Chapter 15 Part I The World as Representation §15

We have been convinced that intuition is the highest source of all evidence, and that no absolute truth can be established except directly or indirectly upon intuition, and that the shortest path is also the surest, since concepts intervening are bound to fail. and when we look at mathematics with this belief, at what Euclid established as a science and which in general has survived to the present day, I say that we cannot avoid thinking that mathematics The road is both strange and upside down.What we want is to reduce a logical basis to an intuitive basis. Mathematics, on the contrary, has to go to great lengths to make things difficult and abandon its own intuitive basis, which is always close at hand, in order to replace it. Take logical evidence.We cannot but think of this as a man who saws off his legs in order to walk with a stick, or as the prince in The Triumph of Sensibility escapes from the real beauty of nature in order to admire and imitate it. Landscape stage set.Here I have to recall what I have said in Chapter 6 of "The Law of Reason", and assume that this is also fresh in the reader's memory, as if it is at present.In this way, my statement here can be linked with what I said there, without pointing out again that the difference between the mere ground of knowledge of a mathematical truth and its ground of existence is that the former can be obtained by logical means, and the latter is space and time. The direct relationship between various parts, which is known by intuitive means.Only the understanding of this connection can be really satisfying and can provide thorough knowledge; if it is only the basis of knowledge, it will always stay on the surface of things. Although it can give people the knowledge that things are so, it cannot give people knowledge. The knowledge of why [things] are the way they are.Euclid took the latter path, which is obviously not conducive to science.For example, he should have shown once and for all from the outset how in a triangle the angles and sides determine each other, how they cause each other; All forms; it should be noted that this form, in the relation of the angles and sides of a triangle, as everywhere, entails the necessity that one thing is so by virtue of something else quite different. Like that.He does not allow people to have a thorough understanding of the nature of triangles in this way, but proposes arbitrarily chosen propositions about some segments of triangles, and provides logical grounds for these propositions through logically difficult proofs obtained according to the law of contradiction. People are not Having acquired all the necessary knowledge of these spatial relations, one obtains only arbitrarily conveyed results from these relations; it is like showing a man a fine machine and only telling him of its different functions, without It is the same as not telling him the internal structure and operation principle of this machine.Everything that Euclid proved is like that, and people are forced to admit it because of contradictions, but why it is like that, it is impossible to know.So people have an uncomfortable feeling almost as if they have seen a magic show. In fact, most of Euclid's proofs are obviously like magic.Truth almost always slips in by the back door, because it emerges by accident from some incidental circumstance.An indirect counter-evidence often closes one door after another, leaving only one door open, which is the door through which people have no choice but to enter.Often in geometry, as in the Pythagorean theorem, straight lines are required to be drawn, without being able to understand why they are done; Convinced by the learner, the learner has to bow down and admit something that he doesn't understand the situation at all.So much for the fact that a student can read Euclid from cover to cover and still not gain any real understanding of the laws of spatial relations, but merely recite some of the results derived from these laws.This kind of empirical, non-scientific knowledge is like a doctor. Although he knows which medicine to use for a certain disease, he does not understand the relationship between the two.All this is due to people's whims, rejecting the own way of seeking proof and evidence of a cognitive type, and arbitrarily replacing it with a way that is incompatible with this type.At the same time, in other respects Euclid's method of carrying out his proposition is admirable, and this has been the case for so many centuries, that it has been declared that his method of governing mathematics is the first of all scientific discourse. It is a model for all other sciences to emulate; but people come back from here later, not knowing why.In my eyes, the method used by Euclid in mathematics can only be regarded as a very "brilliant" error.Any delusion of a large scale, intentional and systematic, and later universally sanctioned, may concern life as well as science; it may almost always find its grounds in the authoritative philosophy of the time.It was the Elijahians who first discovered the distinction, and more often the conflict, between things of intuition and things of thought, and made extensive use of this distinction in their philosophical aphorisms, the Sophistry.After the Elijahs came the Macquaries, the Dialectics, the Sophists, the New Schools, and the Skeptics; they pointed out that it was the illusions, that is, the delusions of the senses, or rather of the understanding. mistaken.The understanding turns the data of the senses into intuitions, and often makes us see things whose unreality is apparent to the reason;Knowing that perceptual intuition is not infallible, people have made premature conclusions that only rational and logical thinking can establish truth; in fact, Plato (in "Parmenides"), the Macquarie