Teaching mathematics with a different philosophy Part 2: Calculus without Limits C. K. Raju School of Mathematical Sciences Universiti Sains Malaysia 11700, Penang, Malaysia cr!craju.net Abstract: "he e#am$le of the calculus is use% to e#$lain ho& sim$le, $ractical math &as ma%e enormously com$le# 'y im$osing on it the (estern (estern religiously)colore% notion of mathematics as *$erfect+. (e (e %escri'e a $e%agogical e#$eriment to mae math easy ' y teaching *calculus &ithout limits+ using the ne& realistic $hiloso$hy of eroism, %ifferent from Platonic i%ealism or formalist meta$hysics. -es$ite its %emonstrate% a%vantages, it is 'eing resiste% 'ecause of the e#isting colonial hangover.
1. Introduction Part 1 e#$laine% ho& (estern (estern mathematics originate% in mathesis an% an % religious 'eliefs a'out the soul. ence, mathematics &as first 'anne% 'y the church, an% later reinter$rete%, in a theologically)correct &ay, &ay, as a *universal+ meta$hysics. "his $ost)Crusa%e reinter$retation /'ase% on the myth of *ucli%+ &as not historically vali%, for the Elements the Elements %i% %i% use em$irical $roofs. $roofs. ventually, ventually, il'ert an% Russell eliminate% em$irical $roofs /in the Elements the Elements an% an% mathematics an% ma%e mathematics fully meta$hysical. ven so, this meta$hysics is not universal, universal, 'ut has a variety of 'iases, as &as $ointe% out. 2t has nil $ractical utility. utility. 2n contrast, most math of $ractical value originate% in the non)(est &ith a %ifferent e$istemology, e$istemology,1 &hich $ermitte% em$irical $roofs /&hich %o not %iminish $ractical value in any &ay. (hile the (est (est a%o$te% this math for its $ractical value, it trie% to force)fit it into its religious 'eliefs a'out math3 first that math must 'e *$erfect+ /since it incor$orates eternal truths, an%, secon%, that this $erfection coul% only 'e achieve% ac hieve% through meta$hysics, since the em$irical &orl% &as consi%ere% im$erfect. /ence, the i%ea that math must 'e 'e meta$hysical. Present)%ay learning %ifficulties in math reflect the historical %ifficulties that arose in this &ay, 'ecause the (est (est im$ose% a religiously 'iase% (es (estern tern meta$hysics on $ractical non)(estern non)(estern mathematics. 4n e#am$le might mae matters clearer.
2. The case of the calculus Contrary to the false history that the calculus originate% &ith 5e&ton an% 6ei'ni, it has no& 'een firmly esta'lishe% that the calculus originate% in 2n%ia. 2n%ia. "he $rocess starte% in the 8th c., &ith 4rya'hata9s attem$ts to calculate $recise trigonometric values. "hat $recision &as nee%e% for practical for practical reasons for accurate astronomical mo%els, an% the calen%ar:essential for monsoon)%e$en%ent agriculture:as also for navigation.; /4griculture an% overseas tra%e &ere the t&o ey sources of &ealth in 2n%ia then. 2n the 1
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>irst their reliance on charts an% straight lines le% to the $ro'lem of lo#o%romes. /2n%o)4ra'ic navigators %i% not have this $ro'lem, for they use% celestial navigation,? an% the 7th c. @hasara ha% alrea%y mentione%8 the o'jection that the s$hericity of the earth must 'e taen into account &hile calculating latitu%e an% longitu%e. Secon%, uro$eans coul% not use techni=ues such as those of @hasara, to %etermine longitu%e, 'ecause Colum'us grossly un%erestimate% the sie of the earth, lea%ing to the 1801 Portuguese 'an on carrying glo'es a'oar% shi$s. "hey coul% not even %etermine latitu%e in %aytime since their religious calen%ar /Aulian calen%ar &as off 'y 11 %ays in the 1 |
"hus, the num'er coul% also 'e finitely un%erstoo% as the ratio of the circumference to the %iameter of a circle. o&ever, -escartes o'jecte% that the ratio of the length of a curve% line &ith a straight line &as 'eyon% the human min%.11 5o& here is such a sim$le thing:a chil% can use a string to measure the length of a curve% line, an% straighten it to com$are it to that of a straight line. "hat &as ho& mathematics &as taught in 2n%ia1 since the %ays of the sulba sutra :or *a$horisms on the string+:'ut a major (estern $hiloso$her asserte% this to 'e im$ossi'leF -escartes9 %ifficulty arose from his religious 'eliefs a'out mathematics as *$erfect+. ence, he naively imagine% that *rigorously+ o'taining the circumference of a circle re=uire% one to 'rea u$ the circumference into straight)line segments, an% $hysically sum u$ the lengths of the segments, lea%ing to the infinite /*6ei'ni+ series. -escartes thought such an infinite sum coul% only 'e %one 'y Go%. Sto$$ing the sum at a finite stage &oul% mean neglecting a small =uantityB though irrelevant for all $ractical a$$lications, it &oul% mae mathematics *im$erfect+, hence not mathematics at all. Galileo concurre%, an% hence left matters to his stu%ent Cavalieri, to avoi% the ris of %isre$ute.
