Chapter 10 Chapter 6 On the Subject's Third Class of Objects and the Forms in which the Principle of Sufficient Reason Governs

Section 35 Interpretation of such objects What constitutes the third object of our faculty of representation is the formal part of total representation, that is to say, the intuition given to us a priori of the forms of our internal and external senses, namely space and time. As pure intuition, these forms are themselves objects of the faculty of representation, without being conditional on a thorough representation and on determining whether the representations were first imposed upon them as empty or full; for even pure points and Lines also cannot provide perceptual intuition, but only a priori intuition, just as the infinite extension and infinite divisibility of space and time are only objects of pure intuition and have nothing to do with empirical intuition.In the third class of representations, space and time are pure intuitions; in the first class of representations, they are interconnected and perceived perceptually; the difference between the two classes of objects is matter, and therefore, on the one hand, I define matter as Awareness of space and time, on the other hand, defines matter as embodied cause and effect.

On the contrary, the causal form belonging to the understanding cannot by itself be the object of our power of representation, nor can we be conscious of it unless it is connected with matter in our cognition. § 36 Law of Sufficient Reason for Existence Space and time are constructed in such a way that all their parts are interrelated, one being a condition of the other, and the other being a condition of the other.We call this relation in space place; in time we call succession.These connections are quite different from all other possible connections in our representations, and are very special; therefore understanding and reason cannot be grasped by pure concepts, and only a priori pure intuition can make us understand them; because concepts alone cannot explain clearly. Up and down, left and right, front and back, past and future.Kant rightly confirms this by asserting that the difference between the left and the right glove cannot be discerned except by intuition.The law by which the parts of space-time are mutually defined by reference to these two relations (position and succession) is what I call the law of sufficient reason of existence.I gave an example of this connection in Section 15, through the relationship between the sides and angles of a triangle, to show that this connection is not only completely different from the relationship between cause and effect, but also different from the relationship between cognition and inference. relation; therefore, the condition in question may be called the ground of existence.Of course, the deep apprehension of a ground of being can become a ground of cognition, just as the comprehension of the law of causality and its other application in special cases is the ground of cognition of consequences; but this does not eliminate the ground of being. , grounds of becoming, and grounds of knowing.It happens from time to time that what appears to one form of the principle of sufficient reason is inference, and another form appears to be ground.For example, according to the law of causality, the rise of mercury in the thermometer is the inference of the increase of heat, while according to the law of sufficient reason for knowledge, it is a ground, the basis of knowing the increase of heat, and the basis of judgment for making this assertion.

Section 37 Grounds for existence in space The position of each part in space with respect to another, say a given line - and the same applies to surfaces, volumes, and points - also completely determines the position of any other possible line; The former has a relationship between inference and evidence.Since the position of the given line and any other possible line determines its position with all other lines as well, and since the position of the first two lines itself is also determined by all other lines, which It does not matter which one is considered to be the first to be ascertained and to determine the position of the other lines, i.e. it does not matter which particular line is called the reason (ratio) and the rest are called inferences (rationata).This is so because there is no succession in space; because it is by combining space and time to form joint representations of complex experience that the coexistence of representations arises.Something similar to the so-called correlation is thus everywhere found in the grounds of being in space, which we will address in Section 48, where we will deal more fully with the correlation of grounds.Since every line is determined by all other lines, and likewise determines all other lines, it is arbitrary to regard any line as merely determining rather than being determined, and a line The position with respect to any other line does not preclude the question that, in its position relative to some other line, this second position necessarily determines and enables the determination of the first position.It is therefore as impossible to find the beginning of the front in the series of links in which there is a chain of grounds, as it is to find the beginning in the series of links in a chain of grounds; nor can we find any rear end, since space is infinite. And the lines in space are also infinite.All possible relative spaces are trajectories, because they are limited; all these trajectories have their existence grounds among each other, because they are connected.Thus the series of reasons in space, like the series of becomings, proceed in infinity; moreover, not only in one direction, but, like the latter, in all directions.

All this cannot be explained; for the truth of these laws is founded directly on the intuitions of space given to us a priori, and is a priori. § 38 Grounds of existence in time, arithmetic Each moment in time is conditioned on the previous moment.The reason why the law of sufficient grounds as inferences is so simple here is that time has only one dimension, and therefore its relations cannot be multiplicity.Each moment is conditioned by the previous moment; we can only reach it through its previous moment: this can only come about in so far as the past moment existed and disappeared.All counting depends on the connection of divisible times, on which numerals serve only to mark a single stage in the succession; thus also the whole of arithmetic, which teaches us only the organized shorthand notation of calculation.Every number presupposes a pre-existing number as the basis of its existence: we can only reach ten through all the numbers before ten. Only by virtue of this knowledge do I know that there must be eight, six, Four.