school, Pyrrhon and the New School Splash have pointed out in some examples (such as those used later by Sextus, Embire Gurkin) that inferences and concepts, on the other hand, also lead to errors and even absurd inferences. And sophistry, saying that these things are easier to produce than illusions in perceptual intuition, but more difficult to explain.At that time, rationalism, which was opposed to empiricism, prevailed, and Euclid followed rationalism in dealing with mathematics, so he only based axioms, helplessly, on intuitive proofs, and everything else on inferences. superior.His method has been authoritative throughout all [past] centuries; and so long as a priori pure intuition is not distinguished from empirical intuition, it must continue to do so.Although Euclidean commentator Proclus seems to have seen this difference, for example, Kepler translated a passage into Latin in his "The Harmony of the World", which is the original work of this commentator in This is shown; but Proclus does not pay much attention to this matter, he brings it up in isolation, he is not noticed, and he does not follow it to the end.So Kant's theory, which was destined to bring about such a great change in the knowledge, thought, and behavior of the peoples of Europe, would not bring about the same change in the field of mathematics until two thousand years later.For it is only from this great philosopher that we understand that the intuition of space and time is entirely different from that of experience, that it is utterly independent of all impressions of the senses, that it determines the senses and is not determined by them, that is, that the intuition of space and time is Transcendental, and therefore absolutely impervious to the delusions of the senses; and only after we have learned this can we understand that Euclid's logical methods in mathematics are superfluous prudence, like a crutch to a sound leg. It's like pedestrians who regard the white dry road as water at night. They are afraid of stepping into the water. They would rather take one step higher and one step lower on the side of the road, and walk one step after another, thinking that they have not encountered the water that did not exist.Only now have we been able to say with certainty that when we directly observe a geometric figure, it necessarily appears to us before us, neither from an imprecise drawing on paper, nor from what we imagine while looking at it. abstraction.Rather, it comes from the form of all a priori knowledge in our consciousness.This form is everywhere the principle of reason; here, as the form of intuition, space is the principle of reason of being.The self-evidence and properness of the law of reason of existence and the self-evidence and properness of the law of reason of cognition, that is, the truth of logic, are of the same magnitude and directness.Therefore, we don't need to, and we can't leave the field of mathematics itself in order to believe in the latter alone, and seek mathematical proofs in two fields that have nothing to do with mathematics, the field of concepts.If we stick to mathematics in its own right, we gain a [great] advantage in that what we know in mathematics as "so" and "why" are now one thing, not Euclid completely split it into two matters, only allowed to know the former, not allowed to know the method of the latter.In fact, Aristotle said very pertinently in Section 27 of the first part of "Post-analysis": "Telling us at the same time that there is 'a thing' and the knowledge of 'why it is' is better than telling us separately how things are and why they are. The knowledge of things should be more accurate and superior." In physics, our satisfaction can only be achieved by the unity of knowledge of how things are and why they are.To know that the column of mercury in Torricelli's tube is higher than twenty-eight inches is insufficient knowledge without also knowing that it is so due to the pressure of the air.But can the hidden properties in the mathematics garden, such as [knowing] that the line segments of two intersecting chords in a circle always form the same rectangle, satisfy us?The "so" here has been proved by Euclid in the thirty-fifth theorem in the third volume, but the "why" is still not explained.Likewise, the Pythagorean theorem tells us about a hidden property of right triangles.Euclid's pretentious and well-thought-out proofs can't be avoided as soon as it comes to "why is this so", but the following simple and familiar figures are much stronger than his proofs at a glance.This figure gives us a penetrating understanding of the matter, a firm understanding of the necessity [above] in our hearts, and the dependence of [above] on the right angle: when the two sides of the Pythagorean are not equal, we must Of course, solving problems can also start from this intuitive understanding.This is true of any possible truth of geometry, simply because every discovery of such a truth proceeds from this intuitive necessity, and the proofs are added after the fact.Therefore, people only need to analyze the thinking process when they first found out a geometric truth, and they can intuitively know its inevitability.I would like mathematics to be taught in an analytical way at all and not in the synthetic way Euclid used.It is true that analytical methods have great difficulties with complex mathematical truths, but they are not insurmountable difficulties.In Germany there have been repeated initiatives to change the way mathematics is taught and to advocate more of this analytical approach.The most resolute in this regard was Mr. Gosack, the teacher of mathematics and physics at the Liberal Arts School in Nordhausen, because he appended a detailed explanation to the syllabus of the school examination on April 6, 1852. , [which is] how to try my principles for geometry.

