Intuition et invention mathématique

105
SEPTIEMBRE
2016
!
Intuition et invention mathématique | Florian Forestier
!
!
!"#$%#%&"'(#'%")("#%&"'*+#,-*+#%.$('
"#$%&'(!"$%)*+&)%!
,(&-)%*&+.!/'%&*0123!4$%5$(()!
!
!
67$58)+!9)!:)!+);+)3!<=&!>%$#$(?)!)+!9.-)#$>>)!=(!:)%+'&(!($@5%)!9)!>&*+)*!$=-)%+)*!
9'(*! ($+%)! '%+&:#)! A!B'+C.@'+&<=)*! )+! :$(:%.+=9)*! >C.($@.($#$?&<=)*DE3! )*+! @$&(*! #'!
>C&#$*$>C&)!9)*!@'+C.@'+&<=)*!<=)!#7);'@)(!>C&#$*$>C&<=)!9)!:)!<=)!#)*!@'+C.@'+&<=)*!
>)=-)(+!($=*!>)%@)++%)!9)!:$@>%)(9%)!*=%!#)*!>%$9=:+&$(*!$=!:%.'+&$(*!9)!#7)*>%&+!C=@'&(F!
6'!>$*&+&$(!<=7&#!)(+)(9!'**=@)%3!'>>=G.!>'%!#)!+);+)!)+!#)*!>'%$#)*!97=(!:)%+'&(!($@5%)!9)!
@'+C.@'+&:&)(*3! )*+! =(! %)H=*! 9)! #'! +CI*)! H$%@'#&*+)! )+! #'! (.:)**&+.! 97=(! <=)*+&$(()@)(+!
H$%@=#.!)(!+)%@)!9)!*)(*!)+!97);>.%&)(:)F!!!
!
1#!*7'?&+!)(!)HH)+!>$=%!($=*!9)!>%)(9%)!':+)!97)@5#.)!9)!:)!<=)!#)*!@'+C.@'+&<=)*!()!
*$(+!>'*!=(!>=%!#'(?'?)!H$%@)#F!J)!<=&!%)#I-)!)(!)##)*!9)!#'!H$%@'#&*'+&$(!)+!9=!:'#:=#!(7)(!
:$(*+&+=)!>'*!#'!9&@)(*&$(!=(&<=)3!(&!@K@)!#'!9&@)(*&$(!H$(9'@)(+'#)3!:'%!A!=(!*)(*3!=()!
*&?(&H&:'+&$(3!=(!:)%+'&(!&@'?&('&%)!'::$@>'?(L'&M)(+!#)!+%'-'&#!9=!:C)%:C)=%F!N!#'!#&@&+)3!#'!
#$?&<=)!(7L)*+M!=+&#)!<=7'>%I*!:$=>3!>$=%!()++$G)%!#'!@'+C.@'+&<=)!9)!+$=+)!:)++)!?'(?=)!<=&!
)*+!:)>)(9'(+!#7C=@=*!@K@)!9)!#'!>)(*.)!@'+C.@'+&<=)O!E!!
!
J)++)!>$*&+&$(!%)#I-)!5&)(!>#=+P+!97'5$%9!9)*!>)%*>):+&-)*!)+!9)*:%&>+&$(*!'@$%:.)*!
>'%! #)*! @'+C.@'+&:&)(*! )=;0@K@)*! #$%*<=7&#*! *7&(+.%)**)(+3! >#=+P+! <=7'=;! H$(9)@)(+*! 9)!
#)=%!9&*:&>#&()3!$=3!>%$5#I@)!<=&!#)*!+$=:C)!>#=*3!<=7Q!#7$%9%)!97'%+&:=#'+&$(!9)*!*+%=:+=%)*!
<=)! :)##)0:&! 9.>#$&)3! Q! #'! H'R$(! 9$(+! &#*! +%'-'&##)(+F! S)! <=)##)!H'R$(! >%$:I9)(+0&#*! >$=%!
*7'::$%9)%! *=%! 9)*! >%$5#I@)*3! 9.H&(&%! >)=! Q! >)=! 9)*! $58)+*3! 9)*! *+%=:+=%)*3! 9)*! *+%'+.?&)*!
9.@$(*+%'+&-)*3!$=!>$=%!*7)(+)(9%)!*=%!9)*!>%$?%'@@)*!$=!9)*!>%$8)+*!>'%+'?.*3!'#$%*!<=)!
#)*!$58)+*!)+!:$(:)>+*!Q!9.-)#$>>)%!*$(+!)(:$%)!>$=%!=()!?%'(9)!>'%+!Q!9.-)#$>>)%F!!
!
J)++)! )(<=K+)! (7)*+! >'*! 97'5$%9! >C&#$*$>C&<=)3!<=)! :)! *$&+! 9'(*! #7$>+&<=)! 97=()!
&(+)%%$?'+&$(!*=%!#)*!H$(9)@)(+*!9)*!@'+C.@'+&<=)*3!$=3!&(-)%*)@)(+3!97=()!&(+)%%$?'&+$(!
*=%!#)=%!:'>':&+.!Q!'++)&(9%)3!%.-.#)%!$=!9.-)#$>>)%!9)*!-.%&+.*!@.+'>CG*&<=)*!-$&%)!9&-&()*F!
T##)! (7)*+! >'*! ($(! >#=*! >*G:C$#$?&<=)3! :'%! &#! *7'?&+! 5&)(! @$&(*! 97);>#&<=)%! >'%! 9)*!
