phenomenology of mind-第13章

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supposed　to　be　concerned　about。

As　to　mathematical　truths；　we　should　be　still　less　inclined　to　consider　anyone　a　geometer　who　had
got　Euclid's　theorems　by　heart　（auswendig）　without　knowing　the　proofs；　without；　if　we　may　say
so　by　way　of　contrast；　getting　them　into　his　head　（inwendig）。　Similarly；　if　anyone　came　to　know
by　measuring　many　right…angled　triangles　that　their　sides　are　related　in　the　way　everybody　knows；
we　should　regard　knowledge　so　obtained　as　unsatisfactory。　All　the　same；　while　proof　is　essential　in
the　case　of　mathematical　knowledge；　it　still　does　not　have　the　significance　and　nature　of　being　a
moment　in　the　result　itself　；　the　proof　is　over　when　we　get　the　result；　and　has　disappeared。　Qua
result　the　theorem　is；　no　doubt；　one　that　is　seen　to　be　true。　But　this　eventuality　has　nothing　to　do
with　its　content；　but　only　with　its　relation　to　the　knowing　subject。　The　process　of　mathematical
proof　does　not　belong　to　the　object；　it　is　a　function　that　takes　place　outside　the　matter　in　hand。
Thus；　the　nature　of　a　right…angled　triangle　does　not　break　itself　up　into　factors　in　the　manner　set
forth　in　the　mathematical　construction　which　is　required　to　prove　the　proposition　expressing　the
relation　of　its　parts。　The　entire　process　of　producing　the　result　is　an　affair　of　knowledge　which
takes　its　own　way　of　going　about　it。　In　philosophical　knowledge；　too；　the　way　existence；　qua
existence；　comes　about　（Werden）　is　different　from　that　whereby　the　essence　or　inner　nature　of　the
fact　comes　into　being。　But　philosophical　knowledge；　for　one　thing；　contains　both；　while
mathematical　knowledge　sets　forth　merely　the　way　an　existence　comes　about；　i。e。　the　way　the
nature　of　the　fact　gets　to　be　in　the　sphere　of　knowledge　as　such。　For　another　thing；　too；
philosophical　knowledge　unites　both　these　particular　movements。　The　inward　rising　into　being；　the
process　of　substance；　is　an　unbroken　transition　into　outwardness；　into　existence　or　being　for
another；　and　conversely；　the　coming　of　existence　into　being　is　withdrawal　into　the　inner　essence。
The　movement　is　the　twofold　process　in　which　the　whole　comes　to　be；　and　is　such　that　each　at　the
same　time　posits　the　other；　and　each　on　that　account　has　in　it　both　as　its　two　aspects。　Together
they　make　the　whole；　through　their　resolving　each　other；　and　making　themselves　into　moments　of
the　whole。　

In　mathematical　knowledge　the　insight　required　is　an　external　function　so　far　as　the　subject…matter
dealt　with　is　concerned。　It　follows　that　the　actual　fact　is　thereby　altered。　The　means　taken；
construction　and　proof；　contain；　no　doubt；　true　propositions；　but　all　the　same　we　are　bound　to　say
that　the　content　is　false。　The　triangle　in　the　above　example　is　taken　to　pieces；　and　its　parts　made
into　other　figures　to　which　the　construction　in　the　triangle　gives　rise。　It　is　only　at　the　end　that　we
find　again　reinstated　the　triangle　we　are　really　concerned　with；　it　was　lost　sight　of　in　the　course　of
the　construction；　and　was　present　merely　in　fragments；　that　belonged　to　other　wholes。　Thus　we
find　negativity　of　content　coming　in　here　too；　a　negativity　which　would　have　to　be　called　falsity；　just
as　much　as　in　the　case　of　the　movement　of　the　notion　where　thoughts　that　are　taken　to　be　fixed
pass　away　and　disappear。　

