What's so sexy about math? | Cédric Villani

Prevodilac: Dragana Radmanovic Lektor: Tijana Mihajlović

What is it that French people do better than all the others?

Šta to Francuzi rade bolje od svih drugih?

If you would take polls,

Ako biste sproveli istraživanje,

the top three answers might be:

najčešća tri odgovora bi možda bila:

love, wine and whining.

ljubav, vino i žaljenje.

(Laughter)

(Smeh)

Maybe.

Možda.

But let me suggest a fourth one:

Ali, dozvolite mi da predložim i četvrti:

mathematics.

matematika.

Did you know that Paris has more mathematicians

Da li znate da Pariz ima više matematičara

than any other city in the world?

od bilo kog drugog grada u svetu?

And more streets with mathematicians' names, too.

I više ulica sa imenima matematičara, takođe.

And if you look at the statistics of the Fields Medal,

I ukoliko pogledamo statistiku Fildsove medalje,

often called the Nobel Prize for mathematics,

često nazivane Nobelovom nagradom za matematičare,

and always awarded to mathematicians below the age of 40,

koja se uvek dodeljuje matematičarima ispod 40 godina,

you will find that France has more Fields medalists per inhabitant

otkrićete da Francuska ima više dobitnika ove medalje po glavi stanovnika

than any other country.

od bilo koje druge države.

What is it that we find so sexy in math?

Šta nam je tako privlačno u matematici?

After all, it seems to be dull and abstract,

Naposletku, deluje da je dosadna i apstraktna;

just numbers and computations and rules to apply.

samo brojevi, računanje i pravila koja treba primeniti.

Mathematics may be abstract,

Matematika je možda apstraktna,

but it's not dull

ali nije dosadna

and it's not about computing.

i nije u vezi sa proračunima.

It is about reasoning

U vezi je sa razmišljanjem

and proving our core activity.

i dokazivanjem naše suštinske aktivnosti.

It is about imagination,

U vezi je sa maštom,

the talent which we most praise.

talentom koji cenimo više od svega.

It is about finding the truth.

U vezi je sa nalaženjem istine.

There's nothing like the feeling which invades you

Nema ničega boljeg od osećaja koji te prožme

when after months of hard thinking,

kada, posle meseci teškog razmišljanja,

you finally understand the right reasoning to solve your problem.

konačno razumeš kako da rešiš svoj problem.

The great mathematician André Weil likened this --

Veliki matematičar, Andre Vejl, je uporedio to -

no kidding --

bez šale -

to sexual pleasure.

sa seksualnim zadovoljstvom.

But noted that this feeling can last for hours, or even days.

S tim da ovaj osećaj može da traje satima ili čak danima.

The reward may be big.

Nagrada može biti velika.

Hidden mathematical truths permeate our whole physical world.

Skrivene matematičke istine prožimaju sav naš fizički svet.

They are inaccessible to our senses

Do njih se ne može doći putem čula,

but can be seen through mathematical lenses.

ali se mogu uočiti kroz matematička sočiva.

Close your eyes for moment

Zatvorite oči na trenutak

and think of what is occurring right now around you.

i mislite o tome šta se dešava trenutno oko vas.

Invisible particles from the air around are bumping on you

Nevidljive čestice iz vazduha se sudaraju sa vama,

by the billions and billions at each second,

milijarde i milijarde svake sekunde,

all in complete chaos.

potpuno haotično.

And still,

Ali, ipak,

their statistics can be accurately predicted by mathematical physics.

njihova statistika se može tačno predvideti putem matematičke fizike.

And open your eyes now

Otvorite oči sada

to the statistics of the velocities of these particles.

da vidite statistiku brzina ovih čestica.

The famous bell-shaped Gauss Curve,

Poznata Gausova kriva u obliku zvona,

or the Law of Errors --

ili zakon grešaka -

of deviations with respect to the mean behavior.

odstupanja koja se odnose na srednjestatističko ponašanje.