@.:'(&*@)*! >*G:C$#$?&<=)*3! ()=%$#$?&<=)*3! )+:F3! *&($(! #)*! $58)+*! @'+C.@'+&<=)*! )=;0
@K@)*3!9=!@$&(*!#7':+&-&+.!:$(:)>+=)##)!9)*!@'+C.@'+&:&)(*3!<=)3!*=&-'(+!'=!>#=*!>%I*!:)++)!
':+&-&+.!)(!+%'&(!9)!*)!H'&%)3!9)!+)(+)%!9)!@&)=;!:$@>%)(9%)!#'!H'R$(!9$(+!($=*!>)(*$(*!)+!:)!
<=)! ($=*! H'&*$(*! #$%*<=)! ($=*! >)(*$(*F! J)! (7)*+! >'*! =(! C'*'%93! %.>.+$(*0#)3! *&! =(! =()!
?%'(9)! >'%+&)! 9)*! %.H.%)(:)*! @$5&#&*.)*! 9'(*! :)++)! $>+&<=)! *$(+! &**=)*! 9)! #'! >#=@)! 9)!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
D!"F!"UVT4W1TV3!A!B'+C.@'+&<=)*!)+!:$(:%.+=9)*!>C.($@.($#$?&<=)*!E3!!""#$%&'(%')*+",-+",$,./%'
"01123413F!
O!XF!JYNVVN,S!)+!/F!JNVW1TV3!5%'6+%$'%"'-#7*+-#7/89%&3!/'%&*3!N?'#@'3!OZZ[3!A!1(+%$9=:+&$(!E3!>F!DDF!

106
SEPTIEMBRE
2016
!
Intuition et invention mathématique | Florian Forestier
!
!
@'+C.@'+&:&)(*F! 1#! >)=+! *7'?&%! 9)*! .:%&+*! 9)! /$&(:'%.3! 9=! +%'-'&#! >#=*! *G*+.@'+&<=)!
97Y'9'@'%9! &(+)%%$?)'(+! #)*! @'+C.@'+&:&)(*! '=+$=%! 9)! #=&! *=%! #)*! &@'?)*! )+! #)! #'(?'?)!
=+&#&*.)*! 9'(*! #'! H$%@=#'+&$(! 9)! #)=%*! %.*=#+'+*! )+! %)@'%<='(+! #7&@>$%+'(:)! 9)!#7&@'?)!
?.$@.+%&<=)\3!$=3!Q!($+%)!.>$<=)3!9)!+%'-'=;!97&(*>&%'+&$(!+$=+!'=**&!5&)(!>C&#$*$>C&<=)!9)!
:)%+'&(*!@'+C.@'+&:&)(*!9)!>%$H)**&$(!L>'%!);)@>#)!]F!6$(?$3!"F!/'+%'*M!=*'(+!9)!:$(:)>+*!
>C&#$*$>C&<=)*!>$=%!.:#'&%)%!#7);'@)(!9)!#)=%!>%'+&<=)!)+!9)!*'!H.:$(9&+.F!!
!
6)*! +%'-'=;! 9)! ]F! 6$(?$! ($=*! &(+.%)**)(+! +$=+! >'%+&:=#&I%)@)(+! >'%! #)=%! H'R$(! 9)!
%)@)++%)3! '-):! /$&(:'%.3! @'&*! '=**&! B)%#)'=0/$(+G3! #)! :$%>*! 9=! :C)%:C)=%! '=! :^=%! 9)!
#7&(+)%%$?'+&$(F! J)! H'&*'(+3! :)+!'=+)=%! (7)(+)(9! :)%+)*! >'*! .:%'*)%! #)! *)(*! 9)! #'! >%'+&<=)!
@'+C.@'+&<=)! *=%! 97'=+%)*! >%'+&<=)*! >#=*! 9&%):+)@)(+! ?)*+=)##)*! )+! %=9&@)(+'&%)*3! @'&*!
%'>>)#)%! <=)! #7':+&-&+.! @'+C.@'+&<=)! )*+! +$=8$=%*! :)##)! 97=(! *=8)+! H&(&3! &(:'%(.3! *)! *&+='(+!
H':)! '=! @$(9)! )+! H':)! '=;! $58)+*! 9)! *'! 9&*:&>#&()! '-):! =()! @.@$&%)! *>.:&H&<=)3! )+! #)*!
:$(*+%=&*'(+!*)#$(! =()! +)@>$%'#&+.! *>.:&H&<=)F!6'! +)@>$%'#&+.! -.:=)! 9)!#7);>.%&)(:)! 9)! #'!
:%.'+&$(!@'+C.@'+&<=)!)*+!@K@)!>$=%!]F!6$(?$!:)!<=&!>#'&9)!#)!>#=*!()++)@)(+!:$(+%)!=()!
%.9=:+&$(!#$?&:&*+)! 9)*! @'+C.@'+&<=)!_! #'! :$(:)>+&$(! :$(*+%=:+&-&*+)! 9=! +)@>*! :C)`!
Y)%5%'(93!W=%&(?3!JC=%:C!)+!Y)G+&(?3!&@>#&<=)%'&+!'&(*&3!.:%&+0&#3!=()!@.@$&%)!9&?&+'#)!*'(*!
