Advertisement for post-doctoral Research Associate for my INSA project 1998(on Historia Matematica, currently archived at http://mathforum.org/kb/message.jspa?messageID=1174792

[HM] RA in History of the Calculus Posted: Jul 16, 1998 11:00 PM

Wanted: Research Associate in the History of Mathematics

This is a unique opportunity to work in a high-profile area for the project

Madhava and the Origin of the Differential Calculus,

sponsored by the Indian National Science Academy, coordinated by C. K. Raju. The project seeks to revise the current history of the calculus. It will focus on early developments in the calculus 1400-1720, and will cover all aspects of the Madhava/Gregory/Taylor series expansion, and its transmission from the Malabar coast to Europe, especially through the manuscripts of Jyeshtadeva's Yuktibhasha (1501), and the roughly contemporary Tantrasangraha of Neelkantha Somayajulu. Only incomplete and rough English translations of the above manuscripts are today available, and the work will involve a close comparison of the contents of these manuscripts with some of the work of Kepler, Cavalieri, Fermat, Pascal, Wallis, Gregory, Newton, Leibniz, and Taylor.

Qualifications: Ph.D. in mathematics, physics or allied field. Should be thorough with calculus, basic analysis, and its use in classical dynamics and planetary motion. Knowledge of at least one of Malayalam, Sanskrit, or Latin. Some exposure to the philosophy of non-formal mathematics would be preferred.

Location: National Institute of Science, Technology, and Development Studies, Pusa, New Delhi, 110012, India. This institution functions under the umbrella of the Council of Scientific and Industrial Research, an organization of the Government of India.

Duration: Initially for one year, but likely to be extended to three years.

Contact: ckraju@csnistad.ren.nic.in OR c_k_raju@hotmail.com. Avoid sending applications by post because a postal strike is on. Applications may also be couriered to Dr. C. K. Raju at NISTADS at the above address, or faxed to 91-(0)11-575-4640 or 91-(0)11-221-6895 (redial later in case of voice response).

Applications open until: 29 July 98. Subsequently until the position is filled.

Ghadar Jari Hai, Vol 2, Issue 1 200726

Book Review

Cultural Foundations of MathematicsIt has been common understanding that mathematical proof based on deduction is

universal and is the ultimate proof and also that mathematical truths are eternal universal truths. C K Raju argues that this is a narrow European view of mathematics and the Indian view was very different and empirical. Thus he has raised the important issue of cultural

foundation of mathematics. We present here a summary of his startling new book.

British scholars have known since 1832 that traditional In-dian mathematicians had de-

veloped a way to handle infinite se-ries, a key component of the calculus. However, Western historians have denied that this amounted to the calculus proper, and many aspects of this fascinating Indian contribution to science have remained unclear for the last 175 years.

In this book, Raju asks four questions that have not been asked before. (1) How were infinite series useful to Indian society? (2) Did the Indian infinite series amount to the calculus? (3) Was this Indian math-ematics transmitted to Europe be-fore Newton and Leibniz? (4) Does the traditional Indian approach to mathematics have any practical ap-plications today? Raju’s answers are as follows:

1) The main source of wealth in India is agriculture which depends on the monsoons. The monsoons are “erratic”, so a good calendar is indis-pensable to Indian agriculture. The traditional Indian calendar identi-fies the months of Sawan and Bha-don as the rainy season, unlike the

common Gregorian calendar which has no rainy season. Traditional In-dian festivals like Rakhi and Holi do not occur on “fixed” days of the Gre-gorian calendar (such as 25 Decem-ber or 15 August), and are related to agriculture. Constructing this specialised Indian calendar required complex planetary models. Calibrat-ing this calendar across the length and breadth of India required precise knowledge of the size of the earth, and of ways of determining latitude and longitude of any place and all this required precise trigonometric values. (This knowledge was use-ful also for navigation and overseas trade with Alexandria, Arabs, Africa, and China was also a key source of wealth in India.) The required trigo-nometric values were developed in India since the Surya Siddhanta (3rd c.) and Aryabhata (5th c.). Over the next thousand years these trigono-metric values were gradually made more precise, and that led to the development of the Indian infinite series. Thus, Raju concludes that the social utility for agriculture and navigation drove the development of the Indian infinite series.