5e&ton thought he ha% resolve% these %ifficulties of -escartes an% Galileo, a'out infinity, 'y a clever a$$eal to Go%F e nee%e% the notion of time %erivative for his secon% *la&+ of motion. e thought this notion of %erivative coul% 'e ma%e *rigorous+ 'y his %octrine of flu#ions. "his re=uire% that time itself must *flo&+ /*smoothly+, or *e=ua'ly+.1; 5o&, &hile things may flo& in time, the slightest thought sho&s that this i%ea that time itself flo&s is meaningless, an% has long 'een recognie% as such.1? 5evertheless, 5e&ton thought mathematics &as the *$erfect+ language in &hich Go% ha% &ritten the *la&s+ of nature. e a%mitte% that time coul% not 'e $ro$erly measure% 'y $hysical $henomena &hich &ere *im$erfect+. @ut he $ostulate% a $erfect, *a'solute, true, an% mathematical time+, &hich *flo&s e=ua'ly+ 'ut *&ithout relation to anything e#ternal+.18 ach a%jective, *a'solute+, *true+, *mathematical+, sho&s that 5e&ton thought time &as meta$hysical an% no&n only to Go%, an% if anyone still ha% a %ou't, he a%%e% the last clause *&ithout relation to anything e#ternal+. Peo$le often =uote 5e&ton on this &ithout un%erstan%ing that maing time meta$hysical &as the &eaest $oint of his $hysics, an% his $hysics faile% e#actly for that reason, an% ha% to 'e re$lace% 'y relativity.1< "his sho&s ho& meta$hysical consi%erations regar%ing mathematics have im$e%e% science. "hese (estern %ifficulties &ith the 2n%ian calculus continue% &ith @ereley9s o'jections17 to the illogical $roce%ures a'out infinitesimalsHflu#ions use% 'y 6ei'ni an% 5e&ton, to &hich 5e&ton9s su$$orters ha% no serious ans&er.1E ventually, -e%ein% 'rought in formal reals, R, 'ut this re=uire% the meta$hysical mani$ulation of infinity ena'le% 'y Can tor9s set theory. "hat, in turn, &as sus$ect an% &as formalise% only in the 1;09s. >ormal set theory is so %ifficult that only a fe& mathematicians 'other to learn it:the hea% of the math %e$artment of an 22" coul% not even state the formal %efinition of a set, &hen $u'licly challenge% to %o so 'y this author. 5aturally, stu%ents &ho learn the *ne& math+, &hich 'egins &ith set theory, fin% math %ifficult. 2ronically, this formalisation le% to the 'elate% realiation that calculus can also 'e %one over *non)4rchime%ean+ fiel%s, larger than R, such as the fiel% of rational functions use% 'y 2n%ian mathematicians &ho treate% rational functions much lie or%inary fractions. 2n such a fiel%, limits are not uni=ue, unless one %iscar%s infinitesimals, an% that $roce%ure /e#actly &hat 2n%ian mathematicians a%o$te% is e=uivalent to limits 'y or%er counting. 5ote that all this theologising a'out the *$erfect+ &ay to han%le infinity, &hich &ent through curve% lines, flu#ions, formal reals, limits, an% sets, an% has returne% to infinitesimals, has a%%e% not a n iota to the practical value of calculus. 4rya'hata9s numerical metho%1 /e=uivalent to &hat is to%ay calle% *uler9s metho%+ for or%inary %ifferential e=uations is still a%e=uate for all $ractical a$$lications of calculus to 5e&tonian $hysics. /If course, the metho% can 'e an% has 'een im$rove%. 5aturally, engineering stu%ents still as to%ay *&hat is the the $oint of %oing limitsJ+, an% the teachers have no ans&er e#ce$t to recite the magical &or% *rigor+. 4s &e have alrea%y seen, this claim of rigor has no su'stance, 'ut incor$orates merely an unreasona'le %eman% that mathematics must conform to a $articular, religiously)'iase% meta$hysics. "his theological (estern vie& of math &as glo'alise% 'y the political force of colonialism. 