$=5#&3!'#$%*!<=)!#7C$@@)!8=*+)@)(+!$=5#&)F!J)++)!*+%=:+=%)!9)!H%'?@)(+'+&$(!)+!9)!>)%+)!9)!
*'!>)(*.)!)+!9)!*$(!);>.%&)(:)!(7)(!)*+!>'*!=()!#&@&+'+&$(3!@'&*3!aF0bF!]&%'%9!G!&(*&*+)!'=**&3!
*'!9&@)(*&$(!)**)(+&)##)[F!!
!
6)! >%.*)(+! +);+)! )(+)(9! 9'(*! #'! >)%*>):+&-)! <=)! ($=*! -)($(*! 9)! >%.*)(+)%!
*7&(+.%)**)%! Q! <=)#<=)*! <=)*+&$(*! >#=*! >%.:&*)*!_! )(! >%)@&)%! #&)=3! Q! #'! <=)*+&$(!9)! #'!
?.(.%'#&*'+&$(!)+!9=!>%$:)**=*!9)!?.(.%'#&*'+&$(!'=!*)&(!9)*!@'+C.@'+&<=)*!c!)(*=&+)!Q!:)##)!
9)! #7&9.)! >%$5#.@'+&<=)! 97=(! *$#! *)(*&5#)! 9)! #7':+&-&+.! @'+C.@'+&<=)!c! >=&*! Q! :)##)! 9)! #'!
:%.'+&-&+.!)+!9)!#7&(-)(+&$(!@'+C.@'+&<=)!_!)(H&(3!Q!#'!<=)*+&$(!9)!#7$58):+&-&+.3!$=!>#=+P+!9=!
%.)#!@'+C.@'+&<=)3!)+!9)!#'!H'R$(!9$(+!:)#=&0:&!'HH):+)!#7&(+=&+&$(!9=!@'+C.@'+&:&)(F!!
Généralisation et abstraction
Le triomphe de la raison
!
J$@@)! ($=*! #)! %'>>)#$(*! 9'(*! A!B'+C.@'+&<=)*! )+! :$(:%.+=9)*!
>C.($@.($#$?&<=)*d!E3!#)*!@'+C.@'+&<=)*!+)(9)(+3!9)>=&*!#)!9.5=+!9=!e1e)3!Q!9.?'?)%!9)*!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
\!aF!YNSNBNVS3!:&&#/'&96'$#')&;<*,$,./%'(%'$=/">%"7/,"'(#"&'$%'(,-#/"%'-#7*+-#7/89%!LDfdfM!c!!YF!/U1XJNVT3!
5=/">%"7/,"'-#7*+-#7/89%3!DfZg3!.9&+&$(!a)'(!]'5'G!
[!J&+.!>'%!BF0"F!VUb3!A!6)!%.)#!9=!:'#:=#E3!>F!OZZ3!&(!XF!JYNVVN,S!)+!/F!JNVW1TV3!5%'6+%$'%"'-#7*+-#7/89%&3!
,)?</7?!
d!"F! "UVT4W1TV3! A!B'+C.@'+&<=)*! )+! :$(:%.+=9)*! >C.($@.($#$?&<=)!E3! !""#$%&' (%' )*+",-+",$,./%'
"01123413F!!

107
SEPTIEMBRE
2016
!
Intuition et invention mathématique | Florian Forestier
!
!
*+%=:+=%)*!H$%@)##)*!);>#&:&+'(+!#)*!%)#'+&$(*!)(+%)!9)*!$58)+*!<=7)##)*!9.:%&-)(+!:$@@)!9)*!
:'*!9)!#$&*!>#=*!?.(.%'#)*F!!
!
S$(($(*!9h'5$%9!=(!);)@>#)!9)!:)!>%$:)**=*!)(!>%$>$*'(+!=(!5%)H!C&*+$%&<=)!9)!#'!
%.*$#=+&$(!9)*!.<='+&$(*!9)!9)?%.!*=>.%&)=%!$=!.?'#!Q!+%$&*iF!J$@@)!#)!%'>>)##)!aF!2=&##)@&(3!
#)*! .<='+&$(*! 9)! 9)?%.! \! L9)! #'! H$%@)! ';\!j! 5;O!j! :;! j! 9M! *)! %.*$#-)(+! :#'**&<=)@)(+! )(!
>#=*&)=%*!.+'>)*F!1#!*h'?&+!+$=+!9h'5$%9!9)!H'&%)!9&*>'%'k+%)!#)!+)%@)!)(!;O!>'%!=()!>%)@&I%)!
*=5*+&+=+&$(!9$(+!#)!>%&(:&>)!)*+!:$((=!9)>=&*!#)*!'#?.5%&*+)*!&+'#&)(*!9=!e12)!L>$=%!$5+)(&%!
=()!.<='+&$(!9)!#'!H$%@)!'D;\!j!5O;!j!:DMF!S'(*!=(!*):$(9!+)@>*3!&#!*h'?&+3!>'%!=()!*):$(9)!
*=5*+&+=+&$(3!9h$5+)(&%! Q! >'%+&%! 9)! #h.<='+&$(! >%$>$*.)! =()! *):$(9)! .<='+&$(!'=;&#&'&%)F!
J)##)0:&!)*+!9)!9)?%.!i3!@'&*!*)!#'&**)!+$=8$=%*!%.9=&%)!Q!=()!.<='+&$(!9)!9)?%.!O3!:'%!)##)!)*+!
