Hegel’s Philosophy of Nature

Mathematics

§ 197.

(1) The first or immediate determination of nature is the abstract generality of its self-externality,-its unmediated indifference, space. It is the wholly ideal juxtaposition, because it is being outside of itself and absolutely continuous, because this being apart from itself is still entirely abstract, and has no specific difference within itself.

Much has been said, from different theoretical positions, about the nature of space. I will mention only the Kantian determination that space is, like time, a form of sensory intuition. It has also become customary to establish fundamentally that space must be regarded only as something subjective in representation. Disregarding what, in the Kantian conception, belongs to subjective idealism and its determinations (cf § 5), the correct determination remains that space is a mere form, i.e., an abstraction, that of immediate externality. - To speak of points of space, as if they constituted the positive element of space, is inadmissible, since space, on account of its lack of differentiation, is only the possibility and not the positing of that which is negative and therefore absolutely continuous. The point is therefore rather the negation of space.-This also settles the question of the infinitude of space. Space is in general pure quantity (§ 53f), though no longer as a logical determination, but rather as existing immediately and externally. Nature, consequently, does not begin with quality but with quantity, because its determination is not, like logical being, the absolute first and immediate, but essentially a mediated being, a being external to and other than itself

§ 198.

Space has, as the concept in general (and more determinate than an indifferent self-externality) its differences within it: (a) in its indifference these are immediately the three dimensions, which are merely diverse and quite indeterminate.

But geometry is not required to deduce that space necessarily has precisely three dimensions, for it is not a philosophical science, and may therefore presuppose space as its object. Moreover, even apart from this, no thought is given to the demonstration of such a necessity. The necessity rests on the nature of the concept, whose determinations, however, because they depict themselves in these first elements of being apart from themselves, in abstract quantity, are only entirely superficial and a completely empty difference. One can also, therefore, not say how height, length, and width differ from each other, because they only ought to be different, but are not yet differences.-Height has its more precise determination as direction according to the center of the earth, but this does not at all concern the nature of space for itself Following from this point it is equally as indifferent whether this direction is called height or depth, or length or breadth, which is also often called depth.

§ 199.

(b) But the difference of space is essentially a determinate, qualitative difference. As such it is (a) first, the negation of space itself because this is immediate and undifferentiated self-externality, the point. (b) The negation as negation, however, is itself spatial, and the relation of the point to space is the line, the first otherness of the point. (c) The truth of the otherness is, however, the negation of the negation. The line, therefore, passes over into the plane, which on the one hand is a determinacy opposed to line and point, and thus is plane in general, but on the other hand is the suspended negation of space, and thus the re-establishment of spatial totality, which, however, now contains the negative moment within itself an enclosing surface, which splits off an individual, whole space.

That the line does not consist of points, nor the plane of lines, follows from their concepts, for the line is the point existing outside of itself relating itself to space, and suspending itself and the plane is just as much the suspended line existing outside of itself.-Here the point is represented as the first and positive entity, and taken as the starting point. The converse, though, is also true: in as far as space is positive, the plane is the first negation and the line is the second, which, however, is in its truth the negation relating self to self the point. The necessity of the transition is the same.-

The other configurations of space considered by geometry are further qualitative limitations of a spatial abstraction, of the plane, or of a limited spatial whole. Here there occur a few necessary moments, for example, that the triangle is the first rectilinear figure, that all other figures must, to be determined, be reduced to it or to the square, and so on.-The principle of these figures is the identity of the understanding, which determines the figurations as regular, and in this way grounds the relationships and sets them in place, which it now becomes the purpose of science to know.

It may be noted in passing that it was an extraordinary notion of Kant's to claim that the definition of the straight line as the shortest distance between two points is a synthetic proposition, for my concept of straightness contains nothing of size, but only a quality. In this sense every definition is a synthetic proposition. What is defined, the straight line, is in the first place the intuition or representation, and the determination that it is the shortest distance between two points constitutes in the first place the concept (namely, as it appears in such definitions, cf. § 110). That the concept is not already given by the intuition constitutes precisely the difference between the two, and is what calls for a definition. That something seems to the representation to be a quality, though its specificity rests on a quantitative determination, is something very simple, and also the case for example with the right angle, the straight line, and so on.

§ 200.

(2) Negativity, which as point relates itself to space and in space develops its determinations as line and plane, is, however, in the sphere of self-externality equally for itself and appearing indifferent to the motionless coexistence of space. Negativity, thus posited for itself is time.

§ 201.

Time, as the negative unity of being outside of itself, is just as thoroughly abstract, ideal being: being which, since it is, is not, and since it is not, is.