2) Western scholars have dubbed the Indian infinite series as “pre-cal-culus”, claiming that the calculus proper emerged with the “funda-mental theorem of calculus” which was absent in India. Raju responds to this criticism in various ways. (a) First, he questions the premise that mathematics means theorem-prov-ing rather than calculation. This re-quires a re-examination of all West-ern history and philosophy. Raju argues that “Euclid” is a historical concoction, and that the Elements, attributed to “Euclid”, is actually a Neo-Platonic religious book. It was radically reinterpreted by Christian rational theologians after the 12th c. CE, to support their agenda of con-verting Arabs, during the Crusades. To this end they declared reason (and mathematics) to be universal. However, since Buddhists and Jains have used a different logic from that used in mathematical proof today, this notion of proof can never be uni-versal. Different principles of proof or pramana were used in Indian tradition, where mathematics is not singled out as requiring a special sort of proof. Raju concludes that

27Ghadar Jari Hai, Vol 2, Issue 1 2007

Book Review

the current Western conception of mathematics as theorem-proving is loaded with religious beliefs unlike calculation which is secular. Hence, the alleged superiority of present-day formal mathematics ultimately rests on religious prejudices which, though deep seated, must be reject-ed. In particular, Raju reaches the radical conclusion that deduction is more fallible than induction:

a conclusion which stands most of Western philosophy on its head. (b) Raju completes the “missing links” in past studies of Indian infinite series to show that there actually was valid pramana for the derivation of the Indian infinite series. This pramana differed from formal mathematical proof, but it was not inferior for that reason.

(c) Finally, Raju argues that Ary-abhata had already developed a neat technique, similar to what is today known as Euler’s method of solv-ing ordinary differential equations, and that this calculation technique is a superior substitute for the fun-damental theorem of calculus even today. Thus, Raju concludes that In-dians did have the calculus, and that denying this merely amounts to an imposition of Western religious be-liefs.

3) Raju points out that informa-tion often flows towards the military aggressor. Examples are (a) Alexan-der’s loot of books, (b) Hulegu and the Samarkand observatory, (c) the Latin translations of Arabic books at Toledo, during the Crusades, and (d) the British colonialists. This happens because the military aggressor is of-ten in a lower state of development. (Toynbee calls these “barbarian in-cursions”). Specifically, Raju points to Vasco da Gama’s and Columbus’ ignorance of celestial navigation: Vasco was brought to India from Af-

rica by an Indian navigator whose instrument the befuddled Vasco carried back to Europe to study. Eu-rope then dreamt of wealth through overseas trade, so several European nations instituted huge prizes for a good technique of navigation. In 16th c. Europe, precise trigonometric val-ues were critical to navigation: and the problems of determining latitude, longitude and loxodromes. Catholic missionaries were in Cochin, since 1500, and had started a college for the local Syrian Christians. The Por-tuguese shared a common patron in the Raja of Cochin with the authors of key Indian texts on the Indian in-finite series, such the Yuktidipika of Sankara Variyar. Later Jesuits got numerous Indian texts translated and despatched them to Europe on the model followed earlier at Toledo. Jesuits like Matteo Ricci specifically looked for Indian astronomy texts, to assist with the Gregorian calendar reform. Thus, Europeans had ample opportunity and motivation to ob-tain the relevant Indian mathemati-cal texts. A trail of circumstantial evidence is visible as the contents of these Indian texts start appearing implicitly or explicitly in European astronomical and mathematical works from the mid-16th c: Merca-tor’s chart, Tycho Brahe’s planetary model, Clavius’ trigonometric tables, Kepler’s planetary orbits, Cavalieri’s indivisibles, Fermat’s challenge prob-lem, Pascal’s quadrature, “Newton’s” sine series.