This curve tells about the statistics of velocities of particles

Kriva pokazuje statistiku brzina čestica

in the same way as a demographic curve

na isti način kao što i demografska kriva

would tell about the statistics of ages of individuals.

pokazuje statistiku godina uzrasta pojedinaca.

It's one of the most important curves ever.

To je jedna od najvažnijih krivih u istoriji.

It keeps on occurring again and again,

Ponavlja se iznova i iznova

from many theories and many experiments,

u mnogim teorijama i eksperimentima,

as a great example of the universality

kao divan primer univerzalnosti

which is so dear to us mathematicians.

koja je nama, matematičarima, tako draga.

Of this curve,

O ovoj je krivoj

the famous scientist Francis Galton said,

poznati naučnik Francis Galton rekao:

"It would have been deified by the Greeks if they had known it.

„Da su znali za nju, Grci bi je obogotvorili.

It is the supreme law of unreason."

To je viši zakon iracionalnosti.“

And there's no better way to materialize that supreme goddess than Galton's Board.

Nema boljeg načina za materijalizovanje te visoke boginje nego kroz Galtonovu tablu.

Inside this board are narrow tunnels

Unutar ove table su uski tuneli

through which tiny balls will fall down randomly,

kroz koje će male loptice padati nadole nasumično,

going right or left, or left, etc.

idući desno, ili levo, ili levo itd.

All in complete randomness and chaos.

U potpunoj nasumičnosti i haosu.

Let's see what happens when we look at all these random trajectories together.

Da vidimo šta se dešava kada pogledamo sve ove nasumične putanje zajedno.

(Board shaking)

(Tabla se trese)

This is a bit of a sport,

Ovo je pomalo poput sporta,

because we need to resolve some traffic jams in there.

zato što se moramo rešiti saobraćajne gužve ovde.

Aha.

Aha.

We think that randomness is going to play me a trick on stage.

Mislimo da će nasumičnost da me prevari na sceni.

There it is.

Evo je.

Our supreme goddess of unreason.

Naša visoka boginja iracionalnosti.

the Gauss Curve,

Gausova kriva,

trapped here inside this transparent box as Dream in "The Sandman" comics.

zarobljena ovde unutar ove providne kutije poput Sna u stripu „Sandman“.

For you I have shown it,

Vama sam je pokazao,

but to my students I explain why it could not be any other curve.

ali svojim studentima objašnjavam zašto to ne može biti neka druga kriva.

And this is touching the mystery of that goddess,

I tu se približavamo misteriji te boginje,

replacing a beautiful coincidence by a beautiful explanation.

zamenjujući prelepu slučajnost prelepim objašnjenjem.

All of science is like this.

Sva nauka je takva.

And beautiful mathematical explanations are not only for our pleasure.

A prelepa matematička objašnjenja nisu tu samo zbog našeg zadovoljstva.

They also change our vision of the world.

Ona takođe menjaju naš pogled na svet.

For instance,

Na primer,

Einstein,

Ajnštajn, Perin, Smolučovski,

Perrin,

Smoluchowski,

they used the mathematical analysis of random trajectories

koristili su matematičku analizu nasumičnih putanja

and the Gauss Curve

i Gausovu krivu

to explain and prove that our world is made of atoms.

da objasne i dokažu da je naš svet izgrađen iz atoma.

It was not the first time

To nije prvi put

that mathematics was revolutionizing our view of the world.

da je matematika iz korena menjala naš pogled na svet.

More than 2,000 years ago,

Pre više od 2 000 godina,

at the time of the ancient Greeks,

u vreme antičke Grčke,

it already occurred.

to se već dogodilo.

In those days,

U to vreme,

only a small fraction of the world had been explored,

samo mali procenat sveta je bio otkriven,

and the Earth might have seemed infinite.

a Zemlja je možda delovala kao da je beskonačna.

But clever Eratosthenes,

Ali, umni Eratosten,

using mathematics,

koristeći se matematikom,

was able to measure the Earth with an amazing accuracy of two percent.

bio je u stanju da izmeri Zemlju

sa sjajnom preciznošću od dva procenta greške.