Section 39 Geometry Likewise, the whole of geometry depends on the connection of divisible spatial positions.Geometry is thus the cognition of this connection; but, as we have said, it is impossible to arrive at this cognition by mere concepts, or by any other means than intuition, and every proposition of geometry must Reduction to sense intuitions, and the proof is nothing more than the clarification of the particular relation in question; nothing else matters.We find, however, that geometry is treated quite differently.Only the twelve axioms of Euclidean geometry are considered to be based on pure intuition, and even only three, the ninth, eleventh, and twelfth, are admitted to be based on different intuitions. while others are supposed to be based on the recognition that in science, as in experience, we do not deal with real things in themselves juxtaposed and subject to endless variations, but that, on the contrary, we It deals with concepts, and in mathematics pure intuition, number and shape, its laws are valid for all experience, and combine the synthesis of concepts with the clarity of single representations.For, as intuitive representations, their certainty is extremely precise—in which case there is nothing undetermined—but they are still general, because they are empty forms of all phenomena, and thus these forms apply to Of all the real objects to which these forms belong, so what Plato says of the "ideas" applies to concepts as well to these pure intuitions, even in geometry, that is, neither It may be completely similar, otherwise there would be no distinction between form and object.It seems to me that it also applies to pure intuition in geometry, without which these objects, as proper spaces, would differ from one another by their spatial arrangement, that is, their position.Plato said this long ago, as Aristotle said: "He further said that besides sensible things and ideas, there is mathematics, which is different from sensible things because it is eternal. What is immovable is also different from ideas, because many things in them are like each other; but ideas are absolutely unique."2

-------- ①Plato's "ideas" can ultimately be said to be pure intuitions, which apply not only to the formal but also to the material part of total representations—thus can be described as total representations, which are completely determined, but At the same time it includes many things, such as concepts—that is, as the embodiment of concepts, but fully suitable for them, see what I have explained in the twenty-eighth section. ② Aristotle's "Metaphysics" I. 6. Compare X. 1. Since the difference in position does not cancel the rest of the commonality, the So I think it is more in line with the nature of science to replace the other nine axioms with this knowledge, because the purpose of science is to understand the particular through general knowledge, so it is not so appropriate to express the nine axioms separately on the basis of the same concept .Moreover, Aristotle said: "It is equality that constitutes unity" can also be applied to geometric figures.

But pure intuition in time, that is, mathematics, has no distinctions in spatial arrangement, and here nothing but the identity of different things belongs likewise to concepts and nothing else: for there is only one 5 and one 7.We may also discover here the grounds for why 7+5=12 is an a priori synthetic proposition, as Kant profoundly discovered that this proposition is based on intuition rather than the law of identity, as Herder in his Critique of Metaphysics said. 12=12 is the same proposition. In geometry, therefore, we resort to intuition only in dealing with axioms.All other axioms are to be demonstrated, that is, to be given a basis for knowledge, the truth of which is to be accepted by everyone.This would show the logical truth of the theorem, but not its a priori truth (cf. Sections 30 and 32), since the latter resides in the ground of being and not in the ground of knowledge, therefore, except by Intuition can figure out nothing else.This explains why geometrical arguments of this type, while explicitly expressing the belief that the proven theorems are true, still fail to show why the theorems it proves are so.In other words, we have not found its reason for existence, but usually this will arouse our strong desire to find out the reason for its existence.Since a demonstration by showing grounds for knowing can only produce belief, not knowledge, it might be more accurately called an index rather than an argument, which is why, in most cases, when it is intuitive, the A sense of discomfort is brought about by a complete lack of knowledge; and here the desire to know why becomes all the more acute as soon as one knows exactly what it is.This impression is a lot like the feeling we get when something changes in or out of our pocket and we don't know how.In this type of argument, the grounds for knowledge established without grounds for existence are very similar to certain physical theories that only provide phenomena without explaining their causes. success in the platinum crucible; and the grounds of existence of geometrical propositions discovered by intuition, like every cognition we acquire, satisfy us.Once we have found a ground for existence, we base our belief in the truth of the theorem only on that ground, and not on the grounds of knowledge given to us by the argument.Let us look, for example, at the sixth proposition of the first book of Euclid:—

"If two angles of a triangle are equal, then the corresponding sides are also equal." Euclid's argument is as follows:— "Let abc be a triangle in which Eabc = Eacb, then side ab must be equal to side ac. "For, if the side ab is not equal to the side ac, then one of the two sides must be greater than the other. If the side ab is greater than the side ac; take bd from ba to be equal to ca, and join dc. Thus, in Fdbc and Fabc, since db Equal to ac, and bc is the common side of these two triangles, the two sides db and bc are equal to side ac and side bc respectively; Edbc is equal to Eacb, therefore, the base dc is equal to the base ab, Fdbc is equal to Fabc, the smaller The triangle is equal to the larger triangle—that is absurd. So it is not that ab is not equal to ac, but that ab is equal to ac."

