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conception; as it is presented to the mind; may contain a number of

obscure representations; which we do not observe in our analysis;

although we employ them in our application of the conception; I can

never be sure that my analysis is complete; while examples may make

this probable; although they can never demonstrate the fact。 instead

of the word definition; I should rather employ the term exposition…

a more modest expression; which the critic may accept without

surrendering his doubts as to the completeness of the analysis of

any such conception。 As; therefore; neither empirical nor a priori

conceptions are capable of definition; we have to see whether the only

other kind of conceptions… arbitrary conceptions… can be subjected

to this mental operation。 Such a conception can always be defined; for

I must know thoroughly what I wished to cogitate in it; as it was I

who created it; and it was not given to my mind either by the nature

of my understanding or by experience。 At the same time; I cannot say

that; by such a definition; I have defined a real object。 If the

conception is based upon empirical conditions; if; for example; I have

a conception of a clock for a ship; this arbitrary conception does not

assure me of the existence or even of the possibility of the object。

My definition of such a conception would with more propriety be termed

a declaration of a project than a definition of an object。 There

are no other conceptions which can bear definition; except those which

contain an arbitrary synthesis; which can be constructed a priori。

Consequently; the science of mathematics alone possesses

definitions。 For the object here thought is presented a priori in

intuition; and thus it can never contain more or less than the

conception; because the conception of the object has been given by the

definition… and primarily; that is; without deriving the definition

from any other source。 Philosophical definitions are; therefore;

merely expositions of given conceptions; while mathematical

definitions are constructions of conceptions originally formed by

the mind itself; the former are produced by analysis; the completeness

of which is never demonstratively certain; the latter by a

synthesis。 In a mathematical definition the conception is formed; in a

philosophical definition it is only explained。 From this it follows:



  *The definition must describe the conception completely that is;

omit none of the marks or signs of which it composed; within its own

limits; that is; it must be precise; and enumerate no more signs

than belong to the conception; and on primary grounds; that is to say;

the limitations of the bounds of the conception must not be deduced

from other conceptions; as in this case a proof would be necessary;

and the so…called definition would be incapable of taking its place at

the bead of all the judgements we have to form regarding an object。



  (a) That we must not imitate; in philosophy; the mathematical

usage of commencing with definitions… except by way of hypothesis or

experiment。 For; as all so…called philosophical definitions are merely

analyses of given conceptions; these conceptions; although only in a

confused form; must precede the analysis; and the incomplete

exposition must precede the complete; so that we may be able to draw

certain inferences from the characteristics which an incomplete

analysis has enabled us to discover; before we attain to the

complete exposition or definition of the conception。 In one word; a

full and clear definition ought; in philosophy; rather to form the

conclusion than the commencement of our labours。* In mathematics; on

the contrary; we cannot have a conception prior to the definition;

it is the definition which gives us the conception; and it must for

this reason form the commencement of every chain of mathematical

reasoning。



  *Philosophy abounds in faulty definitions; especially such as

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 prio

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