GENESIS OF A PYTHAGOREAN UNIVERSE

F. Acerbi À 04 survey Byzantine recensions of greek mathematical and astronomical texts: e Estudios bizanticos. 2016, 4, p 133 - 213 Lela Alexidze (Tbilisi) AZ co 5 A Lo 6 Phasis 13-14, 2010-2011 p 9-26 ORPHISCHE THEOGONIE UND PLATONISCHE KOSMOLOGIE IN DEN PROKLOSKOMMENTAREN http://www.poetry.net/poem/16414 Hans Christian Andersen “Formens evige Magie 3p Berges, S. Why don't women philosophers count as philosophers? A? 0 Y Posted on June 12, 2013 by Sandrine Berges Feminist history of philosophy. NL GENESIS OF A PYTHAGOREAN UNIVERSE Alexey Burov* and Lev Burov* FERMILAB-PUB-13-533-ad 8p Chapter Trick or Truth? Part of the series The Frontiers Collection pp 157-170 \ as Importance of Mathematics and Geometry in Architecture p 1-11 Ashish Choudhary1 4 Journal of Recent Activities in Architectural Sciences Volume 1 Issue 1. 2016 ds À LI Collier, J. THE Mythological Picture OF CEBES THE THEBAN. The emperor Marcus Antonius his conversation with himself 1708

FERMILAB-PUB-13-533-ad GENESIS OF A PYTHAGOREAN UNIVERSE Alexey Burov* and Lev Burov° *FNAL, Batavia, IL °Scientific Humanities LLC, San Francisco, CA Abstract Wide   range,   high   precision   and   simplicity   of   the   fundamental   laws   of   nature   rule   out   the   possibility   for   them   to   be   randomly   generated   or   selected.   Therefore   purpose  is  present  in  their  selec:on.   Introduc)on      The  task  of  science,  as  it  is  generally  assumed,  is  to   find   the   laws   of   nature   allowing   both   to   explain   the   diversity   of   observa:ons   as   well   as   to   predict   new   ones.   Science   seeks   to   discover   the   logic   that   is   hidden   beneath   phenomena   and   which   determines   their  flow  and  quali:es.  The  understanding  of  truth  as   uncovering   of   hidden   essence,   as   dis-­‐covery,   is   embedded   in   the   Greek   word   αλήθεια   (truth),   consis:ng   of   nega:on   (α-­‐)   and   λήθη,   which   means   a   veil   or   concealment.   Pythagoras   taught   that   this   essence  is  the  harmony  of  hidden  unity  which  can  be   expressed  in  the  language  of  numbers.  When  Galileo   stated  that  nature  is  a  book  wriNen  in  the  language  of   mathema:cs,   he   was   expressing   this   ancient   Pythagorean   credo.   The   same   can   be   said   about   Dirac,  whose  fundamental  belief  was  that  “the  laws  of   nature   should   be   expressed   in   beau:ful   equa:ons”,   and   about   Einstein   who   believed   that   the   strongest   and  noblest  mo:ve  for  the  scien:fic  search  is  a  deep   convic:on  of  the  ra:onality  of  the  universe,  saturated   with  the  cosmic  religious  feeling.     When   theories   that   exhaust   phenomena   are   formulated  and  logically  unified  into  a  single  theory  of   everything,   the   task   of   fundamental   science   is   finished.  Whatever    this  theory  of  everything    may  be,   other   theories   in   physics   will   be   its   consequences   as   limit   cases   or   asymptotes.   Although   humanity   does   not  now  and  may  possibly  never  have  such  a  theory  in   its  fullness,  many  of  its  limit  cases  are  known  to  us  as   concrete   theories,   such   as   classical   and   quantum   mechanics,   general   theory   of   rela:vity,   the   standard   model,  and  others.      The  laws  of  nature  are  discovered  as  composite  and   specific   mathema:cal   structures.   As   these   structures   are   revealed,   we   unavoidably   come   to   a   certain   ques:on  regarding  the  structures  themselves.  First  of   all,   why   does   any   law   expressed   by   one   or   another   mathema:cal   formula   structure   our   world   at   all? While  it  is  thinkable  for  a  universe  to  be  structured  by   any    logically  consistent  system,  out  of  this  infinite  set   of  structures  only  one  determines  our  universe.  Why   this   structure   and   not   another?   Why   are   the   laws   simple   enough   to   be   discovered?   Why   are   they   mathema:cally  beau:ful?    Who  or  what  singled  it  out   and  on  what  ground?       In   this   way   the   laws   of   nature   become   a   problem,   though  not  in  the  usual  scien:fic  context  of  searching   them   out,   but   as   something   that   requires   its   own   explana:on.   The   illusory   nature   of   an   explana:on   that  does  not  go  beyond  natural  laws  was  pointed  out   by  Ludwig  WiNgenstein[1]:        The  whole  modern  concep:on  of  the  world   is   founded   on   the   illusion   that   the   so-­‐called   laws  of  nature  are  the  explana:ons  of  natural   phenomena.   Thus   people   today   stop   at   the   laws   of   nature,   trea:ng   them   as   something   inviolable,  just  as  God  and  Fate  were  treated   in   past   ages.   And   in   fact   both   are   right   and   both  wrong:  though  the  view  of  the  ancients   is   clearer   in   so   far   as   they   have   a   clear   and   acknowledged   terminus,   while   the   modern   system   tries   to   make   it   look   as   if   everything   were  explained.   Operated by Fermi Research Alliance, LLC under Contract No. De-AC02-07CH11359 with the United States Department of Energy.

