Number the Language of Science.pdf

8/19/2019 Number the Language of Science.pdf

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Page numbers followed by n indicate endnotes.


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TobiasDantzig

NUMBERThe Language of Science

Edited by

Joseph Mazur Foreword by

Barry Mazur


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The Masterpiece Science Edition

Pi Press New York

PI PRESS

An imprint of Pearson !ucation" #nc$

%%&' A(enue of the Americas" New York" New York %))*+,orewor!" Notes" Afterwor! an! ,urther -ea!ings . /))' by Pearson

!ucation" #nc$. %0*)" %0**" %0*0" an! %0'1 by the Macmi22an3ompany

This e!ition is a repub2ication of the 1th e!ition of Number " origina22y pub2ishe! by Scribner" an #mprint of Simon 4 Schuster #nc$

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Pearson !ucation 3ana!a" :t!$Pearson !ucati


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Appen!i= A$ n the -ecor!ing of Numbers /+%

Appen!i= B$ Topics in #ntegers /;;

Appen!i= 3$ n -oots an! -a!ica2s *)*

Appen!i= D$ n Princip2es an! Arguments */;

Afterwor! *1*

Notes *'%

,urther -ea!ings *;*

#n!e= *&'


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,orewor!

he book you ho2! in your han!s is a many7stran!e!me!itation on Number" an! is an o!e to the beauties ofmathematics$

TThis c2assic is about the e(o2ution of the Number concept$

Yes9 Number has ha!" an! wi22 continue to ha(e" anevolution $Cow !i! Number begin e can on2y specu2ate$

Di! Number make its initia2 entry into 2anguage as ana!Eecti(e Three cows" three !ays" three mi2es$ #magine thee=hi2aration you wou2! fee2 if you were the first human to be

struck with the start2ing thought that a unifying threa! bin!sFthree cowsGto Fthree !ays"G an! that it may be worthwhi2e to!ea2 with their common three7ness$ This" if it e(er occurre! to asing2e person at a sing2e time" wou2! ha(e been a monumenta22eap forwar!" for the !isembo!ie! concept of three7ness" the nounthree " embraces far more than cows or !ays$ #t wou2! a2so ha(eset the stage for the comparison to be ma!e between" say" one !ay

an! three !ays" thinking of the 2atter !uration as trip2e the former"ushering in yet another (iew ofthree " in its ro2e in the acti(ity of trip2ingHthree embo!ie!" if you wish" in the (erbto triple $

r perhaps Number emerge! from some other route9 a formof incantation" for e=amp2e" as in the chi2!renIs rhyme F ne" two" buck2e my shoe $G

Cowe(er it began" this story is sti22 going on" an! Number"humb2e Number" is showing itse2f e(er more centra2 to our un!erstan!ing of what is$ The ear2y Pythagoreans must be!ancing in their ca(es$


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#f # were someone who ha! a yen to 2earn about math" but

ne(er ha! the time to !o so" an! if # foun! myse2f maroone! onthat pro(erbia2 F!esert is2an!"G the one book # wou2! hope to ha(ea2ong is" to be honest" a goo! swimming manua2$ But the secon! book might (ery we22 be this one$ ,or Dantzig accomp2ishes theseessentia2 tasks of scientific e=position9 to assume his rea!ers ha(eno more than a genera2 e!ucate! backgroun!H to gi(e a c2ear an!(i(i! account of materia2 most essentia2 to the story being to2!H to

te22 an important storyH an!?the task most rare2y achie(e! of a22 ? to e=p2ain i!eas an! not mere2y a22u!e to them$

ne of the beautifu2 stran!s in the story of Number is themanner in which the concept change! as mathematicianse=pan!e! the repub2ic of numbers9 from the counting numbers

%" /" *"to the rea2m that inc2u!es negati(e numbers" an! zero

K*" K/" K%" )" L%" L/" L*" an! then to fractions" rea2 numbers" comp2e= numbers" an!" (ia a!ifferent mo!e of co2onization" to infinity an! the hierarchy of infinities$ Dantzig brings out the moti(ation for each of theseaugmentations9 There is in!ee! a unity that ties these separatesteps into a sing2e narrati(e$ #n the mi!st of his !iscussion of the

e=pansion of the number concept" Dantzig uotes :ouis#>$ hen aske! what the gui!ing princip2e was of hisinternationa2 po2icy" :ouis #> answere!"FAnne=ationO ne cana2ways fin! a c2e(er 2awyer to (in!icate the act$GBut Dantzighimse2f !oes not re2egate anything to 2ega2 counse2$ Ce offersintimate g2impses of mathematica2 birth pangs" whi2e constant2yfocusing on the (ita2 uestion that ho(ers o(er this story9 hat

!oes it mean for a mathematica2 obEect to e=ist Dantzig" in hiscomment about the emergence of comp2e= numbers muses thatF,or centuries the concept of comp2e= numbersQ figure! as a sort


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Foreword 9

of mystic bon! between reason an! imagination$G Ce uotes

:eibniz to con(ey this turmoi2 of the inte22ect9F TQhe Di(ine Spirit foun! a sub2ime out2et in that won!er of ana2ysis" that portent of the i!ea2 wor2!" that amphibian between being an! not7being" which we ca22 the imaginary root of negati(eunity$G R/%/

Dantzig a2so te22s us of his own ear2y moments of perp2e=ity9

F# reca22 my own emotions9 # ha! Eust been initiate! into themysteries of the comp2e= number$ # remember my bewi2!erment9here were magnitu!es patent2y impossib2e an! yet susceptib2e of manipu2ations which 2ea! to concrete resu2ts$ #t was a fee2ing of !issatisfaction" of rest2essness" a !esire to fi22 these i22usorycreatures" these empty symbo2s" with substance$ Then # was taught

to interpret these beings in a concrete geometrica2 way$ There camethen an imme!iate fee2ing of re2ief" as though # ha! so2(e! anenigma" as though a ghost which ha! been causing meapprehension turne! out to be no ghost at a22" but a fami2iar part of my en(ironment$G R/'1

The interp2ay between a2gebra an! geometry is one of thegran! themes of mathematics$ The magic of high schoo2 ana2yticgeometry that a22ows you to !escribe geometrica22y intriguingcur(es by simp2e a2gebraic formu2as an! tease out hi!!en properties of geometry by so2(ing simp2e e uations has f2owere! ?in mo!ern mathematics?into a powerfu2 interming2ing of a2gebraic an! geometric intuitions" each fortifying the other$ -enDescartes proc2aime!9 F# wou2! borrow the best of geometry an!

of a2gebra an! correct a22 the fau2ts of the one by the other$G Thecontemporary mathematician Sir Michae2 Atiyah" in comparingthe g2ories of geometric intuition with the e=traor!inary efficacyof a2gebraic metho!s" wrote recent2y9


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FA2gebra is the offer ma!e by the !e(i2 to the mathematician$ The

!e(i2 says9 # wi22 gi(e you this powerfu2 machine" it wi22 answer any uestion you 2ike$ A22 you nee! to !o is gi(e me your sou29gi(e up geometry an! you wi22 ha(e this mar(e2ous machine$RAtiyah" Sir Michae2$Special Article: Mathematics in the 20 th

Century $ Page ;$ Bu22etin of the :on!on Mathematica2 Society" *1R/))/ %K%'$ G

#t takes DantzigIs !e2icacy to te22 of the mi22ennia72ongcourtship between arithmetic an! geometry without smoothing outthe ,austian e!ges of this 2o(e story$

#n uc2i!Is lements o! "eometry " we encounter uc2i!Is!efinition of a 2ine9 FDefinition /$ A 2ine is brea!th2ess 2ength$G Nowa!ays" we ha(e other perspecti(es on that stap2e of p2anegeometry" the straight 2ine$ e ha(e the number 2ine" represente!as a horizonta2 straight 2ine e=ten!e! infinite2y in both !irectionson which a22 numbers?positi(e" negati(e" who2e" fractiona2" or irrationa2?ha(e their position$ A2so" to picture time (ariation" weca22 upon that cru!e mo!e2" the time2ine" again represente! as ahorizonta2 straight 2ine e=ten!e! infinite2y in both !irections" tostan! for the profoun!" e(er7baff2ing" e(er7mo(ing frame of pastUpresentUfutures that we think we 2i(e in$ The story of howthese !ifferent conceptions of strai#ht line negotiate with eachother is yet another stran! of DantzigIs ta2e$

Dantzig tru2y comes into his own in his !iscussion of there2ationship between time an! mathematics$ Ce contrasts 3antorIstheory" where infinite processes aboun!" a theory that he maintainsis Ffrank2y !ynamic"G with the theory of De!ekin!" which herefers to as Fstatic$G Nowhere in De!ekin!Is !efinition of rea2number" says Dantzig" !oes De!ekin! e(en Fuse the wor!in!initee=p2icit2y" or such wor!s astend " #row " beyond measure "conver#e " limit " less than any assi#nable $uantity " or other substitutes$G


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Foreword 11

At this point" rea!ing DantzigIs account" we seem to ha(e

come to a resting p2ace" for Dantzig writes9FSo it seems at first g2ance that here in De!ekin!Is formu2ation of rea2 numbersQ we ha(e fina22y achie(e! a comp2ete emancipationof the number concept from the yoke of time$G R%&/

To be sure" this Fcomp2ete emancipationG har!2y ho2!s up toDantzigIs secon! g2ance" an! the eterna2 issues regar!ing time an!its mathematica2 representation" regar!ing the continuum an! itsre2ationship to physica2 time" or to our 2i(e! time?prob2ems weha(e been ma!e aware of since Veno?remain constantcompanions to the account of the e(o2ution of number you wi22rea! in this book$

Dantzig asks9 To what e=tent !oes the wor2!" the scientific

wor2!" enter crucia22y as an inf2uence on the mathematica2 wor2!"an! (ice (ersa

FThe man of science wi22 actsas i! this wor2! were an abso2utewho2e contro22e! by 2aws in!epen!ent of his own thoughts or actH but whene(er he !isco(ers a 2aw of striking simp2icity or one of sweeping uni(ersa2ity or one which points to a perfect harmony inthe cosmos" he wi22 be wise to won!er what ro2e his min! has p2aye! in the !isco(ery" an! whether the beautifu2 image he sees inthe poo2 of eternity re(ea2s the nature of this eternity" or is but aref2ection of his own min!$G R/1/Dantzig writes9

FThe mathematician may be compare! to a !esigner of garments"who is utter2y ob2i(ious of the creatures whom his garments mayfit$ To be sure" his art originate! in the necessity for c2othing such

creatures" but this was 2ong agoH to this !ay a shape wi22occasiona22y appear which wi22 fit into the garment as if thegarment ha! been ma!e for it$ Then there is no en! of surprise an!of !e2ightOGR/1)


