Seduced by Calculus

7/25/2019 Seduced by Calculus

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Seduced by calculus

The 2010 Fields Medal was won by a French

mathematician captivated by the crowning

mathematical achievement of the Enlightenment.

Alex Bellos explains.

The French mathematician Cdric Villani is no ordinary lookinguniversity professor. Handsome and slender, with a boyish face and

a wavy, neck length bob, he looks more like a dandy from the Belle

Epoque, or a member of an avant garde student rock band.

He always wears a three-piece suit, starched white collar, lavaliere

cravat the kind folded etravagantly in a giant bow and asparkling, tarantula-si!ed spider brooch. "#omehow $ had to do it,%

he said of his appearance. "$t was instinctive.%

$ first met &illani in Hyderabad, $ndia, at the '()( $nternational

*ongress of +athematicians, or $*+, the four-yearly gathering of

the tribe. f the ,((( delegates, &illani was the focus of mostattention, not because he was the most elaborately dressed, but

because he received the ields +edal at the opening gala.


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/he ields is the highest honour in maths and is awarded at each

$*+ to two, three or four mathematicians under the age of 0(. /he

age rule recognises the original motivation behind the pri!e, which

was conceived by the *anadian mathematician 1. *. ields. He

wanted not only to recognise work already done, but also to

encourage future success. #uch is the acclaim afforded by a ields

+edal, however, that since the first two were awarded in )23, they

have helped establish a cult of youth, implying that once you hit 0(

you4re past it. /his is unfair. +any mathematicians produce their

best work after the age of 0(, although ields medallists can struggleto regain focus, since fame brings with it other responsibilities.

+athematicians gather at the $*+ to take stock of their

achievements, and the ields +edal citations provide the clearest

snapshot of the most eciting recent work. 5nlike the citations for

the other three winners in '()(, which were impenetrable to me andeven to many of the mathematicians present, &illani4s citation was

understandable to the non-specialist. He won "for his proofs of

nonlinear 6andau damping and convergence to equilibrium for the

Bolt!mann equation%.

/he Bolt!mann equation, devised by the 7ustrian physicist 6udwigBolt!mann in )89', concerns the behaviour of particles in a gas, and

is one of the best known equations in classical physics. :ot only is

&illani a devotee of the )2th century4s neckwear, he is also a world

authority on its applied mathematics.


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/he Bolt!mann equation is what is known as a partial differential

equation, or ;


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his fingers he drew lines upon his naked body, so far was he taken

from himself, and brought into ecstasy or trance, with the delight he

had in the study of geometry%.

/he initial task of geometry was the calculation of area. ?7ccording

to Herodotus, geometry began as a practice devised by Egyptian ta

inspectors to calculate areas of land destroyed by the :ile4s annual

floods.@ 7s we all know, the area of a rectangle is the width

multiplied by the height, and from this formula we can deduce that

the area of a triangle is half the base times the height. /he Areeksdevised methods to calculate the areas of more complicated shapes.

f these, the most impressive achievement was 7rchimedes4s

"quadrature of the parabola%, by which is meant calculation of the

area bounded by a line and a parabola, which is a specific type of 5-

shaped curve. 7rchimedes first drew a large triangle inside the

parabola, as illustrated below, then on either side of this he drewanother triangle. n each of the two sides of these smaller triangles,

he drew an even smaller triangle, and so on, such that all three

points of each triangle were always on the parabola. /he more

triangles he drew, the closer and closer their combined area was to

the area of the parabolic section. $f the process was allowed to carry

on forever the infinite number of triangles would perfectly cover thedesired area.


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The quadrature of the parabola.

7rchimedes4 quadrature of the parabola is the most sophisticated

eample from the classical age of the method of ehaustion, the

technique of adding up a sequence of small areas that converge

towards a larger one. /he proof is considered his finest moment

because it represents the first "modern% view of mathematical

infinity. 7rchimedes was the earliest thinker to develop the

apparatus of an infinite series with a finite limit. /his was important

not only for conquering the areas of shapes significantly more eotic

than the parabola, but also for starting on the conceptual path

towards calculus. f the giants on whose shoulders $saac :ewton

would eventually perch, 7rchimedes was the first.


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$nfinity is a number bigger than any other. $t has a twin concept, the

infinitesimal, which is a number smaller than any other, yet still

larger than !ero.

$n the )9th century, mathematicians realised how useful the

infinitesimal was, even though it was a concept that didn4t make

much sense it was the mathematical equivalent of having your

cake and eating it. /he infinitesimal was both something and

nothing= large enough to be of mathematical use, but small enough

to disappear when you needed it to.

Calculating the area of a circle with

infinitesimals.

or eample, consider the circle illustrated here. $nside is adodecagon, a )'-sided shape made up of )' identical triangles

sharing a common verte, or point. /he combined area of the

triangles is approimately the area of the circle. $f $ drew a polygon

with more sides within the circle, containing more, thinner triangles,


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their combined area would approimate the circle more closely. 7nd

if $ kept on increasing the number of sides, in the limit $ would have

a polygon with an infinite number of sides containing an infinite

number of infinitely thin triangles. /he area of each triangle is

infinitesimal, yet their combined area is the area of the circle, as

illustrated below left.

