304
TOPIC1876287010064133194229286
2.218428701005982
PHILOSOPHY2289287010061133168223292343413475546607
OF2919287010069124
THE3066287010064136193
ANCIENT3281287010065134190226283351415
WORLD3717287010096165232287359
12628878014010860120180
«Achilles921771012670172234312351389429491545
and1502771012691170248
the1785771012669147209
Tortoise»20297710126116185248295364403457520589
:2620769012839
Zeno26957690128108169239309
argued3040769012884131200270333402
that3477769012861131194233
the3745769012861131194
swift3974769012876178216263301
hero4311769012892155202271
Achilles46167690128124186256296335374436490
will51417690128124163202241
never5671011012870133202265312
be9141011012875138
able10871011012868137177239
to13611011012844115
catch15111011012868131169231301
up18471011012876145
to20281011012845115
a21791011012868
slow2282101101286099168270
tortoise25871011012844115162200270309363425
that30471011012845115178216
has32981011012876139192
been35261011012876139200270
given38321011012875114184246315
a41821011012868
head42851011012876139201271
start.4592101101286098161208246281
Analyze491410110128101171233272343404466
the5671253012839110172
reasoning7751253012846107170224294364403473542
behind13531253012870132202240311380
this17691253012837108147202
claim.20061253012862100163201310343
«The9211496012670164242305
Falling12651496012686154194232271349419
Grain»172314960126109172241280358427
:21511494012839
A222914940128101
single2365149401285997168238276339
falling27391494012851112151190229299368
grain31431494012873121182221291
makes347314940128110169240303357
no38661494012873144
sound,40441494012859129199269339374
but44541494012873143182
a46711494012867
falling47721494012851113151191229299369
bag51771494012873136205
of5671735012870117
grain7191735012872119181219290
produces10441735012871118187258327390453506
an15861735012864133
audible17551735012863133202242311351413
sound.22031735012856126196266336371
This26111735012886156194249
leads28951735012841102165235289
to32201735012841110
a33661735012864
paradox:34651735012871134181242312382452491
if3991173501284087
a41141735012863
part42121735012872135181219
(one44671735012849118188250
grain)47521735012872119181220290337
does51241735012871141203257
not5671977012870141179
possess7821977012878148203257320373428
a12461977012870
property13511977012877124194264327374412481
(sound),18671977012854109180250320390437472
then23741977012847116179249
the26591977012846116179
whole288119770128102170241279342
(a32581977012855117
bag34101977012878140210
of36561977012877124
grain)38151977012878126188226295343
should41931977012862133203272312381
not46101977012878148187
possess48331977012877147201256318373427
it5295197701284786
either.5672219012862101141210273320355
Explain9572219012886155225264326364435
the14272219012838108171
philosophical16322219012870140178218287342412482552591654716754
problem24212219012870116187256295357467
this29202219012839111150204
poses.31602219012870139194256311347
3.A9212633012870106192
special11472633012854125187249288358396
task15792633012839108163226
for18402633012837107162
the20382633012837108169
«Pythagorean22432633012870156218257326396467536591653723793
School».30702633012870133203273342382451487
Explain9212874012886155226265327366436
the13922874012897167229
philosophical165628740128129198237276346401471540611649712774812
and250428740128121190259
religious279928740128105167205245314354423493548
significance338228740128113152222291331378416479541610673735
of415128740128129176
mathematics436228740128168228267337399508568607647710764
for516228740128105174221
Pythagoras5673116012878148185256319389459506568622
and12243116012863133202
answer14623116012862131186288348395
the18923116012839109172
following20993116012846116155194264366404474543
questions.26783116012870139202256295334404474528563
The9213357012886155218
Pythagoreans117433570128124194231302364433504550613675744799
discovered200833570128115154209271340410473520582651
that26953357012883153215254
the29843357012884153216
side32353357012899138208274
and354333570128107176247
diagonal382433570128116154217286357426489527
of438733570128114161
a458233570128107
square472433570128100170240302349411
are517033570128107154216
incommensurable5673599012839110172241351459519589644715762825894933995
–159735990128103
their17363599012870141204242289
ratio20603599012879141179218288
cannot23833599012895157226296366405
be282335990128103164
expressed30213599012895165236283344399453516585
as36413599012895149
a38263599012894
fraction39553599012879126187249287327396467
(a44573599012879140
ratio46333599012878141180218288
of495635990128103150
two51413599012871173241
integers).5673840012839110148211280343389443490525
This112738400128114184223277
discovery14403840012898137191254324394456503573
(such20473840012875129200263332
numbers24143840012899169278346409456510
are29603840012891137200
now31953840012898168270
called34993840012891153192231294363