Here's another example.

Evo ga još jedan primer.

In 1673, Jean Richer noticed

Godine 1673. Žan Rišer je primetio

that a pendulum swings slightly slower in Cayenne than in Paris.

da se klatno malčice sporije ljulja u Kajenu nego u Parizu.

From this observation alone, and clever mathematics,

Iz samo ove observacije i uz lukavu matematiku,

Newton rightly deduced

Njutn je pravilno zaključio

that the Earth is a wee bit flattened at the poles,

da je Zemlja malo ravnija na polovima,

like 0.3 percent --

negde oko 0,3 procenta -

so tiny that you wouldn't even notice it on the real view of the Earth.

toliko malo da to ne možete ni primetiti na pravoj slici Zemlje.

These stories show that mathematics

Ovakve priče pokazuju da uz matematiku

is able to make us go out of our intuition

možemo da idemo dalje od naše intuicije,

measure the Earth which seems infinite,

da izmerimo Zemlju koja deluje kao beskonačna,

see atoms which are invisible

da vidimo atome koji su nevidljivi

or detect an imperceptible variation of shape.

ili da detektujemo nevidljive varijacije oblika.

And if there is just one thing that you should take home from this talk,

I ako biste izdvojili samo jednu stvar koju treba da zapamtite iz ovog govora,

it is this:

to je ovo:

mathematics allows us to go beyond the intuition

matematika nam dopušta da idemo dalje od naše intuicije

and explore territories which do not fit within our grasp.

i da istražujemo teritorije koje ne možemo da zamislimo.

Here's a modern example you will all relate to:

Evo modernog primera koji je blizak svima:

searching the Internet.

pretraživanje interneta.

The World Wide Web,

Svetska mreža,

more than one billion web pages --

više od milijardu veb-stranica -

do you want to go through them all?

da li biste pretraživali kroz sve njih?

Computing power helps,

Računarska moć pomaže,

but it would be useless without the mathematical modeling

ali bi bila beskorisna bez matematičkog modelovanja

to find the information hidden in the data.

kojim se nalazi informacija skrivena u podacima.

Let's work out a baby problem.

Hajde da prođemo kroz mali zadatak.

Imagine that you're a detective working on a crime case,

Zamislite da ste detektiv koji radi na slučaju zločina,

and there are many people who have their version of the facts.

i imate mnogo ljudi koji imaju svoju verziju toga šta se dogodilo.

Who do you want to interview first?

Koga biste prvo intervjuisali?

Sensible answer:

Razuman odgovor:

prime witnesses.

glavne svedoke.

You see,

Vidite,

suppose that there is person number seven,

pretpostavimo da vam osoba broj sedam

tells you a story,

ispriča priču,

but when you ask where he got if from,

ali kada je upitate odake ona to zna,

he points to person number three as a source.

ona vam kaže da je njen izvor osoba broj tri.

And maybe person number three, in turn,

I možda osoba broj tri dalje

points at person number one as the primary source.

istakne osobu broj jedan kao primarni izvor.

Now number one is a prime witness,

Sada je osoba broj jedan glavni svedok.

so I definitely want to interview him -- priority.

tako da definitivno hoću nju da intervjuišem - prioritet.

And from the graph

I iz grafika

we also see that person number four is a prime witness.

možemo da pročitamo da je osoba broj četiri takođe među glavnim svedocima.

And maybe I even want to interview him first,

I možda ja mogu i nju prvo da intervjuišem,

because there are more people who refer to him.

zato što ima više osoba koje upućuju na nju.

OK, that was easy,

U redu, to je bilo lako,

but now what about if you have a big bunch of people who will testify?

ali šta ako imate gomilu ljudi koji treba da svedoče?

And this graph,

Ovaj grafik

I may think of it as all people who testify in a complicated crime case,

može da se razume kao da su to svi svedoci ovog komplikovanog kriminalnog slučaja,

but it may just as well be web pages pointing to each other,

ali to isto tako mogu biti i veb-stranice koje ukazuju jedna na drugu,

referring to each other for contents.