9)!#'!H$%@)!'O;i!j!5O;\!j!:OF!U(!.+'5#&+!9)!:)++)!H'R$(!=()! %)#'+&$(!)(+%)!#)*!*$#=+&$(*!9)*!
9)=;! .<='+&$(*! Q! >'%+&%! 9)! #'! %)#'+&$(! 9)*! +)%@)*! *=5*+&+=.*F! /'%! =(! '=+%)! >%$:.9.3! $(!
*=5*+&+=)!'=;!.<='+&$(*!9)!9)?%.![!=()!.<='+&$(!9)!9)?%.!DO!<=)!#h$(!>)=+!%.9=&%)!Q!=()!
.<='+&$(!9)!9)?%.!\F!S'(*!#)*!9)=;!:'*3!:h)*+!>%.:&*.@)(+!#)!H'&+!9)!>'%-)(&%!Q!=()!.<='+&$(!
9)! 9)?%.! *=>.%&)=%3! @'&*! )(! -.%&+.! %.9=:+&5#)! Q! =(! 9)?%.! &(H.%&)=%! Q! #'! >%$>$*.)3! <=&! )(!
>)%@)+! #'! %.*$#=+&$(F! 6'! <=)*+&$(! )*+! '#$%*! 9)! *'-$&%! >$=%<=$&! $(! ()! >)=+! +%$=-)%! 9)!
*)@5#'5#)!H$%@=#)!>$=%!#)*!.<='+&$(*!9)!9)?%.!*=>.%&)=%!$=!.?'#!Q!d3!>$=%<=$&!#7.<='+&$(!9)!
9)?%.!*=>.%&)=%!$5+)(=)!Q!>'%+&%!97)##)*!*7'-I%)!&%%.9=:+&5#)F!
!
J$@@)! #)! @$(+%)! aF! 2=&##)@&(3! 6'?%'(?)! );'@&()! :)++)! <=)*+&$(! )(! >'%+'(+! 9)!
#h.<='+&$(! '=;&#&'&%)F! 1#! @$(+%)!>$=%<=$&! #'! +):C(&<=)! 9)! %.*$#=+&$(! %.=**&+! >$=%! #)*!
.<='+&$(*!9)!9)?%.!\!)+![3!>$=%<=$&!)(!9h'=+%)*!+)%@)*!$(!>)=+!)HH):+&-)@)(+!$5+)(&%!=()!
+)##)!.<='+&$(!'=;&#&'&%)!%.9=:+&5#)!Q!=()!.<='+&$(!9)!9)?%.!&(H.%&)=%)!Q!#h.<='+&$(!>%$>$*.)3!
)+!>$=%<=$&!#h.<='+&$(!'=;&#&'&%)3!#$%*<=)!#'!>%$>$*.)!)*+!9)!9)?%.!d!$=!>#=*3!)*+!9)!9)?%.!
&%%.9=:+&5#)@)(+! *=>.%&)=%F! J)! %)(-)%*)@)(+! 9)! >)%*>):+&-)! )*+! :'%':+.%&*+&<=)! 97=()!
.-$#=+&$(! 9)!#'! %):C)%:C)! @'+C.@'+&<=)3! <=&!*7&(+.%)**)! 9I*! #$%*! '=;!*+%=:+=%)*!
@'+C.@'+&<=)*! ?.(.%'#)*! :':C.)*! *$=*! #)*! >%$5#I@)*3! )+! <=&! >)%@)++)(+!9h)(! >%$9=&%)!
#h&(+)##&?&5&#&+.!>%$>%)F!!
!
J)!@$=-)@)(+!*)!>$=%*=&+!'-):!#)*!+%'-'=;!9)!]'=**3!)+!+%$=-)!*$(!'::$@>#&**)@)(+!
9'(*! #'! %.-$#=+&$(! ?'#$&*&)(()F! J)##)0:&! >%$9=&+! )(H&(! #'! *+%=:+=%)! ?#$5'#)! #&'(+! #)*! %':&()*!
9h=()! .<='+&$(! Q! *)*! :$)HH&:&)(+*F!T##)! @$(+%)!#)*! %'&*$(*! '%&+C@.+&<=)*! >%$H$(9)*! >$=%!
#)*<=)##)*!#'!@.+C$9)!9)!*=5*+&+=+&$(3!<=&!%.=**&+!>$=%!#)*!.<='+&$(*!9)!9)?%.!&(H.%&)=%!$=!
.?'#!Q![3!.:C$=)!>$=%!#)*!.<='+&$(*!9)!9)?%.!*=>.%&)=%F!6)*!:$(:)>+*!9)!?%$=>)3!9)!:$%>*3!
97'(()'=;3!97'#?I5%)*3!&9.'=;3!*$(+!&**=*!9)!:)++)!%.-$#=+&$(!_!&#*!>)%@)++)(+!9)!:'%':+.%&*)%!
9&%):+)@)(+!#)*!%)#'+&$(*!9)*! .#.@)(+*!9h=(!)(*)@5#)!)+! #)*! $>.%'+&$(*! '=;<=)##)*! $(! #)*!