Here  WiNgenstein  cri:cizes  a  silent  acceptance  of  a   composite  and  special  mathema:cal  structure  as  the   ul:mate   explana:on   of   the   world.   Such   explana:on   barred   from   further   ques:oning   and   not   subject   to   reasonable   ground   of   its   own   existence   is   an   affirma:on   of   unreasonableness   of   this   ground.   In   other   words,   it   is   an   acceptance   of   absurdity   as   the   ul:mate   founda:on   of   existence.   Such   supers::on   destroys   the   meaning   of   fundamental   science   by   undermining  the  importance  of  reason,  subjected    by   this  supers::on  to  the  absurd.      What  can  be  the  answers  concerning  the  source  of   the   laws   of   nature?   Is   there   any   way   of   choosing   or   rejec:ng   one   or   another?   That   is   the   topic   of   discussion  in  the  present  ar:cle.   The  Fine  Tuning  Ques)on       “There   is   now   broad   agreement   among   physicists   and   cosmologists”,   writes   Paul   Davis   [2],   “that   the   universe   is   in   several   respects   ‘fine-­‐tuned'   for   life”.   Similarly,  Stephen  Hawking  has  noted:         The   laws   of   science,   as   we   know   them   at   present,   contain   many   fundamental   numbers,   like   the   size   of   the   electric   charge   of   the   electron  [fine  structure  constant]  and  the  ra:o   of   the   masses   of   the   proton   and   the   electron.   ...   The   remarkable   fact   is   that   the   values   of   these   numbers   seem   to   have   been   very   finely   adjusted   to   make   possible   the   development  of  life.[3].     Another   crucial   point   is   ar:culated   by   Alexei   Tsvelik   [4]:   [since]   the   number   of   exis:ng   life-­‐imposing   condi:ons   by   far   exceeds   the   number   of   constants,   their   fulfillment   could   not   be   achieved  by  fine  tuning  of  these  constants  and   required   also   the   right   choice   of   the   fundamental  principles  of  physical  laws.   The   premise   of   the   fine-­‐tuned   universe   revived   the   old   metaphysical   problem   of   the   source   of   order   in   the  world  as  the  problem  of  fine-­‐tuning:  who  or  what   tuned  the  universe  so  fine?  A  pure  scien:fic  approach   required   finding   an   objec:ve   answer:   not   “somebody”  but  “something”  as  the  cause  of  tuning.       Order  From  Chaos   It   is   thought   that   this   “something”   could   be   any   combina:on   of   laws   of   nature   provided   by   one   or   another   general   theory   and   chao:c   factors;   or,   using   the  terms  of  Platonic  philosophy,  any  combina:on  of   forms   and   chaos.   However,   as   it   was   noted   by   WiNgenstein,   any   theory   used   in   that   respect   itself   requires   to   be   explained.   John   А.   Wheeler   expressed   the   same   as   a   ques:on:   why   is   this   very   theory   structuring   everything   existent?   Why   doesn’t   some   other  theory  instead?  In  other  words,  the  use  of  any   theory   for   this   does   not   solve   the   problem   of   fine-­‐ tuning,  but  moves  it  to  a  higher  level.  The  only  way  to   solve  this  problem  totally  in  the  framework  of  science   is   to   show   a   possibility   of   appearance   of   being   from   nothing,   or   chaosogenesis,   the   appearance   of   order   from   chaos.   Indeed,   theories,   being   specific   formal   structures,   are   limited   and   composite   en::es,   and   thus   lead   to   the   ques:on   “why   this   theory   and   not   other?”.  Chaos  per  se  is  limitless  and  structureless,  a   totality   intrinsically   undivided   into   “this”   and   “that”,   whose   various   manifesta:ons   differ   from   each   other   due  only  to  the  variety  of  doors  that  one  or  another   theory  opens  for  chaos  to  enter.    Historically,  the  idea   of   chaosogenesis   is   very   old,   having   been   traced   down   to   Hesiod   and   pre-­‐Socra:cs,   and   it   had   been   opposed   by   the   Pythagoreans   and   Platonics.   For   instance,   Plo:nus   wrote:   “Any   aNempt   to   derive   order,   reason,   or   the   direc:ng   soul   from   the   unordered  mo:on  of  atoms  or  elements  is  absurd  and   impossible.”[5]   Not   all   contemporary   cosmologists   share   Plo:nus’   views   on   the   chaosogenesis,   so   the   idea  is  frequently  pronounced.     Max   Tegmark   has   formulated   the   "Ul:mate   ensemble   theory   of   everything”,   whose   main   mo:va:on  is  clearly  expressed  [6]:     If  the  TOE  [theory  of  everything]  exists  and  is   one   day   discovered,   then   an   embarrassing   ques:on   remains,   as   emphasized   by   John   Archibald   Wheeler:   Why   these   par:cular   equa:ons,  not  others?  Could  there  really  be  a   fundamental,   unexplained   ontological   asymmetry  built  into  the  very  heart  of  reality,

