Singulars are united through essence with the highest One. All diversity of the Plural resides merely in the form, limitation or manifestation. The one unchanging essence possesses one ideal form, which is that of pure unity, and the same essence has a limitation, a real form, which is that of subdivision. There is only one essence in all things, the 0, the highest identity; but there are infinitely numerous forms.

Numbers are naught else than different forms of the one unchangeable essence, namely, the 0.

If, then, all numbers are only zero in a state of extension, and are consequently identical with it, the question arises, what are the first finitings of zero, or as what does it appear when it is no longer merely ideal or indefinite; in short, what is the first form of the real zero, or of the essence in general?

ESSENCE OF NOTHING.

41. The ideal zero is absolute unity, or monas; it is not a singularity, such as one individual thing, or as the number 1; but an indivisibility, a numberlessness, in which neither 1 nor 2, neither a line nor a circle can be found; in short, an unity without distinction, an homogeneity, brightness, or translucency, a pure identity.

42. The mathematical monas is eternal. It succumbs to no definitions of time and space, is neither finite nor infinite, neither great nor small, neither quiescent nor moved; but it is and it is not all this. That is the conception of eternity.

Mathematics is thus in possession of an eternal principle.

43. Since all the sciences are equivalent to mathematics, nature must also possess an eternal principle.

The principle of nature, or of the universe, must be of one and the same kind with the principle of mathematics. For there cannot be two kinds of monades, nor of eternities, nor of certainties. The highest unity of the universe is thus the Eternal. The Eternal is one and the same with the zero of mathematics. The Eternal and zero are only denominations differing in accordance with their respective sciences, but which are essentially one.

44. The Eternal is the nothing of Nature.

As the whole of mathematics emerges out of zero, so must everything which is a Singular have emerged from the Eternal or nothing of Nature.

The origin of the Singular is nothing else than a manifestation of the Eternal. Thereby unity, brightness, homogeneity are lost, and converted into multiplicity, obscurity, diversity.

Unity posited manifoldly is an expansion without termination, but one that always remains the same.

Realization or manifestation is an expansion of the Eternal.

FORMS OF NOTHING.

45. The first form of the expansion or manifestation of the mathematical monas, or of 0 is + -. The + - is nothing else than the definition of 0. 0 is the reduction of the positive and negative series of numbers, upon which the whole of arithmetic depends. A series of numbers is, however, nothing else than a repetition of a + 1 or a-1; consequently, the whole of arithmetic reduces itself to + 1-1.

What, however, is a + 1, or-1? Obviously nothing else than a single + or-. The figure is quite superfluous, and only indicates how often + or-has been assumed; instead, therefore, of + 1 we can posit +; instead of-1 simply-. The series + 1 + 1 + 1 is synonymous with + + +; or instead of 3 we may posit + + +, and so on for every figure ad libitum. The figures are nothing more than shorter signs for the two highest mathematical forms or ideas of numbers. Numbers are nothing different from the ideas of numbers; they are the latter themselves, only several times posited. Essentially numbers do not exist, but only their two ideas. These ideas, however, exist an infinite number of times.

Multiplicity or real infinity is, accordingly, nothing special or particular, but only an arbitrary repetition of the Ideal, an incessant positing of the idea. The idea posited is reality, non-posited it is = 0.

46. The first multiplicity is duality, + -. This duality alters nothing in the essence of the monas, for + - is = 0. It is the monas itself only under another form. In multiplication it is thus the form alone that changes.

There are many forms, but not many essences.

47. The first or primary duality is not, however, a double unity, both members of which are of equal rank, but an antagonism, disunion, or diversity. Many diversities are multiplicity. The Many is thus complex. The first form is not therefore a simple division of zero or the primary unity, but an antagonistic positing of itself, a becoming manifold.

48. Every Finite is in the same manner only the self-definition of the Eternal. The Eternal becomes, accordingly, real, by binary self-division. When the Eternal is manifested, it is either a positive or negative. The whole of arithmetic is nothing else than a ceaseless act of positing and negating, of affirming and denying.

All realization is nothing else than the act of positing and negating. The act of positing and negating of the Eternal is called realization.

49. Positing and negating is, however, an act or function. Arithmetic is, therefore, a ceaseless process of acting or performing. Numbers are acts of the primary idea, or, properly speaking, stationary points of its function, and hence proceeds a division into the two ideas + and -. If these remain always alone nothing is added to them. They alone produce the whole science of arithmetic, and simply because they are never exhausted by the act of positing themselves repeatedly, but capable after this of again becoming suppressed. Since + is in essence nothing else than a simple positing, a mere affirmation, and-a mere suppression of this affirmation, a negation; so is the positive unity = 1 nothing but an affirmation once declared, and the whole series of numbers is a reiterated affirmation. The act of affirmation alone gives the number, and the latter is thus the definite quantity devoid of intrinsic value. Bare affirmation alone without reference to any substance is unity, duality, &c.

SOMETHING.

50. Still, however, there must be something, which is posited and negatived. The form must have a substance.

This something is the primary idea, or the very Eternal of mathematics, the zero; for + - is = 0. The + is naught else than zero affirmed; the-naught else than this + 0 negatived = - 0. Now since an affirmation once declared is = 1, so are unity and zero identical. Zero differs only from finite unity in that it is not affirmed.

51. The - is not simply the want of affirmation, but its explicit abstraction. The + presupposes the 0; the - the + and 0; the 0, however, presupposes neither + nor -. Purely negative quantities are, as is known, a nonentity, because they can only bear reference to positive quantities. The - is, indeed, the retroversion of + into 0; yet alone, therefore, it is not perfectly equal to 0. It is a retrovertent, and consequently the second act, which presupposes the positive. By the - we know what is not; the 0 is, however, a nothing in every respect. The-is the copula between 0 and +.

52. If the + is the 0 posited, so is it a nothing posited or determined. This position is, however, a number, and therefore a mathematical something. The nothing thus becomes a something, a Finite, a Real, through the simple positing of itself, and the something becomes a nothing by the removal of this self-position. The nothing itself is, however, the mere neglect of its self-position. The something, the + -, has consequently not arisen or been evolved out of nothing, or been produced from it by addition; but it is nothing itself; the whole undivided nothing has become unity. The nothing once posited as nothing is = 1. We cannot speak of production or evolution in this case; but of the complete identity and uniformity of the nothing with the something; it is a virgin product or birth.

53. Zero must be endlessly positing itself, for in every respect it is indefinite or unlimited, eternal. The number of finite singularities must, therefore, pass into the Infinite.

54. The whole of Arithmetic is nothing but the endless repetition of nothing, an endless positing and suppressing of nothing.

We can become acquainted with nothing but the nothing, for the Original of our knowledge is the 0.

There is no other science than that which treats of nothing.

Every Real, if it were such in itself, could not be known, because the possibilities of its properties would pass into the Infinite. The nothing alone is cognizable, because it has only a single property, namely, that of having none; concerning which knowledge no doubt can be entertained.