In order to improve the method of mathematics, it is first of all required to abandon the preconception that proven truths are somehow superior to intuitive truths, or that logical truths based on the law of contradiction are superior to metaphysical truths; [In fact] the latter is immediately self-evident, and the pure intuition of space also belongs to the [self-evident] truth. What is most true and which cannot be explained in any way is the content of the principle of sufficient reason.For the principle of sufficient reason, in its individual forms, signifies the general form of all our representations and "knowledges."All explanations are reduced to the principle of sufficient reason, pointing out the relationship between appearances in individual cases, and these relationships are basically expressed by the law of sufficient reason.Therefore, the principle of sufficient reason is the principle on which all explanations [are based], so that it cannot explain itself, and does not need an explanation.Every explanation presupposes it and has meaning only through it.But there is no distinction between its forms; it is equally true as the principle of being, or of becoming, or of action, or of knowing, The same is not provable.In all its forms, the relation of ground and consequence is a relation of necessity; this relation is at all the highest source of the concept of "necessity," that is, its only meaning.If there is a ground, then there is no necessity but that of the consequence, and there is no ground which does not lead to the necessity of the consequence.Therefore, deriving the consequences in the conclusion from the existing cognition grounds in the premise is equally reliable as determining the spatial consequences from the spatial existence grounds.If I intuitively understand the relationship between the basis of existence and its consequences in this space, then this truth is equal to the truth of logic.And every theorem of geometry is an expression of this relation, and is as true as any of the twelve axioms.This expression is a metaphysical truth, and as such it is as immediately true as the contradiction itself.The law of contradiction is a super-logical truth and the universal basis of all logical proofs.Whoever denies the necessity shown in intuition of the spatial relations expressed by the theorems of geometry has an equal right to deny those axioms, the consequences deduced from their premisses, and even the contradiction itself; for all these are equally Some relationships that cannot be proved, are directly self-evident, and can be known a priori.Therefore, the relationship of space has a necessity that can be directly recognized, but people have to use a logical proof to derive this necessity from the law of contradiction; It was as if the lord leased the land to him.But that's what Euclid did.He was only compelled to ground his axioms on direct evidence, after which all geometrical truths were to be proved logically, that is to say, from the agreement of axioms and theorems on the premise of those axioms. Prove the assumptions made in , or the previous theorems, or prove the contradictions of the assumptions, the axioms, the previous theorems, or even the theorems themselves from the opposite side of the theorems.But the axioms themselves do not have any more direct proofs than any other theorems of geometry, but are simpler because they are less content.

When a prisoner is interrogated, his confession is always recorded in order to judge the authenticity of the confession from its consistency.But this is only a measure of last resort; it would not be possible to study the veracity of each confession directly, because the prisoner can also justify his lies from beginning to end.But [by the consistency of confession alone,] this is the way Euclid was supposed to study space.Though he proceeds from the correct premise that nature, being uniform everywhere, must also be uniform in space in its elementary forms; and since the parts of space are In the relationship of mutual ground and consequence, therefore, there is no determination of space that can be different from its original form without contradicting all other determinations.But this is a burdensome and unsatisfactory detour, which regards indirect knowledge as preferable to equally true direct knowledge; Big bad for science.In the end, it completely blocks the beginner's understanding of the laws of space, and does not even make him used to the real search for the basis and the inner connection of things; instead, it induces him to take the historical knowledge of "things are like this" as his own. foot.This method is often praised for exercising discrimination, but it is really nothing more than the memory exertion of the student in order to retain all that material, [because] the correspondence between the materials is to be compared.

