**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 understood^{4}
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
counters^{7}
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 tables^{8}
(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)
stated^{9}
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 children^{10}).
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.

More details about II PEC can be downloaded from http://www.hbcse.tifr.res.in/download.

Contact: (Please use feedback form)

C. K. Raju

Chair

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

**Notes**

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!”,
http://ckraju.net/papers/MathEducation1Euclid.pdf
(b) “Teaching Racist History”, I*ndian Journal
of Secularism*, **11** (2008) 25-28.
http://ckraju.net/papers/Teaching-racist-history.pdf
(c) “Itihas ke vichalan” (“Distortion of history")
*Jansatta* 23 Jan 2008, op-ed page,
http://ckraju.net/papers/Jansatta-Euclid.jpg.

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”,
http://faculty.kfupm.edu.sa/phys/alshukri/PHYS215/Islamic_astronomy.htm

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
http://ckraju.net/IndianCalculus/Hawaii.pdf

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
32^{nd} Indian Social Science Congress, session on
mathematics education, Mumbai, Dec 2007.
http://ckraju.net/papers/MathEducation2RopeTrick.pdf.

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
http://ckraju.net/calculus/calculus.html,
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 4^{th}
Dialogue on Buddhism and Physical Science, Nalanda, 22-24 October
2008.