the critique of pure reason-第44章

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diagonal；　do　not　necessarily　belong　to　each　other；　and　of　this　kind　is

the　synthesis　of　the　homogeneous　in　everything　that　can　be

mathematically　considered。　This　synthesis　can　be　divided　into　those　of

aggregation　and　coalition；　the　former　of　which　is　applied　to

extensive；　the　latter　to　intensive　quantities。　The　second　sort　of

combination　（nexus）　is　the　synthesis　of　a　manifold；　in　so　far　as　its

parts　do　belong　necessarily　to　each　other；　for　example；　the　accident

to　a　substance；　or　the　effect　to　the　cause。　Consequently　it　is　a

synthesis　of　that　which　though　heterogeneous；　is　represented　as

connected　a　priori。　This　combination…　not　an　arbitrary　one…　I

entitle　dynamical　because　it　concerns　the　connection　of　the

existence　of　the　manifold。　This；　again；　may　be　divided　into　the

physical　synthesis；　of　the　phenomena　divided　among　each　other；　and　the

metaphysical　synthesis；　or　the　connection　of　phenomena　a　priori　in　the

faculty　of　cognition。



　　　　　　　　　　　　　　　　　1。　AXIOMS　OF　INTUITION。

　　　　　The　principle　of　these　is：　All　Intuitions　are　Extensive

　　　　　　　　　　　　　　　　　　　　　　Quantities。



　　　　　　　　　　　　　　　　　　　　　　　　　PROOF。



　　All　phenomena　contain；　as　regards　their　form；　an　intuition　in

space　and　time；　which　lies　a　priori　at　the　foundation　of　all　without

exception。　Phenomena；　therefore；　cannot　be　apprehended；　that　is；

received　into　empirical　consciousness　otherwise　than　through　the

synthesis　of　a　manifold；　through　which　the　representations　of　a

determinate　space　or　time　are　generated；　that　is　to　say；　through　the

composition　of　the　homogeneous　and　the　consciousness　of　the

synthetical　unity　of　this　manifold　（homogeneous）。　Now　the

consciousness　of　a　homogeneous　manifold　in　intuition；　in　so　far　as

thereby　the　representation　of　an　object　is　rendered　possible；　is　the

conception　of　a　quantity　（quanti）。　Consequently；　even　the　perception

of　an　object　as　phenomenon　is　possible　only　through　the　same

synthetical　unity　of　the　manifold　of　the　given　sensuous　intuition；

through　which　the　unity　of　the　composition　of　the　homogeneous　manifold

in　the　conception　of　a　quantity　is　cogitated；　that　is　to　say；　all

phenomena　are　quantities；　and　extensive　quantities；　because　as

intuitions　in　space　or　time　they　must　be　represented　by　means　of　the

same　synthesis　through　which　space　and　time　themselves　are　determined。

　　An　extensive　quantity　I　call　that　wherein　the　representation　of

the　parts　renders　possible　（and　therefore　necessarily　antecedes）　the

representation　of　the　whole。　I　cannot　represent　to　myself　any　line；

however　small；　without　drawing　it　in　thought；　that　is；　without

generating　from　a　point　all　its　parts　one　after　another；　and　in　this

way　alone　producing　this　intuition。　Precisely　the　same　is　the　case

with　every；　even　the　smallest；　portion　of　time。　I　cogitate　therein

only　the　successive　progress　from　one　moment　to　another；　and　hence；　by

means　of　the　different　portions　of　time　and　the　addition　of　them；　a

determinate　quantity　of　time　is　produced。　As　the　pure　intuition　in　all

phenomena　is　either　time　or　space；　so　is　every　phenomenon　in　its

character　of　intuition　an　extensive　quantity；　inasmuch　as　it　can

only　be　cognized　in　our　apprehension　by　successive　synthesis　（from

part　to　part）。　All　phenomena　are；　accordingly；　to　be　considered　as

aggregates；　that　is；　as　a　collection　of　previously　given　parts；

which　is　not　the　case　with　every　sort　of　quantities；　but　only　with

those　which　are　represented　and　apprehended　by　us　as　extensive。

　　On　this　successive　synthesis　of　the　productive　imagination；　in　the

generation　of　figures；　is　founded　the　mathematics　of　extension；　or

geometry；　with　its　axioms；　which　express　the　conditions　of　sensuous

intuition　a　priori；　under　which　alone　the　schema　of　a　pure

conception　of　external　intuition　can　exist；　for　example；　〃be　tween　two

points　only　one　straight　line　is　possible；〃　〃two　straight　lines　cannot

enclose　a　space；〃　etc。　These　are　the　axioms　which　properly　relate　only

