the critique of pure reason-第138章

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correspond　to　the　conception　of　reality；　except　in　experience；　it

cannot　be　presented　to　the　mind　a　priori　and　antecedently　to　the

empirical　consciousness　of　a　reality。　We　can　form　an　intuition；　by

means　of　the　mere　conception　of　it；　of　a　cone；　without　the　aid　of

experience；　but　the　colour　of　the　cone　we　cannot　know　except　from

experience。　I　cannot　present　an　intuition　of　a　cause；　except　in　an

example　which　experience　offers　to　me。　Besides；　philosophy；　as　well　as

mathematics；　treats　of　quantities；　as；　for　example；　of　totality；

infinity；　and　so　on。　Mathematics；　too；　treats　of　the　difference　of

lines　and　surfaces…　as　spaces　of　different　quality；　of　the

continuity　of　extension…　as　a　quality　thereof。　But；　although　in　such

cases　they　have　a　common　object；　the　mode　in　which　reason　considers

that　object　is　very　different　in　philosophy　from　what　it　is　in

mathematics。　The　former　confines　itself　to　the　general　conceptions；

the　latter　can　do　nothing　with　a　mere　conception；　it　hastens　to

intuition。　In　this　intuition　it　regards　the　conception　in　concreto；

not　empirically；　but　in　an　a　priori　intuition；　which　it　has

constructed；　and　in　which；　all　the　results　which　follow　from　the

general　conditions　of　the　construction　of　the　conception　are　in　all

cases　valid　for　the　object　of　the　constructed　conception。

　　Suppose　that　the　conception　of　a　triangle　is　given　to　a

philosopher　and　that　he　is　required　to　discover；　by　the

philosophical　method；　what　relation　the　sum　of　its　angles　bears　to　a

right　angle。　He　has　nothing　before　him　but　the　conception　of　a

figure　enclosed　within　three　right　lines；　and；　consequently；　with

the　same　number　of　angles。　He　may　analyse　the　conception　of　a　right

line；　of　an　angle；　or　of　the　number　three　as　long　as　he　pleases；　but

he　will　not　discover　any　properties　not　contained　in　these

conceptions。　But；　if　this　question　is　proposed　to　a　geometrician；　he

at　once　begins　by　constructing　a　triangle。　He　knows　that　two　right

angles　are　equal　to　the　sum　of　all　the　contiguous　angles　which　proceed

from　one　point　in　a　straight　line；　and　he　goes　on　to　produce　one

side　of　his　triangle；　thus　forming　two　adjacent　angles　which　are

together　equal　to　two　right　angles。　He　then　divides　the　exterior　of

these　angles；　by　drawing　a　line　parallel　with　the　opposite　side　of　the

triangle；　and　immediately　perceives　that　be　has　thus　got　an　exterior

adjacent　angle　which　is　equal　to　the　interior。　Proceeding　in　this　way；

through　a　chain　of　inferences；　and　always　on　the　ground　of

intuition；　he　arrives　at　a　clear　and　universally　valid　solution　of　the

question。

　　But　mathematics　does　not　confine　itself　to　the　construction　of

quantities　（quanta）；　as　in　the　case　of　geometry；　it　occupies　itself

with　pure　quantity　also　（quantitas）；　as　in　the　case　of　algebra；

where　complete　abstraction　is　made　of　the　properties　of　the　object

indicated　by　the　conception　of　quantity。　In　algebra；　a　certain

method　of　notation　by　signs　is　adopted；　and　these　indicate　the

different　possible　constructions　of　quantities；　the　extraction　of

roots；　and　so　on。　After　having　thus　denoted　the　general　conception

of　quantities；　according　to　their　different　relations；　the　different

operations　by　which　quantity　or　number　is　increased　or　diminished

are　presented　in　intuition　in　accordance　with　general　rules。　Thus；

when　one　quantity　is　to　be　divided　by　another；　the　signs　which

denote　both　are　placed　in　the　form　peculiar　to　the　operation　of

division；　and　thus　algebra；　by　means　of　a　symbolical　construction　of

quantity；　just　as　geometry；　with　its　ostensive　or　geometrical

construction　（a　construction　of　the　objects　themselves）；　arrives　at

results　which　discursive　cognition　cannot　hope　to　reach　by　the　aid

of　mere　conceptions。

　　Now；　what　is　the　cause　of　this　difference　in　the　fortune　of　the

philosopher　and　the　mathematician；　the　former　of　whom　follows　the　path

of　conceptions；　while　the　latter　pursues　that　of　intuitions；　which