In addition, it is worth noting that this method of proof is only used in geometry and not in arithmetic.In arithmetic one really only uses intuitions to clarify truths, and intuitions here are mere counting.Because the intuition of number is only in time, it cannot be expressed by perceptual graphics like geometry, which removes a worry, [don't worry] the intuition is only empirical, so it is inevitable to be confused by illusions.It turned out that it was only this concern that could bring the logical method of proof into geometry.Because time has only one direction, counting is the only arithmetic operation, and all other operations must be reduced to this operation.This counting is nothing but a priori intuition.One can invoke this intuition here without hesitation; only by virtue of this intuition is everything else, every calculation, every equation finally confirmed.For example, instead of proving, people use pure intuition in time, counting, which turns every individual proposition into an axiom.So the whole of arithmetic and algebra is not full of proofs in geometry, but just a way of simplifying counting.The intuition of numbers we have obtained in time is as mentioned above, probably only up to "ten", and cannot be more; beyond this, there must be an abstract concept of "number", a concept fixed in a word, rise instead of intuition.The intuition is therefore no longer really complete, but is only marked with complete exactness.In this case, due to the important auxiliary tool of the natural order of numbers, it is still possible to use the same small number to replace a larger number [with the same value], and it is still possible to make any calculation intuitively obvious.This is the case even when people use abstraction to a high degree; not only numbers are thought in abstraction, but also indefinite quantities or whole calculus, which can be marked by symbols in this sense, for example; No more calculations, just a hint.

With the same right and with the same validity in geometry as in arithmetic, one can only take as grounds of truth a priori pure intuition.In fact, what endows geometry with greater self-evidence is always this intuitively recognized necessity according to the law of reason of existence.It is on this self-evident ground that the truth of the theorems of geometry is established in everyone's consciousness, and not at all on artificial logical proofs.Logical proofs are always too distant from the matter, and are mostly soon forgotten; but forgetting does not impair [one's] certainty.Even if there is no logical proof at all, it will not reduce the self-evidence of geometry. This is because the self-evidence of geometry does not need to be proved by logic. s things.This is equivalent to a cowardly soldier stabbing an enemy killed by others, and bragging that he killed the enemy.

With all the above, it is hoped that people will no longer doubt that since the self-evident truth in mathematics has become the model and symbol of all self-evident truths, it is not based on proofs in essence but on direct intuition. .So here, as everywhere, intuition is always the source and final ground of all truth.And the intuition on which mathematics is based has a great advantage over any other intuition, viz. empirical intuition; that is, the intuition on which mathematics is based is a priori and therefore independent of experience; It is part by part, obtained sequentially. For the transcendental intuition, [no distinction is made between distant and near], everything is present at the same time, and people can start from the basis or the consequence at will.This gives a full and infallible validity to the a priori intuition on which mathematics is based, since in this intuition the effect is derived from the knowledge of the cause, and this is the only knowledge that has necessity.For example, the equality of three sides in a triangle is considered to be based on the equality of angles.On the contrary, all empirical intuitions and most of the experiences only recognize the cause from the effect in reverse. This method of knowing cannot be said to be without errors, because the effect can be said to be necessary only after the cause has been given; There can be no such necessity in assuming the cause from the effect, since the same effect may arise from different causes.The latter way of knowing is always just inductive, that is, pointing to a cause from many consequences and assuming that the cause is correct.But since individual cases can never be collected in one place, such truths are by no means infallible.However, all perceptual and intuitive cognitions and most experiences have only this kind of truth.The perception of the senses prompts the understanding to draw a conclusion from the effect to the cause, but inferences from the [effect] produced by the cause to the cause are by no means reliable, so that a false appearance as perceptual delusions is possible; and if As mentioned above, it also occurs frequently.As long as some or all of the five senses are directed to the same cause, the possibility of false appearance is reduced to a minimum, but not completely eliminated.For in some cases, as with counterfeit coins, one deceives all senses.All empirical knowledge, and thus the whole of natural science, apart from its pure (what Kant called metaphysical) parts, are also in the same case.Here too, the cause is recognized from the effect, so all theories about nature are based on assumptions.Assumptions are often wrong, and wrong assumptions can only gradually give way to more correct assumptions.It is only in experiments carried out deliberately that the process of cognition proceeds from cause to effect, that is, the sure path followed; but these experiments themselves proceed on hypotheses.Hence no branch of natural science, such as physics, astronomy, or physiology, can be discovered at once, like mathematics or logic, but has required, and still requires, centuries of collected, compared experience. .Only after repeated empirical verification can the induction on which the hypothesis is based be brought so near to a degree of completeness that this degree of completeness can in practice take the place of exactness.This source of completeness, then, is hardly thought to be of any disadvantage to hypotheses, any more than the incommensurability of straight lines and curves is thought to be of any disadvantage to the application of geometry, or that "logarithms" never reach Perfect precision does no harm to arithmetic.It turns out that just as people [can] use infinite fractions to make the circle infinitely close to the square, and make