Why were the Indian texts not acknowledged? To understand this, Raju first points to the Hellenisa-tion of history that took place at To-ledo. During the religious fanaticism of the crusades, all secular world knowledge in Arab libraries up to the 11th c. was appropriated to the West by attributing it to the theologically

correct “Greeks”. The Arabs, against whom the crusades were on, were de-clared to be mere intermediaries who helped to restore this “European in-heritance” to Europe. (The arrival of Byzantine Greek texts in the 15th c. further confounded matters.)

This claim of transmission of hy-pothetical “Greek” knowledge to Ar-abs and all others, involves double standards of evidence. To expose this, Raju considers, as an example, the Almagest. The text dates from post-9th c., but is attributed to a “Claudius Ptolemy” from the 2nd c. However, Raju argues, the Almagest is an accretive text which was repeat-edly updated with inputs from Indian astronomy and mathematics at both Jundishapur (6th c.) and Baghdad (early 9th c.) where Indian texts on astronomy are known to have been imported and translated into Pahla-vi and Arabic. Thus, transmission of trigonometric values took place from Indian texts to the Almagest, rather than the other way round as is usu-ally claimed by stock histories today, without any evidence. In support of this claim, Raju contrasts the rela-tive sophistication of the Almagest text with the non-textual evidence of the crudeness of the Greek and

During the religious fanaticism of the Crusades, all secular world knowledge

in Arab libraries up to the 11th c. was appropriated to

the West by attributing it to the theologically correct

“Greeks”.

Ghadar Jari Hai, Vol 2, Issue 1 200728

Book Review

Roman calendars, which could not get the length of the year right, de-spite repeated attempts at calendar reform in the 4th through 6th c. CE. This could hardly have happened if the Almagest had then existed in its present form.

This European tradition of sup-pressing non-Christian sources was continued during the Inquisition, when it was dangerous to acknowl-edge anything theologically incor-rect. For example, a key navigational breakthrough in Europe was Mer-cator’s chart (common “map of the world”) which shows loxodromes as straight lines. This required precise trigonometric values and a technique equivalent to the fundamental theo-rem of calculus. Mercator, however, was arrested by the Inquisition, and his sources remain a mystery to this day. Similarly, Clavius, a high reli-gious official, could hardly be expected to acknowledge his debt to non-Chris-tians for trigonometric values, or the Gregorian calendar reform. This ha-bitual nonacknowledgment of non-Christians led to the proliferation of claims of “independent rediscovery” by Europeans. Raju also points to the papal “Doctrine of Christian dis-covery”, promulgated in that period, according to which only Christians could be “discoverers” - hence it was said that Vasco da Gama “discovered” India or that Columbus “discovered” America, since this “discovery” im-plied ownership, and the existing in-habitants did not count.

To correct this long-standing rac-ist bias, and given the difficulty of obtaining suppressed documents, Raju hence proposes new standards of evidence to decide transmission. This includes the consideration of opportunity, motivation, and circum-stantial evidence, as above. It also in-cludes the common-sense “epistemic

test”: when two students turn in identical answer sheets, the one who does not understand what he claims to have authored is the one who has copied. Thus, lack of understanding is proof of transmission. Hence, proof of transmission of the calculus comes from the fact that for centuries it re-mained half-understood in Europe: given the notorious difficulties with Newton’s fluxions, and Leibniz’s in-finitesimals both of which were even-tually abandoned. While Clavius published elaborate trigonometric ta-bles, he did not know the elementary trigonometry needed to use them to determine the size of the earth. (This was then badly needed for naviga-tion, especially since Columbus had underestimated the size of the earth by 40 percent to support his project of reaching East by going West.)