splirng   mathema:cal   structures   into   two   classes,   those   with   and   without   physical   existence?   Aser   all,   a   mathema:cal   structure   i s   n o t   “c r e a t e d ”   a n d   d o e s n ’ t   e x i s t   “somewhere”.   It   just   exists.   As   a   way   out   of   this   philosophical   conundrum,   I   have   suggested   that   complete   mathema:cal   democracy  holds:  that  mathema:cal  existence   and   physical   existence   are   equivalent,   so   that   all   mathema:cal   structures   have   the   same   ontological  status.     Thus,   for   Tegmark   the   terminus   ul:mately   explaining   everything   exis:ng   is   the   totality   of   all   mathema:cal   forms,   the   platonic   world.   To   “just   exist”,   the   mathema:cal   structure   has   to   be   self-­‐ consistent,   logically   acceptable.   What   he   doesn’t   men:on   is   the   unity   of   these   forms.   This   unity   must   not   only   somehow   bind   every   one   of   them   together   but   it   has   to     guarantee   their   self-­‐consistency.   The   forms   though   are   mental   en::es.   They   are   not   thinkable   without   a   mind   which   contains   them   as   truly  self–consistent.  Thus,  we  have  to  conclude  that   this   unity,   the   terminus   of   Tegmark’s   being,   is   an   absolute  mind,  even  if  it  is  not  men:oned  at  all.  What   makes   this   mind   special   and   dis:nc:ve   from   its   various  platonic  versions  is  its  total  indifference  to  the   forms   it   contains.   That   is   what   Tegmark   calls   “the   mathema:cal  democracy”.     It   has   to   be   stressed,   that   purely   by   itself,   without   any   forms   involved,   chaos   cannot   produce   anything,   and   Tegmark’s   model   is   not   an   excep:on   from   this   rule:  it  assumes  that  all  possible  worlds  are  based  on   mathema:cal   structures,   such   as   groups,   algebras,   fields,  sets  of  equa:ons,  and  other  formal  systems.  It   also  assumes  that  there  is  a  way  for  these  structures   to   show   themselves   as   phenomena,   and   to   be   observed  both  as  mathema:cal  and  physical  objects.   Chaos   comes   in   this   picture   as   a   randomness   of   a   universe   we   happen   to   be   born   in,   with     the   only   limita:on   that   the   laws   of   this   universe   are   compa:ble  with  life  and  consciousness.    What  makes   Tegmark’s   model   very   special   is   its   minimal   involvement   of   a   priori   concre:za:on   or   selec:on   principles,  which  is  why  we  are  equa:ng  this  model  of   “mathema:cal  democracy”  with    chaosogenesis.     A   possibility   for   the   structure   of   the   fundamental   laws  of  nature  to  be  random  to  some  unclear  degree   and   beyond   that   to   be   non-­‐randomly   selected   by   some  unpronounced  en:ty  was  expressed  by  several   leading  scien:sts,  e.  g.  by  Andrei  Linde  (see  a  cita:on   below)  and  Steven  Weinberg  [7]:     …we   have   to   keep   in   mind   the   possibility   that  what  we  now  call  the  laws  of  nature  and   the  constants  of  nature  are  accidental  features   of   the   big   bang   in   which   we   happen   to   find   ourselves,   though   constrained   (as   is   the   distance   of   the   Earth   from   the   Sun)   by   the   requirement   that   they   have   to   be   in   a   range   that  allows  the  appearance  of  beings  that  can   ask  why  they  are  what  they  are.     The   Darwinian   theory   of   evolu:on   is   widely   believed  to  explain  the  birth  of  order  from  chaos.  To   follow  its  line  of  thought,  our  universe  is  considered  a   member  of  a  huge  or  infinite  ensemble  of  universes,   one  generated  by  the  other,  with  daughter  universes   mostly   inheri:ng   the   logical   structure   of   the   mother   ones,  adding  some  muta:ons  on  top  of  it  [8,9].  Aser   the   heredity   and   varia:on   of   the   mul:plying   logical   structures   is   seNled,   the   third   Darwinian   principle,   selec:on,   can   be   introduced   as   well.   This   role   is   played   by   the   so   called   weak   anthropic   principle,   or   WAP  [10],  poin:ng  out  that  only  those  universes  can   be   observed   where   observers   can   appear,   which   selects  a  narrow  class  of  fine-­‐tuned  universes  as  it  is   noted   in   Weinberg’s   quota:on   above.   