Heres another way the infinitesimal was useful in determining

gradients. or readers who have forgotten what a gradient is, it is the

measure of the slope, calculated bydividing the distance moved upby the distance moved along. #o, in the illustration below right, the

gradient of the road is )C0 because the distance moved up is )((m

and the distance along is 0((m. +athematicians, however, wanted to

find a method to calculate the gradient of tangents, which are those

lines that touch a curve at a single point.

A gradient


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the tangent

/he trick to finding the gradient of a tangent at point ; is to make an

approimation of the tangent, and then to improve the

approimation until it coincides with the desired line. De do this by

drawing a line through ; that cuts the curve at nearby point , and

then we bring closer and closer to ;. Dhen hits ;, the line is the

tangent.

/he gradient of the line through ; and is FyCF. ?/he Areek letter

delta, F, is a mathematical symbol meaning a small increment@. 7s

closes in on ;, the value FyCF approaches the gradient of the

tangent at ;. But we have a problem. $f we let actually reach ;,

then Fy G ( and F G (, meaning that the gradient of the curve at ;

is (C(. Bad maths alert /he rules of arithmetic prohibit division by

!ero /he solution is to keep at an infinitesimal distance from ;. $f

we do, we can say that when becomes infinitesimally close to ;,


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the value FyCF is infinitesimally close to the gradient of the curve

at ;.

Approximating a tangent.

$n )33I, $saac :ewton, recently graduated from *ambridge,

returned to live with his mother in their 6incolnshire farmhouse. /he

Areat ;lague was devastating towns across the country. /he

university had closed down to protect its staff and students. :ewton

made himself a small study and started to fill a giant Jotter he calledthe Daste Book with mathematical thoughts. ver the net two

years the solitary scribbler, undistracted, devised new theorems that

became the foundations of the ;hilosophiK :aturalis ;rincipia

+athematica, his )389 treatise that, more than any work before or

since, transformed our understanding of the physical universe. /he

;rincipia established a system of natural laws that eplained whyobJects, from apples falling off trees to planets orbiting the #un,

move as they do. >et :ewton4s breakthrough in physics required an

equally fundamental breakthrough in maths. He formali!ed the

previous half-century4s work on infinity and infinitesimals into a


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general system with a unified notation. He called it the method of

fluions, but it became better known as the "calculus of

infinitesimals4, and now, simply, "calculus4.

7 body that moves changes its position, and its speed is the change

in position over time. $f a body is travelling with a fied speed, it

changes its position by a fied amount every fied period. 7 car with

constant speed that covers 3( miles between 0pm and Ipm is

travelling at 3( miles per hour. :ewton wanted to solve a different

problem= how does one calculate the speed of a body that is nottravelling at a constant speedL or eample, let4s say the car above,

rather than travelling consistently at 3(mph, is continually slowing

down and speeding up because of traffic. ne strategy to calculate

its speed at, say, 0.(pm, is to consider how far it travels between

0.(pm and 0.)pm, which will give us a distance per minute. ?De

Just need to multiply the distance by 3( to get the value in mph.@ Butthis figure is Just the average speed for that minute, not the

instantaneous speed at 0.(pm. De could aim for a shorter interval

say, the distance travelled between 0.(pm and ) second later, which

would give us a distance per second. ?De4d then multiply by ,3((

to get the value in mph@. But again this value is the average for that

second. De could aim for smaller and smaller intervals, but we arenever going to get the instantaneous speed until the interval is tinier

than any other when it is !ero, in other words. But when the

interval is !ero, the car does not move at all


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/his line of reasoning should sound familiar, because $ used it two

paragraphs ago when eplaining how to calculate the gradient of a

tangent. /o find the gradient we divide an infinitesimally small

quantity ?length@ by another infinitesimally small quantity ?another

length@. /o get the instantaneous speed we also divide an

infinitesimally small quantity ?distance@ by another infinitesimally

small quantity ?time@. /he problems are mathematically equivalent.

:ewton4s method of fluions was a method to calculate gradients,

which enabled him to calculate instantaneous speeds.

*alculus allowed :ewton to take an equation that determined the

position of an obJect, and from it devise a secondary equation about

the obJect4s instantaneous speed. $t also allowed him to take an

equation determining the obJect4s instantaneous speed, and from it

devise a secondary equation about position, which, as it turned out,

was equivalent to the calculation of areas using infinitesimals*alculus, therefore, gave him the mathematical tools to develop his

laws of motion. $n his equations, he called the variables and y

"fluents% and the gradients "fluions4, written by the "pricked

letters% and .

Dhen :ewton returned to *ambridge after two years avoiding theplague in 6incolnshire, he did not tell anyone about the method of

fluions. n the continent, Aottfried 6eibni! was developing an

equivalent system. 6eibni! was Aerman by birth but a man of the

world a lawyer, diplomat, alchemist, engineer and philosopher.


