Zero point

“ The + - or, in other words, numbers are acts or functions. Zero is, consequently, the primary act. Zero is, therefore, no absolute nothing, but an act without substratum. Generally speaking there is, therefore, no nothing; the mathematical nothing is itself an act, consequently a something. The nothing is only postulate. ”

“ Theoretically, and in one, and by no means an unimportant sense, the Zero-Element or Nothing-side of the Universe or of a given Department of Being, is one whole half, or an equal hemisphere of the Totality of Being. ”

“ It is true, in arithmetic, that zero is a very important member of the number series. But it gets its whole importance by its contrasts and its definite quantitative relations with the other numbers. Just so here, if the Absolute is not only zero, but also real, also the goal, also the valuable, it is so by contrast with the finite search for that goal. But to suppose, as the mystic does, that the finite search has of itself no Being at all, is illusory, is Mâya, is itself nothing, this is also to deprive the Absolute of even its poor value as a contrasting goal. ”

“ As Oken says, "The ideal zero is absolute unity; not a singularity, as the number one, but an indivisibility, a numberlessness, a homogeneity, a translucency, a pure identity. It is neither great nor small, quiescent nor moved ”

“ If I had but time I could develop the whole into a science, whose certainty and accuracy would be assured. But your pedant, even if he knew its rules, would infallibly break down; because, like the skilful general, there are moments when you must fling away rule and trust to instinct—a glorious instinct, quite as infallible. I felt it all to-day, and scarcely ever was at fault. The strangest "power" I see is that of Zero, and there is one man present, who I admit, has some of this instinct with a true knowledge of the laws and seasons that relate to this Zero. ”

“ But if you ask me what I regard as the range of the realm of validity, and what I think to be the extent of possible experience, and of the truth of ideas, then I can only say that the range of valid possible experience is viewed by me as infinitely more extended than my actual human experience. From the mathematical point of view the realm of truth is in fact explicitly infinite. ”

“ You think yourself the perfection of existence, while you are in reality the most imperfect and imbecile. You profess to see, whereas you can see nothing but a Point! You plume yourself on inferring the existence of a Straight Line; but I can see Straight Lines and infer the existence of Angles, Triangles, Squares, Pentagons, Hexagons, and even Circles. ”

“ But as soon as the logical reason dominates the situation, in place of this palpable plunge into a real concrete experience, with its accompanying sensation of appalling wonder and terrible freedom, we are offered nothing but a thin, dry, barren mathematical formula called "infinity," the mere mention of which freezes the imagination at its source. ”

“ After all, there is some logic—a peculiar logic of today—in this state of being a point. A point has more unknowns than any other entity. If a point should start to move, it might become thousands of curves, or hundreds of solids. ”

“ 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. ”

“ We know the conditions which must be fulfilled, but at the same time, we see the impossibility of fulfilling them. When any one tries to persuade us of the contrary, we reflect on the idea of the infinite, and say: "No; it is a contradiction of infinity; it is not infinite, but finite." We distinguish perfectly well between the absence of the perception of the limit and its non-existence. ”

“ It should rather be realized that a fiction containing an initial but only provisional contradiction serves as a starting-point for just those investigations that are most subtle and that have far-reaching consequences. We should have no Infinitesimal Calculus, no Algebra, no Atomic Theory, no Theory of Gravitation if, to avoid all initial contradictions, we surrender the fiction of differentials, of imaginary quantities, of the atom, of action at a distance. ”

“ Human history moves upward in circles, like an aero. The circles are at times golden, sometimes they are bloody, but all have 360 degrees. They go from 0° to 10°, 20°, 200°, 360°—and then again 0°. Yes, we have returned to zero. But for a mathematically working mind it is obvious that this zero is different: it is a perfectly new zero. We started from zero to the right and came to zero on the left. Hence instead of plus zero we are at minus zero. Do you understand?
This zero appears to me now as a silent, immense, narrow rock, sharp as a blade. ”

“ 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. ”

“ Hence to assert the impenetrability of matter is simply to recognize the interconnexion between the notions of number and space, it is to state a property of number rather than of matter.—Yet, it will be said, do we not count feelings, sensations, ideas, all of which permeate one another, and each of which, for its part, takes up the whole of the soul?—Yes, undoubtedly; but, just because they permeate one another, we cannot count them unless we represent them by homogeneous units which occupy separate positions in space and consequently no longer permeate one another. ”

“ Observe in the first place, that every one of the arguments by which the relativity of our knowledge is demonstrated, distinctly postulates the positive existence of something beyond the relative. To say that we cannot know the Absolute, is, by implication, to affirm that there is an Absolute. In the very denial of our power to learn what the Absolute is, there lies hidden the assumption that it is; and the making of this assumption proves that the Absolute has been present to the mind, not as a nothing, but as a something. ”

“ In other words, the minimum in each sphere of being contains implicitly in itself the whole reality of that sphere. The minimum is its substance, not merely the ultimate of analysis, but the actual source, the [Pg 232] dynamic origin of reality, as God is implicitly the whole universe and also the source of the universe as it actually exists. It is because the minimum is all reality, is the maximum, that the knowledge of it gives us that of the whole. ”