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This bears some resemb2ance in tone to the famous essay of

the physicist ugene igner" FThe 5nreasonab2e ffecti(eness of Mathematics in the Natura2 Sciences"G but Dantzig goes on" byoffering us his high2y persona2 notions of sub%ective reality an!ob%ective reality$ bEecti(e rea2ity" accor!ing to Dantzig" is animpressi(e2y 2arge receptac2e inc2u!ing a22 the !ata that humanityhas ac uire! Re$g$" through the app2ication of scientificinstruments $ Ce a!opts Poincar Is !efinition of obEecti(e rea2ity"

Fwhat is common to many thinking beings an! cou2! be commonto a22"G to set the stage for his ana2ysis of the re2ationship between Number an! obEecti(e truth$

Now" in at 2east one of #mmanue2 WantIs reconfigurations of those two mighty wor!s sub%ectan! ob%ect&a !ominating ro2e is p2aye! by WantIs !e2icate concept of the sensus communis $ This sensus communis is an inner Fgenera2 (oice"G somehowconstructe! within each of us" that gi(es us our e=pectations of how the rest of humanity wi22 Eu!ge things$

The obEecti(e rea2ity of Poincar an! Dantzig seems tore uire" simi2ar2y" a kin! of inner (oice" a facu2ty resi!ing in us"te22ing us something about the rest of humanity9 The Poincar 7Dantzig obEecti(e rea2ity is a fun!amenta22y subEecti(e consensusof what is common2y he2!" or whatcould be he2!" to be obEecti(e$This (iew a2rea!y a2erts us to an un!er2ying circu2arity 2urking behin! many !iscussions regar!ing obEecti(ity an! number" an!"in particu2ar behin! the sentiments of the essay of igner$ Dantzigtrea!s aroun! this 2ight2y$

My brother Joe an! # ga(e our father" Abe" a copy of Number:'he (an#ua#e o! Science as a gift when he was in his ear2y ;)s$Abe ha! no mathematica2 e!ucation beyon! high schoo2" butretaine! an ar!ent 2o(e for the a2gebra he 2earne! there$ nce"when we were uite young" Abe imparte! some of the mar(e2s of a2gebra to us9 F#I22 te22 you a secret"G he began" in a conspiratoria2(oice$ Ce procee!e! to te22 us how" by making use of the magic power of the cipher ) " we cou2! fin!that number which when you


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Foreword 1*

double it and add one to it you #et 11 $ # was uite a 2itera27min!e!

ki! an! rea22y thought of ) as our fami2yIs secret" unti2 # was!isabuse! of this attribution in some math c2ass a few years 2ater$ur gift of DantzigIs book to Abe was an astoun!ing hit$ Ce

worke! through it" b2ackening the margins with notes"computations" e=egesesH he rea! it o(er an! o(er again$ Ceengage! with numbers in the spirit of this bookH he teste! his own(ariants of the 6o2!bach 3onEecture an! ca22e! them his

"oldbach +ariations $ Ce was" in a wor!" enrapture!$But none of this is surprising" for DantzigIs book captures

both sou2 an! inte22ectH it is one of the few great popu2ar e=pository c2assics of mathematics tru2y accessib2e to e(eryone$

,-arry Ma.ur


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!itorIs Note to the MasterpieceScience !ition

he te=t of this e!ition of Number is base! on the fourthe!ition" which was pub2ishe! in %0'1$ A new

forewor!" afterwor!" en!notes section" an! annotate! bib2iography are inc2u!e! in this e!ition" an! the

origina2 i22ustrations ha(e been re!rawn$TThe fourth e!ition was !i(i!e! into two parts$ Part %"F (o2ution of the Number 3oncept"G comprise! the %/ chaptersthat make up the te=t of this e!ition$ Part /" FProb2ems 2! an! New"G was more technica2 an! !ea2t with specific concepts in!epth$ Both parts ha(e been retaine! in this e!ition" on2y Part / isnow set off from the te=t as appen!i=es" an! the FpartG 2abe2 has been !roppe! from both sections$

#n Part /" DantzigIs writing became 2ess !escripti(e an! moresymbo2ic" !ea2ing 2ess with i!eas an! more with metho!s" permitting him to present technica2 !etai2 in a more concise form$

Cere" there seeme! to be no nee! for en!notes or further commentaries$ ne might e=pect that a ha2f7century of a!(ancement in mathematics wou2! force some changes to asection ca22e! FProb2ems 2! an! New"G but the tit2e is mis2ea!ingHthe prob2ems of this section are not o2! or new" but are a co22ectionof c2assic i!eas chosen by Dantzig to show how mathematics is!one$

#n the pre(ious e!itions of Number " sections were numbere!within chapters$ Because this numbering scheme ser(e! nofunction other than to in!icate a break in thought from the


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pre(ious paragraphs" the section numbers were !e2ete! an!rep2ace! by a sing2e 2ine space$

Preface to the ,ourth !ition

uarter of the century ago" when this book wasfirst written" # ha! groun!s to regar! the work as a pioneering effort" inasmuch as theevolution o! thenumber concept ? though a subEect of 2i(e2y

!iscussion among professiona2 mathematicians" 2ogicians an! phi2osophers?ha! not yet been presente! to the genera2 pub2icas a cu2tura2 issue$ #n!ee!" it was by no means certain at the

time that there were enough 2ay rea!ers intereste! in suchissues to Eustify the pub2ication of the book$ The receptionaccor!e! to the work both here an! abroa!" an! the numerous books on the same genera2 theme which ha(e fo22owe! in itswake ha(e !ispe22e! these !oubts$ The e=istence of a sizab2e bo!y of rea!ers who are concerne! with the cu2tura2 aspects of mathematics an! of the sciences which 2ean on mathematics is

to!ay a matter of recor!$

A

#t is a stimu2ating e=perience for an author in the autumn of 2ife to 2earn that the sustaine! !eman! for his first 2iteraryeffort has warrante! a new e!ition" an! it was in this spirit that# approache! the re(ision of the book$ But as the work progresse!" # became increasing2y aware of the pro!igiouschanges that ha(e taken p2ace since the 2ast e!ition of the book

appeare!$ The a!(ances in techno2ogy" the sprea! of thestatistica2 metho!" the a!(ent of e2ectronics" the emergence of nuc2ear physics" an!" abo(e a22" the growing importance of automatic computors? ha(e swe22e! beyon! a22 e=pectationthe ranks of peop2e who 2i(e on the fringes of mathematica2


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acti(ityH an!" at the same time" raise! the genera2 2e(e2 of mathematica2 e!ucation$ Thus was #=(i N5MB -

confronte! not on2y with a (ast2y increase! au!ience" but witha far more sophisticate! an! e=acting au!ience than the one #ha! a!!resse! twenty o!! years ear2ier$ These soberingref2ections ha! a !ecisi(e inf2uence on the p2an of this newe!ition$ As to the e=tent # was ab2e to meet the cha22enge of these changing times?it is for the rea!er to Eu!ge$

=cept for a few passages which were brought up to !ate"the volution o! the Number Concept " Part ne of the presente!ition" is a (erbatim repro!uction of the origina2 te=t$ Bycontrast" Part Two? /roblems& ld and New ?is" for a22 intentsan! purposes" a new book$ ,urthermore" whi2e Part ne !ea2s2arge2y with concepts an! i!eas$ Sti22" Part Two shou2! not beconstrue! as a commentary on the origina2 te=t" but as anintegrate! story ofthe development o! method and ar#ument inthe !ield o! number $ ne cou2! infer from this that the four chapters of /roblems& ld and New are more technica2 incharacter than the origina2 twe2(e" an! such is in!ee! the case$

n the other han!" uite a few topics of genera2 interest wereinc2u!e! among the subEects treate!" an! a rea!er ski22e! in theart of FskippingG cou2! rea!i2y circum(ent the more technica2sections without straying off the main trai2$

Tobias Dantzig /aci!ic /alisadesCali!orniaSeptember 1& 19 *


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Preface to the ,irst !ition

his book !ea2s with i!eas" not with metho!s$A22irre2e(ant technica2ities ha(e been stu!ious2y

a(oi!e!" an! to un!erstan! the issues in(o2(e! noother mathematica2 e uipment is re uire! than thatoffere! in the a(erage highschoo2 curricu2um$TBut though this book !oes not presuppose on the part of therea!er a mathematica2 e!ucation" it presupposes something Eustas rare9 a capacity for absorbing an! appraising i!eas$

,urthermore" whi2e this book a(oi!s the technica2 aspects

of the subEect" it is not written for those who are aff2icte! withan incurab2e horror of the symbo2" nor for those who areinherent2y form7b2in!$ This is a book on mathematics9 it !ea2swith symbo2 an! form an! with the i!eas which are back of thesymbo2 or of the form$

The author ho2!s that our schoo2 curricu2a" by strippingmathematics of its cu2tura2 content an! 2ea(ing a bare ske2eton

of technica2ities" ha(e repe22e! many a fine min!$ #t is the aimof this book to restore this cu2tura2 content an! present thee(o2ution of number as the profoun!2y human story which it is$

This is not a book on the history of the subEect$ Yet thehistorica2 metho! has been free2y use! to bring out the rX2eintuition has p2aye! in the e(o2ution of mathematica2 concepts$An! so the story of number is here unfo2!e! as a historica2 pageant of i!eas" 2inke! with the men who create! these i!easan! with the epochs which pro!uce! the men$=(iii N5MB -


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3an the fun!amenta2 issues of the science of number be presente! without bringing in the who2e intricate apparatus of the science This book is the authorIs !ec2aration of faith that it

can be !one$ They who rea! sha22 Eu!geO Tobias Dantzigashin#ton& 34C4

May *& 19*0


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3 C A P T - %

,ingerprints'en cycles o! the moon the 5oman year comprised:'his number then was held in hi#h esteem&

-ecause& perhaps& on !in#ers we are wont to count&r that a woman in twice !ive months brin#s !orth&

r else that numbers wa6 till ten they reach And then !rom one be#in their rhythm anew4

, (i! & Fasti& 7774

an" e(en in the 2ower stages of !e(e2opment" possesses a facu2ty which" for want of a better name" # sha22 ca22 Number Sense $ This facu2ty

permits him to recognize that something haschange! in a sma22 co22ection when" without his !irect know2e!ge"an obEect has been remo(e! from or a!!e! to the co22ection$M Number sense shou2! not be confuse! with counting" which is probab2y of a much 2ater (intage" an! in(o2(es" as we sha22 see" arather intricate menta2 process$ 3ounting" so far as we know" is anattribute e=c2usi(e2y human" whereas some brute species seem to possess a ru!imentary number sense akin to our own$ At 2east"such is the opinion of competent obser(ers of anima2 beha(ior" an!the theory is supporte! by a weighty mass of e(i!ence$

Many bir!s" for instance" possess such a number sense$ #f anest contains four eggs one can safe2y be taken" but when two areremo(e! the bir! genera22y !eserts$ #n some unaccountab2e way

the bir! can !istinguish two from three$ But this facu2ty is by no%


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1 5 2 0 )

