Continuity Definition Math
The fabric of the world
II. continuity and discontinuity
World mathematics
If we now want to start to transfer our results to the universe, we have to gather all contemplation. It is necessary to summarize from a formal point of view that whose contradiction is not a formal one. Mathematics operates with ideal entities, and there it is easy to uncover connections; the world equation directs material factors – how can the two be compared? –
Thoughts live close together, but things collide hard in space.says Schiller.
The step we dare to take is a fatal one. We will have to strain our imagination and power of abstraction to the utmost, and who knows whether we will succeed anyway? – At the beginning of this chapter I sought the objectivity
of mathematics as it presents itself to our point of view. Let us now remember our results of that time: the human concepts are mere pictures of the events, whose contradictions in themselves have no more to mean than color contrasts. The constructions of the concepts, whose formal connection reflects our laws of thought, have objective validity, because they are the expression of the laws, which govern, delimit and close the human being from within – the expression of that, what characterizes the human being as a product of nature; what he does not create, but receives from the universe, of which he is a part, or more exactly: what he is, as a member of the universe. Therefore those laws are not human in contrast to the universal, but only human in so far as they are universal at the same time. But we must imagine nature uniform in its laws, whether their manifestations are equally manifold and incomparable. If therefore the thinking may be incommensurable with substance and force, it obeys from inside the same basic equation, which also governs the inorganic events. How would it be now, if the laws of mathematics, which bring continuity and discontinuity in formal regard in connection, would be consubstantial with those, which form the discontinuity of the universe continuously? Should the discontinuity of the substance relate to the continuity of the force in the end as arithmetic does to geometry? Should perhaps substance and force not be merged for the same reason which separates the arithmetic discontinuum from the geometric continuum? Here it is necessary to be careful; even if our assumption is correct, it will be difficult to make a final judgment, because nowhere the connection might be more difficult to reconstruct from the projection than just here. The whole antinomy continuity-discontinuity is formally based on that of being and becoming; but this belongs exclusively to life. Chamberlain has convincingly shown that the substance only is, the force only becomes, being and becoming, however, belong exclusively to life.^{1}. Now, however, the world equation, as we recognized, must suffice substance, force and life, thus being, becoming and (being and becoming); it would consist, mathematically speaking, of the independent variables x, y and z. Is it possible to calculate x and y alone from the given z? For this it would need a second equation. But now we must remember that life is expressed only in substance and force, that these are known to us at least as a function of life. For life means to thinking a formal synthesis over and of force and substance, and there is not a single vital reaction – even the psychic ones not excluded – which could not be proved to have a physical basis. Thus we have the further equation x, y =. φ (z). Hereby, the greatest difficulty falls away: if x, y and z obey the same law, z is given, but x and y on the other hand are known as a function of z, then those must obviously be capable of a true expression in z, of an undistorted projection in geometry. Referred back to the conceptual: the antinomy of the universe of discontinuity and continuity in the form of substance and force must be capable of a reflection in that of being and becoming. So it is not that the universe both is and becomes – x and y are independent variables – but the projection of x and y on x, y, being and becoming, must give an undistorted picture, since the laws which cause and govern both antinomies are the same. This is what we wanted to know; let us now proceed to carry out the reconstruction of the connection from the projection! In the course of the first chapter we recognized that the conception of the substance, however we may place ourselves, must necessarily be an atomistic one; and yet a really satisfactory unification cannot be achieved by ever progressive analysis: whether we consider the starry sky, crystal molecules, or the atoms of the chemist, whether we sublimate the substance to the ether and dissolve it into the absurd sum of force plus empty space – always the same picture confronts the eye, which the planetary systems shine to us in the large. Does not the arithmetic teach us now exactly the same? – A line is divisible up to the infinity, and there is no part which would be so small that it would not allow again an unlimited analysis, thus could be considered again as a line; the infinitely small reflects the infinitely large, yes it is identical with it, since with formal consideration all size differences are inconsequential. And if we set as limits of all possible analysis mathematical points, discrete, identical schemes, atoms of thinkability, and consider them for themselves, we recognize that each contains in itself the whole synthesis: the point is projection of the infinite, it is beginning of an infinite continuum, yes, it contains it, since a circle of infinitely large diameter finds place in a point. This is the transition from arithmetic to geometry. And exactly in the same sense the last analysis of the substance leads to force centra, which reflect in the empty space the whole synthesis, which we tried to dissolve: the transition from substance to force proceeds, at the expense of imaginability admittedly, exactly in the same way as that from arithmetic to geometry. The discontinuity of the substance and the arithmetic mirror each other in all pieces. The essence of the force now, from our point of view, is its continuity. It may act from boundary to boundary, from atom to atom, in itself it has none. Whether it propagates with infinite speed, like gravity, or with finite speed, like electricity, whether all bodies and obstacles are transparent to it, as it is the case with the former, or whether it, like radiant energy, can be stopped and thrown back by resisting media, in itself it knows no distances nor borders. It is indivisible, indissoluble; it can be transformed, but never lost, as the law of conservation of energy teaches, and if it reaches the zero point in one direction, this end means at the same time the beginning of a new infinite continuum. One may confine it within limits, restrict its sphere of action to infinity, its essence is nevertheless never determined by these external limits; as Stallo^{2} says:
Every force is remote force, or it is not.