Thus,   though   our  universe  is  thought  of  in  this  Darwinian  approach   as   ul:mately   generated   by   chaos,   its   fine   tuning   apparently  receives  a  scien:fic  explana:on  as  a  result   of  a  Darwinian  chaosogenesis.  Although  in  the  infinite   megaverse   only   a   :ny   por:on   of   universes   is   fine-­‐ tuned   for   life   and   consciousness,   the   probability   for   any  observer  to  see  the  universe  as  fine-­‐tuned  is  one   hundred   percent.   Nothing   seemingly   contradicts   the   assump:on   that   our   universe   is   a   random   representa:ve   of   WAP-­‐selected   subset   of   Tegmark’s   mul:verse,   but   is   that   really   so?   Does   the   universe   indeed   have   no   clear   signature   excluding   any   possibility   of   it   having   been   randomly   selected   from   this  totality  of  all  possible  mathema:cal  structures?  Is   the   concept   of   chaosogenesis   irrefutable   by   any   thinkable   observa:on,   i.   e.   is   it   not   a   scien:fic   hypothesis?  Apparently,  it  is  considered  as  irrefutable

by   some   leading   experts.   For   instance,   Brian   Greenе   clearly  says  that  [11]:     I   draw   the   line   at   ideas   that   have   no   possibility   of   being   confronted   meaningfully   by  experiment  or  observa:on,  not  because  of   human   frailty   or   technological   hurdles,   but   because  of  the  proposals’  inherent  nature.  Of   the   mul:verses   we’ve   considered,   only   the   full-­‐blown  version  of  the  Ul:mate  Mul:verse   falls   into   this   netherland.   If   absolutely   every   possible  universe  is  included,  then  no  maNer   what   we   measure   or   observe,   the   Ul:mate   Mul:verse   [i.e.   Tegmark’s   one]   will   nod   and   embrace  our  result.   Contrary   to   B.   Greenе,   we   are   showing   below   that   Tegmark’s   hypothesis   runs   counter   to   certain   observa:ons,  so  it  fails,  and  fails  as  a  scien:fic  theory.   A  Cosmic  Observer   “Observers”  in  WAP  are  not  normally  specified;  it  is   not   taken   into   account   what   it   is   namely   they   do   observe.   We   suppose   that   to   be   qualified   as   “observers”  they  at  least  have  to  be  conscious,  as  it  is   also   reasonable   to   assume   that   conscious   creatures   observe   their   immediate   space   of   life   support   and   have  access  to  at  least  empirical  knowledge  about  it.   However,   this   sort   of   knowledge   has   nothing   to   do   with   theore:cal   knowledge   of   the   big   cosmos;   the   first   by   no   means   entails   the   second.   Let   us   fix   this   point   of   an   important   dis:nc:on,   a   dis:nc:on   between   those   simple,   minimal,   empirical   observers   and  cosmic  observers,  who  are  discovering  theories  of   big   cosmos,   seeing   their   universe   both   at   extremely   large   and   extremely   small   scales,   far   exceeding   the   scale   of   immediate   life   support.   To   become   cosmic   observers,   minimal   ones   must   live   in   a   very   specific   world  among  the  populated  worlds.  Specifically,  their   universe  has  to  be  theore:cally  comprehensible  on  a   big  cosmic  scale;  their  world  has  to  be  theore<zable,   so  to  say.  In  other  words,  the  possibility  for  observers   to   be   not   just   simple   but   cosmic   requires   their   universe  to  have  a  very  special  logical  structure:  it  has   to  be  described  by  elegant  laws  covering  many  orders   of   magnitude   of   their   parameters.   Contemporary   humanity  is  indeed  a  cosmic  observer.  For  today,  our   scale   of   scien:fic   cogni:on   is   described   by   an   enormous  dimensionless  parameter  ~1045;  that  big  is   the  ra:o  of  the  sizes  of  largest  object  of  physics,  the   universe,  ~1026m,  to  the  smallest  ones,  the  top  quark   and  the  Higgs  boson,  corresponding  to  ~10-­‐19m.   The  Condi)on  of  Elegance   This   condi:on   of   theore<zability   apparently   is   extraneous   to   the   selec:ve   anthropic   principle,   that   is,   theore:zability   seems   unnecessary   for   the   universes   to   be   populated   by   conscious   creatures   or   to   be   observed.   In   fact,   the   laNer   condi:on   is   essen:ally   local;   it   requires   something   like   a   life-­‐ friendly   planet   inside   any   universe.   The   former   condi:on,   though,   is   global;   it   requires   the   laws   of   nature  to  be  elegant  on  a  big  cosmic  scale,  a  scale  by   far  exceeding  that  of  the  life  on  the  planet.  