“ Reality exists now, but the infinite has disappeared. To describe a figure in space, we are obliged to limit infinite space; to represent to ourselves a change in time, we are obliged to divide the totality of time. Thus we only arrive at reality by limitation, at the positive, at a real position, by negation or exclusion; to determination, by the suppression of our free determinableness. ”

“ Every mathematician is acquainted with the contradictions immanent in our ideas of infinity. For example, the familiar proposition that parallel straight lines meet at infinity, is a contradiction. Again, a decreasing geometrical progression can be added up to infinity, the infinite number of its terms adding up in the sum-total to a finite number. The idea of infinite space itself is a contradiction. You can say of it exactly what Zeno said of the many. There must be in existence as much space as there is, no more. ”

“ We know, in reality, that if, wishing to determine the infinite, we conceive a point above our head, this point, whatever it may be, will never appear to us to have the character in question: otherwise, that which would be situated above the point so conceived as the limit of the infinite, would be at the same moment, and by virtue of its relation to the same point, both high and low ”

“ THE inevitable rational consequences which follow with mathematical certainty when once the existence of space-time as a model of the universe is conceded, are often puzzling to beginners, who feel that these consequences are too paradoxical to be acceptable. Of course, we must distinguish between a paradox of feeling, and a paradox of logic or an inconsistency. ”

“ Sometimes, when I am going to sleep, I think of Space and wonder whether it is circumnavigated by the curves that go away to Negative Infinity and come back again from Positive Infinity, as if the two Infinities met. Sometimes I snap at people for saying Two and Two make Four as if it were an axiom, instead of being a result attained by rigid proof. ”

“ For the possibilities of thought being infinite, either all thought is excluded once for all from the possibility of error, or else to every possible truth there can be opposed an infinite mass of error. All this infinite mass is at stake upon the issue of our investigation. Total relativity, or else an infinite possibility of truth and error; that is the alternative before us. And total relativity of thought involves self-contradiction.
Every way but one has been tried to lead us out of our difficulty. ”

“ For as the absolute infinite number implies the absolute negation of all limit, when, in treating of the realization of this conception, I meet with things that are contradictory, I say that the realization of the conception is contradictory, because the general and indeterminate conception is more extended than all possible number. ”

“ The pragmatist doctrine and attitude will perhaps be easier to understand if we take it in regard to a particular instance of truth and error in regard to fundamental notions. In the last four or five years a new principle has been formulated in Physics, named the Principle of Relativity. It revolutionises the current conceptions of space and time. It is so recent that probably some of my readers now hear of it for the first time, and therefore before I refer to its formulation by mathematicians I will give a simple illustration to explain what it is. ”

“ Supposing you knew-not by sight or by instinct, but by sheer intellectual knowledge, as I know the truth of a mathematical proposition—that what we call empty space was full, crammed. Not with lumps of what we call matter like hills and houses, but with things as real—as real to the mind. ”

“ But to this idea of relativity, both in the case of goodness and every other order of phenomena, popular philosophy remains blind. Everything, for it, is either a delusion, and so nothing at all, or is on the other hand a fact, and, because it exists, therefore, as such, real. ”

“ But are not people always mistaken who think to find the perfect comprehended in the imperfect, the infinite enchained and made tangible in the finite? Bah! The same old story, the same old vicious circle, the everlastingly recurring mathematical view of things that cannot be treated mathematically; the fruitless attempt to measure the harmonious circle of the soul by the angular square of the book. What poor things our minds are after all. ”

“ The study of ancient and modern philosophic theories is useful as showing how impossible it is, for even the greatest thinkers of any age, to grasp the Absolute with our understanding or to measure the Infinite with our finite units. The propounders of all these theories seem to me to be, without exception, looking in the wrong direction for the "Reality of Being" ”

“ Because, of two propositions, one of which must be true, it proves both impossible—it proves that space cannot have a limit, because a limit is inconceivable, and yet that it must have a limit, because unlimited space is inconceivable; it proves, therefore, that space has a limit and has no limit, which is absurd. ”

“ But still this world that one thus sets up must be the true world. Else is there no meaning in the doctrine of relativity. Twist as one will, one gets not out of the whirlpool of thought. Error must be real, and yet, as common sense arranges these judgments and their relations to one another, error cannot be real. ”

“ The need of the absolute was not mathematical or scientific, [148] but emotional. Man was still tortured by his determination to be the center of things, himself the fixed absolute! The need of a familiar, fixed cave where he might run and hide, close himself in securely when the chaos of storm outside became too frightening to bear. The need of a fixed absolute, whether in philosophy or science, a fixed spot that would not shift.
The science of psi, then, was based in a willingness to shift?
He looked down at the equation, to see if he were still on the track. ”

“ It will be seen that if his way of dealing with these topics is capable of being successfully carried through, a wholly new light is thrown on the time-honoured controversies of realists and idealists, and a method is obtained of solving all that is soluble in their problem.
The speculations of the past as to the reality or unreality of the world of physics were baffled, at the outset, by the absence of any satisfactory theory of the mathematical infinite. ”