Tune, like space, is a pure form of sensuousness, or intuition; but, as with space, the difference between objectivity and a contrastingly subjective consciousness does not matter to time. If these determinations are applied to space and time, then space is abstract objectivity, whereas time is abstract subjectivity. Time is the same principle as the I = I of pure self-consciousness; but the same principle or the simple concept still in its entire externality, intuited mere becoming, pure being in itself as sheer coming out of itself. Time is just as continuous as space, for it is abstract negativity relating itself to itself and in this abstraction there is as yet no real difference.

In time, it is said, everything arises and passes away, or rather, there appears precisely the abstraction of arising and falling away. If abstractions are made from everything, namely, from the fullness of time just as much as from the fullness of space, then there remains both empty time and empty space left over; that is, there are then posited these abstractions of exteriority.-But time itself is this becoming, this existing abstraction, the Chronos who gives birth to everything and destroys his offspring.-That which is real, however, is just as identical to as distinct from time. Everything is transitory that is temporal, that is, exists only in time or, like the concept, is not in itself pure negativity. To be sure, this negativity is in everything as its immanent, universal essence, but the temporal is not adequate to this essence, and therefore relates to this negativity in terms of its power. Time itself is eternal, for it is neither just any time, nor the moment now, but time as time is its concept. The concept, however, in its identity with itself I= 1, is in and for itself absolute negativity and freedom. Time, is not, therefore, the power of the concept, nor is the concept in time and temporal; on the contrary, the concept is the power of time, which is only this negativity as externality.-The natural is therefore subordinate to time, insofar as it is finite; that which is true, by contrast, the idea, the spirit, is eternal. Thus the concept of eternity must not be grasped as if it were suspended time, or in any case not in the sense that eternity would come after time, for this would turn eternity into the future, in other words into a moment of time. And the concept of eternity must also not be understood in the sense of a negation of time, so that it would be merely an abstraction of time. For time in its concept is, like the concept itself generally, eternal, and therefore also absolute presence.

§ 202.

The dimensions of time, the present, future, and past, are only that which is becoming and its dissolution into the differences of being as the transition into nothingness, and of Nothingness as the transition into being. The immediate disappearance of these differences into individuality is the present as now, which is itself only this disappearance of being into nothingness, and of nothingness into being.

(1) The finite present is differentiated from the infinite in that the finite is the moment now and hence as its abstract moments, as past and future, which is different from the infinite as from the concrete unity. Eternity as concept, here contains these moments in itself and its concrete unity is therefore not the moment now, because it is motionless identity, concrete being as universal, and not that which is disappearing into nothingness, as becoming.-Furthermore in nature, where time is now, there does not occur the subsisting difference of these dimensions; they are necessarily only in subjective representation, in memory, fear, or hope. The abstract past, however, and future of time is space, as the suspended space is at first the point and time.

(2) There is no science of time in opposition to the finite science of space, geometry, because the differences of time do not have the indifference of being outside of itself which constitutes the immediate determinacy of space, and therefore they can not be expressed as spatial configurations. The principle of time only reaches this ability when the understanding has paralysed it and reduced its negativity to the unit. This motionless unit, as the sheer carnality of thought, can be used to form external combinations, and these, the numbers of arithmetic, can themselves be brought under the categories of the truth as intuition or as understanding merely for itself because the latter is only abstract, whereas the former is concrete. This dead unit, now the highest externality of thought, can be used to form external combinations, and these combinations, the figures of arithmetic, can in turn be organised by the determination of the understanding in terms of equality and inequality, identity and difference. The science which has unity as its principle is therefore constituted in opposition to geometry.

(3) The name of mathematics has moreover been used for the philosophical observation of space and time, because it lies close to this observation, despite the fact that mathematics, as noted, considers strictly the determinations of magnitude of its objects and not time itself but only the unit in its configurations and connections. To be sure, time becomes in the theory of movement an object of science, but applied mathematics is generally not an immanent science, precisely because it involves the application of pure mathematics to a given material and its determinations as derived from experience.

