Jacques Hadamard
One of the last universal mathematicians, Jacques Hadamard (December 8, 1865 October 17, 1963) was calle !the livin" le"en of mathematics#$ Durin" his lon" life, he wor%e in man& areas of mathematics but is 'robabl& best remembere for his 1896 'roof of the 'rime number theorem, which was in e'en entl& 'rove that same &ear b& (el"ian anal&st an number theorist )harles e la *all+e ,oussin# -he theorem was a con.ecture ma e b& /auss in 1790# 1t states that if 2(n) is the number of 'rimes not "reater than n, then 2(n) is as&m'totic to n3ln n, that is, the limit of their ratio is 1, as n 4 5# 6a amar also intro uce what have come to be %nown as 6a amar matrices that have foun im'ortant a''lications in creatin" error correctin" co es, e7'erimental an combinatorial esi"ns, mas%s for s'ectrosco'ic anal&sis, an in encr&'tion# ,rivate %e& encr&'tion has become im'ortant in the mo ern era as the stan ar encr&'tion for e8commerce, e8ban%in", an e8mail#
6a amar was born in *ersailles, 9rance, where his father was a :atin teacher at the :&c+e 1m'+rial an his mother was a istin"uishe 'iano teacher# -he famil& later move to ,aris where out sub.ect live as an infant throu"h the 'rolon"e an "ruelin" sie"e of ,aris urin" the 9ranco8,russian ;ar# -he )it& of :i"hts was bombar e with new heav& caliber sie"e "uns es'eciall& ma e for the battle b& the <ru'' )or'oration# -he 'o'ulation be"an to suffer from famine an was force to %ill their horses, cats, o"s an animals in =oos for foo # On >anuar& 08, 1871, the 9rench an the 1?6,@@@ efen ers of ,aris surren ere # (etween the time of the humiliatin" 9rench surren er an the si"nin" of -he -reat& of 9ran%furt in Aa& 1871, civil war bro%e out in ,aris an the house occu'ie b& the 6a amar famil&
was estro&e in a fire# Durin" this 'erio the famil& suffere further 'ersonal tra"e ies with the eath of two of >acBuesC sisters# 1n 188? 6a amar entere the Dcole Eormale Fu'+rieure, "ra uatin" in 1888# Durin" the ne7t five &ears he tau"ht mathematics at various schools# 6e receive his docteur s sciences in 1890 for a thesis on functions efine b& -a&lor series# -hat same &ear, 6a amar marrie :ouise8Gnna -r+nel who, li%e her husban , came from a >ewish famil&# -he& ha five chil ren#
-he &ear 1890 was ca''e when 6a amar receive the /ran ,ri7 es Fciences Aath+matiBues for his 'a'er !Determination of the number of 'rimes less than a "iven number#$ 6a amar lecture at the 9acult+ es Fciences of (or eau7 from 1893 to 1897, urin" which time he 'ublishe 09 'a'ers on a wi e ran"e of to'ics, inclu in" his 'roof of the ,rime Eumber -heorem# 6e move to ,aris in 1897 where he was a lecturer at the Forbonne until 19@9, the &ear he became 'rofessor at the )ollH"e e 9rance# 9or the ne7t twent& &ears he ran the hu"el& successful Sminaire Hadamard# 1n 1910 6a amar succee e )amille >or an as 'rofessor of anal&sis at the Dcole ,ol&techniBue# -ra"e & struc% urin" ;;1 when 6a amar Cs two ol est sons were %ille at the (attle of *er un# 6a amar later wrote that his man& &ears of !'ure .o&$ came to an en in that &ear# 6a amar retire from his various 'ositions in 1937 at a"e 70#
;hen 9rance fell to the /erman inva ers in 19?@, he an his famil& esca'e to the Inite Ftates, where he was a''ointe a visitin" 'rofessor at )olumbia Iniversit& but coul not fin a 'ermanent 'ost in the Inite Ftates# 1n 19??, he receive the sa news that his thir son ha been %ille in the war, while servin" with the 9ree 9rench forces in Eorth Gfrica# Jeturnin" to 9rance at the en of the hostilities 6a amar 'ursue his interests in music, ferns an fun"i# 6e was such an active 'eace cam'ai"ner that it reBuire the active lobb&in" of I#F# mathematicians to overcome 'olitical o''osition to his atten ance at the 1nternational Aathematical )on"ress hel in )ambri "e, Aassachusetts in 195@# 6e was ma e honorar& 'resi ent of the )on"ress# G &ear before his eath,