Generally,   local   condi:ons   do   not   entail   global   consequences,   and   since   theore:zability   is   a   specific   func:onal   requirement   detached   from   WAP   selec:on,   we   have   to   conclude   that   it   is   highly   unlikely   for   an   observed   universe   to   be   theore:zable.   Since   we   know,   aser   Isaac   Newton,   that   our   universe   is   theore:zable,   chaosogenesis  theory  is  apparently  refuted.  However,   this   refuta:on,   being   qualita:ve   only,   leaves   a   possibility   to   object.   Its   core   statement,   that   theore:zability   is   a   specific   requirement   detached   from   the   anthropic   condi:on   for   universes   “to   be   observed”   can   be   ques:oned.   How   can   we   be   sure   that   theore:zability   is   logically   independent   from   WAP?   It   would   not   be   independent,   if   WAP   did   not   allow  for  our  theore:zable  laws  of  nature  any  visible   modifica:ons  even  at  extremes  of  very  large  and  very   small   scales,   modifica:ons   that   might   exclude   the   appearance   of   conscious   beings   for   one   or   another   reason.   The   very   concept   of   a   fine-­‐tuned   universe   is   sugges:ng   to   us   that   sort   of   an   idea   concerning   the   fundamental   constants,   and   so,   we   may   ask:   what   if   the  same  is  true  concerning  the  very  structure  of  the   laws   of   nature?   Although   it   is   hard   to   believe   that   moderate   modifica:on   of,   say,   General   Rela:vity   at   the   distances   exceeding   the   solar   system,   can   drama:cally   reduce   the   possibility   of   consciousness   on   our   planet,   we   should   consider   a   chance   that   it   cannot  be  excluded.  This  very  argument  for  the  strong   rela:on   between   the   weak   anthropic   principle   and

theore:zability   was   recently   suggested   by   A.   Linde     ...   the   infla:onary   mul:verse   consists   of   myriads  of  'universes'  with  all  possible  laws  of   physics  and  mathema:cs  opera:ng  in  each  of   them.   We   can   only   live   in   those   universes   where  the  laws  of  physics  allow  our  existence,   which  requires  making  reliable  predic:ons.   Let’s   accept   this   arguable   hypothesis,   and   suppose   that   somehow   WAP   does   not   allow   significant   devia:ons   of   the   laws   of   nature   locally   compa:ble   with  conscious  beings  from  the  globally  theore:zable   form.   Then,   the   ques:on   is:   which   devia:ons   from   the   exis:ng   laws   are   allowed   by   the   anthropic   principle?   If   the   world   is   generated   by   chaos,   all   imaginable   addi:onal   terms   to   the   life-­‐selected   ones   are   coming   into   play;   the   amplitude   or   width   of   the   resulted   devia:on   is   limited   by   the   anthropic   principle,   but   func:onal   behavior   of   the   devia:on   is   arbitrary.   We   have   some   es:ma:ons   about   the   allowed   devia:on   in   the   context   of   fine-­‐tuning   as   rela:ve   varia:ons   of   the   fundamental   constants   compa:ble  with  WAP,  and  the  most  stringent  of  them   are   at   the   order   of   10-­‐3,   i.e.   0.1%   [10].   Since   we   are   considering   here   the   problem   of   func:onal   accuracy,   the     enormously   stringent   requirements   on   some   constants,   like   the   ini:al   condi:ons   at   the   big   bang   [13],  do  not  reduce  the  amplitude  of  these  func:onal   varia:ons.   Thus,   working   on   the   Linde   argument,   we   may  roughly  es:mate  the  sensi:vity  of  the  anthropic   selec:on   to   the   rela:ve   func:onal   varia:ons   of   the   fundamental  laws  to  not  be  finer  than  0.1%    or  so.    If   the   laws   of   nature   were   generated   by   a   random   choice   from   Tegmark’s   mul:verse,   they   would   be   expressed   by   irregular   func:ons   possibly   following   elegant   ones   within   a   rela:ve   width   of   ~0.1%   or   more.   In   this   respect,   it   does   not   maNer   whether   chaos   reveals   itself   through   arbitrary   func:ons   or   arbitrary   mathema:cal   structures;   with   Tegmark’s   “mathema:cal  democracy”  func:onal  representa:ves   of   the   two   families   are   indis:nguishable   and   are   dominated   by   extremely   complicated,   prac:cally   irregular   func:ons.   The   elegant   formulas   might   be   approxima:ons  to  the  real  irregular  fundamental  laws   with  that  WAP-­‐determined  accuracy,  but  not  beNer.     Moreover,   measurements   of   the   fundamental   constants  in  this  world  would  be  reproducible  only  at   the   anthropic   level,   not   beNer.   