or¡¡opposites¡¡¡all¡¡these¡¡form¡¡part¡¡of¡¡the¡¡cognition¡¡of¡¡reason¡¡on¡¡the
ground¡¡of¡¡conceptions£»¡¡and¡¡this¡¡cognition¡¡is¡¡termed¡¡philosophical¡£¡¡But
to¡¡determine¡¡a¡¡priori¡¡an¡¡intuition¡¡in¡¡space¡¡£¨its¡¡figure£©£»¡¡to¡¡divide
time¡¡into¡¡periods£»¡¡or¡¡merely¡¡to¡¡cognize¡¡the¡¡quantity¡¡of¡¡an¡¡intuition
in¡¡space¡¡and¡¡time£»¡¡and¡¡to¡¡determine¡¡it¡¡by¡¡number¡¡¡all¡¡this¡¡is¡¡an
operation¡¡of¡¡reason¡¡by¡¡means¡¡of¡¡the¡¡construction¡¡of¡¡conceptions£»¡¡and
is¡¡called¡¡mathematical¡£
¡¡¡¡The¡¡success¡¡which¡¡attends¡¡the¡¡efforts¡¡of¡¡reason¡¡in¡¡the¡¡sphere¡¡of
mathematics¡¡naturally¡¡fosters¡¡the¡¡expectation¡¡that¡¡the¡¡same¡¡good
fortune¡¡will¡¡be¡¡its¡¡lot£»¡¡if¡¡it¡¡applies¡¡the¡¡mathematical¡¡method¡¡in
other¡¡regions¡¡of¡¡mental¡¡endeavour¡¡besides¡¡that¡¡of¡¡quantities¡£¡¡Its
success¡¡is¡¡thus¡¡great£»¡¡because¡¡it¡¡can¡¡support¡¡all¡¡its¡¡conceptions¡¡by¡¡a
priori¡¡intuitions¡¡and£»¡¡in¡¡this¡¡way£»¡¡make¡¡itself¡¡a¡¡master£»¡¡as¡¡it
were£»¡¡over¡¡nature£»¡¡while¡¡pure¡¡philosophy£»¡¡with¡¡its¡¡a¡¡priori¡¡discursive
conceptions£»¡¡bungles¡¡about¡¡in¡¡the¡¡world¡¡of¡¡nature£»¡¡and¡¡cannot¡¡accredit
or¡¡show¡¡any¡¡a¡¡priori¡¡evidence¡¡of¡¡the¡¡reality¡¡of¡¡these¡¡conceptions¡£
Masters¡¡in¡¡the¡¡science¡¡of¡¡mathematics¡¡are¡¡confident¡¡of¡¡the¡¡success
of¡¡this¡¡method£»¡¡indeed£»¡¡it¡¡is¡¡a¡¡common¡¡persuasion¡¡that¡¡it¡¡is¡¡capable
of¡¡being¡¡applied¡¡to¡¡any¡¡subject¡¡of¡¡human¡¡thought¡£¡¡They¡¡have¡¡hardly
ever¡¡reflected¡¡or¡¡philosophized¡¡on¡¡their¡¡favourite¡¡science¡¡¡a¡¡task
of¡¡great¡¡difficulty£»¡¡and¡¡the¡¡specific¡¡difference¡¡between¡¡the¡¡two¡¡modes
of¡¡employing¡¡the¡¡faculty¡¡of¡¡reason¡¡has¡¡never¡¡entered¡¡their¡¡thoughts¡£
Rules¡¡current¡¡in¡¡the¡¡field¡¡of¡¡common¡¡experience£»¡¡and¡¡which¡¡common
sense¡¡stamps¡¡everywhere¡¡with¡¡its¡¡approval£»¡¡are¡¡regarded¡¡by¡¡them¡¡as
axiomatic¡£¡¡From¡¡what¡¡source¡¡the¡¡conceptions¡¡of¡¡space¡¡and¡¡time£»¡¡with
which¡¡£¨as¡¡the¡¡only¡¡primitive¡¡quanta£©¡¡they¡¡have¡¡to¡¡deal£»¡¡enter¡¡their
minds£»¡¡is¡¡a¡¡question¡¡which¡¡they¡¡do¡¡not¡¡trouble¡¡themselves¡¡to¡¡answer£»
and¡¡they¡¡think¡¡it¡¡just¡¡as¡¡unnecessary¡¡to¡¡examine¡¡into¡¡the¡¡origin¡¡of
the¡¡pure¡¡conceptions¡¡of¡¡the¡¡understanding¡¡and¡¡the¡¡extent¡¡of¡¡their
validity¡£¡¡All¡¡they¡¡have¡¡to¡¡do¡¡with¡¡them¡¡is¡¡to¡¡employ¡¡them¡£¡¡In¡¡all¡¡this
