the importance of mathematics

8/14/2019 The Importance of Mathematics

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THE IMPORTANCE OF MATHEMATICSW. T. GowersIt is with some disbelief that I stad here ad !re!are to addressthis "atheri" o thes#b$e%t of the im!orta%e of mathemati%s. For a start& it is a

e'traordiar( hoo#r tobe i)ited to "i)e the *e(ote address at a millei#m meeti" iParis. Se%odl(& "i)i"a le%t#re o the si"i+%a%e of mathemati%s demads wisdom&$#d"met ad mat#rit(&ad there are ma( mathemati%ias far better edowed tha I amwith these ,#alities&i%l#di" se)eral i this a#die%e. I ho!e therefore that (o# will#derstad that m(tho#"hts are ot f#ll( formed- if I had bee as*ed to s!ea* o thiss#b$e%t +)e (ears a"o&I wo#ld ha)e "i)e a %om!letel( di+eret le%t#re& ad I am %o+detthat i +)e (earstime it wo#ld a"ai ha)e %ha"ed.M( title /whi%h I did ot a%t#all( %hoose m(self& tho#"h I willi"l(a"reed to it0 also!la%es o me a "reat b#rde of res!osibilit(. After all& I ams!ea*i" to a a#die%e whi%h%otais ot $#st mathemati%ias b#t $o#ralists ad other i#etialo1mathemati%ias.If I fail to %o)i%e (o# that mathemati%s is im!ortat ad

worthwhile& I will be letti"dow the mathemati%al %omm#it(& ad also letti" dow Mr Cla(&whose "eerosit( hasmade this e)et !ossible ad is bee+ti" mathemati%s i ma(other wa(s as well.2fort#atel(& if oe s#r)e(s i a s#!er+%ial wa( the )ast a%ti)it( ofmathemati%iasaro#d the world& it is eas( to %ome awa( with the im!ressio thatmathemati%s is ota%t#all( all that im!ortat. The !er%eta"e of the worlds!o!#latio& or e)e of the

worlds #i)ersit(1ed#%ated !o!#latio& who %o#ld a%%#ratel( state asi"le mathemati%altheorem !ro)ed i the last +ft( (ears& is small& ad smaller still ifFermats last theorem ise'%l#ded. If (o# as* a mathemati%ia to e'!lai what he or shewor*s o& (o# will #s#all(be met with a shee!ish "ri ad told that it is ot !ossible to do soi a short time. If(o# as* whether this m(sterio#sl( %om!li%ated wor* has !ra%ti%ala!!li%atios /ad we all

"et as*ed this from time to time0& the there are )ario#s t(!i%alres!oses& oe of them


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immediatel( im!ressi)e.Oe is the lie ta*e b( the famo#s Cambrid"e mathemati%ia G. H.Hard(& whowas !erfe%tl( %otet& ideed almost !ro#d& that his %hose +eld&N#mber Theor(& had

o a!!li%atios& either the or i the foreseeable f#t#re. For him& themai %riterio ofmathemati%al worth was bea#t(. At the other ed of the s!e%tr#mthere are mathemati13%ias who to wor* i areas s#%h as theoreti%al %om!#ter s%ie%e&+a%ial mathemati%s orstatisti%s& areas of a%*owled"ed !ra%ti%al im!orta%e.Mathemati%ias i these areas %a!oit to ideas that ha)e had a bi" im!a%t& s#%h as the 4la%*1S%holese,#atio for deri)ati)e!ri%i"& whi%h has trasformed the o!eratio of the +a%ialmar*ets& ad the !#bli%1*e(%r(!tos(stem i)eted b( Ri)est& Shamir ad Adelma& whi%h isow the basis for se%#rit(o the iteret& ad whi%h& as has bee !oited o#t ma( times& isa a!!li%atio of#mber theor( that Hard( %ertail( did ot e'!e%t.Also at the a!!lied ed of the s!e%tr#m there are ma(mathemati%ias whose wor*has itimate %oe%tios with theoreti%al !h(si%s. A%t#all(& it is ot

ob)io#s that #if(i"Geeral Relati)it( ad 5#at#m Me%hai%s wo#ld ha)e dire%t!ra%ti%al a!!li%atios& si%etoda(s !h(si%s alread( !ro)ides #s with !redi%tios that area%%#rate to withi the limitswe %a meas#re. 4#t oe e)er *ows& ad i a( %ase s#%h abrea*thro#"h wo#ld be ofabsol#tel( f#dametal iterest to s%ie%e& or ideed a(bod( withthe sli"htest itelle%t#al%#riosit(. If mathemati%ias %a ma*e a %otrib#tio to this area&the the( will at least

