Second People's Education Congress

Focal Theme: Science Education in India.

Venue: Homi Bhabha Centre for Science Education, Mumbai, 11-14 March 2009.

Panel on History and Philosophy of Science in Science Education

When religion mixes with state, that manifestly influences history-writing, and we have seen this happening recently in India. However, this proposition applies also to the Western history and philosophy of science as was brought out recently in a book, Cultural Foundations of Mathematics.1

Briefly, during the religious fanaticism of the Crusades, a Hellenic origin was concocted for all pre-Crusade world-knowledge accumulated in Arabic books captured at Toledo. Late and obviously accretive texts were uncritically attributed in their entirety to theologically-correct early Greeks, even when contrary to common sense and non-textual evidence. There is no serious evidence that figures such as Euclid or Claudius Ptolemy even existed.2 During the Inquisition (and the prevailing atmosphere of religious intolerance in the rest of Europe) this process of making the origins of knowledge theologically correct was carried forward, by claiming “independent rediscovery” by European sources. A well-known scandal here is the case of Copernicus,3 but, of course, there are numerous other cases. Later-day racist and colonial historians built upon this process of self-glorification to arrive at the present shape of the Western history of science, which attributes almost anything worthwhile in science either to a Hellenic source or to later-day Europeans.

This motivation also impacted the philosophy of science: acceptable knowledge had to be theologically correct. Attributing the Elements to an unknown “Euclid” helped to reinterpret it in a way acceptable to post-Crusade theology. This reinterpretation of the Elements differed from its earlier Neoplatonic understanding (to which Origen's pre-Nicene theology was close) and is best understood4 as an adaptation of Islamic rational theology (aql-i-kalam), and al Ghazali's criticism, to suit post-Nicene Christian theology.

This process of theological purification made mathematics metaphysical. That attitude persists in present-day formal mathematics which not only deprecates the empirical but is divorced from it, and hence is entirely metaphysical. This metaphysics involves an obvious religious bias: most theorems of present-day formal mathematics would fail if one used Buddhist catuskoti or Jain syadavada instead of 2-valued logic,5 or rejected deduction as unreliable like the Lokayata. Obviously enough, formalism cannot resort to empirical methods to decide the nature of logic, and even if it did, the outcome is not guaranteed (e.g. one might have to contend with quantum logic). Noticeably, this argument, which has been around for a decade,6 has gone unanswered though it destroys a central tenet of Western philosophy of science along with the philosophy of mathematics.

For purposes of pedagogy, a more honest history and a less-biased philosophy is required to enable one to learn from the past. On the epistemic test, mistakes expose false claims of independent rediscovery. But such mistakes are also useful for pedagogy through the principle that “phylogeny is ontogeny”. For example, Gerbert of Aurillac (Pope Sylvester II) imported “Arabic numerals” in Europe. Accustomed to the abacus and failing to understand the place-value system, he made a blunder. He inscribed the new numerals at the back of his old counters7 hoping that would help arithmetic along! Such mistakes help to understand the difficulties that today arise in the minds of children in transition from abacus to arithmetic algorithms.

Such mistakes abound. Clavius published elaborate trigonometric tables8 (remarkably similar to those derived by calculus techniques, and readily available for the preceding half-century to his Jesuit brethren in their Cochin college). However, Clavius did not know the elementary trigonometry needed to use those tables to measure the size of the earth (a critical input which could have solved the longitude problem of European navigation).

Descartes (presumably in response to Pascal and Fermat who were enthusiastic about the imported calculus) stated9 that it was beyond the capacity of the human mind to determine the ratios of curved and straight lines (although this can be easily accomplished by using a flexible string, as was traditionally taught to Indian school children10). Galileo concurred and hence left it to Cavalieri to claim credit for the calculus. Apart from the cultural preference for straight lines, Descartes’ and Galileo’s strange objections to the calculus can be understood by the human mind only in the context of theological correctness: not only did they mistrust empirical procedures in preference to a biased metaphysics, they sought some imagined perfection or certainty in mathematics, and hence thought the calculus involved supertasks. Newton's fluxions and Leibniz's differences attempted to assimilate the calculus to this European understanding of mathematics, and some of their mistakes were pointed out by Berkeley. These mistakes, and the religious beliefs (about the perfection of mathematics) which led to them, persisted at least until the present day theory of the continuum and limits, which relies on set theory to provide a metaphysical mechanism for performing supertasks.11 These mistakes are useful to understand why school children today find the calculus a difficult subject.