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6eibni! was also the mathematician most obsessed with notation.

/he symbols he used for his system of calculus were clearer than

:ewton4s, and are the ones we use today.

6eibni! introduced the terms d and dy for the infinitesimal

differences in and y. /he gradient, which is one infinitesimal

difference divided by the other, he wrote dyCd. /hanks to his use of

the word "difference4, the calculation of gradient became known as

"differentiation4. 6eibni! also introduced the distinctive stretched

"s4, M, as the symbol for the calculation of area. $t4s an abbreviationof summa, or sum, since the calculation of area is based on infinite

sums of infinitesimals. n the suggestion of his friend 1ohann

Bernoulli, 6eibni! called his technique calculus integralis, and the

calculation of area became known as "integration4. 6eibni!4s M is the

most maJestic symbol in maths, reminiscent of the f-hole of a cello

or violin.

*alculus comprises differentiation ?computation of gradient@ and

integration ?computation of area@. $n general terms, gradient is the

rate of change of one quantity over another, and area is the measure

of how much one quantity accumulates with respect to another.

*alculus thus provided scientists with a way to model quantities thatvaried in relation to each other. $t is a formidable instrument to

eplain the physical world because everything in the universe, from

the tiniest atoms to the largest galaies, is in a state of permanent

flu.


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Dhen we know the relationship between two varying quantities, we

can describe them in an equation using the symbols for

differentiation and integration. 7n equation in and y that includes

the term dyCd is called a "simple differential equation4. $f there aremore than two variables, say , y and t, the rates of change are

written NyCN, or NyCNt, with the rounded N. /he equation is called a

"partial differential equation4, or ;


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us how one variable changes with respect to another one, but not to

all of them. ;


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/he second time $ met &illani was in ;aris. #ince '((2 he has been

director of the $nstitut Henri ;oincarQ, rance4s elite maths institute,

which is situated among the universities of the 6atin uarter. His

office is a comfortable clutter of books, paper, coffee mugs, awards,

pu!!les and geometrical shapes.

&illani4s appearance was unchanged since we met in $ndia at the

$nternational *ongress of +athematicians= burgundy cravat, blue

three-piece suit, and a metal spider glistening on his lapel. He said

his look emerged when he was in his twenties. He wore shirts withlarge sleeves, then with lace, then a top hatS "$t was like a scientific

eperiment, and gradually it was Tthis is me4.% 7nd the spiderL He

enJoys its ambiguity. "#ome people think the spider is a maternal

symbol. thers think that the web is a symbol for the universe, or

that the spider is the big architect of the world, like a way to

personify Aod. #piders don4t leave people indifferent. >ouimmediately have a reaction.% /he spider is an archetype rich with

interpretations, $ thought, Just like mathematics is an abstract

language with innumerable applications. &illani4s field is ;


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of the subsequent work that led to his ields +edal. He now views it

with tenderness and devotion. "$t4s like the first girl you fall in love

with,% he confided. "/he first equation you see you think it is the

most beautiful in the world.% east your eyes on her again=

/he Bolt!mann equation belongs to the field of statisticalmechanics= the branch of mathematical physics that investigates how

the behaviour of individual molecules in a cloud of gas influences

macroscopic properties like temperature and pressure. /he equation

describes how a gas disseminates by considering the likelihood of

any of its molecules being in any particular spot, with a particular

speed, at a particular time. U/he f is a "probability density function4,that gives the probability of particles having a position near and a

speed near v at time t.V /he model assumes that particles in a gas

bounce around according to :ewton4s laws, but in random

directions, and describes the effects of their collisions using the

maths of probability. &illani pointed at the left side of the equation=

"/his is Just particles going in straight lines.% He pointed to the right

side of the equation= "7nd this is Just shock. /ik-ding /ing-dik% He

bumped his fists together several times. "ften in ;


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phenomena and also live in completely different mathematical

worlds.%

$f you filmed a single gas particle bouncing off another gas particle,

and showed it to a friend, there is no way he or she would know

whether you were playing the film forwards or backwards, since

:ewton4s laws are time-reversible. But if you filmed a gas spreading

from a beaker to its surroundings, a viewer would instantly be ableto tell which way the film was being played, since gases do not suck

themselves back into beakers. Bolt!mann established a mathematical

foundation for the apparent contradiction between micro- and

macroscopic behaviour by introducing a new concept, entropy. /his


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is the measure of disorder in theoretical terms the number of

possible positions and speeds of the particles at any time. Bolt!mann

then showed that entropy always increases. &illani4s breakthrough

paper concerned Just how fast entropy increases before reaching the

totally disordered state.

/he Bolt!mann equation has straightforward applications, such as in

aeronautical engineering, to determine what happens to planes when

they fly through gases. $ts usefulness is what first appealed to &illani

when he embarked on his ;h

Seduced by Calculus

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