2t &as sta'ilise% 'y Macaulay9s &ell no&n intervention &ith the e%uc ation system, an% the continue% su$$ort for it is rea%ily un%erstoo% on untington9s %octrine of soft)$o&er.0 4n% this &ay of teaching math continues to 'e uncritically follo&e% to this %ay even after in%e$en%ence. "his is the first attem$t to try to re)e#amine an% critically re)evaluate the (estern $hiloso$hy of math1 an% suggest an alternative to uro$ean ethnomathematics. "he ne& $hiloso$hy $ro$ose% 'y this author has no& 'een rename% *eroism+,; to em$hasie that it is 'eing use% for its $ractical value, an% %oes not %e$en% u$on /the inter$retation of any @u%%hist te#ts a'out sunyavada. 4 ey i%ea is that of mathematics as an a%junct $hysical theory. 4nother ey i%ea is that, lie infinitesimals, small num'ers may 'e neglecte%, as in a com$uter calculation, 'ut on the ne&
groun%s that ideal representations are erroneous, for they can never 'e achieve% in reality /&hich is continuously changing. /#actly &hat constitutes a %iscar%a'le *small+ num'er, or a *$ractical infinitesimal+, is %eci%e% 'y the conte#t,? as &ith formal infinitesimals or or%er)counting. "his is the antithesis of the (estern vie& that mathematics 'eing *i%eal+ must 'e *$erfect+, an% that only meta$hysical $ostulates for mani$ulating infinity /as in set theory, lai% %o&n 'y authoritative (estern mathematicians, are relia'le, an% all else is erroneous. 4s this %e'ate 'et&een realism an% i%ealism is an ol% one, &e &ill not go into further %etails here.
3. The experiment 2f the learning %ifficulties &ith math arise from the theological com$le#ities that the (est has &oven into math, an% if that math &as universalise% 'y the $olitical force of colonialism, then the natural reme%y is to %ecolonise math 'y %is$ensing &ith those theological com$le#ities, &hich any&ay a%% nothing to the $ractical value of math. 4n% the fact is that the mass of stu%ents to%ay learn math for its $ractical value. "hese consi%erations le% to the ne& course on calculus &ithout limits, &hich aims to teach calculus using /a eroism, /' com$uters, an% /c 'y follo&ing the actual historical trajectory of the %evelo$ment of calculus as concerning the numerical solution of or%inary %ifferential e=uations. 4 recent e#$eriment, over the last cou$le of years, has teste% the feasi'ility an% %esira'ility of teaching this ne& course. "he e#$eriment involve% five grou$s till no&, one at the Central University of "i'etan Stu%ies, Sarnath,8 an% four grou$s at the School of Mathematical Sciences, Universiti Sains Malaysia /USM. "he grou$ sies varie% from < to ;8. "he ? grou$s at the USM consiste% of one grou$ of $ost) gra%uate math stu%ents, one grou$ of un%ergra%uate $ure math stu%ents, one grou$ of un%ergra%uate a$$lie% math stu%ents, an% one grou$ of non)math stu%ents. "he availa'ility of four grou$s at USM allo&e% one to test separately the various claims a'out the course as follo&s. 4mong the a%vantages claime% for the ne& course are the follo&ing. /1 "he ne& $hiloso$hy maes the calculus easier to un%erstan%. "hus, any calculus stu%ent to%ay d
x x can $arrot off that dx e = e , 'ut, fe& /even among 22" stu%ents have even a rough i%ea of the