Against this background, the panel will consider the following questions among others. (1) Whose interests are served by promoting racist history through school texts? (Current Indian school texts display racist images of imaginary figures like Euclid and neither the authors of these texts, nor the Indian government agency responsible for publishing them have been able to respond appropriately to the repeated public demand for the primary evidence on which these historical claims are based.) (2) In a secular country like India, is it appropriate to continue teaching formal mathematics which has no practical value except to acculture the student into a religiously biased metaphysics? Specifically, is it appropriate to do so at the school (K-12) level where no choices are available to students? (3) Most K-12 math (arithmetic, algebra, trigonometry, calculus) originated in a non-European cultural setting. On the principle that phylogeny is ontogeny, the confusion in math teaching can be understood as arising because present-day math teaching retraces the trajectory of the absorption of these ideas in the West, and hence recreates the confusion that accompanied this process (of making mathematics theologically correct). This confusion can be avoided by rejecting the cultural beliefs superposed on this mathematics in Europe, and going back to the original practical context in which that mathematics developed: Aryabhata’s approach to calculus is far easier to understand and offers far greater practical value today (since it is better suited to the technology of computation) than the religiously tinted hence enormously complex approach of formal mathematics via continuum, limits and axiomatic set theory. Should math teaching, at least at school level, not be based on practical value as in the ongoing course on “Calculus without Limits”12? (4) If engaging with the philosophy of mathematics (“rigour”) is a must, would it not be more appropriate to base math teaching on alternative, realistic philosophies of mathematics, such as zeroism, implicit in its developmental context?

Those who disagree with the above articulation are, of course, also welcome to join in and debate. If there are enough panelists who wish to take up other related topics with specific pedagogical implications, it should be possible to have the panel in more than one session.

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C. K. Raju

Panel on History and Philosophy of Science in Science Education, II PEC


1.C. K. Raju, Cultural Foundations of Mathematics, Pearson Longman, 2007.

2.Details about Euclid are in chp. 1 of the above book, while some details about Ptolemy are in chp. 5, “Models of Information Transmission”. Some details accessible online are in (a) “Towards Equity in Mathematics Education 1: Good-Bye Euclid!”, (b) “Teaching Racist History”, Indian Journal of Secularism, 11 (2008) 25-28. (c) “Itihas ke vichalan” (“Distortion of history") Jansatta 23 Jan 2008, op-ed page,

3.E.g., George Saliba, “Arabic Astronomy and Copernicus”, A History of Arabic Astronomy, New York, 1994, chp. 15. The claim of independent rediscovery is by Owen Gingerich, “I personally believe he [Copernicus] could have invented the method independently.” “Islamic astronomy”,

4.C. K. Raju, “The Religious Roots of Mathematics”, Theory, Culture and Society, 23 (2006) 95-97.

5.For an easy account of various systems of logic see C. K. Raju, “Logic”, in Encyclopedia of Non-Western Science, Technology and Medicine, Springer, 2008.

6.C. K. Raju, “Computers, Mathematics Education and the Alternative Epistemology of the Calculus in the Yuktibhasa” Philosophy East and West, 51 (2001) 325-62, and earlier references cited there. Draft available online at

7.Karl Menninger, Number Words and Number Symbols: A Cultural History of Numbers, trans. Paul Broneer, MIT Press, Cambridge, Mass., 1969, p. 325.

8.Christophori Clavii Bambergensis, Tabulae Sinuum, Tangentium et Secantium ad partes radij 10,000,000..., Ioannis Albini, 1607.

9.René Descartes, The Geometry, trans. David Eugene and Marcia L. Latham, Encyclopaedia Britannica, Chicago, 1990, Book 2, p. 544.

10.C. K. Raju, “Indian Rope Trick”. Paper presented at the 32nd Indian Social Science Congress, session on mathematics education, Mumbai, Dec 2007.

11.As is clear from the issue of renormlization in quantum field theory, and the related question of shock waves, there is no guarantee that mathematical analysis, or the Schwartz theory of distributions, or Non-Standard analysis can, even now, satisfactorily handle all relevant calculus supertasks, without explicit empirical inputs. See “Renormalization and shocks”, appendix to ref 1.

12.For more details of this new calculus course, see, from where a project document can be downloaded. For the relation to sunyavada, see “Number Representations in Calculus, Algorismus and Computers: sunyavada vs formalism”, chp. 8 in Cultural Foundations of Mathematics, cited above. For an elaboration of zeroism, as resting on practical value rather than any textual authority, see C. K. Raju, “Zeroism and Calculus without Limits”, paper to be presented at the 4th Dialogue on Buddhism and Physical Science, Nalanda, 22-24 October 2008.