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第141章

the critique of pure reason-第141章

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contain some of the elements requisite to form a complete

definition。 If a conception could not be employed in reasoning

before it had been defined; it would fare ill with all philosophical

thought。 But; as incompletely defined conceptions may always be

employed without detriment to truth; so far as our analysis of the

elements contained in them proceeds; imperfect definitions; that is;

propositions which are properly not definitions; but merely

approximations thereto; may be used with great advantage。 In

mathematics; definition belongs ad esse; in philosophy ad melius esse。

It is a difficult task to construct a proper definition。 Jurists are

still without a complete definition of the idea of right。



  (b) Mathematical definitions cannot be erroneous。 For the conception

is given only in and through the definition; and thus it contains only

what has been cogitated in the definition。 But although a definition

cannot be incorrect; as regards its content; an error may sometimes;

although seldom; creep into the form。 This error consists in a want of

precision。 Thus the common definition of a circle… that it is a curved

line; every point in which is equally distant from another point

called the centre… is faulty; from the fact that the determination

indicated by the word curved is superfluous。 For there ought to be a

particular theorem; which may be easily proved from the definition; to

the effect that every line; which has all its points at equal

distances from another point; must be a curved line… that is; that not

even the smallest part of it can be straight。 Analytical

definitions; on the other hand; may be erroneous in many respects;

either by the introduction of signs which do not actually exist in the

conception; or by wanting in that completeness which forms the

essential of a definition。 In the latter case; the definition is

necessarily defective; because we can never be fully certain of the

completeness of our analysis。 For these reasons; the method of

definition employed in mathematics cannot be imitated in philosophy。

  2。 Of Axioms。 These; in so far as they are immediately certain;

are a priori synthetical principles。 Now; one conception cannot be

connected synthetically and yet immediately with another; because;

if we wish to proceed out of and beyond a conception; a third

mediating cognition is necessary。 And; as philosophy is a cognition of

reason by the aid of conceptions alone; there is to be found in it

no principle which deserves to be called an axiom。 Mathematics; on the

other hand; may possess axioms; because it can always connect the

predicates of an object a priori; and without any mediating term; by

means of the construction of conceptions in intuition。 Such is the

case with the proposition: Three points can always lie in a plane。

On the other hand; no synthetical principle which is based upon

conceptions; can ever be immediately certain (for example; the

proposition: Everything that happens has a cause); because I require a

mediating term to connect the two conceptions of event and cause…

namely; the condition of time…determination in an experience; and I

cannot cognize any such principle immediately and from conceptions

alone。 Discursive principles are; accordingly; very different from

intuitive principles or axioms。 The former always require deduction;

which in the case of the latter may be altogether dispensed with。

Axioms are; for this reason; always self…evident; while

philosophical principles; whatever may be the degree of certainty they

possess; cannot lay any claim to such a distinction。 No synthetical

proposition of pure transcendental reason can be so evident; as is

often rashly enough declared; as the statement; twice two are four。 It

is true that in the Analytic I introduced into the list of

principles of the pure understanding; certain axioms of intuition; but

the principle there discussed was not itself an axiom; but served

merely to present the principle of the possibility of axioms in

general; while it was really nothing more than a principle based

upon conceptions。 For it is one part of the duty of transcendental

philosophy to establish the possibility of mathematics itself。

Philosophy possesses; then; no axioms; and has no right to impose

its a priori principles upon thought; until it has established their

authority and validity by a thoroughgoing deduction。

  3。 Of Demonstrations。 Only an apodeictic proof; based upon

intuition; can be termed a demonstration。 Experience teaches us what

is; but it cannot convince us that it might not have been otherwise。

Hence a proof upon empirical grounds cannot be apodeictic。 A priori

conceptions; in discursive cognition; can never produce intuitive

certainty or evidence; however certain the judgement they present

may be。 Mathematics alone; therefore; contains demonstrations; because

it does not deduce its cognition from conceptions; but from the

construction of conceptions; that is; from intuition; which can be

given a priori in accordance with conceptions。 The method of

algebra; in equations; from which the correct answer is deduced by

reduction; is a kind of construction… not geometrical; but by symbols…

in which all conceptions; especially those of the relations of

quantities; are represented in intuition by signs; and thus the

conclusions in that science are secured from errors by the fact that

every proof is submitted to ocular evidence。 Philosophical cognition

does not possess this advantage; it being required to consider the

general always in abstracto (by means of conceptions); while

mathematics can always consider it in concreto (in an individual

intuition); and at the same time by means of a priori

representation; whereby all errors are rendered manifest to the

senses。 The former… discursive proofs… ought to be termed acroamatic

proofs; rather than demonstrations; as only words are employed in

them; while demonstrations proper; as the term itself indicates;

always require a reference to the intuition of the object。

  It follows from all these considerations that it is not consonant

with the nature of philosophy; especially in the sphere of pure

reason; to employ the dogmatical method; and to adorn itself with

the titles and insignia of mathematical science。 It does not belong to

that order; and can only hope for a fraternal union with that science。

Its attempts at mathematical evidence are vain pretensions; which

can only keep it back from its true aim; which is to detect the

illusory procedure of reason when transgressing its proper limits; and

by fully explaining and analysing our conceptions; to conduct us

from the dim regions of speculation to the clear region of modest

self…knowledge。 Reason must not; therefore; in its transcendental

endeavours; look forward with such confidence; as if the path it is

pursuing led straight to its aim; nor reckon with such security upon

its premisses; as to consider it unnecessary to take a step back; or

to keep a strict watch for errors; which; overlooked in the

principles; may be detected in the arguments themselves… in which case

it may be requisite either to determine these principles with

greater strictness; or to change them entirely。

  I divide all apodeictic propositions; whether demonstrable or

immediately certain; into dogmata and mathemata。 A direct

synthetical proposition; based on conceptions; is a dogma; a

proposition of the same kind; based on the construction of

conceptions; is a mathema。 Analytical judgements do not teach us any

more about an object than what was contained in the conception we

had of it; because they do not extend our cognition beyond our

conception of an object; they merely elucidate the conception。 They

cannot therefore be with propriety termed dogmas。 Of the two kinds

of a priori synthetical propositions above mentioned; only those which

are employed in philosophy can; according to the general mode of

speech; bear this n

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