phenomenology of mind-第14部分

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real　meaning；　precisely　because　we　want　to　designate　the　aspect　or　moment　of　complete
otherness；　the　terms　true　and　false　must　no　longer　be　used　where　their　otherness　has　been
cancelled　and　superseded。　Just　as　the　expressions　〃unity　of　subject　and　object〃；　of　〃finite　and
infinite〃；　of　〃being　and　thought〃；　etc。；　are　clumsy　when　subject　and　object；　etc。；　are　taken　to　mean
what　they　are　outside　their　unity；　and　are　thus　in　that　unity　not　meant　to　be　what　its　very
expression　conveys；　in　the　same　way　falsehood　is　not；　qua　false；　any　longer　a　moment　of　truth。　

Dogmatism　as　a　way　of　thinking；　whether　in　ordinary　knowledge　or　in　the　study　of　philosophy；　is
nothing　else　but　the　view　that　truth　consists　in　a　proposition；　which　is　a　fixed　and　final　result；　or
again　which　is　directly　known。　To　questions　like；　〃When　was　Caesar　born？〃。　〃How　many　feet
make　a　furlongs〃；　etc。；　a　straight　answer　ought　to　be　given；　just　as　it　is　absolutely　true　that　the
square　of　the　hypotenuse　is　equal　to　the　sum　of　the　squares　of　the　other　two　sides　of　a　right…angled
triangle。　But　the　nature　of　a　so…called　truth　of　that　sort　is　different　from　the　nature　of　philosophical
truth。

　　　　　　　　　　　　　　　　　　　　　12。　　Historical　and　mathematical　truth

As　regards　truth　in　matters　of　historical　fact…to　deal　briefly　with　this　subject　—　so　far　as　we
consider　the　purely　historical　element；　it　will　be　readily　granted　that　they　have　to　do　with　the　sphere
of　particular　existence；　with　a　content　in　its　contingent　and　arbitrary　aspects；　features　that　have　no
necessity。　But　even　bare　truths　of　the　kind；　say；　like　those　mentioned；　are　impossible　without　the
activity　of　self　…consciousness。　In　order　to　know　any　one　of　them；　there　has　to　be　a　good　deal　of
comparison；　books　must　be　consulted；　or　in　some　way　or　other　inquiry　has　to　be　made。　Even　in　a
case　of　direct　perception；　only　when　we　know　it　along　with　the　reasons　behind　it；　is　it　held　to　be
something　of　real　value；　although　it　is　merely　the　naked　fact　itself　that　we　are；　properly　speaking；
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；　const