FRO M A M ANUAL P UBLIS HED IN

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Fin#erprints *

means confine! to bir!s$ #n fact the most striking instance we

know is that of the insect ca22e! the Fso2itary wasp$G The mother wasp 2ays her eggs in in!i(i!ua2 ce22s an! pro(i!es each eggwith a number of 2i(e caterpi22ars on which the young fee! whenhatche!$ Now" the number of (ictims is remarkab2y constant for a gi(en species of wasp9 some species pro(i!e '" others %/"others again as high as /1 caterpi22ars per ce22$ But mostremarkab2e is the case of the"enus umenus " a (ariety in which

the ma2e is much sma22er than the fema2e$ #n some mysteriousway the mother knows whether the egg wi22 pro!uce a ma2e or afema2e grub an! apportions the uantity of foo! accor!ing2yH she!oes not change the species or size of the prey" but if the egg isma2e she supp2ies it with fi(e (ictims" if fema2e with ten$

The regu2arity in the action of the wasp an! the fact that thisaction is connecte! with a fun!amenta2 function in the 2ife of theinsect make this 2ast case 2ess con(incing than the one whichfo22ows$ Cere the action of the bir! seems to bor!er on theconscious9

A s uire was !etermine! to shoot a crow which ma!e itsnest in the watch7tower of his estate$ -epeate!2y he ha! trie! tosurprise the bir!" but in (ain9 at the approach of the man the

crow wou2! 2ea(e its nest$ ,rom a !istant tree it wou2!watchfu22y wait unti2 the man ha! 2eft the tower an! then returnto its nest$ ne !ay the s uire hit upon a ruse9 two men entere!the tower" one remaine! within" the other came out an! went on$But the bir! was not !ecei(e!9 it kept away unti2 the man withincame out$ The e=periment was repeate! on the succee!ing !ayswith two" three" then four men" yet without success$ ,ina22y" fi(e

men were sent9 as before" a22 entere! the tower" an! oneremaine! whi2e the other four came out an! went away$ Cere the


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crow 2ost count$ 5nab2e to !istinguish between four an! fi(e it

prompt2y returne! to its nest$Two arguments may be raise! against such e(i!ence$ The first isthat the species possessing such a number sense are e=cee!ing2yfew" that no such facu2ty has been foun! among mamma2s" an!that e(en the monkeys seem to 2ack it$ The secon! argument isthat in a22 known cases the number sense of anima2s is so 2imite!in scope as to be ignore!$

Now the first point is we22 taken$ #t is in!ee! a remarkab2efact that the facu2ty of percei(ing number" in one form or another" seems to be confine! to some insects an! bir!s an! tomen$ bser(ation an! e=periments on !ogs" horses an! other !omestic anima2s ha(e fai2e! to re(ea2 any number sense$

As to the secon! argument" it is of 2itt2e (a2ue" because thescope of the human number sense is a2so uite 2imite!$ #n e(ery practica2 case where ci(i2ize! man is ca22e! upon to !iscernnumber" he is conscious2y or unconscious2y ai!ing his !irectnumber sense with such artifices as symmetric pattern rea!ing"menta2 grouping or counting$Countin# especia22y has becomesuch an integra2 part of our menta2 e uipment that psycho2ogica2tests on our number perception are fraught with great

!ifficu2ties$ Ne(erthe2ess some progress has been ma!eHcarefu22y con!ucte! e=periments 2ea! to the ine(itab2econc2usion that the !irectvisual number sense of the a(erageci(i2ize! man rare2y e=ten!s beyon! four" an! that thetactilesense is sti22 more 2imite! in scope$

Anthropo2ogica2 stu!ies on primiti(e peop2es corroboratethese resu2ts to a remarkab2e !egree$ They re(ea2 that those

sa(ages who have not reached the sta#e o! !in#er countin# area2most comp2ete2y !epri(e! of a22 perception of number$ Such isthe case among numerous tribes in Austra2ia" the South Sea


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#s2an!s" South America" an! Africa$ 3urr" who has ma!e an

e=tensi(e stu!y of primiti(e Austra2ia" ho2!s that but few of thenati(es are ab2e to !iscern four" an! that no Austra2ian in his wi2!state can percei(e se(en$ The Bushmen of South Africa ha(e nonumber wor!s beyon!one" two an! many " an! these wor!s areso inarticu2ate that it may be !oubte! whether the nati(es attacha c2ear meaning to them$

e ha(e no reasons to be2ie(e an! many reasons to !oubt

that our own remote ancestors were better e uippe!" since practica22y a22 uropean 2anguages bear traces of such ear2y2imitations$ The ng2ishthrice " Eust 2ike the :atinter " has the!oub2e meaning9 three times" an! many$ There is a p2ausib2econnection between the :atintres " three" an!trans " beyon!H thesame can be sai! regar!ing the ,renchtr8s " (ery" an!trois " three$

The genesis of number is hi!!en behin! the impenetrab2e(ei2 of count2ess prehistoric ages$ Cas the concept been born of e=perience" or has e=perience mere2y ser(e! to ren!er e=p2icitwhat was a2rea!y 2atent in the primiti(e min!9 Cere is afascinating subEect for metaphysica2 specu2ation" but for this(ery reason beyon! the scope of this stu!y$

#f we are to Eu!ge of the !e(e2opment of our own remote

ancestors by the menta2 state of contemporary tribes we cannotescape the conc2usion that the beginnings were e=treme2ymo!est$ A ru!imentary number sense" not greater in scope thanthat possesse! by bir!s" was the nuc2eus from which the number concept grew$ An! there is 2itt2e !oubt that" 2eft to this !irectnumber perception" man wou2! ha(e a!(ance! no further in theart of reckoning than the bir!s !i!$ But through a series of

remarkab2e circumstances man has 2earne! to ai! hise=cee!ing2y 2imite! perception of number by an artifice whichwas !estine! to e=ert a tremen!ous inf2uence on his future 2ife$


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This artifice is counting" an! it is tocountin# that we owe that

e=traor!inary progress which we ha(e ma!e in e=pressing our uni(erse in terms of number$There are primiti(e 2anguages which ha(e wor!s for e(ery co2or of the rainbow but ha(e no wor! for co2orH there are otherswhich ha(e a22 number wor!s but no wor! for number$ The sameis true of other conceptions$ The ng2ish 2anguage is (ery rich innati(e e=pressions for particu2ar types of co22ections9 !loc "herd "

set " lot an! bunch app2y to specia2 casesH yet the wor!scollection an! a##re#ate are of foreign e=traction$

The concrete prece!e! the abstract$ F#t must ha(e re uire!many ages to !isco(er"G says Bertran! -usse22" Fthat a brace of pheasants an! a coup2e of !ays were both instances of thenumber two$G To this !ay we ha(e uite a few ways of e=pressing the i!eatwo9 pair" coup2e" set" team" twin" brace" etc$"etc$

A striking e=amp2e of the e=treme concreteness of the ear2ynumber concept is the Thimshian 2anguage of a British3o2umbia tribe$ There we fin! se(en !istinct sets of number wor!s9 one for f2at obEects an! anima2sH one for roun! obEectsan! timeH one for counting menH one for 2ong obEects an! treesH

one for canoesH one for measuresH one for counting when no!efinite obEect is referre! to$ The 2ast is probab2y a 2ater !e(e2opmentH the others must be re2ics of the ear2iest !ays whenthe tribesmen ha! not yet 2earne! to count$

#t is counting that conso2i!ate! the concrete an! thereforeheterogeneous notion of p2ura2ity" so characteristic of primiti(eman" into thehomo#eneous abstract number concept " whichma!e mathematics possib2e$


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Yet" strange though it may seem" it is possib2e to arri(e at a

2ogica2" c2ear7cut number concept without bringing in theartifices of counting$e enter a ha22$ Before us are two co22ections9 the seats of

the au!itorium" an! the au!ience$ithout countin# we canascertain whether the two co22ections are e ua2 an!" if not e ua2"which is the greater$ ,or if e(ery seat is taken an! no man isstan!ing" we now without countin# that the two co22ections are

e ua2$ #f e(ery seat is taken an! some in the au!ience arestan!ing"we now without countin# that there are more peop2ethan seats$

e !eri(e this know2e!ge through a process which!ominates a22 mathematics an! which has recei(e! the name of one;to;one correspondence $ #t consists in assigning to e(eryobEect of one co22ection an obEect of the other" the process beingcontinue! unti2 one of the co22ections" or both" are e=hauste!$

The number techni ue of many primiti(e peop2es isconfine! to Eust such such a matching or ta22ying$ They keep therecor! of their her!s an! armies by means of notches cut in atree or pebb2es gathere! in a pi2e$ That our own ancestors werea!ept in such metho!s is e(i!ence! by the etymo2ogy of the

wor!s tally an! calculate " of which the first comes from the:atin talea " cutting" an! the secon! from the :atincalculus " pebb2e$

#t wou2! seem at first that the process of correspon!encegi(es on2y a means for comparing two co22ections" but isincapab2e of creating number in the abso2ute sense of the wor!$Yet the transition from re2ati(e number to abso2ute is not!ifficu2t$ #t is necessary on2y to createmodel collections " eachtypifying a possib2e co22ection$ stimating any gi(en co22ectionis then re!uce! to the se2ection among the a(ai2ab2e mo!e2s of


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one which can be matche! with the gi(en co22ection member by

member$Primiti(e man fin!s such mo!e2s in his imme!iateen(ironment9 the wings of a bir! may symbo2ize the number two" c2o(er72ea(es three" the 2egs of an anima2 four" the fingerson his own han! fi(e$ (i!ence of this origin of number wor!scan be foun! in many a primiti(e 2anguage$ f course" once thenumber word has been create! an! a!opte!" it becomes as goo!

a mo!e2 as the obEect it origina22y represente!$ The necessity of !iscriminating between the name of the borrowe! obEect an! thenumber symbo2 itse2f wou2! natura22y ten! to bring about achange in soun!" unti2 in the course of time the (ery connection between the two is 2ost to memory$ As man 2earns to re2y morean! more on his 2anguage" the soun!s superse!e the images for which they stoo!" an! the origina22y concrete mo!e2s take theabstract form of number wor!s$ Memory an! habit 2en!concreteness to these abstract forms" an! so mere wor!s becomemeasures of p2ura2ity$

The concept # Eust !escribe! is ca22e!cardinal number$ Thecar!ina2 number rests on the princip2e of correspon!ence9 itimp2iesno countin#4 To create a counting process it is notenough to ha(e a mot2ey array of mo!e2s" comprehensi(e thoughthis 2atter may be$ e must !e(ise a number system 9 our set of mo!e2s must be arrange! in an or!ere! se uence" a se uencewhich progresses in the sense of growing magnitu!e" thenatural

se$uence 9 one" two" three $ nce this system is create!"countin# a collection means assigning to e(ery member a termin the natura2 se uence inordered succession unti2 the co22ectionis e=hauste!$ The term of the natura2 se uence assigne! to the


