xi i
PREFACE
dean migh t no t giv e yo u a raise ; yo u migh t no t b e invite d t o spea k a t
that bi g conferenc e comin g up .
Now le t u s loo k throug h th e othe r en d o f th e telescope . Th e hars h
reality i s this : I f yo u prov e th e Rieman n hypothesis , o r th e three -
dimensional Poincar e conjecture , o r Fermat' s Las t Theorem , the n th e
world is willing to forgive yo u a lot. I t woul d b e nice if your pape r wer e
well written , fo r the n mor e peopl e coul d benefi t fro m i t mor e quickly .
But—even i f the pape r i s abysmally written— a handfu l o f experts wil l
be abl e t o slu g thei r wa y throug h it , the y wil l teach ' it t o others , per -
haps mor e transparen t proof s ma y come out, textbook s wil l eventuall y
appear. Scienc e i s a proces s tha t tend s t o wor k itsel f out .
In fac t mos t o f us d o not produc e wor k that i s at th e hig h leve l just
described—certainly no t consistently so. I f your work is not written i n a
clear fashion, s o that th e reader ma y quickly apprehen d wha t th e pape r
is about , wha t th e mai n result s are , an d ho w th e argument s proceed ,
then ther e i s a considerabl e likelihoo d tha t h e wil l se t th e pape r asid e
before readin g muc h of it. You r work will not hav e the impac t tha t yo u
had hope d o r intended .
I a m certainl y no t writin g thi s boo k t o advocat e tha t yo u se t asid e
each o f you r paper s fo r a year , i n th e fashio n o f Heilbronn , an d the n
rewrite it . Rather , I a m askin g yo u to conside r th e valu e o f learning t o
write. Heilbron n ha d hi s techniques fo r sharpenin g u p hi s prose . Eac h
of us mus t lear n hi s own .
I know many example s of mathematicians A an d S , o f roughly sim -
ilar talent , wit h th e propert y tha t A ha s enjoye d muc h greate r succes s
than 5 , an d considerabl y mor e recognitio n fo r hi s ideas , becaus e A
wrote hi s wor k i n a n appealin g an d readabl e fashio n an d B di d not .
The A's an d S' s tha t I am thinkin g abou t ar e no t a t th e Field s Meda l
level; Field s Medalist s ar e exceptiona l i n almos t ever y respect , an d
probably d o no t nee d m y advice . Instead , th e example s o f whic h I
speak ar e severa l notche s dow n fro m tha t augus t level , lik e most o f us .
Even if you accep t m y thesis—that i t i s worthwhile to lear n to writ e
mathematics well—yo u ma y fee l tha t fine writin g i s no t a n avocatio n
that yo u wis h t o pursue . Fai r enough : i f yo u ha d wante d t o becom e a
writer, the n probabl y tha t i s what yo u woul d hav e done . Bu t I submi t
that a reasonabl e alternativ e migh t b e t o spen d a n hou r o r tw o wit h
this book, an d perhaps another hour or two considering how its precepts
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