čiji sadržaj usmerava sa jedne na drugu.

Which ones are the most authoritative?

Koje stranice su najautoritativnije?

Not so clear.

Nije sasvim jasno.

Enter PageRank,

Pristupimo „Pejdž ranku“,

one of the early cornerstones of Google.

jednom od kamena temeljaca Gugla.

This algorithm uses the laws of mathematical randomness

Ovaj algoritam koristi zakone matematičke nasumičnosti

to determine automatically the most relevant web pages,

da automatski odredi najrelevantnije veb-stranice,

in the same way as we used randomness in the Galton Board experiment.

na isti način kako smo koristili nasumičnost

u eksperimentu Galtonove table.

So let's send into this graph

Onda, propustimo kroz ovaj grafik

a bunch of tiny, digital marbles

gomilu malih, digitalnih klikera

and let them go randomly through the graph.

i pustimo ih da idu nasumično kroz grafik.

Each time they arrive at some site,

Svaki put kad su na nekom sajtu,

they will go out through some link chosen at random to the next one.

izaći će sa tog sajta i preći na drugi koristeći se nekim nasumičnim linkom.

And again, and again, and again.

I ponovo, i ponovo, i ponovo.

And with small, growing piles,

I malim, rastućim gomilama,

we'll keep the record of how many times each site has been visited

merićemo koliko puta su ovi digitalni klikeri

by these digital marbles.

posetili svaki sajt.

Here we go.

Krećemo.

Randomness, randomness.

Nasumičnost, nasumičnost.

And from time to time,

I s vremena na vreme,

also let's make jumps completely randomly to increase the fun.

da bi bilo zabavnije, preskačimo sasvim proizvoljno.

And look at this:

I pogledajte ovo:

from the chaos will emerge the solution.

iz haosa se pojavljuje rešenje.

The highest piles correspond to those sites

Najviše hrpe odgovaraju onim sajtovima

which somehow are better connected than the others,

koji su na neki način bolje povezani od drugih,

more pointed at than the others.

koji češće od drugih upućuju na druge stranice.

And here we see clearly

I ovde jasno vidimo

which are the web pages we want to first try.

koje veb-stranice hoćemo prvo da isprobamo.

Once again,

Opet,

the solution emerges from the randomness.

rešenje se pojavljuje iz nasumičnosti.

Of course, since that time,

Naravno, od tada,

Google has come up with much more sophisticated algorithms,

Gugl je smislio mnogo sofisticiranije algoritme,

but already this was beautiful.

ali je već tada ovo bilo prelepo.

And still,

Ipak,

just one problem in a million.

to je samo jedan problem iz milion problema.

With the advent of digital area,

Sa razvojem digitalne oblasti,

more and more problems lend themselves to mathematical analysis,

sve više i više problema se može podvrgnuti matematičkoj analizi,

making the job of mathematician a more and more useful one,

čineći posao matematičara sve korisnijim,

to the extent that a few years ago,

do tog nivoa da je pre par godina,

it was ranked number one among hundreds of jobs

bio broj jedan na listi među stotinama poslova

in a study about the best and worst jobs

u istraživanju o najboljim i najgorim poslovima

published by the Wall Street Journal in 2009.

koje je „Vol strit džurnal“ objavio 2009. godine.

Mathematician --

Matematičar -

best job in the world.

najbolji posao na svetu.

That's because of the applications:

To je zbog primena:

communication theory,

komunikaciona teorija,

information theory,

informaciona teorija,

game theory,

teorija igara,

compressed sensing,

kompresija signala,

machine learning,

mašinsko učenje,

graph analysis,

teorija grafova,

harmonic analysis.

harmonijska analiza.

And why not stochastic processes,

I zašto da ne, stohastički procesi,

linear programming,

linearno programiranje

or fluid simulation?

ili simulacija dinamike tečnosti?

Each of these fields have monster industrial applications.

Svaka od ovih oblasti ima ogromnu primenu u industriji.