*$=@)+!*'(*!<=)!:)##)*0:&!(h'&)(+!Q!K+%)!<='#&H&.)*F!6)*!?%$=>)*!()!*$(+!>'*!*)=#)@)(+!9)*!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
i!aF! 2,166TB1X3! @*/$,&,)*/%' (%' $A#$.BC6%D' E,-%' )6%-/%6' F' G%<*%6<*%&' &96' 89%$89%&' <,"<%)7&' %7' -+7*,(%&' (%'
$A#$.BC6%'-,(%6"%3!/'%&*!_!/%)**)*!,(&-)%*&+'&%)*!9)!"%'(:)3!DfiOF!

108
SEPTIEMBRE
2016
!
Intuition et invention mathématique | Florian Forestier
!
!
$58)+*! ?.(.%'=;! =+&#&*.*! 9)! H'R$(! $>.%'+$&%)! #$%*! 9)! #'! %.*$#=+&$(! 9)! >%$5#I@)*! c! &#*!
9)-&)(()(+!9)*!,CH%7&'I')#67'%"7/B6%'89/'&,"7'+79(/+&'),96'%9JK-L-%&?''
'
/$=%! aF! 2=&##)@&(3! )(! :$(:#=*&$(3! #)! @$=-)@)(+! 97'5*+%':+&$(! <=&! :'%':+.%&*)! #)*!
@'+C.@'+&<=)*!)*+!=(!@$=-)@)(+!9)!#7&(+)##&?)(:)!*)!:$@>%)('(+!)##)0@K@)!)+!.#&@&('(+!
>)=!Q!>)=! #)! C'*'%9! )+! #7'%5&+%'&%)!9)! *)*! @$9)*! 97'>>#&:'+&$(*! _! !)##)! ()!%)#I-)!9$(:! >'*!
9h=()! );+)(*&$(! >%$?%)**&-)! 9=! :C'@>! 9)! :'*! )@5%'**.*! >'%! =()! +C.$%&)3! @'&*! 9h=()!
);>#$%'+&$(!9)*!%'&*$(*!9)! *+%=:+=%)! <=&! );>#&<=)(+! #)! :$@>$%+)@)(+! 9)*! $58)+*! *)#$(!
#)=%*!9&HH.%)(+)*!*>.:&H&:'+&$(*F!!
!
4)#$(!aF!2=&##)@&(3!=(!>'%'##I#)!*)@5#)!>$**&5#)!)(+%)!#)*!>%&(:&>)*!Q!#h$)=-%)!9'(*!
#'! %):C)%:C)! @'+C.@'+&<=)! )+! :)=;! );>#&:&+.*! >'%! #)*! >C&#$*$>C&)*! 9)*! .>$<=)*!
:$%%)*>$(9'(+)*F! 6'! >C&#$*$>C&)! :'%+.*&)(()! :$%%)*>$(9%'&+! >'%! );)@>#)! Q! =()!
:$(:)>+&$(!)(:$%)!$>.%'+$&%)!9)*!@'+C.@'+&<=)*3!';.)!*=%!#'!9.:$=-)%+)!>%$?%)**&-)!9)!
*+%'+.?&)*3!9$(+! #)!@'+C.@'+&:&)(!S)*:'%+)*! *)%'&+!.?'#)@)(+!#h&##=*+%'+&$(!c! #)!>%)@&)%!
+$=%('(+! 9)! #'! ?.(.%'#&*'+&$(! L$>.%'(+3! *)#$(! #h'=+)=%3! )(+%)! ]'=**! )+! N5)#l! ]'##$&*M3! Q!
>'%+&%! 9=<=)#! &#! *h'?&+! 9)! 9.>#$G)%! #' )6/,6/! #)! >%&(:&>)! 9h)(?)(9%)@)(+! )+! 9)!
9&HH.%)(:&'+&$(! 9)*! :'*3! >$=%%'&+! K+%)! @&*! )(! %)?'%9! 9)! #'! >C&#$*$>C&)! :#'**&<=)!
'##)@'(9)!c!#)!*):$(9!+$=%('(+!9)!#'!?.(.%'#&*'+&$(3!:$(9=&*'(+3!#=&3!Q!#'!9&HH.%)(:&'+&$(!
9)*!';&$@'+&<=)*!)+!9)!#)=%*!@$9I#)*3!:$%%)*>$(9%'&+!5&)(!*m%!'=!@$@)(+!'=+%&:C&)(!9)!
#'! H&(! 9=! e1e)! *&I:#)! )+! Q! #'! ('&**'(:)! 9)! #'! >C.($@.($#$?&)! )+! 9)! #'! >C&#$*$>C&)!
'('#G+&<=)MF!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
n!J$@@)! ($=*! #)! %'>>)#&$(*! 9'(*! A! B'+C.@'+&<=)*! )+! :$(:%.+=9)*! >C.($@.($#$?&<=)*! E3! !""#$%&' (%'
)*+",-+",$,./%'"011234133!#'!@&*)!)(!>#':)!9)!#'!7*+,6/%'(%&'-9$7/)$/</7+&'9)!V&)@'((!8$=)!=(!%P#)!&@>$%+'(+!
9'(*!:)!@$=-)@)(+!9)!?.(.%'#&*'+&$(!)(!@$9&H&'(+!%'9&:'#)@)(+!#)!*)(*!9)!#'!9.@'%:C)!@'+C.@'+&<=)nF!T##)!
@$9&H&)!)(!>'%+&:=#&)%!:)#=&!9)!#'!9&*+&(:+&$(!9)!#7'%&+C@.+&<=)!)+!9)!#'!?.$@.+%&)!)+!#)*!H'R$(*!9)!+%'&+)%!:)*!9)=;!