be able to !oit to a h#"e e'teral a!!li%atio of mathemati%s.Most mathemati%ias& i%l#di" me& lie somewhere i the middle ofthe s!e%tr#m&whe it %omes to o#r attit#de to a!!li%atios. We wo#ld bedeli"hted if we !ro)ed atheorem that was fo#d to be #sef#l o#tside mathemati%s 1 b#t wedo ot a%ti)el( see* todo so. Gi)e the %hoi%e betwee a iteresti" b#t !#rel(mathemati%al !roblem ad a#iteresti" !roblem of !otetial bee+t to e"ieers& %om!#ter

s%ietists or !h(si%ists&


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we will o!t for the former& tho#"h we wo#ld %ertail( feel aw*ward ifobod( wor*ed o!ra%ti%al !roblems.A%t#all(& this attit#de is held e)e b( ma( of those who wor* imore !ra%ti%alseemi"

areas. If (o# !ress s#%h a !erso& as*i" for a s!e%i+% e'am!le of aa!!li%atioi b#siess& id#str( or s%ie%e of their ow wor* as o!!osed to aa!!li%atio of a res#lt itheir "eeral area& (o# will ofte& tho#"h ot i)ariabl(& witess a#%omfortable rea%tio.It t#rs o#t that a "reat deal of the resear%h i e)e the so1%alled!ra%ti%al areas is i fa%tot !ra%ti%al at all. I am ot tr(i" to draw attetio to some sort ofs%adal b( sa(i"this 1 as I ho!e to demostrate& this !heomeo is a at#ral addesirable %ose,#e%eof what it meas to )iew the world mathemati%all(.The reaso for it is that mathemati%s is a two1sta"e !ro%ess. Rathertha st#d(i"the world dire%tl(& mathemati%ias %reate so1%alled models of theworld& ad st#d( them.6This a!!lies e)e to the sim!lest mathemati%s. After the a"e of fo#ror +)e we do otst#d( additio b( a%t#all( %ombii" "ro#!s of ob$e%ts ad %o#ti"

them. Istead we#se a abstra%t mathemati%al %ostr#%tio& or model& *ow as the!ositi)e ite"ers /thatis& the #mbers 3&6&7&8&9 ad so o0. Similarl(& we do ot do basi%"eometr( b( %#tti"sha!es o#t of !a!er& !artl( be%a#se it is ot e%essar( ad !artl(be%a#se i a( %ase theres#lti" sha!es wo#ld ot be e'a%t s,#ares& tria"les or whate)erthe( were s#!!osed tobe. O%e a"ai& we st#d( a model& a sort of ideali:ed world that%otais thi"s that we do

ot %ome a%ross i e)er(da( life& s#%h as i+itel( thi lies thatstret%h awa( to i+it(&or absol#tel( !erfe%t %ir%les& ad does ot %otai #tid(& worldl(thi"s li*e hamb#r"ers&%hairs or h#ma bei"s.If oe wor*s i a !ra%ti%al area of mathemati%s& the there will betwo %oi%ti"%riteria for what ma*es a "ood model. O the oe had& the modelsho#ld be a%%#rateeo#"h to be #sef#l& ad o the other& it sho#ld be sim!le ad

ele"at eo#"h to "eerate


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realisti% ad iteresti" mathemati%al !roblems. It is tem!ti"& as amathemati%ia& toatta%h far more im!orta%e to the se%od %riterio 1 mathemati%aliterest ad ele"a%e 1tha to the +rst 1 a%%#ra%( 1 e)e if this meas ot immediatel(