the logarithm infinitely close to the exact, similarly, people [can] use the induction method—that is, to recognize the cause from the consequences knowledge—though not infinite, yet so close to mathematical self-evidence—that is, knowledge from cause to effect—that the possibility of error is so small as to be negligible.But the possibility of error, though small, still exists; for example, to infer all the circumstances from the innumerable circumstances, and in fact to infer the unknown cause from which all the circumstances are based, is an inductive inference.Is there a more reliable statement in this kind of statement than "people's hearts are all on the left"?Yet, on the rarest of occasions, in rare exceptions, there are people whose hearts are on the right side. —Therefore, both sensuous intuition and empirical science have evidence of the same kind.Compared with sensible intuition and empirical science, mathematics, pure natural science and logic, have the advantage as a priori knowledge only in that the formal aspects of cognition in which all a priori are derived are given completely and simultaneously; therefore , In mathematics, pure natural science and logic can often go from cause to consequence; while in perceptual intuition and empirical science, most of them can only go from consequence to cause.In other respects, the law of causality itself, that is, the variable principle of sufficient reason which guides empirical knowledge, is as valid as the other forms of the principle of sufficient reason to which the above-mentioned [pure] sciences are subject a priori. - Logical proofs or inferences derived from concepts have the same advantage as a priori intuitive cognitions in deriving consequences from causes, so that these inferences are also incapable of error in themselves, i.e., in form .It helps that Proof enjoys such a high rating at all.But the infallibility of logical proof is only relative.These proofs are just generalizations under the highest propositions of a science, and these highest propositions are the summation of all the truths of this science, so they cannot be just proofs, but must be based on intuition.This intuition is pure in the aforementioned few a priori sciences, otherwise always empirical, and can only be raised to the general by induction.Therefore, although the particular can be proved from the universal in empirical science, the universal obtains its truth from the particular. The universal is a storehouse for storing equipment, but it is not the soil that can produce itself.

Much has been said about the proof of truth.As to the source and possibility of error, attempts have been made since Plato to explain them.Plato's answer is figurative, saying that a fallacy is like catching a wrong pigeon in a pigeon coop;Regarding the source of fallacies, Kant’s explanations are empty and vague. He used the graph of diagonal movement as an illustration. You can refer to page 294 of the first edition and page 350 of the fifth edition. ——Since truth is the correlation between a judgment and its basis for cognition, how can the person who makes the judgment really believe that there is such a basis but actually does not have such a basis, that is, a fallacy, how is this rational deception possible? is a problem now.I think that the possibility of error is exactly analogous to the possibility of illusion, or the possibility of deception of the understanding, mentioned above.It is my opinion (so here is the right place to insert this statement) that every fallacy is an inference from a conclusion to a ground; and the ground is sound if it is known that the conclusion can have only one ground and never another. , otherwise it is inappropriate.The man who falls into error either assigns to the result a ground which it cannot have at all, which expresses his true lack of understanding, that is, the inability to directly recognize the connection of cause and effect; The conclusion designates a possible ground, and at the same time adds a major premise to his inference from the conclusion to the ground, saying that the conclusion can only be produced by the ground he proposed at any time.In fact, he has the right to say this only after he has done a thorough induction, but he presupposes this without doing it.Therefore, the concept of "whenever" is very broad, and should be replaced by "sometimes" or "mostly": such a conclusion proposition is unresolved, and it cannot be wrong.But since the person who falls into error acts only in the above-mentioned way, he either acts too hastily, or his knowledge of possibility is so limited that he does not know the need for induction.Therefore, fallacies and illusions are exactly alike.Both are deduction from conclusion to basis.Illusions always come from the understanding, that is, directly from intuition itself according to the law of causality; falsehoods always come from reason, that is, the forms that reason has in genuine thinking according to the law of reason, and most of them can also be Caused by the law of causality.There are three examples of fallacies caused by the law of causality, which one may regard as typical or representatives of three types of fallacies: 1) Perceptual illusions (deception of the understanding) contribute to fallacies (deception of reason), for example, people regard paintings as reliefs, and I thought it was a relief.This is a consequence of such a major premise: "If dark gray passes through all the color differences point by point to white, then the cause is light whenever it shines, because light shines on high places and low places. Different, so...." 2) "If there is less money in my cash drawer, then the reason is whenever my servant has an imitation key, so...." 3) "If the prism The sun shadow that is refracted, that is, moved up or down, is no longer round and white as before, but long and colorful. Then, the reason for this, once or even thousands of times, is that there is something hidden in the eyes. The rays of the same quality but of different colors and degrees of refraction are now separated by the different degrees of refraction, and thus appear as long, mottled bands of light; so—let us have a drink!"— —Any fallacy must be attributed to such an inference, that is, an inference based on a major premise that is often generalized wrong, hypothetical, from a certain ground to a certain conclusion.Only calculus errors are not included in this list. This kind of error is not a fallacy but a mistake: that is, the calculus process specified by the concept of number is not completed in pure intuition, not in counting, but another calculus process is completed.