And just because of this its essence it cannot be unified with the substance, which is bound to spatial limits; the force remains just as infinite in its steadiness, whether it occupies space or lies buried in the tiny electron; the extension plays no role for it. Precisely for this reason, however, it can never replace the substance, no matter how we may stand; if we switch off the concept of substance, then – as we saw in the example of electricity – the force finally becomes the substance, but then also all understanding ceases: a substance which acts in the distance, which acts where it is not, which is inert in its essence and yet runs through the universe in eternal change – in this no honest man can think of anything; again, unification means only a cycle. Now, is not the essence of the force, as I present it here, completely identical with that of the geometrical continuum? The geometrical continuum is indivisible, inwardly unlimited, by no analysis whatsoever to decompose, since the synthetic unity of the structure constitutes the atom of geometry. The infinity lies in its essence, it is immanent to it. The geometric infinity can be locked up in an arithmetic point, arbitrarily set outer limits may restrict its appearance, but its inner essence remains completely untouched. And as the passage of the force through the zero point represents only the beginning of a new, differently directed continuum, so the zero point of geometry means only the beginning, from which is constructed, and the passage through it merely a change of direction of the continuum, not its end. Beginning and end coincide for geometry as well as for force. On the other hand, O and ∞, the most concrete symbols of that science, have no comprehensible sense in arithmetic. What should the infinite there, where the limitedness, the finiteness belongs to the essence, the continuum there, where there are only discrete parts, what can O have for a comprehensible sense, where it negates this sense itself? Here all understanding ceases. On the other hand, we now understand all the more clearly the formal equality of essence of force and geometry. Thus the equivalence of substance and force on the one hand and arithmetic and geometry on the other hand would be established. But the parallel goes further: modern physics, as I explained in the first chapter, tries to unify the concepts of force and matter, to merge them in that of the ether. In itself, this is certainly possible, insofar as one can operate formally with the ether just as well as with any other symbol. In non-formal respect, however, this reduction means a death leap: the ether, as a boundary term between physics and metaphysics, is an imaginary quantity for the latter. We illuminated in the first chapter how fatal this step is, because by the assumption of the ether as foundation of the universe physics would become metaphysics, whereby any understanding of nature is made impossible. Now we can consider the same facts from another point of view: a formal unification of force and substance is possible, a transition between both can be thought – but impossible to understand; there are equations in which a symbol represents both terms, only this symbol concerns an imaginary quantity. Similarly, geometry can be transferred by analysis, if not into algebra, then at least into a continuous connection with it, the arithmetic discontinuum can be transferred into the geometric continuum. The operations proceed without difficulties, only they imply the assumption of completely meaningless terms of the type . means in mathematics what the ether means in physics. We see, the parallel fits in all pieces: in formal respect substance and force relate to each other just like arithmetic and geometry; in formal relation both can be brought into each other, but in material understanding any tracing back of both terms to each other is excluded. We can no more understand substance as force than we can understand the discontinuum as continuum, the instant as eternity, the point as infinite magnitude, although the equation in mathematics is a consequence of the laws of thought themselves. So far, I think, everything would be understandable. But now it is necessary to carry out the great reinterpretation, to which our last considerations paved the way, and here it will have its difficulty with the understanding. Indeed, how can the formal unity of arithmetic and geometry be understood as a mirror image of that of matter and force? The complex numbers of the type are incomprehensible in themselves, but, quite apart from the fact that they acquire a quite concrete meaning in geometry, we are not particularly bothered by them, because mathematics is a purely formal science and operates with conventional signs to which nothing in reality needs to correspond; and, moreover, the imaginary expressions of an equation regularly cancel each other out when they are solved. The aether now – the physical equivalent of the – may also play a formal role in the equations of physics, but for these itself it means something material, and an imaginary expression which is supposed to have a concrete content – this seems an absolute absurdity. Conceptually, this is undeniable; but now we must remember two things: first, that. , the arithmetical and conceptual absurdity, acquires a completely concrete meaning in geometry, and secondly, that the human concepts, the human understanding, from the cosmic point of view means a badly subjective phenomenon, a color sensation. Should not the ether, which is an absurdity for thinking, if one remembers all its contradictory properties, possess a reality in the cosmos, just like , the arithmetic absurdum, in the constructions of geometry? Should not the human necessity of thinking, in spite of all its absurdity, be the consequence of a natural law, which actually creates such an ether? Shouldn’t the ether play a transitory role in the economy of nature as well as in the economy of mathematics? in the economy of mathematics, in such a way that the imaginary expressions of the world equation annihilate each other as soon as it gets a solution, from which it would become understandable why we always encounter in nature only firmly defined substances and forces and never ether? – I know, these questions are unanswerable; I wrote them down, because they may stimulate one or the other physicist to a more philosophical consideration of the modern theories, but mainly in order to reach quickly, by this bold leap into the