Similarly, there is the issue of “epistemic discontinuity”: the Indian infinite series developed gradually over a thousand years; but in Europe they appeared almost overnight. From Cavalieri to Newton it hardly took 50 years, and this whole devel-opment started just 50 years after Europe first learnt of decimal arith-metic. Interestingly, in an appendix on the “transmission of the transmis-sion thesis”, Raju applies this “epis-temic test” to prove the transmission of his own ideas to Europe, in a re-markable case of history repeating itself! Thus Raju concludes that the calculus was transmitted from India to Europe, and that similar processes of transmission of information are going on to this very day.

4) Raju points out applications of traditional Indian mathematics to mathematics education, computers, and frontier areas of physics today.

The European difficulties in un-derstanding arithmetic and the cal-culus, both imported from India, are

today replayed in the classroom, in “fast forward” mode, and this is what makes it difficult for students to un-derstand mathematics. The solution is to go back to the understanding within which that mathematics orig-inated.

For example, Indians used a flex-ible rope (= sulba) to measure curved lines since the days of the sulba sutras. However, exactly measur-ing curved lines was declared to be “beyond the capacity of the human mind” by Descartes, since Western geometry was based on the straight line. Hence, Galileo left it to his student Cavalieri to articulate the calculus. These conceptual difficul-ties can be avoided, Raju argues, by switching back to the rope as a su-perior substitute to the compass box today used in schools.

Raju explains how the Indian un-derstanding of number, in the con-text of sunyavada, is related to the representation of numbers on pres-ent-day computers, and why this should be taught in preference to the impractical and formal notion of number taught today in schools.

Raju also explains how to apply this understanding to tackle un-solved problems related to the infini-ties and infinitesimals that arise in the extensions of the calculus today used in frontier areas of physics, in the theory of shock waves, and the renormalisation problem of quantum field theory.

Thus, in the process of answering these four questions, Raju has chal-lenged the entire Western tradition of mathematics, both its history and its philosophy. Especially interesting is the conclusion that the believed cer-tainty of mathematical proof and the infallibility of deduction are based on religious dogma which is remedied by allowing the empirical in mathemat-

29Ghadar Jari Hai, Vol 2, Issue 1 2007

ics as in the alternative philosophy of mathematics Raju proposes. The formal approach to mathematics, Raju argues, is based on the wrong belief that logic is a metaphysical and metamathematical matter on which it is possible to impose a “uni-versal” social consensus. Given the social disagreement over logic, even logic ought to be decided empirically, he maintains.

This alternative philosophy fol-lows realistic sunyavada thinking on representability. (In Buddhist thought, the problem of represent-ability arose because of the denial of the soul, and the consequent prob-lem of representing an “individual” when nothing “essential” stays con-stant for even two instants.) Raju in-terprets sunya as the non-represent-able, “something” which is neglected in a calculation. For example, the number today called π can never be fully specified or distinguished from a potential infinity of other nearby numbers. In any actual calculation, one can only write down its decimal expansion up to a trillion digits, say, beyond which one does not care what

happens. This problem of represent-ability is today made manifest by the finitary thinking of computers, increasingly used for complex math-ematical tasks: surprisingly, even integers cannot be “correctly” repre-sented on a computer. This, argues Raju, is not any sort of limitation or “error” of representation, but is in the nature of things. The error, to the contrary, is in the idealistic approach to mathematics based on a wrong belief in the possibility of upertasks (an infinite series of tasks): the mere name π does not represent a unique number any more than the name of a person represents a unique indi-vidual. Given the paucity of names, a person’s name is just a practical de-vice by means of which we refer to a whole procession of individuals, from birth to death, who differ from each other so “slightly” that we “don’t care” about the difference. The “slight” dif-ference may vary with the context, and may even be manifest as the difference between a child and the (same) child as an adult. The same thing applies to representations of numbers on a computer (or in any

Book Review

practical rule-based calculation).Apart from being very useful for

computation, the mere existence of such an alternative philosophy of mathematics poses a challenge to Western thought, which has never conceived of the possibility that mathematics might be ultimately based on physical hypotheses about logic, and that it might be realistic, fallible and less than perfect.

Summary of : “Cultural Foundations of Mathematics: the Nature of Mathematical Proof and the Transmission of the Cal-culus from India to Europe in the 16th c. CE (Pearson Long-man, 2007).