9$@'&()*F!,()!@=#+&>#&:&+.!)*+!=(!,CH%7'-#7*+-#7/89%'(,7+'(%'$,/&'N,6-%$$%&'89/')%9>%"7')#67/<9$#6/&%6'(=#976%&'
,CH%7&'-,/"&'.+"+6#9J!_!$(!9&*+&(?=)!'&(*&!#)*!@=#+&>#&:&+.*!(/&<6B7%&!L$=!#'!:$@>'%'&*$(!9)*!?%'(9)=%*!*7)HH):+=)!
>'%!(+",-C6%-%"7M!)+!#)*!@=#+&>#&:&+.*!<,"7/"9%&!L$o!:)++)!:$@>'%'&*$(!*7)HH):+=)!>'%!#'!-%&96%MF!6)*!)*>':)*!
.+,-+76/89%&!()!*$(+!>#=*!<=7=(!:'*!>'%+&:=#&)%!9)*!@=#+&>#&:&+.*!:$(+&(=)*3!#7)*>':)!76/K(/-%"&/,"%$!<=7=(!:'*!
>'%+&:=#&)%! 9)! @=#+&>#&:&+.*! "K(/-%"&/,""%$$%&3! #7)*>':)! %9<$/(/%"!=(! :'*! >'%+&:=#&)%! 97)*>':)*! Q! <,96C96%'
<,"&7#"7%F! J)*! :'+.?$%&)*! 9)! @=#+&>#&:&+.*! (7$(+! >'*3! )(! )##)*0@K@)*3! 9)! *)(*! ?.$@.+%&<=)! (&! 9)! *)(*!
'%&+C@.+&<=)!c!)##)*!9.:%&-)(+!$=%".%"(6%-%"7'(%'&769<796%&'<#)#C$%&'(%'(+<6/6%'$%&'$,/&'(%'<%'89%'",9&'<,""#/&&,"&'
/"79/7/>%-%"7' <,--%' #6/7*-+7/89%' ,9' .+,-+76/%F! T(! 97'=+%)*! +)%@)*3! )##)*! :'%':+.%&*)(+! #)*! $>.%'+&$(*3! 9)!
9.9=:+&$(*3!9)!%'&*$(()@)(+*!<=)!#7$(!>)=+!'::$@>#&%!*=%!=(!:)%+'&(!+G>)!9)!@=#+&>#&:&+.3!:7)*+0Q09&%)!9)*!7;)%&'
(%' 7*+,6/%&' <,"&769<7/C$%&F! ! 5=#J/,-#7/&#7/,"3! >%$#$(?)! )+! %'9&:'#&*)! :)! @$=-)@)(+F! T(! )##)! *7$>I%)! =(!
<*#".%-%"7'(%'&7#797'(%&'#J/,-%&D!#)*<=)#*!()!*$(+!>#=*!9)*!&@>.%'+&H*!#')6/,6/3!@'&*!9)*!,CH%7&!9.H&(&**'(+!9)*!
*G*+I@)*!9.9=:+&H*!'=!*)&(!9)*<=)#*!#'!9.@$(*+%'+&$(!*)!*$=@)+!'=;!*)=#)*!%I?#)*!9)!9.%&-'+&$(!'('#G+&<=)F!J)!
:C'(?)@)(+!>$*)!97'&##)=%*!9)*!<=)*+&$(*!:%=:&'#)*!Q!#'!9.@'%:C)!@'+C.@'+&<=)!+$=+!)(+&I%)!>=&*<=)!:7)*+!$#'
N,6-%' -L-%' (%' $#' (+-,"&76#7/,"' %7' $#' (+-,"76#C/$/7+' 89/' %&7' )6/&%' <,--%' ,CH%7F! ,()! %.-&*&$(! ?.(.%'#)! 9)!
#7)(*)@5#)!9)!:)!<=&!>'%'&**'&+!8=*<=7'#$%*!:$@@)!9)*!.-&9)(:)*!'::$@>'?()!:)!+$=%('(+F!p=)#!)*+!)(!)HH)+!#)!
*+'+=+! 9)*! 9.@$(*+%'+&$(*! *7'>>=G'(+! *=%! 9)*! 9$((.)*! &(+=&+&-)*! q! 6'! <=)*+&$(! >)=+! )(! H'&+! K+%)! %)+$=%(.)!
>=&*<=)!#$(?+)@>*3!#)*!9.@$(*+%'+&$(*!*)!*$(+!%)*+%)&(+)*!Q!=(!:C'@>!<=7)##)*!()!>$*'&)(+!>'*!:$@@)!+)#!)+!'=!
*)&(!9=<=)#!)##)*!8$=&**'&)(+!5&)(!97=()!.-&9)(:)!-'#'5#)!c!<=%&7'&%9$%-%"7'I')#67/6'(9'-,-%"7',O'$#'(+N/"/7/,"'
N,6-%$$%' (%&' )6,C$B-%&' ,976%)#&&#/7' $%96'<*#-)' #<79%$$%-%"7')6,(9<7/C$%D' &9&</7#/7'(%&' 89%&7/,"&' ),67#"7'&96' $#'
N,6-9$#7/,"'-L-%'(%'$#'(+N/"/7/,"'P%7'","')$9&'&96'$%&'%J%-)$%&Q'89%'$%'6%.#6('-#7*+-#7/89%'&=%&7'6%7,96"+'(%&'
,CH%7&'(+N/"/&'#9J'$,/&'(%'$%96'(+N/"/7/,"?'5%')6,.6#--%'(%'R/$C%673!)(H&(3!)*+!#'!>#=*!'@5&+&)=*)!+)(+'+&-)!97'**=%)%!