realm of imagination, the level which must form the ground of our construction. The contemplation of the ether shows us all at once the way to the cosmic point of view, from which the realities of nature become abstract symbols in a formal equation, just as the human concepts, without regard to their content, formally related in mathematics, melt together into symbols empty of content and serve only as strings on which the law of the human spirit plays its melodies. We recognized mathematics as function and mirror at the same time of the laws which hold the universe together; it is this connection itself in its only accessible form, and the antinomy between the discrete material and the constant force – we found it reflected in mathematics, as a conflict between the arithmetic and the geometric way of looking at things. Now it is necessary to reduce the material of the world events to its pure form in the same sense as mathematics sublimates the thoughts in order to operate with laws of thought. We imagine the universe as a mathematical entity, the formal of which essentially coincides with the formal of mathematics. World mathematics also postulates two mutually exclusive ways of looking at things: the geometrical one with respect to force, the arithmetical one with respect to substance. Here, too, the contents of the forms exclude each other – substance and force as well as view and concept – and here, too, the forms are uniform in themselves: we know the substance only as the agent of the force, and this only by its effect on the substance, just as it is the same line which appears to us as a sum of points or as an indivisible continuum, depending on the point of view. Only the abstract symbols in human mathematics correspond to concrete realities in the equations of nature. The arithmetic point, for the human being a border concept, in the world mathematics it may correspond to the sun as well as to the smallest atom of matter. And as the same point can contain the geometrical infinite, so the sun holds as well as the atom infinite force, which knows no borders, represents a boundless continuum. The mathematics of the world is geometry, if we consider only the force, and arithmetic, if we limit our view to the substance; the substance cannot create a continuum, and the force does not permit an analysis, and nevertheless the world is a uniform connection, whose formal is faithfully reflected in the forms of thought.
1 | Cf. the whole Plato lecture from Immanuel Kant l. c. |
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2 | l. c. 205. – I would like to use this opportunity to make a short epistemological consideration about the concept of force. Already several times I emphasized that a force only becomes, never is; here an explanation might be appropriate: the concept of force expresses, generally speaking, the possibility of movements – all force effects can be understood kinetically – thus a potency, no fact. In this respect the force is not at all, and in this respect the physicists are right, who want to eliminate their concept completely as empty of content. Nevertheless, the world can be understood in a purely dynamic – i.e. of the mere force, since we can interpret every static fact, every being, also as a balance of forces – so the crystal forms, the chemical substances, etc. – as a dynamic. etc. Let us now compare both views: Kant, the father of modern dynamism, writes about the world view corresponding to this way of looking at things (Negative Größen p. 151 (Rosenkrantz):
In fact, forces which compensate each other must behave mathematically like plus and minus to each other, and these add up to zero; thus the existing equilibrium, the expression of being in nature, the basic fact of every static problem is dynamically equal to zero! This apparent paradox reveals to us in the most drastic way the difference of the dynamic from the static world view: if we start from the concrete being, fixed in material borders, then these borders are the positive, the real. With dynamic questioning, with exclusive attention to becoming, they are not at all; and likewise the reverse is true. We could express ourselves in the following way: arithmetically or statically considered, every movement is a non-being or not-yet-being; only after its cancellation is something. Conversely, the static being, viewed geometrically or dynamically, says only a not-becoming; it does not exist therefore also in the actual literal sense; the becoming alone is the positive. Here we possess a particularly clear picture for the antinomy of being and becoming: the starting point of the static – the being, the equilibrium – means the end for the dynamic. If the equilibrium is reached, nothing happens any more – ends at the same time every possibility of dynamic world view. The latter realization is now of fundamental importance: Forces, as far as they belong to a coherence – like positive and negative electricity – exist as is known only relative to each other – negative electricity without a positive correlate would be an absurdity. The term |
Hermann Keyserling The Structure of the World – 1906 Attempt of a critical philosophy © 1998- School of the Wheel
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9 Answers
SOURCE: WIKIPEDIA Continuity (from Latin continuitas, “synonymous”) denotes a gapless connection, a steadiness, a flowing transition, a connection not interrupted by any boundary; an uninterrupted, uniform progress. It characterizes sequences and processes that run steadily and can change uniformly in one direction. Abrupt, erratic changes are not to be expected as long as the influencing factors remain constant. This results in increased predictability and thus security with respect to the process. In mathematics, “continuous” means the same as “steady”. Outside of technical language usage, “continuous” is synonymous with “ongoing”. The opposite of continuity is discontinuity. The Law of Continuity (Design Law): Stimuli that appear to be a continuation of preceding stimuli are considered to belong together. Continuity, also in the sense of return in contexts. This is extremely important. After all, there is discontinuity! One can also say: continuity is the unity of continuity and discontinuity.
- can google
- under continuity understands times a steitgen process the uniformly and continuously runs..eben kontuinierlich…..also advancement in the planmäßig runs and not sporadically (jumpy)
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