The　real　defect　of　this　kind　of　knowledge；　however；　affects　its　process　of　knowing　as　much　as　its
material。　As　to　that　process；　in　the　first　place　we　do　not　see　any　necessity　in　the　construction。　The
necessity　does　not　arise　from　the　nature　of　the　theorem：　it　is　imposed；　and　the　injunction　to　draw
just　these　lines；　an　infinite　number　of　others　being　equally　possible；　is　blindly　acquiesced　in；　without
our　knowing　anything　further；　except　that；　as　we　fondly　believe；　this　will　serve　our　purpose　in
producing　the　proof。　Later　on　this　design　then　comes　out　too；　and　is　therefore　merely　external　in
character；　just　because　it　is　only　after　the　proof　is　found　that　it　comes　to　be　known。　In　the　same
way；　again；　the　proof　takes　a　direction　that　begins　anywhere　we　like；　without　our　knowing　as　yet
what　relation　this　beginning　has　to　the　result　to　be　brought　out。　In　its　course；　it　takes　up　certain
specific　elements　and　relations　and　lets　others　alone；　without　its　being　directly　obvious　what
necessity　there　is　in　the　matter。　An　external　purpose　controls　this　process。

The　evidence　peculiar　to　this　defective　way　of　knowing　—　an　evidence　on　the　strength　of　which
mathematics　plumes　itself　and　proudly　struts　before　philosophy　—　rests　solely　on　the　poverty　of　its
purpose　and　the　defectiveness　of　its　material；　and　is　on　that　account　of　a　kind　that　philosophy　must
scorn　to　have　anything　to　do　with。　Its　purpose　or　principle　is　quantity。　This　is　precisely　the
relationship　that　is　non…essential；　alien　to　the　character　of　the　notion。　The　process　of　knowledge
goes　on；　therefore；　on　the　surface；　does　not　affect　the　concrete　fact　itself；　does　not　touch　its　inner
nature　or　lotion；　and　is　hence　not　a　conceptual　way　of　comprehending。　The　material　which
provides　mathematics　with　these　welcome　treasures　of　truth　consists　of　space　and　numerical　units
（das　Eins）。　Space　is　that　kind　of　existence　wherein　the　concrete　notion　inscribes　the　diversity　it
contains；　as　in　an　empty；　lifeless　element　in　which　its　differences　likewise　subsist　in　passive；　lifeless
form。　What　is　concretely　actual　is　not　something　spatial；　such　as　is　treated　of　in　mathematics。　With
unrealities　like　the　things　mathematics　takes　account　of；　neither　concrete　sensuous　perception　nor
philosophy　has　anything　to　do。　In　an　unreal　element　of　that　sort　we　find；　then；　only　unreal　truth；
fixed　lifeless　propositions。　We　can　call　a　halt　at　any　of　them；　the　next　begins　of　itself　de　novo；
without　the　first　having　led　up　to　the　one　that　follows；　and　without　any　necessary　connexion　having
in　this　way　arisen　from　the　nature　of　the　subject…matter　itself。　So；　too　—　and　herein　consists　the
formal　character　of　mathematical　evidence　because　of　that　principle　and　the　element　where　it
applies；　knowledge　advances　along　the　lines　of　bare　equality；　of　abstract　identity。　For　what　is
lifeless；　not　being　self…moved；　does　not　bring　about　distinction　within　its　essential　nature；　does　not
attain　to　essential　opposition　or　unlikeness；　and　hence　involves　no　transition　of　one　opposite
element　into　its　other；　no　qualitative；　immanent　movement；　no　self…movement；　It　is　quantity；　a　form
of　difference　that　does　not　touch　the　essential　nature；　which　alone　mathematics　deals　with。　It
abstracts　from　the　fact　that　it　is　the　notion　which　separates　space　into　its　dimensions；　and
determines　the　connexions　between　them　and　in　them。　It　does　not　consider；　for　example；　the
relation　of　line　to　surface；　and　when　it　compares　the　diameter　of　a　circle　with　its　circumference；　it
runs　up　against　their　incommensurability；　i。e。　a　relation　in　terms　of　the　notion；　an　infinite　element；
that　escapes　mathematical　determination。　