And through them,

I kroz primenu,

there is big money in mathematics.

matematika donosi mnogo novca.

And let me concede

I da potvrdim da su,

that when it comes to making money from the math,

kada je reč o zarađivanju od matematike,

the Americans are by a long shot the world champions,

Amerikanci daleko najbolji svetski šampioni u tome,

with clever, emblematic billionaires and amazing, giant companies,

sa mudrim, realnim milijarderima i sjajnim, ogromnim kompanijama,

all resting, ultimately, on good algorithm.

a svi počivaju, na kraju krajeva, na dobrim algoritmima.

Now with all this beauty, usefulness and wealth,

Sada, uzimajući u obzir svu ovu lepotu, praktičnu primenu i bogatstvo,

mathematics does look more sexy.

matematika doista izgleda privlačnije.

But don't you think

Ali, nemojte misliti

that the life a mathematical researcher is an easy one.

da je život matematičara istraživača lak.

It is filled with perplexity,

Ispunjen je zbunjenošću,

frustration,

frustracijom,

a desperate fight for understanding.

očajnom borbom da se nešto razume.

Let me evoke for you

Da vam predstavim

one of the most striking days in my mathematician's life.

jedan od najboljih dana u mom matematičkom životu.

Or should I say,

Ili bolje,

one of the most striking nights.

jednu od mojih najboljih noći.

At that time,

U to vreme, bio sam u Institutu za primenjeno istraživanje u Prinstonu -

I was staying at the Institute for Advanced Studies in Princeton --

koji je mnogo godina bio dom Albertu Ajnštajnu

for many years, the home of Albert Einstein

and arguably the most holy place for mathematical research in the world.

i koji je možda najsvetije mesto za istraživanje matematike u svetu.

And that night I was working and working on an elusive proof,

Te noći sam radio i radio na jednom teškom dokazu,

which was incomplete.

koji je bio nepotpun.

It was all about understanding

Reč je bila o razumevanju

the paradoxical stability property of plasmas,

paradoksalnog svojstva stabilnosti plazmi,

which are a crowd of electrons.

koje su suštinski skupina elektrona.

In the perfect world of plasma,

U savršenom svetu plazmi,

there are no collisions

nema sudara

and no friction to provide the stability like we are used to.

i nema trenja koji bi obezbedili stabilnost na koju smo navikli.

But still,

Ipak, ako se malčice naruši ekvilibrijum plazme,

if you slightly perturb a plasma equilibrium,

you will find that the resulting electric field

primetiće se da električno polje koje se pojavilo kao posledica toga

spontaneously vanishes,

spontano nestaje

or damps out,

i gasi se,

as if by some mysterious friction force.

kao da je pod dejstvom neke tajanstvene sile trenja.

This paradoxical effect,

Ovaj paradoksalni efekat

called the Landau damping,

pod nazivom Landauovo prigušenje

is one of the most important in plasma physics,

jedan je od najvažnijih u fizici plazme

and it was discovered through mathematical ideas.

i otkriven je kroz matematičke ideje.

But still,

Ipak, nedostajalo je potpuno matematičko razumevanje ovog fenomena.

a full mathematical understanding of this phenomenon was missing.

And together with my former student and main collaborator Clément Mouhot,

I zajedno sa svojim bivšim studentom i glavnim saradnikom, Klementom Muoom,

in Paris at the time,

tada u Parizu,

we had been working for months and months on such a proof.

radili smo mesecima i mesecima na tom dokazu.

Actually,

U stvari,

I had already announced by mistake that we could solve it.

ja sam već greškom objavio da možemo to da rešimo.

But the truth is,

Ali, istina je bila

the proof was just not working.

da dokaz jednostavno nije funkcionisao.

In spite of more than 100 pages of complicated, mathematical arguments,

Bez obzira na to što je postojalo

više od 100 stranica komplikovanih matematičkih pretpostavki,

and a bunch discoveries,

kao i niz otkrića

and huge calculation,

i ogromni proračuni,

it was not working.

dokaz nije funkcionisao.