Dr C K Raju is a mathemati-cian, historian and philosopher

and has made important con-tribution through many articles

and monographs towards combating Eurocentric rendi-

tion of the history of Mathemta-ics. He has particularly brought

out the impact of theological controversies in Europe on revi-sionist history of mathematics.

still endures. Today in India, crores upon crores of people continue to live in such conditions of backwardness that they are comparable to those endured by the peasants in pre-rev-olutionary Russia. A colonial justice system, a colonial education system, an army that continues to base itself on colonial traditions, all make a mockery of the struggles our people fought to rid themselves of foreign domination. While Indian capital-ist tycoons crow about the successes they are achieving in a ‘globalizing’ world, tens of thousands of self-re-specting hardworking Indian peas-

ants are committing suicide because they cannot make ends meet. The people are told that they are sov-ereign, but they lack the power to change their conditions for the bet-ter. In these conditions, more and more sections of our people are look-ing for an alternative to the existing system, they are coming around to the view that they have to take mat-ters into their own hands, and that they must get organised for this. In these conditions, the legacy of the October Revolution is more relevant than ever before. Profound, all-sided transformation is an urgent

necessity, and it must begin, as it did in Russia 90 years ago, by sorting out the question of political power - who exercises it, how it is exercised, and what is the purpose for which it is exercised. The sorting out of this question is what will lead to the much-needed renewal of India.

Madhavi Thampi teaches in the department of East Asian

Studies of Delhi University. She has authored a number of works

on the historical relationship between India and China.

Continued from pg 25

MSN Hotmail - Page 1 of 2

.../getmsg?curmbox=F921687081&a=a933e46b497d0686c587b9bf9c018012&msg=MSG916/03/2004

c_k_raju@hotmail.com Printed: Tuesday, March 16, 2004 7:37:42 AM

From : Dennis Francis Almeida <D.F.Almeida@exeter.ac.uk>

Sent : Sunday, November 22, 1998 12:58:51 PM

To : "C. K. Raju" <c_k_raju@hotmail.com>

Subject : URGENT!

MIME-Version: 1.0 Received: from dialup24 [144.173.6.224] by hermes via SMTP (NAA27888); Sun, 22 Nov 1998 13:14:46 GMT Sender: D.F.Almeida@exeter.ac.uk From d.f.almeida@exeter.ac.uk Sun Nov 22 05:14:55 1998 In-Reply-To: <19981105024425.7634.qmail@hotmail.com> Message-ID: <SIMEON.9811221251.B@none.ex.ac.uk> Priority: NORMAL X-Mailer: Simeon for Win32 Version 4.1.5 Build (43) X-Authentication: IMSP

Dear Dr Raju,

This a draft text of my proposal for funds. Please surveyand make any desired changes ASAP.

Note the sections on outcomes - we need to publish 2 papersfor the Research Assessment Exercise in March 2001 butpapers need to have been published by Dec 2000 - is thisfeasible? Your paper on transmission models can serve as abasis for one paper I think. What are your thoughts aboutteam publishing (along the lines of the Bourbaki) as far asthe requirements for the Uni of Exeter are concerned -perhaps we may call ourselves 'Aryahbhata'?!

Also ASAP please send your and Jolly's publication lists.

Dennis

===========================================================

I urgently need the maximum of £5000 for a project on'Proof and Rationale in Medieval Indian Mathematics'.

RATIONALE

The project aims to analyse the manuscript the YUKTIBHASAfrom a modern mathematical viewpoint, study its growth fromthe school of Aryabhata (b. 476 AD, the most well known ofclassical Indian mathematicians) and its epistemologicalsimilarity with 'renaissance' European mathematics, andexplore is conjectured transmission to Europe (inparticular, Spain). The project team consists of myself,Dr. C.K. Raju, and Dr. J. John ( both of the NationalInstitute of Science, technology and Development Studies -NISTADS - New Delhi; both with records of publishing ininternational class journals).