6a amar e7'erience another 'ainful tra"e &# 6is belove "ran son Dtienne was %ille in a climbin" acci ent# -his a''eare to be more than 6a amar coul bear an he ie a few months later, a bit short of his 98th birth a&#
6a amar 'ublishe more than 3@@ novel an hi"hl& creative 'a'ers in anal&tic function theor&, number theor&, anal&tical mechanics, h& ro &namics, calculus of variations, ifferential "eometr&, 'artial ifferential eBuations, 'robabilit& theor&, lo"ic, an the histor& of mathematics# Gt the en of the 19th centur& he 'rove one of the earliest results in what has come to be %nown as )haos -heor&, b& showin" wh& one canCt 're ict what three billiar balls will o when the& careen off each other on the table# 1n 'a'ers of 19@1 an 19@0, 6a amar classifie all mathematical 'roblems into either !well8 'ose $ or !ill8'ose #$ ;hen a new 'roblem is 'ro'ose , the first or er of business is to establish that it has a solution, that the solution is uniBue, an that the solution e'en s in a reasonable wa& on the ata# Fuch a 'roblem is calle !well8'ose ,$ because it is worth 'ursuin", as it is consi ere solvable# 1f a 'roblem is not shown to have an e7istin" an uniBue solution, it is consi ere !ill8'ose ,$ or unsolvable#
One of 6a amar Cs most fascinatin" investi"ations was in the 's&cholo"& of mathematical invention, lea in" to the 'ublication of his fascinatin" boo% The Psychology of Invention in the Mathematical Field# 9irst 'ublishe in 19?5, it was reissue as The Mathematicians Mind in 1973# 1n it 6a amar e7'lores how mathematicians invent new i eas b& relatin" the creative e7'eriences of "reat thin%ers of the era# 6a amar hel the o'inion that mathematical thou"ht is visual an that wor s onl& interfere# 6e inclu e a letter from his close frien Glbert Kinstein anal&=in" his own mechanism of thou"ht# 6a amar etaile how ins'iration mi"ht occur at an& time, often after an in ivi ual ha stru""le for
a&s with a 'roblem# ,uttin" asi e the 'roblem an involvin" the conscious min with other thou"hts, the subconscious continues to wor% on the 'roblem an much to the sur'rise an eli"ht of the
in ivi ual the solution becomes clear# One of the intri"uin" conclusions of 6a amar Cs inBuir& is that most of those he consulte i not !see$ their 'roblems in verbal terms or in al"ebraic s&mbols, but
rather in visual ima"es of a va"ue, ha=& nature# Kinstein wroteL !-he wor s of the lan"ua"e as the& are written or s'o%en o not seem to 'la& an& role in the mechanism of thou"ht M which relies on more or less clear ima"es of a visual an some of a muscular t&'e#$
6a amar liste the sta"es of mathematical creativit& as 're'aration (trial an error), incubation (often subconscious), illumination (freBuentl& su en), an verification (reBuirin" reasonin")# ;hen tr&in" to
'rove a theorem, the 're'aration sta"e involves an intense stu & of the 'roblem, consi erin" the use of various a''roaches common in the fiel # Once the 'roblem is thorou"hl& un erstoo , an %nown a''roaches e7hauste , the mathematician ma& e7'erience a sense of an7iet& or frustration at bein" unable to attac% the 'roblem e uctivel&# 1ncubation be"ins when the mathematician lives with the 'roblem an allows unconscious 'rocesses to o'erate# 1f fortunate, illumination occurs, when a momentar& flash of insi"ht 'rovi es the solution# -he last 'hase of invention is verification, which is when the mathematician writes out the e uctive ar"ument, whose structure a''eare in the moment of illumination# 6a amar Cs contributions were im'ortant not onl& in eterminin" how mathematical reasonin" occurs, but if an how it can be tau"ht# 6e em'hasi=e that in most cases mathematicians canCt foresee whether a tentative line of attac% will be successfulN but the& have a sense of beaut& that irects them# Once a &oun" Gn r+ ;eil s%i''e a 'oint in an ar"ument, maintainin" that it was obvious# 6a amar re'lie that either it was so obvious that it coul be e7'laine in a line, in which case the line shoul be inclu e , or it was not obvious an nee e to be e7'laine #