he　represents；　a　priori；　in　correspondence　with　his　conceptions？　The

cause　is　evident　from　what　has　been　already　demonstrated　in　the

introduction　to　this　Critique。　We　do　not；　in　the　present　case；　want　to

discover　analytical　propositions；　which　may　be　produced　merely　by

analysing　our　conceptions…　for　in　this　the　philosopher　would　have

the　advantage　over　his　rival；　we　aim　at　the　discovery　of　synthetical

propositions…　such　synthetical　propositions；　moreover；　as　can　be

cognized　a　priori。　I　must　not　confine　myself　to　that　which　I

actually　cogitate　in　my　conception　of　a　triangle；　for　this　is

nothing　more　than　the　mere　definition；　I　must　try　to　go　beyond　that；

and　to　arrive　at　properties　which　are　not　contained　in；　although

they　belong　to；　the　conception。　Now；　this　is　impossible；　unless　I

determine　the　object　present　to　my　mind　according　to　the　conditions；

either　of　empirical；　or　of　pure；　intuition。　In　the　former　case；　I

should　have　an　empirical　proposition　（arrived　at　by　actual　measurement

of　the　angles　of　the　triangle）；　which　would　possess　neither

universality　nor　necessity；　but　that　would　be　of　no　value。　In　the

latter；　I　proceed　by　geometrical　construction；　by　means　of　which　I

collect；　in　a　pure　intuition；　just　as　I　would　in　an　empirical

intuition；　all　the　various　properties　which　belong　to　the　schema　of

a　triangle　in　general；　and　consequently　to　its　conception；　and　thus

construct　synthetical　propositions　which　possess　the　attribute　of

universality。

　　It　would　be　vain　to　philosophize　upon　the　triangle；　that　is；　to

reflect　on　it　discursively；　I　should　get　no　further　than　the

definition　with　which　I　had　been　obliged　to　set　out。　There　are

certainly　transcendental　synthetical　propositions　which　are　framed

by　means　of　pure　conceptions；　and　which　form　the　peculiar

distinction　of　philosophy；　but　these　do　not　relate　to　any　particular

thing；　but　to　a　thing　in　general；　and　enounce　the　conditions　under

which　the　perception　of　it　may　become　a　part　of　possible　experience。

But　the　science　of　mathematics　has　nothing　to　do　with　such

questions；　nor　with　the　question　of　existence　in　any　fashion；　it　is

concerned　merely　with　the　properties　of　objects　in　themselves；　only　in

so　far　as　these　are　connected　with　the　conception　of　the　objects。

　　In　the　above　example；　we　merely　attempted　to　show　the　great

difference　which　exists　between　the　discursive　employment　of　reason　in

the　sphere　of　conceptions；　and　its　intuitive　exercise　by　means　of

the　construction　of　conceptions。　The　question　naturally　arises：　What

is　the　cause　which　necessitates　this　twofold　exercise　of　reason；　and

how　are　we　to　discover　whether　it　is　the　philosophical　or　the

mathematical　method　which　reason　is　pursuing　in　an　argument？

　　All　our　knowledge　relates；　finally；　to　possible　intuitions；　for　it

is　these　alone　that　present　objects　to　the　mind。　An　a　priori　or

non…empirical　conception　contains　either　a　pure　intuition…　and　in　this

case　it　can　be　constructed；　or　it　contains　nothing　but　the　synthesis

of　possible　intuitions；　which　are　not　given　a　priori。　In　this　latter

case；　it　may　help　us　to　form　synthetical　a　priori　judgements；　but　only

in　the　discursive　method；　by　conceptions；　not　in　the　intuitive；　by

means　of　the　construction　of　conceptions。

　　The　only　a　priori　intuition　is　that　of　the　pure　form　of　phenomena…

space　and　time。　A　conception　of　space　and　time　as　quanta　may　be

presented　a　priori　in　intuition；　that　is；　constructed；　either　alone

with　their　quality　（figure）；　or　as　pure　quantity　（the　mere　synthesis

of　the　homogeneous）；　by　means　of　number。　But　the　matter　of

phenomena；　by　which　things　are　given　in　space　and　time；　can　be

presented　only　in　perception；　a　posteriori。　The　only　conception

which　represents　a　priori　this　empirical　content　of　phenomena　is　the

conception　of　a　thing　in　general；　and　the　a　priori　synthetical

cognition　of　this　conception　can　give　us　nothing　more　than　the　rule

for　the　synthesis　of　that　which　may　be　contained　in　the

corresponding　a　posteriori　perception；　it　is　utterly　inadeq