In the argument we get the epistemic grounds for the truth of the proposition.But who would place confidence in the truth of geometry on such a proof?Don't we base our trust on the existence grounds of intuitive knowledge?According to the basis of existence (as a kind of necessity that does not need to be further demonstrated, only the evidence provided by intuition is admitted), draw two rays from the two endpoints of another line segment with the same slope to intersect them, and the distance between the intersection point and the two ends of the line segment must be equal; for the two angles thus produced are really but one, and only appear to be two by virtue of their relative positions; and therefore there is no ground for saying that two lines will meet nearer one terminal and farther from the other. .

It is the knowledge of the ground of being that reveals to us the necessary corollary of the conditioned condition from its condition—in this case, equilateral from equiangular—that is, shows their connection; and The basis of knowledge only shows their coexistence.And we go so far as to maintain that the usual method of proof enables us to imagine their coexistence only in an actual figure given to us as an example, and not at any rate always; , our confidence in this truth rests only on induction, on the fact that we find it true in every figure we draw.The ground of existence is not in every case as obvious as in such a simple theorem as Euclid's sixth theorem, but I still believe that it can be made obvious in every theorem, however complex it may be, Propositions can always be reduced to some such simple intuition.Moreover, our a priori awareness of the necessity of this ground of existence of every relation of space is exactly the same as our a priori awareness of the necessity of the cause of every change.Of course, in complex theorems, it is very difficult to reveal the basis of existence, but this kind of research is not for the study of geometry.Therefore, in order to make clearer the meaning of what I have said, I shall now try to find the ground of a proposition of moderate difficulty, which is not quite obvious.As a not very direct theorem, I take Theorem 16 as an example:

"In any triangle, when one side is extended, the exterior angle is greater than either of the other two interior angles." Euclid's proof is as follows:— "Suppose abc is a triangle; extend the side bc to d, then the exterior angle acd will be greater than either of the opposite interior angles bac or cba. Make the middle point e of the side ac, connect be and extend to f, let ef = eb, connect fc. Extend ac to g. Because ae=ec, be=ef; two sides ae, eb are equal to two sides ce, ef respectively; .The remaining two angles corresponding to the equilateral side in the congruent triangle are respectively equal; therefore, Ebae=Eecf. But Eecd＞Eecf, therefore, Eacd＞Ebac." "Similarly, if side bc is equally divided into two, and side ac is extended to g, it can be proved that Ebcg, that is, the opposite angle acd>Eabc." My proof of this proposition is as follows:— If Ebac is equal to Eacd, let alone >Eacd, the line ba is for ca must be in the same direction as bd (because that is what the two angles are equal to), that is, it must be parallel to bd; that is, ba and bd must never intersect; but, to form a triangle, they must be Intersect (existence basis), so it must be the opposite of the condition required by Ebac=Eacd we want to prove. For Eabc to be equal to Eacd, let alone > Eacd, the line ba must be in the same direction as bd and ac (because that is what the angles are equal to), i.e. it must be parallel to ac, that is, ba and ac must be never intersect; but to form a triangle, ba and ac must intersect, which must be the opposite of the condition required by Eabc=Eacd which we want to prove. I have made the above statement without intending to present a new scheme of mathematical argument, nor to replace Euclid's by my proof, since the nature of this proof does not lend itself to it, and in fact it presupposes The concept of parallel lines, the concept of parallel lines appeared later in Euclid.I just wish to show what the ground of existence is, and thus explain its difference from the ground of cognition, which only produces confirmation, which is a completely different thing from the cognition of the ground of existence.The sole purpose of geometry is to produce confirmation, and in this case, as I have said, there is an uneasiness which does not contribute at all to the knowledge of the grounds of being—a knowledge which, like all knowledge, is irritating. Satisfying and pleasurable—this fact is perhaps one of the reasons why otherwise eminent figures dislike mathematics so much. I am tempted to give the diagram again, though it has appeared elsewhere: for sight alone, without words, conveys the truth of Pythagoras' theorem more convincingly than Euclid's trap. The well argument (proof by contradiction) is ten times stronger. Readers who are particularly interested in this chapter can find a more detailed discussion in Section 15 of Volume I and Chapter 13 of Volume II of my masterpieces.