And that night in Princeton,

I te noći u Prinstonu,

a certain gap in the chain of arguments was driving me crazy.

određena nelogičost u lancu pretpostavki me je izluđivala.

I was putting in there all my energy and experience and tricks,

Uložio sam svu svoju energiju, iskustvo i trikove,

and still nothing was working.

a ipak ništa nije funkcionisalo.

1 a.m., 2 a.m., 3 a.m.,

Jedan ujutru, dva ujutru, tri ujutru;

not working.

ne funkcioniše.

Around 4 a.m., I go to bed in low spirits.

Oko četiri ujutru, ležem u rđavom raspoloženju.

Then a few hours later,

Onda, nekoliko sati kasnije,

waking up and go,

budim se i mislim:

"Ah, it's time to get the kids to school --"

„Ah, vreme je da odvedem decu u školu - “

What is this?

Šta je ovo?

There was this voice in my head, I swear.

Čuo sam glas u glavi, kunem se.

"Take the second term to the other side,

„Prebaci drugi član na drugu stranu,

Fourier transform and invert in L2."

Furijeova transformacija i invertuj na L2.“

(Laughter)

(Smeh)

Damn it,

Dođavola,

to je bio početak rešenja!

that was the start of the solution!

You see,

Vidite, mislio sam da sam se odmorio,

I thought I had taken some rest,

but really my brain had continued to work on it.

ali je moj mozak u stvari nastavio da radi na tome.

In those moments,

U tim trenucima,

you don't think of your career or your colleagues,

ne misliš na svoju karijeru ili na kolege;

it's just a complete battle between the problem and you.

to je prosto potpuna bitka između problema i tebe.

That being said,

Imajući to u vidu,

it does not harm when you do get a promotion in reward for your hard work.

ne škodi ni kada se dobije unapređenje kao nagrada za naporan rad.

And after we completed our huge analysis of the Landau damping,

I pošto smo završili našu ogromnu analizu Landauovog prigušenja,

I was lucky enough

bio sam dovoljno srećan

to get the most coveted Fields Medal

da dobijem najpoželjniju Fildsovu medalju

from the hands of the President of India,

koju mi je uručila predsednica Indije,

in Hyderabad on 19 August, 2010 --

u Hajderabadu 19. avgusta 2010. -

an honor that mathematicians never dare to dream,

čast o kojoj se matematičari i ne usuđuju da sanjaju,

a day that I will remember until I live.

dan koji ću pamtiti dok sam živ.

What do you think,

O čemu misliti

on such an occasion?

u takvoj prilici?

Pride, yes?

Ponos, zar ne?

And gratitude to the many collaborators who made this possible.

I zahvalnost mnogim saradnicima koji su učinili to mogućim.

And because it was a collective adventure,

I zato što je to bio zajednički poduhvat,

you need to share it, not just with your collaborators.

potrebno ga je podeliti ne samo sa saradnicima.

I believe that everybody can appreciate the thrill of mathematical research,

Verujem da su svi u stanju da cene uzbuđenje matematičkog istraživanja

and share the passionate stories of humans and ideas behind it.

i da strastveno dele priče o ljudima i idejama iza njih.

And I've been working with my staff at Institut Henri Poincaré,

Moje osoblje pri Institutu Anri Poenkare i ja smo radili,

together with partners and artists of mathematical communication worldwide,

zajedno sa partnerima i umetnicima matematičke komunikacije širom sveta,

so that we can found our own, very special museum of mathematics there.

na osnivanju sopstvenog, veoma specijalnog muzeja matematike tamo.

So in a few years,

Tako, kroz par godina

when you come to Paris,

kada dođete u Pariz,

after tasting the great, crispy baguette and macaroon,

posle isprobavanja sjajnog, hrskavog bageta i makaruna,

please come and visit us at Institut Henri Poincaré,

molim vas dođite i posetite nas u Institutu Anri Poenkare

and share the mathematical dream with us.

i sanjajte matematički san zajedno sa nama.

Thank you.

Hvala.

(Applause)

(Aplauz)