If   physicists   of   that   hypothe:cal   world   tried   making   measurements   of   their   fundamental   constants   at   the   beNer   accuracy,   they   would   realize   that   none   of   the   measurements   are  reproducible  at  that  level;  they  all  contain  space-­‐ :me  noise  with  a  rela:ve  amplitude  of  0.001,  driven   by   infinitely   complicated   terms   of   the   true   laws   of   nature.  So,  physics  in  that  Tegmarkian  universe  would   be   stopped   at   the   anthropic   accuracy   level   simply   because,   with   the   probability   of   100%,   no   reproducible   measurement   would   be   possible   there   with  accuracy  beNer  than  that.     We   know   though,   that   the   real   accuracy   of   our   fundamental   theories   is   not   only   beNer   than   anthropic,  but  many  orders  of  magnitude  beNer;  they   are   absolutely   precise   on   that   scale.   Indeed,   the   General  Rela:vity  test  with  a  double  neutron  star  PSR   1913+16   showed   an   unprecedented   agreement   between  theory  and  observa:on  at  the  level  of  10-­‐14.   Another   impressive   demonstra:on   of   that   extremely   h i g h   p r e c i s i o n   r e l a t e s   t o   t h e   Q u a n t u m   Electrodynamics:   the   theore:cally   predicted   value   of   an   electron’s   magne:c   moment   is   confirmed   by   measurements  with  the  accuracy  ~10-­‐11;  see  e.  g.  Ref.   [13].   Thus,   many   experiments   which   proved   high   precision   of   our   elegant   laws   of   nature,   orders   of   magnitude   beNer   than   the   anthropic   width,   refute   Tegmark’s  hypothesis.   This   considera:on   shows,   by   the   way,   that   cosmological   chaosogenesis   is   a   scien:fic   hypothesis   since  it  is    falsified  by  observa:ons.  Note  that  the  idea   of  the  mul:verse  was  at  least  partly  mo:vated  by  the   wish  to  find  a  pure  scien:fic  explana:on  to  the  fact  of   the   fine-­‐tuned   universe:   if   our   universe   is   the   only   one,   its   fine-­‐tuning   does   not   suggest   any   other   reasonable   explana:on   but   an   act   of   purposeful   crea:on.   For   a   single   universe,   its   fine-­‐tuning   is   too   stringent   for   а   purely   scien:fic   explana:on,   but   the   idea   of   mul:verse   chaosogenesis,   suggested   as   an   aNempt   to   explain   fine   tuning   within   bounds   of   science,   is   refuted   by   the   opposite   reason:   the   es:mated   anthropic   limita:ons   on   fine-­‐tuning   aren’t   anywhere   fine   enough   to   explain   the   experimental

confirma:ons  of  the  extreme  precision  of  the  elegant   forms  as  fundamental  laws.   A  Pythagorean  Universe   A s e r   h a v i n g   a n n o u n c e d   t h e   “c o m p l e t e   mathema:cal   democracy”   at   the   beginning   of   his   ar:cle,   later   on   Tegmark   no:ces   that   “our   physical   laws   appear   rela:vely   simple”.   At   this   point,   to   be   consistent   with   reality,   he   gives   up   the   proclaimed   “mathema:cal   democracy”   in   favor   of   an   aristocracy   of  simple  mathema:cal  forms.  Aser  such  an  overturn,   his   mul:verse   now   has   almost   nothing   to   do   with   chaos;   instead,   it   is   generated   by   some   source   of   elegant   mathema:cal   forms.   As   a   result,   “the   embarrassing   ques:on”   about   the   source   of   this   ontological   inequality   of   the   mathema:cal   forms   remains   as   it   was.   This   contradic:on   of   Tegmark’s   “democra:c”  inten:on  with  his  “aristocra:c”  prac:ce   was  noted  by  Alex  Vilenkin  [14]:     Tegmark’s   proposal,   however,   faces   a   formidable   problem.   The   number   of   mathema:cal   structures   increases   with   increasing   complexity,   sugges:ng   that   “typical”   structures   should   be   horrendously   large   and   cumbersome.   This   seems   to   be   in   conflict   with   the   simplicity   and   beauty   of   the   theories  describing  our  world.   Since   the   laws   of   our   universe   are   not   picked   randomly,  they  can  only  be  purposefully  chosen.  Our   universe  is  special  not  only  because  it  is  populated  by   living   and   conscious   beings   but   also   because   it   is   theore:zable   by   means   of   elegant   mathema:cal   forms,   both   rather   simple   in   presenta:on   and   extremely   rich   in   consequences.   To   allow   life   and   consciousness,   the   mathema:cal   structure   of   laws   has   to   be   complex   enough   so   as   to   be   able   to   generate  rich  families  of  material  structures.  From  the   other   side,   the   laws   have   to   be   simple   enough   to   be   discoverable   by   the   appearing   conscious   beings.   