109
SEPTIEMBRE
2016
!
Intuition et invention mathématique | Florian Forestier
!
!
!
S"%'.+"+6#$/&#7/,"'","'#C&76#<7/>%'
!
J)!*)%'&+!'##)%!-&+)!)(!5)*$?()3!'&(*&3!<=)!9)!:$(*&9.%)%!<=7&#!(7G!-'!9'(*!:)!>%$:)**=*!
9)! ?.(.%'#&*'+&$(! <=)! 97=()! *&@>#)! '5*+%':+&$(! >'%! *=%*$@>+&$(F! 6'! >)%*>):+&-)!
*+%=:+=%'#&*+)!)+!%'+&$('#&*+)!9)!aF!2=&##)@&(3!<=&!-$&+!9'(*!#7.-$#=+&$(!9)*!@'+C.@'+&<=)*!=(!
>%$:)**=*! 9)! @&*)! )(! #=@&I%)! 9)! #'! %'&*$(! %)H$=#'(+! >%$?%)**&-)@)(+! #7'%5&+%'&%)! <=&!
#7C'5&+)3! >)=+! )##)! '=**&! K+%)3! *&($(! @&+&?.)3! 9=! @$&(*! :$@>#);&H&.)F! N-):! NF! 6'=+@'(3! <=)!
($=*! '::$@>'?()%$(*! <=)#<=)*! +)@>*3! &#! H'=+! *$=#&?()%! )(! )HH)+!<=)! #'! ?.(.%'#&*'+&$(!
>%$:I9)! 97=(! +%'-'&#! 97);+%':+&$(! 97&(-'%&'(+*3! Q! >'%+&%! 9)! H$%@)*! 9.:)#'5#)*! Q! =(! :)%+'&(!
(&-)'=3!#)<=)#!:$(*&*+)!)(!>'%+&:=#&)%!9'(*!#'!9.*$#&9'%&*'+&$(!&9.'#)!9)!H$%@)*!<=&!*)@5#)(+!
#&.)*F!N&(*&3!.:%&+!6'=+@'(!
!
AX$=*! '-$(*! -=! 9'(*! +$=*! #)*! :'*3! :)%+'&()*! ($+&$(*! 9)! #h'%&+C@.+&<=)! )+! 9)! #h'#?I5%)!
.#.@)(+'&%)3! <=&! >'%'&**)(+! *&@>#)*! )+! >%&@&+&-)*3! )(-)#$>>)%! =()! >#=%'#&+.! 9)! ($+&$(*!
#$?&<=)*!$=!@'+C.@'+&<=)*3!9.#&:'+)*!Q!>%.:&*)%!@'&*!)(!+$=+!:'*!()++)@)(+!9&*+&(?='5#)*!#)*!
=()*!9)*!'=+%)*F!Jh)*+!'&(*&!<=)!#h.?'#&+.!'%&+C@.+&<=)!)*+!#'!*)=#)!%)#'+&$(!9h.<=&-'#)(:)!+)##)!
<=)!#)!9.($@5%)@)(+!9)*!&(9&-&9=*!9h=(!)(*)@5#)!*)!:$(H$(9)!'-):!#)!9.($@5%)@)(+!9)*!
:#'**)*!9h&(9&-&9=*!.<=&-'#)(+*!'=!*)(*!9)!:)++)!%)#'+&$(c! 9)! @K@)!#h&9.)!9)! @=#+&>#&:'+&$(!
:$(+&)(+! Q! #'! H$&*! #'! H$%@'+&$(! 9)! >%$9=&+*! '%&+C@.+&<=)*! )+! #h':+&$(! 9h$>.%'+)=%*! *=%! =(!
9$@'&()!9h.#.@)(+*!9&*+&(:+*!9)!:)*!$>.%'+)=%*!c!#h&9.)!9)!#h=(&+.!>)=+!K+%)!:$(*&9.%.)!*$&+!9=!
>$&(+!9)!-=)!9)!#h.#.@)(+!=(&+.!9h=(!'(()'=!9)!($@5%)*3!*$&+!9=!>$&(+!9)!-=)!9)!#h$>.%'+)=%!
&9)(+&<=)!9h=(!9$@'&()!9h$>.%'+)=%*!c!#'!#$(?=)=%!9h=(!*)?@)(+!)*+!#&.)!Q!#'!?%'(9)=%!<=h)##)!
@)*=%)! @'&*! )##)! (h)*+! >'*! =(! ($@5%)! '++':C.! >'%! :$(-)(+&$(! Q! :)++)! ?%'(9)=%! c! )(H&(! #'!
-'#)=%! '5*$#=)! 9)! #h'#?I5%)! :#'**&<=)! )(-)#$>>)! Q! #'! H$&*! #h&9.)! 9h$%9$(('(:)! )+! #'!
:$(*+%=:+&$(! 9)! #'! H)%@)+=%)! 9h=(! :$%>*! 9)! ($@5%)*F! 6)! >'**'?)! 9)*! ($+&$(*! 9&+)*! A!