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last member of the co22ection is ca22e! theordinal number of the

co22ection$The or!ina2 system may take the concrete form of a rosary" but this" of course" is not essentia2$ Theordinal system ac uirese=istence when the first few number wor!s ha(e been committe!to memory in theirordered succession& an! a phonetic schemehas been !e(ise! to pass from any 2arger number to its

successor4

e ha(e 2earne! to pass with such faci2ity from car!ina2 toor!ina2 number that the two aspects appear to us as one$ To!etermine the p2ura2ity of a co22ection" i$e$" its car!ina2 number"we !o not bother any more to fin! a mo!e2 co22ection with whichwe can match it"?wecount it$ An! to the fact that we ha(e2earne! to i!entify the two aspects of number is !ue our progressin mathematics$ ,or whereas in practice we are rea22y intereste!in the car!ina2 number" this 2atter is incapab2e of creating anarithmetic$ The operations of arithmetic are base! on the tacitassumption thatwe can always pass !rom any number to its

successor& an! this is the essence of the or!ina2 concept$An! so matching by itse2f is incapab2e of creating an art of

reckoning$ ithout our abi2ity to arrange things in or!ere!

succession 2itt2e progress cou2! ha(e been ma!e$3orrespon!ence an! succession" the two princip2es which permeate a22 mathematics?nay" a22 rea2ms of e=act thought?arewo(en into the (ery fabric of our number system$

#t is natura2 to in uire at this point whether this subt2e !istinction between car!ina2 an! or!ina2 number ha! any part in the ear2yhistory of the number concept$ ne is tempte! to surmise thatthe car!ina2 number" base! on matching on2y" prece!e! the


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or!ina2 number" which re uires both matching an! or!ering$ Yet

the most carefu2 in(estigations into primiti(e cu2ture an! phi2o2ogy fai2 to re(ea2 any such prece!ence$ here(er anynumber techni ue e=ists at a22" both aspects of number arefoun!$

But" a2so" where(er a counting techni ue" worthy of thename" e=ists at a22" !in#er countin# has been foun! to either prece!e it or accompany it$ An! in his fingers man possesses a

!e(ice which permits him to pass imperceptib2y from car!ina2 toor!ina2 number$ Shou2! he want to in!icate that a certainco22ection contains four obEects he wi22 raise or turn !own four fingers simultaneously< shou2! he want to count the sameco22ection he wi22 raise or turn !own these fingersin succession4#n the first case he is using his fingers as a car!ina2 mo!e2" in thesecon! as an or!ina2 system$ 5nmistakab2e traces of this originof counting are foun! in practica22y e(ery primiti(e 2anguage$ #nmost of these tongues the number Ffi(eG is e=presse! by Fhan!"Gthe number FtenG by Ftwo han!s"G or sometimes by Fman$G,urthermore" in many primiti(e 2anguages the number7wor!s upto four are i!entica2 with the names gi(en to the four fingers$

The more ci(i2ize! 2anguages un!erwent a process of

attrition which ob2iterate! the origina2 meaning of the wor!s$ Yethere too FfingerprintsG are not 2acking$ 3ompare the Sanskrit pantcha& fi(e" with the re2ate! Persian pentcha& han!H the-ussian Fpiat"G fi(e" with Fpiast"G the outstretche! han!$

#t is to hisarticulate ten !in#ers that man owes his success inca2cu2ation$ #t is these fingers which ha(e taught him to countan! thus e=ten! the scope of number in!efinite2y$ ithout this

!e(ice the number techni ue of man cou2! not ha(e a!(ance!far beyon! the ru!imentary number sense$ An! it is reasonab2eto conEecture that without our fingers the !e(e2opment of


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number" an! conse uent2y that of the e=act sciences" to which

we owe our materia2 an! inte22ectua2 progress" wou2! ha(e beenhope2ess2y !warfe!$

An! yet" e=cept that our chi2!ren sti22 2earn to count on their fingers an! that we ourse2(es sometimes resort to it as a gestureof emphasis" finger counting is a 2ost art among mo!ern ci(i2ize! peop2e$ The a!(ent of writing" simp2ifie! numeration" an!uni(ersa2 schoo2ing ha(e ren!ere! the art obso2ete an!superf2uous$ 5n!er the circumstances it is on2y natura2 for us toun!erestimate the rX2e that finger counting has p2aye! in thehistory of reckoning$ n2y a few hun!re! years ago finger counting was such a wi!esprea! custom in estern urope thatno manua2 of arithmetic was comp2ete un2ess it ga(e fu22

instructions in the metho!$ RSee page /$The art of using his fingers in counting an! in performingthe simp2e operations of arithmetic" was then one of theaccomp2ishments of an e!ucate! man$ The greatest ingenuitywas !isp2aye! in !e(ising ru2es for a!!ing an! mu2tip2yingnumbers on oneIs fingers$ Thus" to this !ay" the peasant of centra2 ,rance RAu(ergne uses a curious metho! for mu2tip2yingnumbers abo(e '$ #f he wishes to mu2tip2y 0× &" he ben!s !own1 fingers on his 2eft han! R1 being the e=cess of 0 o(er ' " an! *fingers on his right han! R& K ' * $ Then the number of the bent7!own fingers gi(es him the tens of the resu2t R1 L * ; "whi2e the pro!uct of the unbent fingers gi(es him the units R%×/ / $

Artifices of the same nature ha(e been obser(e! in wi!e2yseparate! p2aces" such as Bessarabia" Serbia an! Syria$ Their striking simi2arity an! the fact that these countries were a22 at


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one time parts of the great -oman mpire" 2ea! one to suspect

the -oman origin of these !e(ices$ Yet" it may be maintaine!with e ua2 p2ausibi2ity that these metho!s e(o2(e!in!epen!ent2y" simi2ar con!itions bringing about simi2ar resu2ts$

(en to!ay the greater portion of humanity is counting onfingers9 to primiti(e man" we must remember" this is the on2ymeans of performing the simp2e ca2cu2ations of his !ai2y 2ife$

Cow o2! is our number 2anguage #t is impossib2e to in!icate thee=act perio! in which number wor!s originate!" yet there isunmistakab2e e(i!ence that it prece!e! written history by manythousan!s of years$ ne fact we ha(e mentione! a2rea!y9 a22traces of the origina2 meaning of the number wor!s in uropean2anguages" with the possib2e e=ception of !ive&are 2ost$ An! this

is the more remarkab2e" since" as a ru2e" number wor!s possessan e=traor!inary stabi2ity$ hi2e time has wrought ra!ica2changes in a22 other aspects we fin! that the number (ocabu2aryhas been practica22y unaffecte!$ #n fact this stabi2ity is uti2ize! by phi2o2ogists to trace kinships between apparent2y remote2anguage groups$ The rea!er is in(ite! to e=amine the tab2e atthe en! of the chapter where the number wor!s of the stan!ar!#n!o7 uropean 2anguages are compare!$

hy is it then that in spite of this stabi2ity no trace of theorigina2 meaning is foun! A p2ausib2e conEecture is that whi2enumber wor!s ha(e remaine! unchange! since the !ays whenthey originate!" the names of the concrete obEects from whichthe number wor!s were borrowe! ha(e un!ergone a comp2etemetamorphosis$


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As to the structure of the number 2anguage" phi2o2ogica2

researches !isc2ose an a2most uni(ersa2 uniformity$ (erywherethe ten fingers of man ha(e 2eft their permanent imprint$#n!ee!" there is no mistaking the inf2uence of our ten fingers

on the Fse2ectionG of the base of our number system$ #n a22#n!o uropean 2anguages" as we22 as Semitic" Mongo2ian" an!most primiti(e 2anguages" the base of numeration is ten" i$e$"there are in!epen!ent number wor!s up to ten" beyon! which

some compoun!ing princip2e is use! unti2 %)) is reache!$ A22these 2anguages ha(e in!epen!ent wor!s for %)) an! %)))" an!some 2anguages for e(en higher !ecima2 units$ There areapparent e=ceptions" such as the ng2isheleven an! twelve " or the 6ermanel! an! .w=l!& but these ha(e been trace! toein;li! an! .wo;li!< li! being o2! 6erman forten4

#t is true that in a!!ition to the !ecima2 system" two other bases are reasonab2y wi!esprea!" but their character confirms toa remarkab2e !egree theanthropomorphic nature of our countingscheme$ These two other systems are the uinary" base '" an! the(igesima2" base /)$

#n the$uinary system there are in!epen!ent number wor!sup to !ive&an! the compoun!ing begins thereafter$ RSee tab2e at

the en! of chapter$ #t e(i!ent2y originate! among peop2e whoha! the habit of counting on one han!$ But why shou2! manconfine himse2f to one han! A p2ausib2e e=p2anation is that primiti(e man rare2y goes about unarme!$ #f he wants to count"he tucks his weapon un!er his arm" the 2eft arm as a ru2e" an!counts on his 2eft han!" using his right han! as check7off$ Thismay e=p2ain why the 2eft han! is a2most uni(ersa22y use! by

right7han!e! peop2e for counting$Many 2anguages sti22 bear the traces of a uinary system"

an! it is reasonab2e to be2ie(e that some !ecima2 systems passe!


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through the uinary stage$ Some phi2o2ogists c2aim that e(en the

#n!o7 uropean number 2anguages are of a uinary origin$ They point to the 6reek wor! pempa.ein& to count by fi(es" an! a2soto the un uestionab2y uinary character of the -oman numera2s$Cowe(er" there is no other e(i!ence of this sort" an! it is muchmore probab2e that our group of 2anguages passe! through a pre2iminaryvi#esimal sta#e4

This 2atter probab2y originate! among the primiti(e tribes

who counte! on their toes as we22 as on their fingers$ A moststriking e=amp2e of such a system is that use! by the Maya#n!ians of 3entra2 America$ f the same genera2 character wasthe system of the ancient Aztecs$ The !ay of the Aztecs was!i(i!e! into /) hoursH a !i(ision of the army containe! &)))so2!iers R&))) /)× /) × /) $

hi2e pure (igesima2 systems are rare" there are numerous2anguages where the !ecima2 an! the (igesima2 systems ha(emerge!$ e ha(e the ng2ish score& two;score&an! three;score<the French vin#t R/) an!$uatre;vin#t R1× /) $ The o2! ,renchuse! this form sti22 more fre uent2yH a hospita2 in Paris origina22y bui2t for *)) b2in! (eterans bears the uaint name of>uin.e;+in#t R,ifteen7score H the namen.e;+in#t R 2e(enscore was

gi(en to a corps of po2ice7sergeants comprising //) men$

There e=ists among the most primiti(e tribes of Austra2ia an!Africa a system of numeration which has neither '" %)" nor /)for base$ #t is abinary system" i$e$" of base two$ These sa(agesha(e not yet reache! finger counting$ They ha(e in!epen!entnumbers for one an! two" an! composite numbers up to si=$Beyon! si= e(erything is !enote! by Fheap$G


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3urr" whom we ha(e a2rea!y uote! in connection with the