%otrib#ti" to the "rossatioal !rod#%t of oes %o#tr(.A "ood e'am!le of this attit#de %omes from %om!#ter s%ie%e.Cosider the etwor*show i Fi"#re 3& ad ima"ie that we ha)e bee as*ed to %olo#rthe odes of the etwor*with two %olo#rs& red ad bl#e& i s#%h a wa( that o two odes ofthe same %olo#r aree)er li*ed. S#%h a %olo#ri" is %alled a !ro!er %olo#ri" of theetwor*.If we start with a si"le ode& s#%h as the oe mar*ed A& the itdoest matterwhether we %olo#r it red or bl#e& si%e the roles of the two %olo#rsare iter%ha"eable.Howe)er& o%e we ha)e de%ided that A sho#ld be red& sa(& the thetwo odes mar*ed 4ha)e to be bl#e& si%e the( are li*ed to A ad therefore m#st otha)e the same %olo#r asA. Ha)i" established this& we the see that the odes mar*ed Cm#st all be %olo#red red/si%e the( are li*ed to bl#e odes0& ad the that all odes

mar*ed ; m#st be %olo#redbl#e. 4#t ow we ha)e hit a !roblem& whi%h is that two of the odesmar*ed ; are li*edto oe aother. Si%e all o#r %hoi%es of %olo#r were for%ed from themomet we %olo#redthe ode A& it follows that it is im!ossible to "i)e a !ro!er %olo#ri"of the odes withol( two %olo#rs.Now this ar"#met does more tha merel( establish that oe!arti%#lar etwor* %a17

ot be !ro!erl( %olo#red. We ha)e #sed a "eeral !ro%ed#re& oral"orithm& as it is %alled imathemati%s ad %om!#ter s%ie%e& for testi" whether a( "i)eetwor* %a be !ro!erl(%olo#red with two %olo#rs. 4rie(& this !ro%ed#re %a be des%ribed asfollows- %olo#r oeode arbitraril( ad the %oti#e b( %olo#ri" odes whee)er the%hoi%e of %olo#r isfor%ed. If (o# are e)et#all( for%ed to "i)e two li*ed odes thesame %olo#r the the

etwor* %aot be !ro!erl( %olo#red& ad if (o# are ot the it %a.


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For e'am!le& whe 3


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%a#ses !eo!le to stri)e to be%ome wealth(& ad this bee+ts theetire e%oom( i ma(wa(s& s#%h as i%reasi" the ta' re)e#e for the "o)ermet& whi%h%a the be s!et ohos!itals& s%hools& !#bli% tras!ort ad so o& or %a#si" %om!aies

to be set #!& whi%h!ro)ide li)elihoods for ma( !eo!le. The idi)id#als eed ha)eabsol#tel( o iterest iwhether other members of so%iet( ha)e satisfa%tor( li)es& !ro)idedthat s#+%iet so%ialorder is maitaied& b#t i a idire%t wa( their a%ti)it( does bee+tothers.Of %o#rse& ot e)er(bod( a"rees that e%oomies s#%h as I ha)ebrie( des%ribed are a"ood thi"& ad the last thi" I wish to do is let !oliti%s itr#de o amathemati%al le%t#re.Howe)er& it is s#rel( ot %otro)ersial to state that idi)id#alsel+shess %a lead to !#bli%"ood& ad I wish to ar"#e that somethi" similar ha!!es imathemati%s. Altho#"h?idi)id#al mathemati%ias are moti)ated !rimaril( b( a s#btlemi't#re of ambitio aditelle%t#al %#riosit(& ad ot b( a wish to bee+t so%iet(&e)ertheless& mathemati%s as awhole does bee+t so%iet(. I wo#ld ow li*e to e'amie i more

detail wh( this is.Oe strai"htforward aswer is this- mathemati%s is %hea!& ado%%asioall( !rod#%esbrea*thro#"hs of eormo#s e%oomi% bee+t& either dire%tl(& as ithe %ase of !#bli%1*e(%r(!to"ra!h(& or idire%tl(& as a res#lt of !ro)idi" the e%essar(theoreti%al #der!ii"for s%ie%e. If (o# were to wor* o#t what mathemati%al resear%h has%ost the world ithe last 3


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+a%e miister will oti%e a "a! i the abo)e ar"#met- mi"ht itot be !ossible to a%hie)ethe same bee+ts more %hea!l(= If the bee+ts of mathemati%s%ome from tea%hi" ada few brea*thro#"hs& while most mathemati%ias "et o with their

iteresti" b#t #selessresear%h& the wh( ot %#t the resear%h f#di" to the #selessareas ad $#st s#!!ort thetea%hi" ad the more !ra%ti%all( orieted mathemati%s= Oe of m(mai ob$e%ti)es toda(is to e'!ose the falla%(& or rather falla%ies& that wo#ld lie behids#%h a !ro!osal.The +rst oe is the idea that it is !ossible to idetif( the areas ofmathemati%s thatwill t#r o#t to be #sef#l. I fa%t& it is otorio#sl( hard to !redi%t this&ad the histor( ofmathemati%s is littered with e'am!les of areas of resear%h thatwere iitiall( !#rs#ed fortheir ow sa*e ad later t#red o#t to ha)e a %om!letel(#e'!e%ted im!orta%e. I %o#ldmetio the RSA al"orithm (et a"ai. A more f#dametal e'am!leis the o1E#%lidea"eometr( of Ga#ss& 4ol(ai ad oba%he)s*(& whi%h is iterall(%osistet des!ite s#%ha!!aretl( !arado'i%al !heomea as the e'iste%e of tria"leswith a"les ot addi" to