As for the content of [all] science, fundamentally, it is in fact nothing more than the interrelationships of phenomena in the world, which conform to the law of sufficient reason and are based on the clue that only the law of sufficient reason can make the "why" valid and meaningful. on the mutual relationship.The verification of these relationships is called explanation.If two representations belong to the same class, and this class is governed by a certain form of the principle of sufficient reason, then the so-called explanation can go no further than to point out the mutual relationship of the two representations in this form. .Explain that if this step is reached, there is no need to ask "why" at all: because this verified relationship is a relationship that must never be thought otherwise, that is to say, it is the form of all cognition.So people do not ask why two plus two make four; why the equality of the interior angles of a triangle determines the equality of sides; why any given cause must be followed by its consequences; Make the conclusion self-evident.Any kind of explanation, if it does not reduce to a relationship that can no longer ask "why", can only be based on a hypothetical hidden property.But every primitive force of nature is also of this nature.Any explanation of natural science must end up with such a hidden property, that is, with a black mass.Therefore, the explanation of natural science can only let the inner essence of a stone or a person be explained. The gravity, cohesion, chemical properties, etc. presented by the stone are the same as the cognition and behavior of people. Can't tell why.For example. 'Heavy' is a hidden attribute, because people can imagine that it does not exist, it is not a necessary thing produced from the form of cognition, but the law of inertia is not the case, it is deduced from the law of causality, so it is A sufficient explanation. There are two things that cannot be explained at all, that is, they cannot be reduced to the relationship shown by the principle of sufficient reason; the first is the law of sufficient reason itself in the four forms, because it is all explained. Principle, any explanation is meaningful only in relation to it; the second is that the principle of sufficient reason does not reach the thing-in-itself from which everything inherent in all phenomena comes from, and the knowledge of the thing-in-itself does not obey the principle of sufficient reason at all. Knowing. The thing in itself cannot be obtained and understood, and here we have to let it go; but in the next article, when we re-examine the possible achievements of science, we can understand it. But where natural science, all science, must stop, that is, not only is explanation, where even the principle of explanation—the law of sufficient reason, cannot go a step further, and that is where philosophy [takes the problem] back into its hands and examines it differently from science—. In The Law of Reason 51. I have shown how this or that form of the principle of sufficient reason is the main thread guiding the various sciences.