they¡¡are¡¡perfectly¡¡right£»¡¡if¡¡they¡¡do¡¡not¡¡overstep¡¡the¡¡limits¡¡of¡¡the
sphere¡¡of¡¡nature¡£¡¡But¡¡they¡¡pass£»¡¡unconsciously£»¡¡from¡¡the¡¡world¡¡of
sense¡¡to¡¡the¡¡insecure¡¡ground¡¡of¡¡pure¡¡transcendental¡¡conceptions
£¨instabilis¡¡tellus£»¡¡innabilis¡¡unda£©£»¡¡where¡¡they¡¡can¡¡neither¡¡stand
nor¡¡swim£»¡¡and¡¡where¡¡the¡¡tracks¡¡of¡¡their¡¡footsteps¡¡are¡¡obliterated¡¡by
time£»¡¡while¡¡the¡¡march¡¡of¡¡mathematics¡¡is¡¡pursued¡¡on¡¡a¡¡broad¡¡and
magnificent¡¡highway£»¡¡which¡¡the¡¡latest¡¡posterity¡¡shall¡¡frequent¡¡without
fear¡¡of¡¡danger¡¡or¡¡impediment¡£
¡¡¡¡As¡¡we¡¡have¡¡taken¡¡upon¡¡us¡¡the¡¡task¡¡of¡¡determining£»¡¡clearly¡¡and
certainly£»¡¡the¡¡limits¡¡of¡¡pure¡¡reason¡¡in¡¡the¡¡sphere¡¡of
transcendentalism£»¡¡and¡¡as¡¡the¡¡efforts¡¡of¡¡reason¡¡in¡¡this¡¡direction
are¡¡persisted¡¡in£»¡¡even¡¡after¡¡the¡¡plainest¡¡and¡¡most¡¡expressive
warnings£»¡¡hope¡¡still¡¡beckoning¡¡us¡¡past¡¡the¡¡limits¡¡of¡¡experience¡¡into
the¡¡splendours¡¡of¡¡the¡¡intellectual¡¡world¡¡¡it¡¡becomes¡¡necessary¡¡to
cut¡¡away¡¡the¡¡last¡¡anchor¡¡of¡¡this¡¡fallacious¡¡and¡¡fantastic¡¡hope¡£¡¡We
shall£»¡¡accordingly£»¡¡show¡¡that¡¡the¡¡mathematical¡¡method¡¡is¡¡unattended¡¡in
the¡¡sphere¡¡of¡¡philosophy¡¡by¡¡the¡¡least¡¡advantage¡¡¡except£»¡¡perhaps£»¡¡that
it¡¡more¡¡plainly¡¡exhibits¡¡its¡¡own¡¡inadequacy¡¡¡that¡¡geometry¡¡and
philosophy¡¡are¡¡two¡¡quite¡¡different¡¡things£»¡¡although¡¡they¡¡go¡¡band¡¡in
hand¡¡in¡¡hand¡¡in¡¡the¡¡field¡¡of¡¡natural¡¡science£»¡¡and£»¡¡consequently£»
that¡¡the¡¡procedure¡¡of¡¡the¡¡one¡¡can¡¡never¡¡be¡¡imitated¡¡by¡¡the¡¡other¡£
¡¡¡¡The¡¡evidence¡¡of¡¡mathematics¡¡rests¡¡upon¡¡definitions£»¡¡axioms£»¡¡and
demonstrations¡£¡¡I¡¡shall¡¡be¡¡satisfied¡¡with¡¡showing¡¡that¡¡none¡¡of¡¡these
forms¡¡can¡¡be¡¡employed¡¡or¡¡imitated¡¡in¡¡philosophy¡¡in¡¡the¡¡sense¡¡in
which¡¡they¡¡are¡¡understood¡¡by¡¡mathematicians£»¡¡and¡¡that¡¡the
geometrician£»¡¡if¡¡he¡¡employs¡¡his¡¡method¡¡in¡¡philosophy£»¡¡will¡¡succeed
only¡¡in¡¡building¡¡card¡castles£»¡¡while¡¡the¡¡employment¡¡of¡¡the
philosophical¡¡method¡¡in¡¡mathematics¡¡can¡¡result¡¡in¡¡nothing¡¡but¡¡mere
verbiage¡£¡¡The¡¡essential¡¡business¡¡of¡¡philosophy£»¡¡indeed£»¡¡is¡¡to¡¡mark¡¡out
the¡¡limits¡¡of¡¡the¡¡science£»¡¡and¡¡even¡¡the¡¡mathematician£»¡¡unless¡¡his
talent¡¡is¡¡naturally¡¡circumscribed¡¡and¡¡limited¡¡to¡¡this¡¡particular
department¡¡of¡¡knowledge£»¡¡cannot¡¡turn¡¡a¡¡deaf¡¡ear¡¡to¡¡the¡¡warnings¡¡of