Austra2ian tribes" c2aims that most of these count by pairs$ Sostrong" in!ee!" is this habit of the nati(e that he wi22 rare2ynotice that two pins ha(e been remo(e! from a row of se(enH hewi22" howe(er" become imme!iate2y aware if one pin is missing$Cis sense of parity is stronger than his number sense$

3urious2y enough" this most primiti(e of bases ha! aneminent a!(ocate in re2ati(e2y recent times in no 2ess a person

than :eibnitz$ A binary numeration re uires but two symbo2s" )an! %" by means of which a22 other numbers are e=presse!" asshown in the fo22owing tab2e9Decima2 $ % / * 1 ' + ; &

Binary % %) %% %)) %)% %%) %%% %)))

Decima2 $ 0 %) %% %/ %* %1 %' %+

Binary %))% %)%) %)%% %%)) %%)% %%%) %%%% %))))

The a!(antages of thebase two are economy of symbo2s an!tremen!ous simp2icity in operations$ #t must be remembere! thate(ery system re uires that tab2es of a!!ition an! mu2tip2ication be committe! to memory$ ,or the binary system these re!uce to

% L % %) an! %× % %H whereas for the !ecima2" each tab2ehas %)) entries$ Yet this a!(antage is more than offset by 2ack of compactness9 thus the !ecima2 number 1)0+ /%/ wou2! bee=presse! in the binary system by %")))")))")))")))$

#t is the mystic e2egance of the binary system that ma!e:eibnitz e=c2aim9mnibus e6 nihil ducendis su!!icit unum4 R nesuffices to !eri(e a22 out of nothing$ Says :ap2ace9

F:eibnitz saw in his binary arithmetic the image of 3reation Ce imagine! that 5nity represente! 6o!" an! Vero the (oi!H that


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the Supreme Being !rew a22 beings from the (oi!" Eust as unityan! zero e=press a22 numbers in his system of numeration$ Thisconception was so p2easing to :eibnitz that he communicate! itto the Jesuit" 6rima2!i" presi!ent of the 3hinese tribuna2 for mathematics" in the hope that this emb2em of creation wou2!con(ert the mperor of 3hina" who was (ery fon! of thesciences$ # mention this mere2y to show how the preEu!ices of chi2!hoo! may c2ou! the (ision e(en of the greatest menOG

#t is interesting to specu2ate what turn the history of cu2turewou2! ha(e taken if instea! of f2e=ib2e fingers man ha! ha! Eusttwo Finarticu2ateG stumps$ #f any system of numeration cou2! ata22 ha(e !e(e2ope! un!er such circumstances" it wou2! ha(e probab2y been of the binary type$

That mankin! a!opte! the !ecima2 system is a physiolo#ical

accident4 Those who see the han! of Pro(i!ence in e(erythingwi22 ha(e to a!mit that Pro(i!ence is a poor mathematician$ ,or outsi!e its physio2ogica2 merit the !ecima2 base has 2itt2e tocommen! itse2f$ A2most any other base" with the possib2ee=ception ofnine&wou2! ha(e !one as we22 an! probab2y better$

#n!ee!" if the choice of a base were 2eft to a group of e=perts" we shou2! probab2y witness a conf2ict between the practica2 man" who wou2! insist on a base with the greatestnumber of !i(isors" such astwelve&an! the mathematician" whowou2! want a prime number" such as seven or eleven&for a base$As a matter of fact" 2ate in the eighteenth century the greatnatura2ist Buffon propose! that the !uo!ecima2 system Rbase %/ be uni(ersa22y a!opte!$ Ce pointe! to the fact that %/ has 1

!i(isors" whi2e %) has on2y two" an! maintaine! that throughoutthe ages this ina!e uacy of our !ecima2 system ha! been so


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keen2y fe2t that" in spite of ten being the uni(ersa2 base" most

measures ha! %/ secon!ary units$n the other han! the great mathematician :agrangec2aime! that a prime base is far more a!(antageous$ Ce pointe!to the fact that with a prime base e(ery systematic fractionwou2! be irre!ucib2e an! wou2! therefore represent the number in a uni ue way$ #n our present numeration" for instance" the!ecima2 fraction $*+ stan!s rea22y for many fractions9 *+ Z%))" %&

Z')" an! 0Z/' $ Such an ambiguity wou2! be consi!ere!2essene! if a prime base" such as e2e(en" were a!opte!$

But whether the en2ightene! group to whom we wou2!entrust the se2ection of the base !eci!e! on a prime or acomposite base" we may rest assure! that the numberten wou2!not e(en be consi!ere!" for it is neither prime nor has it asufficient number of !i(isors$

#n our own age" when ca2cu2ating !e(ices ha(e 2arge2ysupp2ante! menta2 arithmetic" nobo!y wou2! take either proposa2serious2y$ The a!(antages gaine! are so s2ight" an! the tra!itionof counting by tens so firm" that the cha22enge seems ri!icu2ous$

,rom the stan!point of the history of cu2ture a change of base" e(en if practicab2e" wou2! be high2y un!esirab2e$ As 2ong

as man counts by tens" his ten fingers wi22 remin! him of thehuman origin of this most important phase of his menta2 2ife$ Somay the !ecima2 system stan! as a 2i(ing monument to the proposition9

Man is the measure o! all thin#s4


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Number or!s of Some #n!o7 uropean :angauges Showing The=traor!inary Stabi2ity of Number or!s

Sanskrit Ancient6reek :atin 6erman ng2ish ,rench -ussian

%/*1'+;

&0

%)%))

%)))

eka!(atricatur pancasassaptaastana(a!acacatasehastr e

en!uotritetra pentehe=hepta

octoennea!ecaecaton=i2ia

unus!uotres

uatuoruin ue

se=septemoctono(em!ecemcentummi22e

einszwei!rei(ierf[nfsechssiebenacht

neunzehnhun!erttausen!

onetwothreefour fi(esi= se(eneightnine tenhun!re!thousan!

un!eu=trois

uatrecinsi=septhuitneuf!i=centmi22e

o!yn!(atrichetyre piatshestsem(osem!e(iat!esiatstotysiaca

A Typica2 \uinary System9 TheAP# :anguage of the New

Cebri!es

or! Meaning

%/*1'

tai2uato2u(ari2una

han!

+;&0

%)

otaio2uaoto2uo(air2ua 2una

other one F two F three F four

two han!sA Typica2 >igesima2 System9 The

Maya :anguage of 3entra2America


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%/)/) /

/) *

/) 1

/) '

/) +

hunka2 bak picca2abkinche2a2ce

%/)1))&)))%+)")))*"/))")))+1")))")))

A Typica2 Binary System9 A estern Tribe of Torres Straits

% urapun/ okosa

* okosa7urapun1 okosa7okosa

' okosa7okosa7urapun+ okosa7okosa7okosa

3 C A P T - / The

mpty 3o2umn

?7t is 7ndia that #ave us the in#enious method o!e6pressin# all numbers by means o! ten symbols& each

symbol receivin# a value o! position as well as anabsolute value< a pro!ound and important idea whichappears so simple to us now that we i#nore its truemerit4 -ut its very simplicity and the #reat ease which it has lent to all computations put our arithmetic in the

!irst ran o! use!ul inventions< and we shall appreciatethe #randeur o! this achievement the more when weremember that it escaped the #enius o! Archimedes and

Apollonius& two o! the #reatest men produced byanti$uity4@

?:ap2ace

s # am writing these 2ines there rings in my ears the o2!refrain9A F-ea!ing" I-iting" I-ithmetic"


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Taught to the tune of a hickory7stickOG

#n this chapter # propose to te22 the story of one of three -Is"the one" which" though o2!est" came har!est to mankin!$

#t is not a story of bri22iant achie(ement"heroic !ee!s"or nob2e sacrifice$ #t is a story of b2in! stumb2ing an! chance!isco(ery" of groping in the !ark an! refusing to a!mit the 2ight$#t is a story rep2ete with obscurantism an! preEu!ice" of soun!

Eu!gment often ec2ipse! by 2oya2ty to tra!ition" an! of reason2ong he2! subser(ient to custom$ #n short" it is a human story$%0


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A S CH EMAT IC DRAW ING O F A

Fin#erprints 21


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ritten numeration is probab2y as o2! as pri(ate property$ There

is 2itt2e !oubt that it originate! in manIs !esire to keep a recor!of his f2ocks an! other goo!s$ Notches on a stick or tree" scratch7es on stones an! rocks" marks in c2ay?these are the ear2iestforms of this en!ea(or to recor! numbers by written symbo2s$Archeo2ogica2 researches trace such recor!s to timesimmemoria2" as they are foun! in the ca(es of prehistoric man in

urope" Africa an! Asia$ Numeration is at 2east as o2! as written

2anguage" an! there is e(i!ence that it prece!e! it$ Perhaps"e(en" the recor!ing of numbers ha! suggeste! the recor!ing of soun!s$

The o2!est recor!s in!icating the systematic use of writtennumera2s are those of the ancient Sumerians an! gyptians$They are a22 trace! back to about the same epoch" aroun! *'))B$3$ hen we e=amine them we are struck with the greatsimi2arity in the princip2es use!$ There is" of course" the possibi2ity that there was communication between these peop2esin spite of the !istances that separate! them$ Cowe(er" it is more2ike2y that they !e(e2ope! their numerations a2ong the 2ines of 2east resistance" i$e$" that their numerations were but anoutgrowth of the natura2 process of ta22ying$ RSee figure page

//$ #n!ee!" whether it be the cuneiform numera2s of the ancientBaby2onians" the hierog2yphics of the gyptian papyri" or the

ueer figures of the ear2y 3hinese recor!s" we fin! e(erywhere a!istinct2ycardinal princip2e$ ach numera2 up to nine is mere2y aco22ection of strokes$ The same princip2e is use! beyon! nine"units of a higher c2ass" such as tens" hun!re!s" etc$" being

represente! by specia2 symbo2s$


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ANCIENT NUMERAT IO NS

S CHEMAT IC DRAW ING O F ENG LIS H T ALLY -S T ICK

Fin#erprints 2*


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The ng2ish ta22y7stick of obscure but probab2y (ery ancient

origin" a2so bears this un uestionab2y car!ina2 character$ Aschematic picture of the ta22y is shown in the accompanyingfigure$ The sma22 notches each represent a poun! ster2ing" the2arger ones %) poun!s" %)) poun!s" etc$

#t is curious that the ng2ish ta22y persiste! for manycenturies after the intro!uction of mo!ern numeration ma!e itsuse ri!icu2ous2y obso2ete$ #n fact it was responsib2e for an

important episo!e in the history of Par2iament$ 3har2es Dickens!escribe! this episo!e with inimitab2e sarcasm in an a!!ress onA!ministrati(e -eform" which he !e2i(ere! a few years after theinci!ent occurre!$