Immanent　or　so…called　pure　mathematics；　again；　does　not　oppose　time　qua　time　to　space；　as　a
second　subject…matter　for　consideration。　Applied　mathematics；　no　doubt；　treats　of　time；　as　also　of
motion；　and　other　concrete　things　as　well；　but　it　picks　up　from　experience　synthetic　propositions
—　i。e。　statements　of　their　relations；　which　are　determined　by　their　conceptual　nature　—　and　merely
applies　its　formulae　to　those　propositions　assumed　to　start　with。　That　the　so…called　proofs　of
propositions　like　that　concerning　the　equilibrium　of　the　lever；　the　relation　of　space　and　time　in
gravitation；　etc。；　which　applied　mathematics　frequently　gives；　should　be　taken　and　given　as　proofs；
is　itself　merely　a　proof　of　how　great　the　need　is　for　knowledge　to　have　a　process　of　proof；　seeing
that；　even　where　proof　is　not　to　be　had；　knowledge　yet　puts　a　value　on　the　mere　semblance　of　it；
and　gets　thereby　a　certain　sense　of　satisfaction。　A　criticism　of　those　proofs　would　be　as　instructive
as　it　would　be　significant；　if　the　criticism　could　strip　mathematics　of　this　artificial　finery；　and　bring
out　its　limitations；　and　thence　show　the　necessity　for　another　type　of　knowledge。

As　to　time；　which；　it　is　to　be　presumed；　would；　by　way　of　the　counterpart　to　space；　constitute　the
object…matter　of　the　other　division　of　pure　mathematics；　this　is　the　notion　itself　in　the　form　of
existence。　The　principle　of　quantity；　of　difference　which　is　not　determined　by　the　notion；　and　the
principle　of　equality；　of　abstract；　lifeless　unity；　are　incapable　of　dealing　with　that　sheer　restlessness
of　life　and　its　absolute　and　inherent　process　of　differentiation。　It　is　therefore　only　in　an　arrested；
paralysed　form；　only　in　the　form　of　the　quantitative　unit；　that　this　essentially　negative　activity
becomes　the　second　object…matter　of　this　way　of　knowing；　which；　itself　an　external　operation；
degrades　what　is　self…moving　to　the　level　of　mere　matter；　in　order　thus　to　get　an　indifferent；
external；　lifeless　content。　

　　　　　　　　　　　　　　13。　　The　nature　of　philosophical　truth　and　its　method

Philosophy；　on　the　contrary；　does　not　deal　with　a　determination　that　is　non…essential；　but　with　a
determination　so　far　as　it　is　an　essential　factor。　The　abstract　or　unreal　is　not　its　element　and
content；　but　the　real；　what　is　self…establishing；　has　life　within　itself；　existence　in　its　very　notion。　It　is
the　process　that　creates　its　own　moments　in　its　course；　and　goes　through　them　all；　and　the　whole
of　this　movement　constitutes　its　positive　content　and　its　truth。　This　movement　includes；　therefore；
within　it　the　negative　factor　as　well；　the　element　which　would　be　named　falsity　if　it　could　be
considered　one　from　which　we　had　to　abstract。　The　element　that　disappears　has　rather　to　be
looked　at　as　itself　essential；　not　in　the　sense　of　being　something　fixed；　that　has　to　be　cut　off　from
truth　and　allowed　to　lie　outside　it；　heaven　knows　where；　just　as　similarly　the　truth　is　not　to　be　held
to　stand　on　the　other　side　as　an　immovable　lifeless　positive　element。　Appearance　is　the　process　of
arising　into　being　and　passing　away　again；　a　process　that　itself　does　not　arise　and　does　not　pass
away；　but　is　per　se；　and　constitutes　reality　and　the　life…movement　of　truth。　The　truth　is　thus　the
bacchanalian　revel；　where　not　a　member　is　sober；　and　because　every　member　no　sooner　becomes
detached　than　it　eo　ipso　collapses　straightway；　the　revel　is　just　as　much　a　state　of　transparent
unbroken　calm。　Judged　by　that　movement；　the　particular　shapes　which　mind　assumes　do　not
indeed　subsist　any　more　than　do　determinate　thoughts　or　ideas；　but　they　are；　all　the　same；　as　much
positive　and　necessary　moments；　as　negative　and　transitory。　In　the　entirety　of　the　movement；　taken
as　an　unbroken　quiescent　whole；　that　which　obtains　distinctness　in　the　course　of　its　process　and
secures　specific　existence；　is　preserved　in　the　form　of　a　self　…recollection；　in　which　existence　is
self…knowledge；　and　self…knowledge；　again；　is　immediate　existence。　

It　might　well　seem　necessary　to　state　at　the　outset　the　chief　points　in　connexion　with　the　method　of
this　process；　the　way　in　which　science　operates。　Its　nature；　however；　is　to　be　found　in　what　has
already　been　said；　while　the　proper　systematic　exposition　of　it　is　the　special　business　of　Logic；　or
ra