philosophy£»¡¡or¡¡set¡¡himself¡¡above¡¡its¡¡direction¡£
¡¡¡¡I¡£¡¡Of¡¡Definitions¡£¡¡A¡¡definition¡¡is£»¡¡as¡¡the¡¡term¡¡itself¡¡indicates£»
the¡¡representation£»¡¡upon¡¡primary¡¡grounds£»¡¡of¡¡the¡¡complete¡¡conception
of¡¡a¡¡thing¡¡within¡¡its¡¡own¡¡limits¡£*¡¡Accordingly£»¡¡an¡¡empirical
conception¡¡cannot¡¡be¡¡defined£»¡¡it¡¡can¡¡only¡¡be¡¡explained¡£¡¡For£»¡¡as
there¡¡are¡¡in¡¡such¡¡a¡¡conception¡¡only¡¡a¡¡certain¡¡number¡¡of¡¡marks¡¡or
signs£»¡¡which¡¡denote¡¡a¡¡certain¡¡class¡¡of¡¡sensuous¡¡objects£»¡¡we¡¡can
never¡¡be¡¡sure¡¡that¡¡we¡¡do¡¡not¡¡cogitate¡¡under¡¡the¡¡word¡¡which¡¡indicates
the¡¡same¡¡object£»¡¡at¡¡one¡¡time¡¡a¡¡greater£»¡¡at¡¡another¡¡a¡¡smaller¡¡number¡¡of
signs¡£¡¡Thus£»¡¡one¡¡person¡¡may¡¡cogitate¡¡in¡¡his¡¡conception¡¡of¡¡gold£»¡¡in
addition¡¡to¡¡its¡¡properties¡¡of¡¡weight£»¡¡colour£»¡¡malleability£»¡¡that¡¡of
resisting¡¡rust£»¡¡while¡¡another¡¡person¡¡may¡¡be¡¡ignorant¡¡of¡¡this
quality¡£¡¡We¡¡employ¡¡certain¡¡signs¡¡only¡¡so¡¡long¡¡as¡¡we¡¡require¡¡them¡¡for
the¡¡sake¡¡of¡¡distinction£»¡¡new¡¡observations¡¡abstract¡¡some¡¡and¡¡add¡¡new
ones£»¡¡so¡¡that¡¡an¡¡empirical¡¡conception¡¡never¡¡remains¡¡within¡¡permanent
limits¡£¡¡It¡¡is£»¡¡in¡¡fact£»¡¡useless¡¡to¡¡define¡¡a¡¡conception¡¡of¡¡this¡¡kind¡£
If£»¡¡for¡¡example£»¡¡we¡¡are¡¡speaking¡¡of¡¡water¡¡and¡¡its¡¡properties£»¡¡we¡¡do
not¡¡stop¡¡at¡¡what¡¡we¡¡actually¡¡think¡¡by¡¡the¡¡word¡¡water£»¡¡but¡¡proceed¡¡to
observation¡¡and¡¡experiment£»¡¡and¡¡the¡¡word£»¡¡with¡¡the¡¡few¡¡signs
attached¡¡to¡¡it£»¡¡is¡¡more¡¡properly¡¡a¡¡designation¡¡than¡¡a¡¡conception¡¡of
the¡¡thing¡£¡¡A¡¡definition¡¡in¡¡this¡¡case¡¡would¡¡evidently¡¡be¡¡nothing¡¡more
than¡¡a¡¡determination¡¡of¡¡the¡¡word¡£¡¡In¡¡the¡¡second¡¡place£»¡¡no¡¡a¡¡priori
conception£»¡¡such¡¡as¡¡those¡¡of¡¡substance£»¡¡cause£»¡¡right£»¡¡fitness£»¡¡and
so¡¡on£»¡¡can¡¡be¡¡defined¡£¡¡For¡¡I¡¡can¡¡never¡¡be¡¡sure£»¡¡that¡¡the¡¡clear
representation¡¡of¡¡a¡¡given¡¡conception¡¡£¨which¡¡is¡¡given¡¡in¡¡a¡¡confused
state£©¡¡has¡¡been¡¡fully¡¡developed£»¡¡until¡¡I¡¡know¡¡that¡¡the
representation¡¡is¡¡adequate¡¡with¡¡its¡¡object¡£¡¡But£»¡¡inasmuch¡¡as¡¡the
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£º
¡¡¡¡*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¡¡be
СÌáʾ£º°´ »Ø³µ [Enter] ¼ü ·µ»ØÊéÄ¿£¬°´ ¡û ¼ü ·µ»ØÉÏÒ»Ò³£¬ °´ ¡ú ¼ü ½øÈëÏÂÒ»Ò³¡£
ÔÞÒ»ÏÂ
Ìí¼ÓÊéÇ©¼ÓÈëÊé¼Ü