—In fact, the most appropriate classification of sciences should also be made in this way. But according to each thread Explanations given, as has been said, are always only relative, always explain things in relation to each other, always leave something unexplained, and this is what every explanation presupposes. For example, in mathematics, it is space and time; in mechanics, physics, and chemistry, it is matter, physical properties, primitive [natural] forces, natural laws, etc.; in botany and zoology, it is the division of species and life itself; In history, all the peculiarities of man and his thoughts and wills;—in all these [sciences] there is also a certain form of application of the principle of sufficient reason according to individual needs.—Philosophy has a characteristic: it不假定任何东西为已知，而是认一切为同样的陌生都是问题；不仅现象间的关系是问题，现象本身也是问题，根据律本身也是问题。别的科学只要把一切还原到根据律，便万事已足；对于哲学这却是一无所获，因为一个系列中此一环节和彼一环节在哲学上都是同样陌生的。此外，这种关联自身和由此而被联结的东西也同样的是问题，而这些东西在其联结被指出以前又和被指出以后同样也还是问题。总之，如已说过，正是科学所假定的，以之为说明的根据和限度的，就正是哲学应有的问题。由此看来，那些科学到此止步的地方，也就正是哲学开步走的地方。证明不能是哲学的基础，因为证明只是从已知的命题演绎未知的命题，而对于哲学来说，一切都是同样的陌生[并无已知未知之别]。不可能有这样一个命题，说由于这一命题始有这世界及其一切现象：因此，不可能象斯宾诺莎所要作的那样，从“一个坚定的原则”进行证明便可引伸出一种哲学来。并且哲学还是最普遍的知识，它的主要命题就不能是从别的更普遍的知识引伸出来的结论。矛盾律不过是把概念问的一致固定下来，但并不产生概念。根据律说明现象间的联系，但不说明现象本身。因此哲学不能从寻找整个世界的一个有效因或一个目的因出发。至少是我的哲学就根本不问世界的来由，不问为何有此世界，而只问这世界是什么。在这里，“为什么”是低于“什么”一级的，因为这“为什么”既只是由于世界的现象[所由呈现]的形式，由于根据律而产生的，并且只在这个限度内有其意义和妥当性，所以早就是属于这个世界的了。人们固然可以说，世界是什么，这是每人无须别的帮助就认识到的[问题]，因为人自己就是认识的主体，世界就是这主体的表象。这种说法在一定限度内也是对的。不过这种认识是一个直观的认识，是具体中的认识；而在抽象中复制这些认识，把先后出现的，变动不居的直观，根本把感这个广泛概念所包括的一切，把只是消极规定的非抽象、非明晰的知识提升为一种抽象的、明晰的、经久的知识，这才是哲学的任务。因此，哲学必须是关于整个世界的本质的一个抽象陈述，既关于世界的全部，又关于其一切部分。但是为了不迷失于无数的个别判断，哲学必须利用抽象作用而在普遍中思维一切个别事物，在普遍中思维个别事物所具的差异；从而它一面要分，一面要合，以便将世界所有纷坛复杂的事物，按其本质，用少数的抽象概念概括起来，提交给知识。哲学既将世界的本质固定于这些概念中，那么，由于这些概念就必须能认识普遍，也要能认识一切特殊，也就是对这两者的认识必须有最准确的联系。因此，在哲学上有天才就在于柏拉图所确定的一点：在多中认一，在一中认多。准此，哲学将是极普遍的判断之总和，而其认识根据直接就是在其完整性中的世界本身，不遗漏任何点滴，也就是在人的意识中呈现出来的一切一切。哲学将是世界在抽象概念中的一个完整的复制，好比明镜中的反映作用似的。而这些抽象概念是由于本质上同一的合为一个概念，本质上相异的分为另一概念才可能的。培根就早已为哲学规定了这个任务，他是这样说的：“最忠实地复述着这世界自己的声音，世界规定了多少，就恰如其分他说出多少；不是别的而只是这世界的阴影和反映，不加上一点自己的东西，而仅仅只是复述和回声；只有这，才是真的哲学。”（《关于广义的科学》第二卷第13页）不过，我们是在培根当时还不能想到的一种更广泛的意义中承认这一点的。

世界各方面、各部分，由于其同属一整体而有的相互一致性也必须重现于世界的抽象复制中。因此在那判断的总和中，此一判断可在某种程度内由彼一判断引伸而来，并且也总是相互引伸的。不过在相互引伸中要使第一个判断有可能，这一些判断都必须齐备才行，也就是要事先把这些判断作为直接建立在对这世界的具体认识上的判断确立起才行：而一切直接的证明都比间接的证明妥当些，所以更应如此。这些判断借助于它们相互之间的谐和甚至汇成一个单一的思想的统一性，而这统一性又来自直观世界本身的谐和与统一，这直观世界又是这些判断共同的认识根据，所以这些判断相互之间的谐和不能作为各判断的最初的东西来为这些判断建立根据，而是只能附带地加强这些判断的真实性而已。——这个问题本身只能由于问题的解决才能完全明白。