FAges ago a sa(age mo!e of keeping accounts on notche! stickswas intro!uce! into the 3ourt of =che uer an! the accounts

were kept much as -obinson 3rusoe kept his ca2en!ar on the!esert is2an!$ A mu2titu!e of accountants" bookkeepers" an!actuaries were born an! !ie! $Sti22 officia2 routine inc2ine! tothose notche! sticks as if they were pi22ars of the 3onstitution"an! sti22 the =che uer accounts continue! to be kept on certainsp2ints of e2m7woo! ca22e!tallies4 #n the reign of 6eorge ### anin uiry was ma!e by some re(o2utionary spirit whether" pens" ink an! paper" s2ates an! penci2s being in e=istence" this obstinatea!herence to an obso2ete custom ought to be continue!" an!whether a change ought not be effecte!$ A22 the re! tape in thecountry grew re!!er at the bare mention of this bo2! an! origina2conception" an! it took unti2 %&/+ to get these sticks abo2ishe!$ #n%&*1 it was foun! that there was a consi!erab2e accumu2ation of themH an! the uestion then arose" what was to be !one with suchworn7out"worm7eaten"rotten o2! bits of woo! The sticks were

house! in estminster" an! it wou2! natura22y occur to anyinte22igent person that nothing cou2! be easier than to a22ow themto be carrie! away for firewoo! by the miserab2e peop2e who2i(e! in that neighborhoo!$ Cowe(er" they ne(er ha! been usefu2"


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an! officia2 routine re uire! that they shou2! ne(er be" an! so theor!er went out that they were to be pri(ate2y an! confi!entia22y burne!$ #t came to pass that they were burne! in a sto(e in theCouse of :or!s$The sto(e" o(er7gorge! with these preposteroussticks" set fire to the pane22ingH the pane22ing set fire to the Couseof 3ommonsH the two houses were re!uce! to ashesH architectswere ca22e! in to bui2! othersH an! we are now in the secon!mi22ion of the cost thereof$G

As oppose! to this pure2y car!ina2 character of the ear2iestrecor!s there is the or!ina2 numeration" in which the numbersare represente! by the 2etters of an a2phabet in their spokensuccession$

The ear2iest e(i!ence of this princip2e is that of thePhoenician numeration$ #t probab2y arose from the urge for

compactness brought about by the comp2e=ities of a growingcommerce$ The Phoenician origin of both the Cebrew an! the6reek numeration is un uestionab2e9 the Phoenician system wasa!opte! bo!i2y" together with the a2phabet" an! e(en the soun!sof the 2etters were retaine!$

n the other han!" the -oman numeration" which hassur(i(e! to this !ay" shows a marke! return to the ear2ier car!ina2 metho!s$ Yet 6reek inf2uence is shown in the 2itera2symbo2s a!opte! for certain units" such as for ten" 3 for hun!re!" M for thousan!$ But the substitution of 2etters for themore pictures ue symbo2s of the 3ha2!eans or the gyptians!oes not constitute a !eparture from princip2e$

The e(o2ution of the numerations of anti uity foun! its fina2e=pression in the or!ina2 system of the 6reeks an! the car!ina2system of -ome$ hich of the two was superior The uestion


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wou2! ha(e significance if the on2y obEect of a numeration were

a compact recor!ing of uantity$ But this is not the main issue$ Afar more important uestion is9 how we22 is the system a!apte!to arithmetica2 operations" an! what ease !oes it 2en! toca2cu2ations

#n this respect there is har!2y any choice between the twometho!s9 neither was capab2e of creating an arithmetic whichcou2! be use! by a man of a(erage inte22igence$ This is why"

from the beginning of history unti2 the a!(ent of our mo!ern positional numeration" so 2itt2e progress was ma!e in the art of reckoning$

Not that there were no attempts to !e(ise ru2es for operatingon these numera2s$ Cow !ifficu2t these ru2es were can be g2eane!from the great awe in which a22 reckoning was he2! in these!ays$ A man ski22e! in the art was regar!e! as en!owe! witha2most supernatura2 powers$ This may e=p2ain why arithmeticfrom time immemoria2 was so assi!uous2y cu2ti(ate! by the priesthoo!$ e sha22 ha(e occasion 2ater to !we22 at greater 2ength on this re2ation of ear2y mathematics to re2igious rites an!mysteries$ Not on2y was this true of the ancient rient" wherescience was bui2t aroun! re2igion" but e(en the en2ightene!

6reeks ne(er comp2ete2y free! themse2(es from this mysticismof number an! form$An! to a certain e=tent this awe persists to this !ay$ The

a(erage man i!entifies mathematica2 abi2ity with uickness infigures$ FSo you are a mathematician hy" then you ha(e notroub2e with your income7ta= returnOG hat mathematician hasnot at 2east once in his career been so a!!resse! There is"

perhaps" unconscious irony in these wor!s" for are not most professiona2 mathematicians spare! a22 troub2e inci!ent toe=cessi(e income


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Fin#erprints 2

There is a story of a 6erman merchant of the fifteenth century"

which # ha(e not succee!e! in authenticating" but it is socharacteristic of the situation then e=isting that # cannot resistthe temptation of te22ing it$ #t appears that the merchant ha! ason whom he !esire! to gi(e an a!(ance! commercia2e!ucation$ Ce appea2e! to a prominent professor of a uni(ersityfor a!(ice as to where he shou2! sen! his son$ The rep2y was thatif the mathematica2 curricu2um of the young man was to be

confine! to a!!ing an! subtracting" he perhaps cou2! obtain theinstruction in a 6erman uni(ersityH but the art of mu2tip2ying an!!i(i!ing" he continue!" ha! been great2y !e(e2ope! in #ta2y"which in his opinion was the on2y country where such a!(ance!instruction cou2! be obtaine!$

As a matter of fact" mu2tip2ication an! !i(ision as practice!in those !ays ha! 2itt2e in common with the mo!ern operations bearing the same names$ Mu2tip2ication" for instance" was asuccession ofduplations& which was the name gi(en to the!oub2ing of a number$ #n the same way !i(ision was re!uce! tomediation& i$e$" Fha2(ingG a number$ A c2earer insight into thestatus of reckoning in the Mi!!2e Ages can be obtaine! from ane=amp2e$ 5sing mo!ern notations9

To!ay Thirteenth century1+ 1+× / 0/%* 1+× 1 0/ × / %&1

%*& 1+× & %&1× 2 = !"1+ !" # 1"$ # $! = 5%"

'0&


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e begin to un!erstan! why humanity so obstinate2y c2ung

to such !e(ices as the abacus or e(en the ta22y$ 3omputationswhich a chi2! can now perform re uire! then the ser(ices of aspecia2ist" an! what is now on2y a matter of a few minutes meantin the twe2fth century !ays of e2aborate work$

The great2y increase! faci2ity with which the a(erage manto!ay manipu2ates number has been often taken as proof of thegrowth of the human inte22ect$ The truth of the matter is that the

!ifficu2ties then e=perience! were inherent in the numeration inuse" a numeration not susceptib2e to simp2e" c2ear7cut ru2es$ The!isco(ery of the mo!ern positiona2 numeration !i! away withthese obstac2es an! ma!e arithmetic accessib2e e(en to the!u22est min!$

The growing comp2e=ities of 2ife" in!ustry an! commerce" of 2an!e! property an! s2a(e7ho2!ing" of ta=ation an! mi2itaryorganization?a22 ca22e! for ca2cu2ations more or 2ess intricate" but beyon! the scope of the finger techni ue$ The rigi!"unwie2!y numeration was incapab2e of meeting the !eman!$Cow !i! man" in the fi(e thousan! years of his ci(i2ize!e=istence which prece!e! mo!ern numeration" counter these!ifficu2ties

The answer is that from the (ery outset he ha! to resort tomechanica2 !e(ices which (ary in form with p2ace an! age butare a22 the same in princip2e$ The scheme can be typifie! by thecurious metho! of counting an army which has been foun! inMa!agascar$ The so2!iers are ma!e to fi2e through a narrow passage" an! one pebb2e is !roppe! for each$ hen %) pebb2esare counte!" a pebb2e is cast into another pi2e representing tens"an! the counting continues$ hen %) pebb2es are amasse! in the


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secon! pi2e" a pebb2e is cast into a thir! pi2e representing

hun!re!s" an! so on unti2 a22 the so2!iers ha(e been accounte!for$,rom this there is but one step to thecountin# board or

abacus which in one form or another has been foun! in practica22y e(ery country where a counting techni ue e=ists$ Theabacus in its genera2 form consists of a f2at boar! !i(i!e! into aseries of para22e2 co2umns" each co2umn representing a !istinct

!ecima2 c2ass" such as units" tens" hun!re!s" etc$ The boar! is pro(i!e! with a set of counters which are use! to in!icate thenumber of units in each c2ass$ ,or instance" to represent ';1 onthe abacus" 1 counters are put on the 2ast co2umn" ; counters onthe ne=t to 2ast an! ' on the thir! to the 2ast co2umn$ RSee figure" page /)$

The many counting boar!s known !iffer mere2y in theconstruction of the co2umns an! in the type of counters use!$The 6reek an! -oman types ha! 2oose counters" whi2e the3hinese Suan7Pan of to!ay has perforate! ba22s s2i!ing ons2en!er bamboo ro!s$ The -ussian Szczety" 2ike the 3hinese(ariety" consists of a woo!en frame on which are mounte! aseries of wire ro!s with s2i!ing buttons for counters$ ,ina22y" it is

more than probab2e that the ancient Cin!udust board was a2soan abacus in princip2e" the part of the counters here being p2aye! by erasab2e marks written on san!$

The origin of the wor! abacus is not certain$ Some trace it tothe Semiticabac& !ustH others be2ie(e that it came from the6reek aba6& s2ab$ The instrument was wi!e2y use! in 6reece"an! we fin! references to it in Cero!otus an! Po2ybius$ The

2atter" commenting on the court of Phi2ip ## of Mace!onia in his istoria makes this suggesti(e statement9


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F:ike counters on the abacus which at the p2easure of theca2cu2ator may at one moment be worth a ta2ent an! the ne=tmoment a cha2cus" so are the courtiers at their WingIs no! at onemoment at the height of prosperity an! at the ne=t obEects of human pity$G

To this !ay the counting boar! is in !ai2y use in the rura2!istricts of -ussia an! throughout 3hina" where it persists inopen competition with mo!ern ca2cu2ating !e(ices$ But in

estern urope an! America the abacus sur(i(e! as a merecuriosity which few peop2e ha(e seen e=cept in pictures$ ,ewrea2ize how e=tensi(e2y the abacus was use! in their owncountries on2y a few hun!re! years ago" where after a fashion itmanage! to meet the !ifficu2ties which were beyon! the power of a c2umsy numeration$

ne who ref2ects upon the history of reckoning up to thein(ention of the princip2e of position is struck by the paucity of achie(ement$ This 2ong perio! of near2y fi(e thousan! years sawthe fa22 an! rise of many a ci(i2ization" each 2ea(ing behin! it aheritage of 2iterature" art" phi2osophy an! re2igion$ But what wasthe net achie(ement in the fie2! of reckoning" the ear2iest art practice! by man An inf2e=ib2e numeration so cru!e as to make progress we227nigh impossib2e" an! a ca2cu2ating !e(ice so2imite! in scope that e(en e2ementary ca2cu2ations ca22e! for theser(ices of an e=pert$ An! what is more" man use! these !e(icesfor thousan!s of years without making a sing2e worth7whi2eimpro(ement in the instrument" without contributing a sing2eimportant i!ea to the systemO