.#.@)(+'&%)*!E!'=;!($+&$(*!'5*+%'&+)*!()!*)!>%.*)(+)!9$(:!>'*!:$@@)!=()!*=5*$@>+&$(!9=!
>'%+&:=#&)%! *$=*! #)! ?.(.%'#! @'&*! :$@@)! #'! 9&-&*&$(! $=! #h'('#G*)! 9h=(! A! @&;+)! E! <=&! +)(9! Q!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
9)! @'(&I%)! *+'5#)! )+! %&?$=%)=*)! #)*! H$(9)@)(+*! 9)*! @'+C.@'+&<=)*F! U(! >)=+3! *)#$(! "%.9.%&:! /'+%'*! 9'(*! A!
6hC$%&`$(!*.@'(+&<=)!)+!:'+.?$%&'#!9)!#'!@.+C$9)!';&$@'+&<=)!E3!T,%&/&3!OZZ\3! 9.+)%@&()%!<='+%)! &(H#=)(:)*!
@'8)=%)*!Q!:)!>%$?%'@@)F!DM!6'!>%)@&I%)!:$(:)%()!$#'$+./7/-#7/,"'(%'$=9&#.%'(%'.+,-+76/%&'","K%9<$/(/%""%&F!
N-):!J'G#)G3!>=&*!r#)&(3!'!#&)=!'&(*&!#7'5*$%>+&$(!9)*!>%$>%&.+.*!@.+%&<=)*!9'(*!#)*!>%$>%&.+.*!>%$8):+&-)*!9)*!
H&?=%)*_! #'! ?.$@.+%&)! )*+! A!'%&+C@.+&*.)!E! )+! *)*! ($=-)'=;! $58)+*! *$(+! )(?)(9%.*! H$%@)##)@)(+3! *'(*! <=)!
#7&(+=&+&$(!(7G!8$=)!>#=*!'=:=(!%P#)!($%@'+&HF!6'!?.(.%'#&*'+&$(!>%$8):+&-)!9)!#7'%&+C@.+&<=)!>)%@)+!9)!#'!*$%+)!9)!
:$(*+%=&%)!'=**&!5&)(!9)*!@$9I#)*!)=:#&9&)(*!<=)!9)*!@$9I#)*!($(0)=:#&9&)(*F!OM!6'!*):$(9)!&(H#=)(:)!-&)(+!9)!
$=#J/,-#7/&#7/,"'(9'<,"7/"93!Q!+%'-)%*!#)*!+%'-'=;!9)!S)9)s&(93!t)&)%*+%'**3!)+!J'(+$%F!\M!6'!+%$&*&I@)!&(H#=)(:)3!
>#=*!*=5+&#)3!)*+!:)##)!9)!#'!+C.$%&)!U#$$,/&F!Y&#5)%+!%)@'%<=)!)(!)HH)+!<=)!#7'5'(9$(!9)!#7CG>$+CI*)!9)!:$(+&(=&+.!
>)%@)+!9)!9.:%&%)3!*=%!$%&'&769<796%&'-/&%&'%"')$#<%')#6'$#'+C.$%&)!9)'U#$$,/&'L#'! :#P+=%)!<='9%'+&<=)!%.)##)!9=!
:$%>*!9)*!%'+&$(()#*M!+$=+)!#'!?.$@.+%&)!)=:#&9&)(()F!/$=%!#'!>%)@&I%)!H$&*3!)(!97'=+%)*!+)%@)*3!$#'6%$#7/,"'%"76%'
9"' &;&7B-%' (=#J/,-%&' %7' $%&' ,CH%7&' 89=/$' (+<6/7' %&7' +C6#"$+%F! ,()! @K@)! ';&$@'+&<=)! >)=+! *)! A!>%$8)+)%!E! *=%!
>#=*&)=%*! *G*+I@)*! 97$58)+*! 9&HH.%)(+*3! :)! <=&! $=-%)! #'! -$&)! Q! #'! +C.$%&)! 9)*! @$9I#)*! )+! Q! #'! 9&*+&(:+&$(! 9)*!
'('#G*)*!*G(+';&<=)*!)+!*.@'(+&<=)*!9)*!+C.$%&)*!@'+C.@'+&<=)*!<=&!8$=)!=(!%P#)!&@>$%+'(+!9'(*!#'!%.H#);&$(!
9)!Y=**)%#F!6'! ($+&$(!9)!:$@>#.+=9)!8$=)!=(! %P#)!&@>$%+'(+!>$=%!.:#'&%)%!#)! *)(*!9)!:)++)!9&*+&(:+&$(F![M!6'!
<='+%&I@)! .-$<=.)! >'%! /'+%'*! )*+! <%$$%' (%' 7%"7#7/>%&' (=#J/,-#7/&#7/,"&' #"7+6/%96%&!LB$%&+`! /'*:C! $=! #7.:$#)!
&+'#&)(()! '-):! ]&=*)>>)! /)'($! )+! ]&=*)>>)! 2)%$()*)MF! T(! 9.>&+! 9)! *$(! 9.*&%! 97.-':=)%! +$=+! %):$=%*! &(+=&+&H3!
Y&#5)%+! 9)@)=%)! +$=+)H$&*! *$=:&)=;! 9)! ()! >'*! %$@>%)! #)! %'>>$%+! &(+=&+&H! $%&?&()##)@)(+! ($=.! '=! *)&(! 9)*!
>%)@&I%)*!:$(*+%=:+&$(*!@'+C.@'+&<=)*F!!
!