definition of e x . 2n the ne& a$$roach, functions are rigorously %efine% as the solution of %ifferential e=uations. "hus y e x is the rigorously %efine% an% e#$laine% as the solution of y ' = y &ith the con%ition that y 0 = 1 . Stu%ents are =uicly a'le to calculate the values of the function, $lot it, an% analyse it in various &ays using soft&are such as this author9s C46CI-. / "he ne& a$$roach allo&s stu%ents to a$$ly calculus to a%vance% $ro'lems. >or e#am$le, the motion of the sim$le $en%ulum involves the Aaco'ian elli$tic functionsB conse=uently, most $eo$le learn only the sim$lifie% theory of the sim$le $en%ulum, an% often confoun% it &ith sim$le harmonic motion.< (ith the ne& a$$roach, the Aaco'ian elli$tic function sn x is not $articularly more com$licate% than sin x, so stu%ents can stu%y the variation in the time $erio% of the sim$le $en%ulum &ith its am$litu%e. Similar remars a$$ly to the $ro'lem of 'allistics &ith air resistance, or &hy a heavier cricet 'all can 'e thro&n further than a tennis 'all. /; "he ne& a$$roach maes calculus so easy that even non)math stu%ents can master it in a short &hile. =
(ith all five grou$s, there &as a $re)test an% a $ost)test. "he $ost)tests inclu%e% < to 10 $ro'lems %ra&n at ran%om /using a $seu%o)ran%om)num'er generator from a $u'lishe% =uestion 'an of 1th stan%ar% calculus. "hose =uestions involve% sym'olic calculation of sym'olically com$le# %erivatives an% integrals, &hich is &hat stu%ents are e#$ecte% to master in current calculus courses. "his &as
inclu%e% in the $ost)test only to %emonstrate that teaching those sills is com$letely $ointless to%ay, &hen it can 'e %one in a jiffy using o$en)source sym'olic mani$ulation $rograms such as M42M4, "hat is not the same as reliance on a calculator for arithmetic sums3 in %aily life one occasionally nee%s to %o arithmetic sums in one9s hea%, 'ut one never nee%s to %o in one9s hea% any com$licate% integrals or %erivatives /involving only elementary functionsF. "he other as$ects of the $ ost)test inclu%e% solving an% analysing the solution of or%inary %ifferential e=uations, since that is the at the heart of a$$lications of the calculus. 4lso inclu%e% &ere some non)elementary elli$tic integrals. "he $erformance on the $ost)test &as uniformly goo% even for the non)math stu%ents /&ith at least mi%%le)school level math. "hus, claims / an% /; &ere vali%ate%. o&ever, the $re)test reveale% that even the $ost)gra%uate stu%ents &ere not &ell)verse% &ith the $hiloso$hy of formal mathematics, an% &ere not comforta'le &ith a%vance% mathematical notions such as the Sch&art %erivative. 4s such it &as not $ossi'le to test &hether they foun% the %ee$er as$ects of the ne& $hiloso$hy easier than the e#isting $hiloso$hy of formalism. "he &hole a$$roach can 'e e#ten%e% to several varia'les an% $artial %ifferential e=uations in an o'vious &ay. @ut that is a future agen%a.
4. The dimension of hegemony 2%entifying the %ifficulties &ith math learning, an% $ro$osing a solution, %oes re$resent a major a%vance. @ut there are %ifficulties in im$lementing the solution. Darious staehol%ers /such as stu%ents afrai% of math, or their $arents are never consulte% to %eci%e &hat sort of math to teach. ven scientists an% engineers are rarely consulte% regar%ing &hat sort of math ought to 'e taught to them. o&ever, if all %ecisions regar%ing the math curriculum are left solely to math *e#$erts+, there is an o'vious conflict of interests3 for these e#$erts &ere 'rought u$ on the ol%er tra%ition of formal mathematics, an% rejecting formal mathematics may &ell mae their $ast &or valueless. Pu'lic %iscussion is one &ay to ensure that the interests of millions of stu%ents are not %isa%vantage%, an% that scientific an% e%ucational activities relate to $u'lic interest.7 Such %iscussions &oul% 'e $articularly &elcome given the other sensitive issue in the $resent case3 namely that im$osing a religiously 'iase% meta$hysics on millions of stu%ents is not only unethical, it is uncon stitutional un%er the 2n%ian constitution &hich guarantees secularism, or un%er any other constitution &hich %oes not $ermit a (estern religious 'ias.