3that the eds of the *otted stri" are f#sed to"ether 1 !erha!s

D*otted loo!s wo#ld bea more a%%#rate term. A !arti%#larl( sim!le *ot is show at thebottom. This is *owas the #*ot& be%a#se& as oe %a easil( see& it %a be #twistedito a sim!le loo!. Theother *ots are more "e#iel( *otted& b#t this is s#r!risi"l( hardto !ro)e& ad to thisda( there is o *ow !ra%ti%al method /that is& al"orithm a"ai0 ofde%idi" whether orot a more %om!li%ated dia"ram of the *id a!!eari" i Fi"#re 7

re!resets a *ot that


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%a be #tied ito a sim!le loo!. Aother %etral !roblem i *ottheor( is to de%idewhe two dia"rams i fa%t re!reset the same *ot& i the sesethat oe %a be twistedito the other. For e'am!le& tho#"h it is ot ob)io#s from the

dia"ram& a "ood metal"(mast %a e)et#all( see that the fo#rth ad +fth dia"rams/readi" from the to!& adfrom left to ri"ht0 re!reset the same *ot.O%e a"ai& these loo*ed li*e am#si" !#::les #til& after wor* ofa#"ha oesad Edward Witte& it was reali:ed that *ot theor( hadf#dametal %oe%tios withtheoreti%al !h(si%s.So 1 mathemati%ias %a tell their "o)ermets 1 if (o# %#t f#di"to !#re mathemati%alresear%h& (o# r# the ris* of losi" o#t o #e'!e%ted bee+ts&whi%h histori%all(ha)e bee b( far the most im!ortat.Howe)er& the miserl( +a%e miister eed ot be %o)i%ed ,#ite(et. It ma( be )er(hard to idetif( !ositi)el( the areas of mathemati%s li*el( to lead to!ra%ti%al bee+ts& b#tthat does ot r#le o#t the !ossibilit( of idetif(i" e"ati)el( theareas that will ,#ite%learl( be #seless& or at least #seless for the e't two h#dred

(ears. I fa%t& the +a%emiister does ot e)e eed to be %ertai that the( will be #seless.If a lar"e area ofmathemati%s has ol( a oe i te tho#sad %ha%e of !rod#%i"e%oomi% bee+t i thee't +ft( (ears& the !erha!s that at least %o#ld be %#t.o# will ot be s#r!rised to hear me sa( that this !oli%( wo#ld stillbe %om!letel(mis"#ided. A ma$or reaso& oe that has bee %ommeted o ma(times ad is im!liedb( the s#btitle of this %ofere%e& A Celebratio of the 2i)ersalit(

of Mathemati%alTho#"htJ& is that mathemati%s is )er( iter%oe%ted& far more sotha it a!!ears o thes#rfa%e. The !i%t#re i the ba%* of the +a%e miisters mid mi"htbe somethi" li*eFi"#re 8. A%%ordi" to this !i%t#re& mathemati%s is di)ided itose)eral s#bdis%i!lies& of)ar(i" de"rees of !ra%ti%alit(& ad it is a sim!le matter to %#tf#di" to the less !ra%ti%aloes.


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A more realisti% !i%t#re& tho#"h still o#tra"eo#sl( sim!li+ed& is "i)ei Fi"#re 9./#st for the !#r!oses of %om!ariso& Fi"#re ? shows Fi"#res 8 ad 9s#!erim!osed.0 Theodes of Fi"#re 9 re!reset small areas of mathemati%al a%ti)it(