This criticism may soun! se(ereH after a22 it is not fair to Eu!ge the achie(ements of a remote age by the stan!ar!s of our own time of acce2erate! progress an! fe(erish acti(ity$ Yet" e(en


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when compare! with the s2ow growth of i!eas !uring the Dark

Ages" the history of reckoning presents a pecu2iar picture of !eso2ate stagnation$hen (iewe! in this 2ight" the achie(ement of the unknown

Cin!u who some time in the first centuries of our era !isco(ere!the principle o! position assumes the proportions of awor2!e(ent$ Not on2y !i! this princip2e constitute a ra!ica2!eparture in metho!" but we know now that without it no

progress in arithmetic was possib2e$ An! yet the princip2e is sosimp2e that to!ay the !u22est schoo2 boy has no !ifficu2ty ingrasping it$ #n a measure" it is suggeste! by the (ery structure of our number 2anguage$ #n!ee!" it wou2! appear that the firstattempt to trans2ate the action of the counting boar! into the2anguage of numera2s ought to ha(e resu2te! in the !isco(ery of the princip2e of position$

Particu2ar2y puzz2ing to us is the fact that the greatmathematicians of c2assica2 6reece !i! not stumb2e on it$ #s itthat the 6reeks ha! such a marke! contempt for app2ie! science"2ea(ing e(en the instruction of their chi2!ren to the s2a(es But if so" how is it that the nation which ga(e us geometry an! carrie!this science so far" !i! not create e(en a ru!imentary a2gebra #s

it not e ua22y strange that a2gebra" that cornerstone of mo!ernmathematics" a2so originate! in #n!ia an! at about the same timewhen positiona2 numeration !i!

A c2ose e=amination of the anatomy of our mo!ern numerationmay she! 2ight on these uestions$ The princip2e of positionconsists in gi(ing the numera2 a (a2ue which !epen!s not on2yon the member of the natura2 se uence it represents" but a2so onthe position it occupies with respect to the other symbo2s of the


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group$ Thus" the same !igit / has !ifferent meanings in the three

numbers *1/" ;/'" /+09 in the first case it stan!s for twoH in thesecon! for twenty" in the thir! for two hun!re!$ As a matter of fact *1/ is Eust an abbre(iation for three hun!re! p2us four tens p2us two units$

But that is precise2y the scheme of the counting boar!"where *1/ is represente! by

An!" as # sai! before" it wou2! seem that it is sufficient totrans2ate this scheme into the 2anguage of numera2s to obtainsubstantia22y what we ha(e to!ay$

TrueO But there is one !ifficu2ty$ Any attempt to make a permanent recor! of a counting7boar! operation wou2! meet theobstac2e that such an entry as = may represent any one of se(era2 numbers9 */" *)/" */)" *))/" an! *)/) among others$ #nor!er to a(oi! this ambiguity it is essentia2 to ha(e some metho!of representing the gaps" i$e$" what is nee!e! is a symbol !or anempty column4

e see therefore that no progress was possib2e unti2 a

symbo2 was in(ente! for anempty c2ass" a symbo2 fornothin#&our mo!ern .ero4 The concrete min! of the ancient 6reeks cou2!not concei(e the (oi! as a number" 2et a2one en!ow the (oi! witha symbo2$

An! neither !i! the unknown Cin!u see in zero the symbo2of nothing$ The #n!ian term for zero was sunya& which meantempty or blan & but ha! no connotation of F(oi!G of Fnothing$GAn! so" from a22 appearances" the !isco(ery of zero was anacci!ent brought about by an attempt to make an unambiguous permanent recor! of a counting boar! operation$


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Cow the #n!ian sunya became the zero of to!ay constitutes one

of the most interesting chapters in the history of cu2ture$ henthe Arabs of the tenth century a!opte! the #n!ian numeration"they trans2ate! the #n!ian sunya by their own" si!r&which meantempty in Arabic$ hen the #n!o7Arabic numeration was firstintro!uce! into #ta2y" si!r was 2atinize! into .ephirum4 Thishappene! at the beginning of the thirteenth century" an! in thecourse of the ne=t hun!re! years the wor! un!erwent a series of

changes which cu2minate! in the #ta2ian .ero $About the same time Jor!anus Nemararius was intro!ucing

the Arabic system into 6ermany$ Ce kept the Arabic wor!"changing it s2ight2y toci!ra4 That for some time in the 2earne!circ2es of urope the wor!ci!ra an! its !eri(ati(es !enote! zerois shown by the fact that the great 6auss" the 2ast of themathematicians of the nineteenth century who wrote in :atin"sti22 use!ci!ra in this sense$ #n the ng2ish 2anguage the wor!ci!ra has becomecipher an! has retaine! its origina2 meaning of zero$

The attitu!e of the common peop2e towar! this newnumeration is ref2ecte! in the fact that soon after its intro!uctioninto urope" the wor!ci!ra was use! as a secret signH but this

connotation was a2together 2ost in the succee!ing centuries$ The(erb decipher remains as a monument of these ear2y !ays$The ne=t stage in this !e(e2opment saw the new art of

reckoning sprea! more wi!e2y$ #t is significant that the essentia2 part p2aye! by zero in this new system !i! not escape the noticeof the masses$ #n!ee!" they i!entifie! the who2e system with itsmost striking feature" theci!ra&an! this e=p2ains how this wor!

in its !ifferent forms" .i!!er& chi!!re&etc$" came to recei(e themeaning of numera2" which it has in urope to!ay$


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This !oub2e meaning" the popu2arci!ra stan!ing for numera2

an! the ci!ra of the 2earne! signifying zero" cause! consi!erab2econfusion$ #n (ain !i! scho2ars attempt to re(i(e the origina2meaning of the wor!9 the popu2ar meaning ha! taken !eep root$The 2earne! ha! to yie2! to popu2ar usage" an! the matter wase(entua22y sett2e! by a!opting the #ta2ian zero in the sense inwhich it is use! to!ay$

The same interest attaches to the wor!al#orithm $ As the

term is use! to!ay" it app2ies to any mathematica2 proce!ureconsisting of an in!efinite number of steps" each step app2ying tothe resu2ts of the one prece!ing it$ But between the tenth an!fifteenth centuriesal#orithm was synonymous with positiona2numeration$ e now know that the wor! is mere2y a corruptionof A2 Wworesmi" the name of the Arabian mathematician of theninth century whose book Rin :atin trans2ation was the firstwork on this subEect to reach estern urope$

To!ay" when positiona2 numeration has become a part of our !ai2y 2ife" it seems that the superiority of this metho!" thecompactness of its notation" the ease an! e2egance it intro!uce!in ca2cu2ations" shou2! ha(e assure! the rapi! an! sweepingacceptance of it$ #n rea2ity" the transition" far from beingimme!iate" e=ten!e! o(er 2ong centuries$ The strugg2e betweenthe Abacists& who !efen!e! the o2! tra!itions" an! the Al#orists&who a!(ocate! the reform" 2aste! from the e2e(enth to thefifteenth century an! went through a22 the usua2 stages of obscurantism an! reaction$ #n some p2aces" Arabic numera2swere banne! from officia2 !ocumentsH in others" the art was prohibite! a2together$ An!" as usua2" prohibition !i! not succee!in abo2ishing" but mere2y ser(e! to sprea!bootle##in#& amp2e


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e(i!ence of which is foun! in the thirteenth century archi(es of

#ta2y" where" it appears" merchants were using the Arabicnumera2s as a sort of secret co!e$Yet" for a whi2e reaction succee!e! in arresting the progress

an! in hampering the !e(e2opment of the new system$ #n!ee!"2itt2e of essentia2 (a2ue or of 2asting inf2uence was contribute! tothe art of reckoning in these transition centuries$ n2y theoutwar! appearance of the numera2s went through a series of

changesH not" howe(er" from any !esire for impro(ement" but because the manua2s of these !ays were han!7written$ #n fact"the numera2s !i! not assume a stab2e form unti2 the intro!uctionof printing$ #t can be a!!e! parenthetica22y that so great was thestabi2izing inf2uence of printing that the numera2s of to!ay ha(eessentia22y the same appearance as those of the fifteenth century$

As to the fina2 (ictory of the A2gorists" no !efinite !ate can beset$ e !o know that at the beginning of the si=teenth centurythe supremacy of the new numeration was incontestab2e$ Sincethen progress was unhampere!" so that in the course of the ne=thun!re! years a22 the ru2es of operations" both on integers an! oncommon an! !ecima2 fractions" reache! practica22y the samescope an! form in which they are taught to!ay in our schoo2s$

Another century" an! the Abacists an! a22 they stoo! for were so comp2ete2y forgotten that (arious peop2es of urope began each to regar! the positiona2 numeration as its ownnationa2 achie(ement$ So" for instance" ear2y in the nineteenthcentury we fin! that Arabic numera2s were ca22e! in 6ermany

3eutsche with a (iew to !ifferentiating them from the 5oman&which were recognize! as of foreign origin$


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*+ N5MB -

As to the abacus itse2f" no traces of it are foun! in estern

urope !uring the eighteenth century$ #ts reappearance ear2y inthe nineteenth century occurre! un!er (ery curiouscircumstances$ The mathematician Ponce2et" a genera2 un!er Napo2eon" was capture! in the -ussian campaign an! spentmany years in -ussia as a prisoner of war$ 5pon returning to,rance he brought among other curios" a -ussian abacus$ ,or many years to come" this importation of Ponce2etIs was regar!e!

as a great curiosity of FbarbaricG origin$ Such e=amp2es of nationa2 amnesia aboun! in the history of cu2ture$ Cow manye!ucate! peop2e e(en to!ay know that on2y four hun!re! yearsago finger counting was the a(erage manIs on2y means of ca2cu2ating" whi2e the counting boar! was accessib2e on2y to the professiona2 ca2cu2ators of the time

3oncei(e! in a22 probabi2ity as the symbo2 for an empty co2umnon a counting boar!" the #n!ian sunya was !estine! to becomethe turning7point in a !e(e2opment without which the progressof mo!ern science" in!ustry" or commerce is inconcei(ab2e$ An!the inf2uence of this great !isco(ery was by no means confine!to arithmetic$ By pa(ing the way to a genera2ize! number concept" it p2aye! Eust as fun!amenta2 a rX2e in practica22y e(ery branch of mathematics$ #n the history of cu2ture the !isco(ery of zero wi22 a2ways stan! out as one of the greatest sing2eachie(ements of the human race$