(4) One could still, however, conceive the thought of a philosophical mathematics, namely, as a science which would recognise those concepts which constitute what the conventional mathematical science of the understanding derives from its presupposed determinations, and according to the method of the understanding, without concepts. However, since mathematics is the science of the finite determinations of magnitude, which remain fixed in their finitude and valid, and should not change in transit, thus it is essentially a science of the understanding. And since it has the ability to express spatial figures and numbers, which gives it an advantage over other sciences of this kind, it ought to retain this ability for itself and to avoid contamination by either concepts, like time, which are heterogeneous to it, or empirical purposes. It therefore remains open for the concept to establish a more fundamental consciousness than has hitherto been shown, both in terms of the leading principles of the understanding and in terms of order and its necessity in arithmetical operations, as well as in the theses of geometry.-If one wanted to treat the forms of space and the unit philosophically, they would lose on these grounds their particular significance, a philosophy of them would become a matter of logic, or would even assume the character of another concrete philosophical science, according to the ways one imparted a more concrete significance to the concepts.-

It would, however, be a superfluous and thankless task to try to use such an unmanageable and inadequate medium as spatial figures and numbers for the expression of thoughts, and to treat them violently for this purpose. For the specific concept would always be related only externally to them. The simple elementary figures and numbers can in any case be used as symbols, which, however, are a subordinate and poor expression for thoughts. The first attempts of pure thought took recourse to such aids: the Pythagorean system of numbers is the famous example of this. But with richer concepts these means became completely unsatisfactory, since their external juxtaposition and contingent combination are not at all appropriate to the nature of the concept, and make it altogether ambiguous which of the many possible relationships in complex numbers and figures should be adhered to. Besides, the fluid character of the concept is dissipated in such an external medium, in which each determination falls into the indifferent being outside the others. This ambiguity could only be removed by an explanation. The essential expression of the thought is in that case this explanation, and this symbolising is an empty superfluity.

Other mathematical determinations, such as infinity and its relationships, the infinitesimal, factors, powers, and so on, have their true concepts in philosophy itself. It is awkward to want to take and derive these from mathematics, where they are employed in a nonconceptual, often meaningless way; rather, they must await their justification and significance from philosophy. The truly philosophical science of mathematics as theory of magnitude would be the science of measures, but this already presupposes the real particularity of things, which is only at hand in concrete nature.

§ 203.

(5) Space and time constitute the idea in and for itself, with space the real or immediately objective side and time the purely subjective side. Space is in itself the contradiction of indifferent being outside of others and undifferentiated continuity, and thereby the pure negativity of itself and the transition into time. Space converts into the individuality of the place. Time is, equally, since its moments held together in unity suspend themselves immediately, the immediate convergence into indifference, into undifferentiated being apart from one another, or into space, so that its place is precisely in that way immediate as sheer indifferent spatiality. This disappearance and regeneration of space in time and of time in space is motion;-a becoming, which, however, is itself just as much immediately the identically existing unity of both, or matter.

The transition from ideality to reality, from abstraction to concrete existence, in this case from space and time to reality, which appears as matter, is incomprehensible to the understanding, and always converts therefore externally for the understanding, and as a given entity. The usual conception is to take space and time as empty and to be filled with matter from the outside. In this way material things are, on the one hand, to be taken as indifferent to space and time, and on the other hand to be taken at the same time as essentially spatial and temporal.

What is usually said of matter is: (a) that it is composite; this refers to its identity with space. Insofar as abstractions are made from time and from all form generally, it is asserted that matter is eternal and immutable. In fact, this follows immediately, but such a matter is also only an untrue abstraction. (b) It is said that matter is impenetrable and offers resistance, is tangible, visible, and so on. These predicates mean nothing else than that matter exists, partly for specific forms of perception, in general for an other, but partly just as much for itself Both of these are determinations which belong to matter precisely because it is the identity of space and time, of immediate being apart from itself or of becoming.

The transition of ideality into reality is demonstrated therefore in the familiar mechanical phenomena, namely, that ideality can take the place of reality and vice versa; and only the usual thoughtlessness of the representation and of the understanding are to blame that, for them, their identity does not derive from the interchangeability of both. In connection with the lever, for example, distance can be posited in the place of mass and vice versa, and a quantum of the ideal moment produces the same effect as the corresponding real moment.

Similarly, velocity, in the magnitude of motion, the quantitative relationship of space and time, represents mass, and conversely, the same real effect emerges if the mass is increased and the velocity proportionately decreased. By itself a brick does not kill a person, but produces this effect only though the velocity it achieves, in other words, the person is killed through space and time.

It is force, a category of reflection fixed by the understanding, which presents itself here as the ultimate, and therefore prevents understanding and lets it seem superfluous to inquire further after the concept. But this at least appears without thought, namely, that the effect of force is something real and appealing to the senses, and in force there is realised that which is in its expression; indeed, it appears that force achieves precisely this force of its expression through the relationship of its ideal moments, of space and time.

Further, it is also in keeping with this nonconceptual reflection that "forces' are seen as implanted in matter, and as originally external to it, so that this very identity of time-and space, which vaguely appears in the reflective category of force, and which in truth constitutes the essence of matter, is posited as something alien to it and contingent, something introduced into it from outside.