The manuscript the YUKTIBHASA written around themiddle of the 17th century was a commentary onpre-calculus mathematical knowledge from the Aryabhataschool - this knowledge pre-dates similar devleopments inEurope (in several cases by 4 centuries). While this hasbeen reported in 1835 it has not been acknowledged becauseearlier Indian MSS such as the TANTA SAMGHRAHA (c 1450) didnot contain the rationale and proofs so cherished byOccidental mathematics and without whichknowledge haslittle credibility in the system of Occidentalmathematics. The YUKTIBHASA DOES contain rationales andproofs. While this has been known for sometime, theYUKTIBHASA has not been analysed in its entirety simplybecause it was written in archaic Malyalam (this is unusalin Indian mathematical MSS which were all written inSanskrit - that it was written in Malyalam, a languageknown to Jesuit missoniaries whose knowledge gathering inwell known, lends credence to the transmission theory).The YUKTIBHASA has only just been literally translated by alanguage scholar (Prof K. V. Sarma) in Madras but has notbeen published - the team has one of the two copies of thetranslation.

PROJECT ACTIONS:

1. In order to study its growth from the Aryabhata school,Sanskrit MSS in Kerala have to be translated and studied.

2. In order to study its transmission to Spain/Portugal,the manuscript libraries at the Universites of Salamanca,Madrid and Barcelona have to be consulted.

3. In order to finalise the construction of the analysis ofthe YUKTIBHASA from a modern mathematical viewpoint, Dr.C.K.Raju and myself need to meet in Exeter to complete thewrite up.

OSTS

c_k_raju@hotmail.com Printed: Thursday, September 6, 2007 12:38 PM

From : George Gheverghese Joseph <Msrbsgj@fs1.ec.man.ac.uk>Reply-To : george.joseph@man.ac.uk

Sent : Friday, July 21, 2000 10:58 AMTo : c_k_raju@hotmail.comSubject : Leverhulme Project

Received: from [130.88.13.7] by hotmail.com (3.2) with ESMTP id MHotMailBB416982001ED820F39E82580D0711F90; Fri Jul 21 02:58:22 2000 Received: from fs1.ec.man.ac.uk ([130.88.27.100])by curlew.cs.man.ac.uk with esmtp (Exim 2.05 #4)id 13FZa3-0001fu-01for c_k_raju@hotmail.com; Fri, 21 Jul 2000 10:58:55 +0100 Received: from UK-AC-MAN-EC-FS1/SpoolDir by fs1.ec.man.ac.uk (Mercury 1.47); 21 Jul 00 10:58:55 BST Received: from SpoolDir by UK-AC-MAN-EC-FS1 (Mercury 1.47); 21 Jul 00 10:58:48 BST From Msrbsgj@fs1.ec.man.ac.uk Fri Jul 21 03:00:59 2000 Organization: Manchester University X-Confirm-Reading-To: george.joseph@man.ac.uk X-pmrqc: 1 Return-receipt-to: george.joseph@man.ac.uk Priority: normal X-mailer: Pegasus Mail for Win32 (v3.12b)

Dear Raju, I understand that there are some difficulties with regard to your participation in the Leverhulme project. I don't know the nature of these difficulties. But I felt that I should make clear my views on the matter. 1) I think your participation is vital in pursuing the project. You and Dennis have put in some very good work and my desire to be involved in it in the first place was a result of the exciting possibilities offered by further research. 2) Having said that, I think a project of this nature which can have a significant impact on both the historiography and narrative of future histories of mathematics is more important than the individuals involved in carrying it out. I pointed out some years ago in the Crest of the Peacock that transmissions were a possibility and I was attacked by a number of reviewers for sticking my neck out. A number accused me of some form of Indian or Keralan aggrandisement. In the new edition, just brought out by Princeton University Press, I have been even more upfront in