1 C. K. Raju, “Computers, Mathematics Education, and the Alternative Epistemology of the Calculus in the YuktiBhâsâ”, Philosophy East and West , 51, 325–362 (2001). C. K. Raju, Cultural Foundations of athematics! "he #ature of athematical $roof and the "ransmission of the Calculus from India to Europe in the %& th c' CE /Pearson 6ongman, -elhi, 007. ; >or a $o$ular account, see Ghadar Jari Hai, 2(1) 26–29 (2007). Full text: htt$3HH2n%ianCalculus.infoHCultural) >oun%ations)Mathematics)revie&)GA.$%f . ? C. K. Raju, “Kamâl or Râpalagâi” in Indo-Portuguese Encounters (ed) L. Varadarajan, (INSA, New Delhi, and Universidade Nova de Lisboa, Lisbon, 2006) vol 2, pp. 483–504. Also, final version in Cultural Foundations of Mathematics, chp. 6. 8 (aghu )haskar*ya, 2.8, e%. an% trans. K. S. Shula /-e$artment of Mathematics an% 4stronomy, 6ucno&, 1<;. < C. K. Raju, Cultural Foundations, ch$. ;. Re$ro%uce% in (ii$e%ia htt$3HHen.&ii$e%ia.orgH&iiHMa%hava9sLsineLta'le 7 "hese te#ts inclu%e the Yuktidipika an% "antrasangrahavyakhya, the Malayalam Yuktibhasa, an% the +ryabhatiyabhasya of 5ilanatha. >or a full account of the manuscri$ts an% other %etails see C. K. Raju, cite% a'ove. E Christo$hori Clavii @am'ergensis, "abulae ,inuum- "angentium et ,ecantium ad partes radi. %/-///-///''' /2oannis 4l'ini, 1<07. "he value of the ra%ius in the title is the one use% in 2n%ian /Rsine values. C. K. Raju, “How and Why the Calculus Was Imported into Europe.” Talk delivered at the International Conference on no!ledge and East-West "ransitions , National Institute of Advanced Studies, Bangalore, Dec 2000. Abstract at htt$3HHcraju.netH2n%ianCalculusH@angalore.$%f . 10 C. K. Raju, Cultural Foundations, ch$. ;. "he 2n%ian mathematicians use% rational functions, &hich form &hat is to%ay calle% a non)4rchime%ean or%ere% fiel%. Such a fiel% formally has infinities an% infinitesimals, so that limits are non uni=ue unless infinitesimals are neglecte% as 2n%ian mathematicians %i%. "his is e=uivalent to or%er counting. 11 Ren -escartes, "he 0eometry, trans. -avi% ugene an% Marcia 6. 6atham, /ncyclo$ae%ia @ritannica, Chicago, 10 @oo , $. 8??. 1 C. K. Raju, The Indian Rope Trick” #hartiya $ama%i& Chintan 7 (4) (New Series), 265–269 (2009). 1; 2saac 5e&ton, athematical $rinciples of #atural $hilosophy, trans. 4. Motte, revise% 'y >lorian Cajori /ncyclo$ae%ia @ritannica, Chicago, 1< $. E. >or the %ifference 'et&een 5e&ton9s metaphysical even tenor hy$othesis, an% @arro&Ns $hysical even tenor hy$othesis, see C. K. Raju, “Time: What is it That it can be Measured” $cience'Education 15, 537–551 (2006). 2. @arro&, *4'solute "ime O6ectiones Geometricae+, in M. Ca$e e%., "he Concepts of ,pace and "ime! their ,tructure and their 1evelopment , @oston Stu%ies in the Philoso$hy of Science, vol. 22 /-. Rei%el, -or%recht, 17< $. 0?. 1? C. K. Raju, *Philosophical Time”, Physics Education 7, 204–17 (1990). Also, Time: Towards a Consistent Theory, (Kluwer Academic, Dordrecht, 1994) chp. 1. The absurdity is particularly brought out by Sriharsa's paradox in his 2handana2handa2hadya- no&a%ays calle% Mc"aggart9s $ara%o#. 