6a amar Cs first became involve in 'olitics urin" the infamous !Dre&fus Gffair#$ 1n 189? Glfre Dre&fus (1859 1935), a 9rench artiller& ca'tain of the /eneral Ftaff, who was >ewish, was falsel& accuse of sellin" militar& secrets to the /ermans# 6e was court8martialle , foun "uilt&, stri''e of
his ran%, an sentence to life im'risonment on DevilCs 1slan in 9rench /uiana# Dre&fus 'roteste his innocence an ha no a''arent motive for treason# Glthou"h his trial ha been hi"hl& irre"ular, anti8 Femitism in 9rance ma e the ver ict 'o'ular# 6a amar was a secon cousin of Dre&fusC wife :ucie, ne :ucie 6a amar # (efore the Gffair, 6a amar an Dre&fus li%e most >ewish89rench bour"eoisie consi ere them com'letel& assimilate into 9rench societ&# 1n Aarch 1896 )olonel /eor"e ,icBuart of 9rench militar& intelli"ence iscovere that the ocument that ha been the 'rinci'al evi ence a"ainst Dre&fus was not written in the han of the accuse , but of another officer, Aa.or Ksterha=&, a man with e7tensive "amblin" ebts# Des'ite an attem't to hush of the iscover&, the Dre&fus famil& became aware of this evi ence an eman e that Ksterha=& be trie for the crime# 6e was, but was
acBuitte b& a militar& tribunal# 6a amar became a lea er in the crusa e to overturn Dre&fusCs conviction an to clear his name# 1n 1898 9rench novelist Kmile Oola es'ouse the cause of Dre&fusN 'ublishin" an o'en letter Jaccuse in which he accuse the arm&Cs /eneral Ftaff of %nowin" that Dre&fus was innocent# 6e accuse them of bein" willin" to see a >ewish victim suffer rather than a mit that the& ha use for"e ocuments to un.ustl& convict Dre&fus# 9rance was s'lit into riotin" cam's#
Oola was fine an sentence to a &ear in 'rison, but esca'e to Kn"lan #
Kventuall& Dre&fus was retrie , foun "uilt& a"ain, an was 'ar one # 6a amar woul not acce't this result an wor%e ili"entl& to clear Dre&fusC name, which occurre on >ul& 0, 19@6 when he was ecorate with the :e"ion of 6onor# -wo a&s after Dre&fusC eath
reinstate to his commission an
his funeral corte"e 'asse the ,lace e )oncor e throu"h the ran%s of troo's assemble for the Eational 6oli a&# 6owever it wasnCt until 1995 that the 9rench arm& officiall& a mitte the& ha been wron" in the case# 9rom the time of Dre&fusC vin ication, 6a amar , who before the Gffair ha seen a'olitical, became active in >ewish causes an the :ea"ue for 6uman Ji"hts, foun e b& Oola, an remaine a 'ublic man for the rest of his life#
One of m& 'rofessors en.o&e sharin" stories of istin"uishe mathematicians he ha %nown# 1n most cases, his stories a''ear in one source or another# Infortunatel&, no confirmation of the followin" tale about 6a amar has been foun # ,erha's, itCs a mathematical urban le"en # ;ith this caveat, itCs worth retellin"# Gfter com'letin" a &ear as a visitin" 'rofessor at )olumbia, 6a amar sou"ht another 'osition in the Inite Ftates while the war ra"e across the oceans# 6e ha no luc%# ;hen he ma e a hast& e7it from 9rance in 19?@, he i nCt ta%e his cre entials with him# Kach time he a''lie for a 'osition, universit& officials were courteous, but insiste that without the necessar& ocumentation the& coul not hire him# Gt one universit&, he recite his e ucational bac%"roun , aca emic 'ositions, 'ublications, etc# to a ean# (ut the a ministrator sai , !1f &ou onl& ha some sort of evi ence of who &ou are an what &ou have one, 1C be ha''& to offer &ou a 'osition#$ 6a amar as%e for time to thin% on the matter an too% a wal% aroun the cam'us# 6e entere the librar&, loo%e aroun , an his face bri"htene # 6e hurrie bac% to the eanCs office# 6e as%e , !Do &ou %now those 'ictures of famous mathematicians on the walls of the librar&P$ -he ean answere , !;h&, &es 1 o#$ 6a amar 'rou l& announce , !1 am the sub.ect of the fourth 'icture to the left of the entrance#$ 6e still i nCt "et the 'osition#
Quotation of the Day: !Fome intervention of intuition issuin" from the unconscious is necessar&
at least to initiate the lo"ical wor%#$ >acBues 6a amar
