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