to　quantities　（quanta）　as　such。

　　But；　as　regards　the　quantity　of　a　thing　（quantitas）；　that　is　to　say；

the　answer　to　the　question：　〃How　large　is　this　or　that　object？〃

although；　in　respect　to　this　question；　we　have　various　propositions

synthetical　and　immediately　certain　（indemonstrabilia）；　we　have；　in

the　proper　sense　of　the　term；　no　axioms。　For　example；　the

propositions：　〃If　equals　be　added　to　equals；　the　wholes　are　equal〃；

〃If　equals　be　taken　from　equals；　the　remainders　are　equal〃；　are

analytical；　because　I　am　immediately　conscious　of　the　identity　of

the　production　of　the　one　quantity　with　the　production　of　the　other；

whereas　axioms　must　be　a　priori　synthetical　propositions。　On　the　other

hand；　the　self…evident　propositions　as　to　the　relation　of　numbers；　are

certainly　synthetical　but　not　universal；　like　those　of　geometry；　and

for　this　reason　cannot　be　called　axioms；　but　numerical　formulae。

That　7　＋　5　=　12　is　not　an　analytical　proposition。　For　neither　in　the

representation　of　seven；　nor　of　five；　nor　of　the　composition　of　the

two　numbers；　do　I　cogitate　the　number　twelve。　（Whether　I　cogitate

the　number　in　the　addition　of　both；　is　not　at　present　the　question；

for　in　the　case　of　an　analytical　proposition；　the　only　point　is

whether　I　really　cogitate　the　predicate　in　the　representation　of　the

subject。）　But　although　the　proposition　is　synthetical；　it　is

nevertheless　only　a　singular　proposition。　In　so　far　as　regard　is

here　had　merely　to　the　synthesis　of　the　homogeneous　（the　units）；　it

cannot　take　place　except　in　one　manner；　although　our　use　of　these

numbers　is　afterwards　general。　If　I　say：　〃A　triangle　can　be

constructed　with　three　lines；　any　two　of　which　taken　together　are

greater　than　the　third；〃　I　exercise　merely　the　pure　function　of　the

productive　imagination；　which　may　draw　the　lines　longer　or　shorter　and

construct　the　angles　at　its　pleasure。　On　the　contrary；　the　number

seven　is　possible　only　in　one　manner；　and　so　is　likewise　the　number

twelve；　which　results　from　the　synthesis　of　seven　and　five。　Such

propositions；　then；　cannot　be　termed　axioms　（for　in　that　case　we

should　have　an　infinity　of　these）；　but　numerical　formulae。

　　This　transcendental　principle　of　the　mathematics　of　phenomena

greatly　enlarges　our　a　priori　cognition。　For　it　is　by　this　principle

alone　that　pure　mathematics　is　rendered　applicable　in　all　its

precision　to　objects　of　experience；　and　without　it　the　validity　of

this　application　would　not　be　so　self…evident；　on　the　contrary；

contradictions　and　confusions　have　often　arisen　on　this　very　point。

Phenomena　are　not　things　in　themselves。　Empirical　intuition　is

possible　only　through　pure　intuition　（of　space　and　time）；

consequently；　what　geometry　affirms　of　the　latter；　is　indisputably

valid　of　the　former。　All　evasions；　such　as　the　statement　that

objects　of　sense　do　not　conform　to　the　rules　of　construction　in

space　（for　example；　to　the　rule　of　the　infinite　divisibility　of

lines　or　angles）；　must　fall　to　the　ground。　For；　if　these　objections

hold　good；　we　deny　to　space；　and　with　it　to　all　mathematics；　objective

validity；　and　no　longer　know　wherefore；　and　how　far；　mathematics　can

be　applied　to　phenomena。　The　synthesis　of　spaces　and　times　as　the

essential　form　of　all　intuition；　is　that　which　renders　possible　the

apprehension　of　a　phenomenon；　and　therefore　every　external　experience；

consequently　all　cognition　of　the　objects　of　experience；　and

whatever　mathematics　in　its　pure　use　proves　of　the　former；　must

necessarily　hold　good　of　the　latter。　All　objections　are　but　the

chicaneries　of　an　ill…instructed　reason；　which　erroneously　thinks　to

liberate　the　objects　of　sense　from　the　formal　conditions　of　our

sensibility；　and　represents　these；　although　mere　phenomena；　as

things　in　themselves；　presented　as　such　to　our　understanding。　But　in

this　case；　no　a　priori　synthetical　cognition　of　them　could　be

possible；　consequently　not　through　pure　conceptions　of　space　and　the

science　which　determines　these　conceptions；　that　is　to　say；

geometry；　wo