ad the lies $oii" themre!reset iterrelatioshi!s betwee those areas. The small areas ofa%ti)it( form %l#sterswhere there are more of these iterrelatioshi!s& ad these %l#sters%a !erha!s be tho#"htof as s#bdis%i!lies. Howe)er& the bo#daries of these %l#sters areot !re%ise& ad ma(of the iterrelatioshi!s are betwee %l#sters rather tha withithem.I !arti%#lar& if mathemati%ias wor* o di+%#lt !ra%ti%al !roblems&the( do ot do soi isolatio from the rest of mathemati%s. Rather& the( bri" to the!roblems se)eral tools1 mathemati%al tri%*s& r#les of th#mb& theorems *ow to be #sef#l/i the mathemati%alsese0& ad so o. The( do ot *ow i ad)a%e whi%h of these toolsthe( will #se& b#t the(ho!e that after the( ha)e tho#"ht hard abo#t a !roblem the( willreali:e what is eeded tosol)e it. If the( are l#%*(& the( %a sim!l( a!!l( their e'isti"e'!ertise strai"htforwardl(.

More ofte& the( will ha)e to ada!t it to some e'tet. Perha!s it willbe hel!f#l if I showaother two !i%t#res& this time ill#strati" what it is li*e to sol)e amathemati%al !roblem.Fi"#re > is a ai)e )iew- (o# start at the bo#dar( of what is *ow&with a de+ite"oal i mid. o# the ha)e a s#%%essio of brilliatl( %le)er ideasafter whi%h the sol#tio!o!s o#t. A more realisti% )iew& whi%h I ha)e tried to re!reset!i%toriall( i Fi"#re &ta*es ito a%%o#t #mero#s im!ortat feat#res of mathemati%al

resear%h s#%h as falsestarts& !romisi" ideas that lead owhere or isi"hts that#e'!e%tedl( sol)e a di+eret!roblem.Th#s& a "ood wa( to thi* abo#t mathemati%s as a whole is that it isa h#"e bod( of*owled"e& a bit li*e a e%(%lo!aedia b#t with a eormo#s#mber of %ross1refere%es.This *owled"e is stored i boo*s& !a!ers& %om!#ters ad the braisof tho#sads of

mathemati%ias ro#d the world. It is ot as %o)eiet to loo* #! a!ie%e of mathemati%s


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as it is to loo* #! a word i a e%(%lo!aedia& es!e%iall( as it is otalwa(s eas( tos!e%if( e'a%tl( what it is that oe wats to loo* #!. Ne)ertheless&this e%(%lo!aediaJ ofmathemati%s is a i%redible reso#r%e. Ad $#st as& if oe were to

tr( to "et rid of all theetries i a e%(%lo!aedia& or& to "i)e a di+eret %om!ariso& allthe boo*s i a librar(&that obod( e)er loo*ed #!& the res#lt wo#ld be a "reatl(im!o)erished e%(%lo!aedia orlibrar(& so& a( attem!t to !#r"e mathemati%s of its less #sef#l !artswo#ld almost %ertail(be )er( dama"i" to the more #sef#l !arts as well.@So far I ha)e sim!l( stated that mathemati%s is f#ll of s#r!risi"%oe%tios. A(mathemati%ia will ha!!il( %o+rm that statemet ad be able to"i)e e'am!les fromhis or her e'!erie%e. Ideed& dis%o)eri" s#r!risi" %oe%tios isoe of the "reat $o(sof the s#b$e%t. I wo#ld li*e to ill#strate the iter%oe%tedess ofmathemati%s with ae'am!le i whi%h I !la(ed a small !art. To do this I m#st brie(des%ribe a few #sol)edmathemati%al !roblems.The +rst oe is sim!le eo#"h that I %a e'!lai it !re%isel(.

Cosider the followi"three se,#e%es of #mbers-9 33 3> 67 6@> 3@ 73 8733 83 >3 3


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of !rime #mbers=Problem 6. Ca arithmeti% !ro"ressios of !rimes ha)e a( /+ite0le"th or is theresome #!!er limit to how lo" the( %a be=I sho#ld sa( i !assi" that if (o# re!la%e Dfo#r b( Dthree i

,#estio /30& the the aswer is*ow to be (es& b#t the !roof is b( o meas eas(. Se%odl(& thelo"est *ow arithmeti%!ro"ressio %osisti" of !rime #mbers has le"th i the earl(tweties ad was fo#d b(a %om!#ter. Of %o#rse& o s#%h e'am!le lea)es #s a( the wiserabo#t ,#estio /60.The e't !roblem I wish to des%ribe is more "eometri%al i%hara%ter. Fi"#re @a showsa tria"le di)ided #! ito ei"ht tall thi tria"les. This is doe i themost ob)io#s wa(-3

the importance of mathematics

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