irrational)38973840012868115162223262300370441502541589
caused45213840012890153222276339408
a49643840012891
deep50963840012899161223292
crisis5674082012862109148202242296
within89840820128154192232302341411
the13444082012892161224
school.160340820128107169238308378417452
Why209240820128185253325
did245040820128123162233
this27184082012892161201255
discovery300840820128122162217278349418481528598
strike363940820128107147194232302365
a403940820128115
blow418840820128123162232333
to45564082012891166
the47574082012892161224
entire501640820128115184224262309370
Pythagorean5674323012878148185256319389459506568631700
system?13024323012871142195234297406466
Why182443230128132201271
did21294323012887127197
the23624323012855125188
Pythagoreans25854323012896166203274336406475522585647716771
impose33914323012856166234304359421
a38474323012880
taboo39614323012857119189259329
on43254323012887157
the45184323012855125188
mention475843230128110169240279318388458
of52524323012886132
irrational567456401283986133195233273342412475514
numbers?11154564012870140249317380427481543
Modern92148070128125195265327374443
science14004807012885147186249318381443
identifies18774807012870139202271310349396434497550
certain24634807012893155202240303341411
numerical290948070128101171280340387426489551590
patterns353848070128100162200240303348418472
in40464807012869139
natural422148070128100162200271317378417
phenomena.467448070128100170233302372481541612674710
Demonstrate56750480128101163273341412467505553615653716
the13185048012858127190
«power15435048012889159228329391438
of20165048012889136
numbers21875048012889159268336399446500
»27345048012870
in28395048012858127
nature.Briefly30015048012889152191260307370405499544583646693731801
explain38365048012881151222260323361432
how43035048012888158260
the45985048012856127189
following48225048012868138176216285387426495565
mathematical56752890128109170209280343452512551590653715753
entities13565289012883153192230270308371425
might181552890128131167238308348
be21985289012891153
related23865289012867130169230269332401
to28235289012859128
the29865289012860130193
structure32145289012874114161230293331401448510
of37595289012890137
the39325289012859129191
world:415752890128122192239278348386
the45795289012859130192
number48055289012892161271339402448
π52885289012898
(pi);5675530012847117157203242
the8445530012838108171
Golden105055300128102170209279342411
Ratio14965530012894156194233303
(φ);18345530012847129177215
the20845530012839108171
exponential22905530012863131201272341404473513551614652
constant29765530012863133203258296359429467
(e).34795530012847109156191
Mathematicians92157730128125187226296359468528567606669708770840894
use18515773012875129192
numbers20785773012876147256324387434488
to26015773012846115
describe27525773012876138192255301339410472
their32595773012844114177216262
properties,35565773012876123192263326371410449512565601
but41925773012876146184
do44125773012876145
not45935773012876146185
give48135773012876115185247
them50955773012845115178287
an5676014012862133
exhaustive7356014012876146215278348402441480550612
philosophical13836014012883152192230301355424495565603666728766
definition.21856014012884146192231301339379417488558593
How281360140128118186288
is31366014012852107
the32786014012852122185
lack34976014012853116177247
of37796014012884131
a39446014012876
universal40556014012883153192262325372425488526
philosophical46166014012883154192232301356426496565605668729767
definition5676255012870133180218288327366405474545
of11466255012870117
number12986255012871140250317380427
related17606255012847109148211249311380
to21766255012839108
the23206255012839110173
depth25286255012870132202240310
of28746255012870116
the30256255012839109172
Pythagorean32326255012878149186256319388458505567629699
idea?39666255012839108171232295
4.A9216669012870106192
special11476669012854125187249288358396
task15796669012839108163226
for18406669012837107162
the20386669012837108169
«Atomist22436669012870156195264366404458498
School».27756669012870133202272342381451489
Look9216911012886154224293
at12496911012870108
the13926911012846115177
illustration1603691101284784123191245282328390428466534603
(Fig.22406911012855132170239274
26).25486911012878146192226
It2808691101285593
depicts29356911012879140210247308346399
a33676911012871
model348169110128109176245307345
of38616911012878124
the40196911012847117178
atom42316911012871107177286
of45486911012879126
the47086911012847116178
simplest49206911012863100209275314375429467
and5677153012862131201
most81371530128109176230268
abundant11167153012873142211281349411480517
chemical16687153012873142203312347409470508
element22107153012873111173282341410448
in26937153012849119
the28477153012849119179
Universe307171530128101169208278339385438500
–36057153012881
one37207153012881149211
could39657153012873142211249318
say,43187153012865127196228
the45797153012850119181
primordial47947153012881127165275341387456495557595