A great !isco(eryO Yes$ But" 2ike so many other ear2y!isco(eries" which ha(e profoun!2y affecte! the 2ife of the race" ?not the rewar! of painstaking research" but a gift from b2in!chance$


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3 C A P T - *

Number72ore? hat is beauti!ul and de!inite and the ob%ect

o! nowled#e is by nature prior to theinde!inite and the incomprehensible and theu#ly4@

?Nicomachus

o two branches of mathematics present agreater contrast than arithmetic an! the'heoryo! Numbers4 NThe great genera2ity an! simp2icity of its ru2es makes

arithmetic accessib2e to the !u22est min!$ #n fact" faci2ityin reckoning is mere2y a matter of memory" an! the

2ightning ca2cu2ators are but human machines" whose onea!(antage o(er the mechanica2 (ariety is greater portabi2ity$

n the other han!" the theory of numbers is by far the most !ifficu2t of a22 mathematica2 !iscip2ines$ #t istrue that the statement of its prob2ems is so simp2e thate(en a chi2! can un!erstan! what is at issue$ But" the

metho!s use! are so in!i(i!ua2 that uncanny ingenuityan! the greatest ski22 are re uire! to fin! a proper a(enueof approach$ Cere intuition is gi(en free p2ay$ Most of the


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properties known ha(e been !isco(ere! by a sort of

induction4 Statements he2! true for centuries ha(e been2ater pro(e! fa2se" an! to this !ay there are prob2emswhich ha(e cha22enge! the power of the greatestmathematicians an! sti22 remain unso2(e!$

*;Arithmetic is the foun!ation of a22 mathematics" pure

or app2ie!$ #t is the most usefu2 of a22 sciences" an! thereis" probab2y" no other branch of human know2e!ge whichis more wi!e2y sprea! among the masses$

n the other han!" the theory of numbers is the branch of mathematics which has foun! the 2east number of app2ications$ Not on2y has it so far remaine! withoutinf2uence on technica2 progress" but e(en in the !omain

of pure mathematics it has a2ways occupie! an iso2ate! position" on2y 2oose2y connecte! with the genera2 bo!y of the science$

Those who are inc2ine! towar!s a uti2itarianinterpretation of the history of cu2ture wou2! be tempte!to conc2u!e that arithmetic prece!e! the theory of numbers$ But the opposite is true$ The theory of integers

is one of the o2!est branches of mathematics" whi2emo!ern arithmetic is scarce2y four hun!re! years o2!$

This is ref2ecte! in the history of the wor!$ The6reek wor! arithmos meant number" an!arithmeticawas the theory of numbers e(en as 2ate as the se(enteenthcentury$ hat we ca22 arithmetic to!ay waslo#istica tothe 6reeks" an! in the Mi!!2e Ages was ca22e!" as wesaw"al#orism4


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But whi2e the spectacu2ar story which # am about to te22

has 2itt2e !irect bearing on the !e(e2opment of other mathematica2 concepts" nothing cou2! ser(e better toi22ustrate the e(o2ution of these concepts$

The in!i(i!ua2 attributes of integers were the obEectof human specu2ation from the ear2iest !ays" whi2e their more intrinsic properties were taken for grante!$ Cow !owe account for this strange phenomenon

The 2ife of man" to borrow a famous ma=im of Montes uieu" is but a succession of (ain hopes an!groun!2ess fears$ These hopes an! fears which to this !ayfin! their e=pression in a (ague an! intangib2e re2igiousmysticism" took in these ear2y !ays much more concretean! tangib2e forms$ Stars an! stones" beasts an! herbs"wor!s an! numbers" were symptoms an! agents of human !estiny$

The genesis of a22 science can be trace! to thecontemp2ation of these occu2t inf2uences$ Astro2ogy prece!e! astronomy" chemistry grew out of a2chemy" an!the theory of numbers ha! its precursor in a sort of numero2ogy which to this !ay persists in otherwise

unaccountab2e omens an! superstitions$F,or se(en !ays se(en priests with se(en trumpetsin(este! Jericho" an! on the se(enth !ay theyencompasse! the city se(en times$G

,orty !ays an! forty nights 2aste! the rain which brought about the great !e2uge$ ,or forty !ays an! fortynights Moses conferre! with Jeho(ah on Mount Sinai$


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,orty years were the chi2!ren of #srae2 wan!ering in the

wi2!erness$Si=" se(en an! forty were the ominous numbers of the Cebrews" an! 3hristian theo2ogy inherite! the se(en9the se(en !ea!2y sins" the se(en (irtues" the se(en spiritsof 6o!" se(en Eoys of the >irgin Mary" se(en !e(i2s castout of Mag!a2en$

The Baby2onians an! Persians preferre! si=ty an! its

mu2tip2es$ er=es punishe! the Ce22espont with *))2ashes" an! Darius or!ere! the 6yn!es to be !ug up into*+) !itches" because one of his ho2y horses ha! !rowne!in the ri(er$

-e2igious (a2ues" says Poincar " (ary with 2ongitu!ean! 2atitu!e$ hi2e *" ;" %)" %*" 1) an! +) wereespecia22y fa(ore!" we fin! practica22y e(ery other number in(este! with occu2t significance in !ifferent p2aces an! at !ifferent times$ Thus the Baby2oniansassociate! with each one of their go!s a number up to +)"the number in!icating the rank of the go! in the hea(en2yhierarchy$

Striking2y simi2ar to the Baby2onian was the number

worship of the Pythagoreans$ #t a2most seems as if for fear of offen!ing a number by ignoring it" they attribute!!i(ine significance to most numbers up to fifty$

ne of the most absur! yet wi!e2y sprea! forms whichnumer2ogy took was the so7ca22e!"ematria4 (ery 2etter in the Cebrew or 6reek a2phabet ha! the !oub2e meaningof a soun! an! of a number$ The sum of the numbersrepresente! by the 2etters of the wor! was thenumber o!


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the word&an! from the stan!point of 6ematria two wor!s

were e ui(a2ent if they a!!e! up to the same number$ Not on2y was 6ematria use! from the ear2iest !ays for the interpretation of Bib2ica2 passages" but there arein!ications that the writers of the Bib2e ha! practice! theart$ Thus Abraham procee!ing to the rescue of his brother 2iasar !ri(es forth *%& s2a(es$ #s it Eust acoinci!ence that the Cebrew wor! 2iasar a!!s up to

*%& Numerous e=amp2es of 6ematria are foun! in 6reek

mytho2ogy$ The names of the heroes Patroc2us" Cector an! Achi22es a!! up to &;" %//'" an! %/;+ respecti(e2y$To this was attribute! the superiority of Achi22es$ A poet!esiring to confoun! his pet enemy" whose name wasThamagoras" pro(e! that the wor! was e ui(a2ent toloimos&a sort of pesti2ence$

3hristian theo2ogy ma!e particu2ar use of 6ematriain interpreting the past as we22 as in forecasting thefuture$ f specia2 significance was +++" the number of the Beast of -e(e2ation$ The 3atho2icsI interpretation of the Beast was the Antichrist$ ne of their theo2ogians"

Peter Bungus" who 2i(e! in the !ays of :uther" wrote a book on numero2ogy consisting of near2y ;)) pages$ Agreat part of this work was !e(ote! to the mystica2 +++"which he ha! foun! e ui(a2ent to the name of :utherHthis he took as conc2usi(e e(i!ence that :uther was theAntichrist$#n rep2y :uther interprete! +++ as the forecast of the!uration of the Papa2 regime an! reEoice! in the fact thatit was so rapi!2y nearing its en!$


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6ematria is a part of the curricu2um of the !e(out

Cebrew scho2ar of to!ay$ Cow ski22e! these scho2ars arein this !ua2 interpretation of Bibi2ica2 wor!s is i22ustrate!in this seeming2y impossib2e feat$ The Ta2mu!ist wi22offer to ca22 out a series of numbers which fo22ow no!efinite 2aw of succession" some running as high as '))an! more$ Ce wi22 continue this perhaps for ten minutes"whi2e his inter2ocutor is writing the numbers !own$ Ce

wi22 then offer to repeat the same numbers without anerror an! in the same succession$ Cas he memorize! theseries of numbers No" he was simp2y trans2ating some passage of the Cebrew scriptures into the 2anguage of 6ematria$

But 2et us return to number worship$ #t foun! its supremee=pression in the phi2osophy of the Pythagoreans$ (ennumbers they regar!e! as so2ub2e" therefore ephemera2"feminine" pertaining to the earth2yH o!! numbers asin!isso2ub2e" mascu2ine" partaking of ce2estia2 nature$

ach number was i!entifie! with some humanattribute$ ne stoo! for reason" because it wasunchangeab2eHtwo for opinionH !our for Eustice" becauseit was the first perfect s uare" the pro!uct of e ua2sH !ivefor marriage" because it was the union of the firstfeminine an! the first mascu2ine number$ R ne wasregar!e! not as an o!! number" but rather as the sourceof a22 numbers$

Strange2y enough we fin! a striking correspon!encein 3hinese mytho2ogy$ Cere the o!! numbers symbo2ize!white" !ay" heat" sun" fireH the e(en numbers" on the other


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han!" b2ack" night" co2!" matter" water" earth$ The

numbers were arrange! in a ho2y boar!" the :o73hou"which ha! magic properties when proper2y use!$FB2ess us" !i(ine number" thou who generatest go!s an!menO ho2y" ho2ytetra tys& though that containest theroot an! the source of the eterna22y f2owing creationO ,or the !i(ine number begins with the profoun!" pure unityunti2 it comes to the ho2y fourH then it begets the mother of a22" the a227compromising" the a227boun!ing" the first7 born" the ne(er7swer(ing" the ne(ertiring ho2y ten" thekeyho2!er of a22$G

This is the prayer of the Pythagoreans a!!resse! tothe tetra tys&the ho2y fourfo2!ness" which was suppose!to represent the four e2ements9 fire" water" air an! earth$The ho2y ten !eri(es from the first four numbers by aunion of %" /" *" 1$ There is the uaint story thatPythagoras comman!e! a new !iscip2e to count to four9

FSee what you thought to be four was rea22y ten an! acomp2ete triang2e an! our passwor!$G

The reference to a comp2ete triang2e is important9 itseems to in!icate that in these ear2y 6reek !ays numberswere recor!e! by !ots$ #n the accompanying figure thetriangu2ar numbers" %" *" +" %)" %'" are shown as we22 asthe s uare numbersH %" 1" 0" %+" /'$ As this was the actua2 beginning of number theory" this re2iance on geometrica2intuition is of great interest$ The Pythagoreans knew that

a s uare number of any rank is e ua2 to the triangu2ar number of the same rank increase! by its pre!ecessor$They pro(e! it by segregating the !ots an! counting


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them" as shown in the figure$ #t is interesting to compare

this metho! with one that a bright highschoo2 boy wou2!use to!ay$ The triangu2ar number of rankn is ob(ious2y %L / L * L L n&the sum of an arithmetic progressionan! e ua2s% Z / nRn L % $


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