18 2. 5e&ton, athematical $rinciples of #atural $hilosophy, cite% a'ove, $. E. See also, C. K. Raju, *"ime3 (hat is it that it can 'e measure%J+, cite% a'ove. 1< S$ecifically, this failure relate% to the concept of time, ha% nothing to %o &ith any e#$eriment. See, C. K. Raju, *In "ime. ;a3 "he Michelson)Morley #$eriment+, $hysics Education, 8/;, 1;)00 /11B an% *In "ime. ;'3 instein9s time+ $hysics Education 8/?, ;);08 /1. 17 George @ereley, "he +nalyst or a 1iscourse +ddressed to an Infidel athematician /6on%on, 17;?, e%. -. R. (ilins, availa'le online at htt$3HH&&&.maths.tc%.ieHQ%&ilinsH@ereleyH 1E Aames Aurin, 0eometry no Friend to Infidelity, 6on%on, 17;?, an% "he inute athematician /6on%on, 17;8. @enjamin Ro'ins, + 1iscourse Concerning the #ature and Certainty of ,ir Isaac #ewton3s ethods of Fluxions- and of $rime and 4ltimate Ratios /6on%on, 17;8. 2t is e#cessively %ifficult to tae some of these res$onses seriously, as $ointe% out in C. K. Raju, Cultural Foundations of athematics, cite% earlier, $. ;E. 1 Since 4rya'hata, 0itika 1 states only sine differences, attem$ts to un%erstan% that in terms of *ucli%ean+ geometry seem to in%icate a colonial hangover. Secon%ly, 4rya'hata9s metho%, tersely %escri'e% in 0anita 1, cannot 'e inter$rete% as an alge'raic e=uation as $ointe% out 'y C. K. Raju, Cultural Foundations, cite% earlier, $$. 1;)1?0. "here can 'e no %ifficulty in a%mitting the use of secon% %ifferences, since a short &hile later, @rahmagu$ta / 2handakhadyaka 22.1.? clearly uses them for =ua%ratic inter$olation. 2n the usual manner of uro$eans claiming *in%e$en%ent re%iscovery+ %uring the 2n=uisition, uler *%iscovere%+ the 2n%ian metho% after stu%ying the 2n%ian techni=ues for an article he &rote on the 2n%ian calen%ar in 1700. 0 C K. Raju, *n%ing 4ca%emic 2m$erialism3 a 'eginning+ 2nternational Seminar on 4ca%emic 2m$erialism at 4l ahra (omensN University, "ehran, May 010. 2n3 Sue)san Ghahremani Ghajar an% Seyye%)4'%olhami% Mirhosse ini, Confronting +cademic 2nowledge /2ran University Press, "ehran, 011, ch$. 7. >ull $a$er at htt$3HHmulti&orl%in%ia.orgH&$)contentHu$loa%sH010H08Hcr)"ehran)tal)on)aca%emicim$erialism. $%f. 1 C. K. Raju, *Com$uters, Mathematics %ucation, an% the 4lternative $istemology of the Calculus in the uti@hTsT+, $hilosophy East and 5est , 51, ;8;< /001. *Mathematics an% Culture+, in 6istory- Culture and "ruth! Essays $resented to 1' $' Chattopadhyaya, -aya Krishna an% K. Satchi%anan%a Murthy e%. /Kali Praash, 5e& -elhi, 1
$$. 171;. Re$rinte% in $hilosophy of athematics Education 11. *"he Mathematical $istemology of Sunya,+ in3 "he Concept of ,unya, 4. K. @ag an% S. R. Sarma e%. /2G5C4, 25S4 an% 4ryan @oos 2nternational, 5e& -elhi, 00 $$. 1un%amental Research, Mum'ai, Ict 00. htt$3HHcraju.netH$a$ersHcalculus)&ithout)limits) $resentation.$%f . < >or a more %etaile% account, see C. K. Raju, “Time: What is it That it can be Measured” $cience'Education 15, 537– 551 (2006). For a student project, see http://ckraju.net/11picsoftime/pendulum.pdf . 7 C. K. Raju, *Kosam'i the mathematician+, Economic and $olitical 5eekly 44/0, ;;?8 /00.