Page 1 of 2MSN Hotmail -

stating the possible transmission, my confidence to an extent being reinforced by the work that Dennis and you have done. It would be a shame if because of personality differences, this worthwhile project can not be completed by the people who have started the process in the firsrt place. 3) From my experience of participating in a number of funded projects of this nature, the identification of individuals and the skills that they will bring to the project are crucial elements in determining whether a project will be funded or not. Once, a decision has been made to fund it, the institution granting the the funds would be willing to listen to any changes that the principal researcher would like to suggest. What is seen as a fatal weakness is not to have identified the individuals who are involved with the project. 3) Collaboration across countries raises some real problems when it comes to funding by British Grant-Making Organisations. No allowance is made for the disparity in incomes, living standards, institutional rigidities etc between Britain and the other countries involved. I was involved many years ago in a collaborative project which involved an African academic from Tanzania. While we were equal in terms of our respective contributions, I found it quite embarrassing to have to take responsibilities for paper clips, taxi receipts used by him. It worked well since we knew each other, were committed and worked to free ourselves of the irksome irritations within the framework of the system. Egos counted for little. The Project ruled supreme. A book came out of the project. So Raju, my plea to you is please take part. The Project is so important and we all feel so passionately about its importance that individuals and their preferences are really of secondary importance. With best wishes, George PS On another matter, I have a few copies of The Crest of the Peacock (last revision) which I wish to donate to institutions/ individuals who cannot afford to buy it and would be interested in doing so. Do you know of any of such and if so, could I send you the copies to be distributed. This revision supercedes the Indian Edition. ----------------------------------- Dr George Gheverghese Joseph, School of Economic Studies, University of Manchester, Oxford Road, Manchester M13 9PL United Kingdom Phone: (0161) 275-4871 Fax: (0161) 275-4812

Page 2 of 2MSN Hotmail -

Dr C. K. Raju

Text Box

Hindustan Times, Bhopal Live, 8 Nov 2004

MESSAGE DISPLAY Message number 6

6 neomail-trash Move

From da2697@tutor.open.ac.uk Sat Dec 03 21:51:41 2005 Return-path: <da2697@tutor.open.ac.uk> Envelope-to: c_k_raju@mcu.ac.in Delivery-date: Sat, 03 Dec 2005 21:51:41 +0000 Received: from mcuac by host.focusindia.com with local-bsmtp (Exim 4.52) id 1EifIB-0003ad-PF for c_k_raju@mcu.ac.in; Sat, 03 Dec 2005 21:51:40 +0000 X-Spam-Level: X-Spam-Checker-Version: SpamAssassin 3.1.0 (2005-09-13) on host.focusindia.com X-Spam-Status: No, score=-2.6 required=5.0 tests=BAYES_00,UNPARSEABLE_RELAY autolearn=ham version=3.1.0 Received: from [137.108.246.32] (helo=venus.open.ac.uk) by host.focusindia.com with esmtp (Exim 4.52) id 1EifIB-0003aZ-JK for c_k_raju@mcu.ac.in; Sat, 03 Dec 2005 21:51:39 +0000 Received: from oufcnt2.open.ac.uk (actually host halley.open.ac.uk) by venus.open.ac.uk via SMTP Local (Mailer 3.1) with ESMTP; Sat, 3 Dec 2005 21:51:24 +0000 Message-id: <fc.004c540103cc4067004c540103cc4067.3cc41ed@oufcnt2.open.ac.uk> Date: Sat, 03 Dec 2005 21:51:22 GMT Subject: An apology X-FC-Form-ID: 141 X-FC-SERVER-TZ: 28837376 To: c_k_raju@mcu.ac.in From: Dennis Almeida <da2697@tutor.open.ac.uk> MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 8bit Status: R