To   sa:sfy   both   condi:ons,   the   laws   must   be   just   right.   The   laws   of   nature   are   fine-­‐tuned   not   only   with   respect   to   the   anthropic   principle   but   to   be   discoverable   as   well.   In   other   words,   the   Universe   is   fine–tuned  with  respect  to  what  can  be  called  as  the   Cosmic   Anthropic   Principle:   its   laws   are   purposefully   chosen  for  the  universe  to  be  cosmically  observed.    It   could   be   even   that   our   laws   are   at   their   simplest   within  our  sort  of  life.  Would  it  be  possible  to  take  any   part   away   from   our   exis:ng   theories   without   compromising  forms  of  life  as  we  know  them?  Such  a   special  universe  deserves  a    proper  term,  and  we  do   not   see   a   beNer   choice   than   to   call   it   Cosmos   or   to   qualify  it  as  Pythagorean,  in  honor  of  the  first  prophet   of   theore:cal   cogni:on,   who   coined   such   important   words  as  cosmos  (order),  philosophy  (love  of  wisdom),   and  theory  (contempla:on).       Since   chaosogenesis,   being   limited   only   by   the   anthropic   principle,   is   the   only   op:on   for   a   completely   scien:fic   solu:on   of   the   problem   of   cosmogenesis,  its  refuta:on  entails  that  the  problem   of   cosmogenesis   cannot   be   solved   within   the   framework  of  science.  Any  scien:fic  approach  to  that   would  require  a  specific  set  of  axioms,  consistent  not   only   with   the   anthropic   principle   but   with   elegant   mathema:cal   forms   truly   underlying   our   world;   however,   we’ve   shown   that   the   ques:on   about   embedment  of  one  instead  of  another  specific  set  as   a   logical   structure   of   the   universe   cannot   be   scien:fically  answered.   Star:ng   with   Pythagoras,   it   was   a   maNer   of   faith   for  sparse  groups  of  few  people  and  lonely  individuals   that   “fundamental   laws   of   nature   are   described   by   beau:ful   equa:ons.”   Theore:cal   science   was   conceived   and   nurtured   by   this   very   faith   with   its   “cosmic   religious   feeling”,   which   inspired   scien:fic   cogni:on   for   twenty-­‐five   centuries.   Without   any   exaggera:on,   all   great   theories,   from   Copernicus,   Kepler  and  Newton  to  Einstein  and  Dirac  happened  as   guesses   on   the   grounds   of   some   fundamentally   simple   ideas   like   symmetry,   conserva:on,   or   equivalence.  The  noted  forty-­‐five  orders  of  magnitude   of   scien:fic   cogni:on,   with   more   than   ten   digits   of   precision  reached  in  some  experimental  verifica:ons,   allow   us   to   conclude     about  a   scien:fic   confirma:on   of  what  was  considered  a  maNer  of  faith  for  two  and   a   half   millennia:   now   it   is   a   maNer   of   fact   that   the   universe   is   indeed   Pythagorean.   In   other   words,   the   existence   of   the   Platonic   world   of   elegant   mathema:cal   forms   structuring   the   physical   world   is   scien:fically   confirmed,   and   the   accuracy   of   this

confirma:on  is  many  orders  of  magnitude  beNer  than   that  of  any  specific  statement  of  physics.     indeterminism,   by   means   of   uncertainty   les   by   the   quantum  laws  of  nature.     Aser   two   and   a   half   millennia   since   its   birth,   fundamental   science   reached   a   grade   of   maturity   allowing   for   a   dual   confirma:on   of   its   faith:   the   Pythagorean   faith   is   confirmed   as   prophecy   coming   true  and  as  a  good  tree  that  brings  forth  good  fruit.     The   very   idea   of   observa:on,   being   so   far   associated   with   material   objects   only,   is   enriched   by   an   even   more   fundamental   meaning   of   the   Platonic   observa:on,   i.   e.   observa:on   of   elements   of   the   Platonic  world  structuring  the  material  world.  Cosmic   observa:on  is  possible  only  due  to  a  combined  vision   of    both  worlds.   Three  Worlds,  Two  Totali)es   Pythagorean  forms  of  the  discovered  laws  of  nature   tell  us  that  the  ul:mate  goal  of  fundamental  physics,   the  theory  of  everything,  either  contains  a  significant   Pythagorean   core,   or,   what   is   more   reasonable   to   assume,  is  totally  Pythagorean.  This  Pythagorean  core   has   to   be   powerful   enough   to   generate   a   sufficiently   rich   set   of   Pythagorean   laws,   as   we   observe,   but   whatever  this  theory  of  everything  is,  it  cannot  be  the   ul:mate   answer   to   the   ques:on   about   the   order   of   being,  because  this  form  is  special  due  to  there  being   other   forms,   and   so,   like   any   other,   it   does   not   cons:tute  a  totality.  For  laws  of  nature,  there  are  only   two   thinkable   explanatory   principles,   opposites   of   each   other,   which   are   totali:es:   chaos   and   mind   as   such.     Because   the   logical   structure   of   our   universe   can   not   be   explained   by   chaos,   and   because   it   can   not   explain   itself,   we   are   les   with   only   one   possible   explana:on   remaining,   that   it   was   conceived   and   realized   by   a   mind.   A.   Vilenkin   prefers   to   formulate   this   apparently   inevitable   conclusion   about   the   cosmic  Mind  as  a  ques:on  [14]:   …   the   laws   should   be   “there”   even   prior   to   the   universe   itself.   Does   this   mean   that   the   laws   are   not   mere   descrip:ons   of   reality   and   can   have   an   independent   existence   of   their   own?   In   the   absence   of   space,   :me,   and   maNer,   what   tablets   could   they   be   wriNen   upon?   The   laws   are   expressed   in   the   form   of   mathema:cal   equa:ons.   If   the   medium   of   mathema:cs  is  the  mind,  does  this  mean  that   mind  should  predate  the  universe?   It   seems   important   to   men:on   here   that   chaos,   refuted  as  a  possible  source  of  the  laws  of  nature,  can   par:cipate,  and  par:cipates,  in  the  physical  world  as   R.  Penrose,  “Three  Worlds,  Three  Mysteries”  [13].   Roger  Penrose  suggested  the  idea  and  the  image  of   “Three   Worlds,   Three   Mysteries”   [13].   The   three   worlds,   Physical,   Platonic,   and   Mental,   differ   :me-­‐ wise.  The  Platonic  world  does  not  have  any  age  at  all;   it  is  atemporal.  The  Physical  world  is  temporal,  and  its   age,  counted  from  the  border  of  all  observa:ons,  the   big  bang,  is  calculated  at  13.798  ±  0.037  billion  years   (note  the  precision!).  The  age  of  humanity  as  cosmic   observers   is   extremely   short   on   that   scale.   Yet   although   the   history   of   many   scien:fic   discoveries   is   known   minutely,   and   although   we   cannot   observe   anything  closer  than  our  thoughts,  the  genesis  of  the   cosmic   observer   remains   no   less   mysterious   to   us   than  the  genesis  of  Physical  and  Platonic  worlds.

Wonder   of   Pythagorean   harmony   of   the   fundamental   laws   of   nature   and   con:nuing   demonstra:ons  of  human  ability  to  discover  them    so   remotely  from  our  own  natural  scale  leads  now  more   than   ever   before   to   deep   ques:ons   about   the   three   mysteries,   whose   entanglement   and   coherence   are   revealing   the   underlying   Unity,   the   ul:mate   transcendental   Source   of   everything   existent,   including  ourselves,  the  growing  cosmic  observers.   The   authors   are   thankful   to   Alexei   Tsvelik   and   Mikhail  Arkadev  for  s:mula:ng  discussions.     [ 1 ]   L .   W i N g e n s t e i n ,   “ Tr a c t a t u s   L o g i c o -­‐ Philosophicus”  (1921)   [ 2 ]   P.   D a v i e s   " H o w   b i o -­‐ f r i e n d l y   i s   t h e   universe".Int.J.Astrobiol  2,  p.115  (2003).   [3]   S.   Hawking,   “A   Brief   History   of   Time”,   Bantam   Books,  p.  125  (1988)   [4]   A.   Tsvelik,   “Life   in   the   Impossible   World”,   Ivan   Limbakh  publ.,  St.-­‐Petersburg,  2012  (in  Russian).   [5]   K.   Jaspers,   “The   Great   Philosophers”,   A   Harvest   Book,  Vol.2.   [6]   M.   Tegmark,   "The   Mathema:cal   Universe”,   Founda:ons  of  Physics  38  (2),  p.  101  (2007).   [7]   S.   Weinberg,   “Lake   Views:   This   World   and   the   Universe”   (Kindle   Loca:ons   337-­‐338).   Harvard   University  Press.  Kindle  Edi:on  (2011).     [8]  L.  Smolin,  “The  Life  of  the  Cosmos”  (1997)   [9]   A.   Linde,   “The   Self-­‐Reproducing   Infla:onary   Universe”,  Sci.  Am.,  Vol.  271,  No.  5,  pp  48-­‐55,  (1994)   [10]   J.D.   Barrow,   F.J.   Tipler   “ The   Anthropic   Cosmological  Principle.”  (1988)   [11]   B.   Greene,   “The   Hidden   Reality:   Parallel   Universes  and  the  Deep  Laws  of  the  Cosmos”  (Kindle   Loca:ons   5943-­‐5947).   Knopf   Doubleday   Publishing   Group.  Kindle  Edi:on  (2011).     [12]  A.  Linde  “Why  Is  Our  World  Comprehensible?”,  in   “This  Explains  Everything”,  Ed.  J.  Brockman,  2012.   [13]   R.   Penrose,   “The   Road   to   Reality”,   First   Vintage   Books  Edi:on,  2007.   [14]  A.  Vilenkin,  “Many  Worlds  in  One:  The  Search  for   Other   Universes”   (Kindle   Loca:ons   3188-­‐3191).   Farrar,  Straus  and  Giroux.  Kindle  Edi:on  (2006).