Simple headers

Dear Professor Raju, I will be brief. I wish to apologise unconditionally for the hurt and distress I have caused you. The content of the complaints you issued against me certainly suggests this and, given the critical retrospection that I have undertaken since leaving Exeter University, some of this have genuine merit. I have no idea what the Vice Chancellor of Exeter University communicated to you after the investigation committee concluded its findings. However nothing he may have said would have conveyed the ugliness that I feel having participated in the actions of that period. Indeed I could not have expressed such sentiments at the time for events dictated otherwise. I do not expect you to accept this apology. It makes little difference to me whether you do or not. The important thing for me is that I have cleared my conscience. For the record I relinquished my post at Exeter in July 2005 and now teach part time on distance learning courses at the OU and for Paul Ernest. However I do intend to continue my research on the Jesuit conduit as an amateur on my own initiative and funds. Yours, Dennis Almeida

Page 1 of 2NeoMail - C K Raju

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You are here: Home > About Us > News > News item

Indians predated Newton 'discovery' by 250 years

13 Aug 2007

A little known school of scholars in southwest India discovered one of the founding principles of modern mathematics hundreds of years before Newton according to new research.

Dr George Gheverghese Joseph from The University of Manchester says the 'Kerala School' identified the 'infinite series'- one of the basic components of calculus - in about 1350.

The discovery is currently - and wrongly - attributed in books to Sir Isaac Newton and Gottfried Leibnitz at the end of the seventeenth centuries.

The team from the Universities of Manchester and Exeter reveal the Kerala School also discovered what amounted to the Pi series and used it to calculate Pi correct to 9, 10 and later 17 decimal places.

And there is strong circumstantial evidence that the Indians passed on their discoveries to mathematically knowledgeable Jesuit missionaries who visited India during the fifteenth century.

That knowledge, they argue, may have eventually been passed on to Newton himself.

Dr Joseph made the revelations while trawling through obscure Indian papers for a yet to be published third edition of his best selling book 'The Crest of the Peacock: the Non-European Roots of Mathematics' by Princeton University Press.

He said: "The beginnings of modern maths is usually seen as a European achievement but the discoveries in medieval India between the fourteenth and sixteenth centuries have been ignored or forgotten.

"The brilliance of Newton's work at the end of the seventeenth century stands undiminished - especially when it came to the algorithms of calculus.

"But other names from the Kerala School, notably Madhava and Nilakantha, should stand shoulder to shoulder with him as they discovered the other great component of calculus- infinite series.

"There were many reasons why the contribution of the Kerala school has not been acknowledged - a prime reason is neglect of scientific ideas emanating from the Non-European world - a legacy of European colonialism and beyond.

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"But there is also little knowledge of the medieval form of the local language of Kerala, Malayalam, in which some of most seminal texts, such as the Yuktibhasa, from much of the documentation of this remarkable mathematics is written."

He added: "For some unfathomable reasons, the standard of evidence required to claim transmission of knowledge from East to West is greater than the standard of evidence required to knowledge from West to East.

"Certainly it's hard to imagine that the West would abandon a 500-year-old tradition of importing knowledge and books from India and the Islamic world.

"But we've found evidence which goes far beyond that: for example, there was plenty of opportunity to collect the information as European Jesuits were present in the area at that time.

"They were learned with a strong background in maths and were well versed in the local languages.

"And there was strong motivation: Pope Gregory XIII set up a committee to look into modernising the Julian calendar.

"On the committee was the German Jesuit astronomer/mathematician Clavius who repeatedly requested information on how people constructed calendars in other parts of the world. The Kerala School was undoubtedly a leading light in this area.

"Similarly there was a rising need for better navigational methods including keeping accurate time on voyages of exploration and large prizes were offered to mathematicians who specialised in astronomy.

"Again, there were many such requests for information across the world from leading Jesuit researchers in Europe. Kerala mathematicians were hugely skilled in this area."

NOTES FOR EDITORS The research was carried out by Dr George Gheverghese Joseph, Honorary Reader, School of Education at The University of Manchester and Dennis Almeida, Teaching Fellow at the School of Education, The University of Exeter.

Dr Joseph and Mr Almeida are available for comment.

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Copies of a related research paper are available.

For more details, contact: Mike Addelman Media Relations Officer Faculty of Humanities The University of Manchester 0161 275 0790 07717 881567

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