Abstracts
ICPR Seminar on
“Science and Technology in the Indic Tradition: Critical
Perspectives and Current Relevance”
at
IISc Bangalore,
Feb 4-5 2017
Axiomatism and Computational Positivism
Two Mathematical Cultures in Pursuit of Exact Sciences
RODDAM NARASIMHA, JNCASR
It is argued here that the mathematical approach to the exact sciences has
historically appeared to contain two largely distinct cultures (which
nevertheless overlap to some extent). One of these takes the deduction of
'certain' conclusions from clearly stated axioms or models as the primary
objective; the other considers number the primary concept, and
emphasises computation and algebra, conforming to unambiguous rules. A
philosophy that may be called computational positivism, whose goal is to
make computation agree with observation, appears to have been
characteristic of Indian (and apparently Babylonian) astronomy. The
interactions between these two cultures have played a key role in the
history of science, and seem set to continue to do so in the future as well.
Linguistic Modeling of the Universe
Prof. V N Jha, Former Director, Centre of Advanced Study in Sanskrit, University of Pune.
Even if we have a cursory look at the history of development of our intellectual and philosophical
culture we can find that all through the history our forefathers have preferred rationality over
emotionality. Our culture, therefore, is a constantly reflecting culture. These reflections kept on
contributing to the development of various scientific and philosophical disciplines.
Kautilya says :
“The science of logic (anviksiki) is the lamp to see i.e. to understand all systems of knowledge; it is
the means to perform all rational human acts; it is the base of all ethical and moral
acts.” (Vidyasamuddesa of Arthasastra)
The system of logic developed by the Nyaya school of thought has been used by all thinkers and
philosophers. The greatest contribution of the Nyaya system of thought is this that it compelled all
thinkers to rationalize their thoughts and to engage in dialogue in order to arrive at the Truth.
The Naiyayikas developed not only a philosophy of life, but also various models, tools and theories
to understand the universe. INDIAN MATHEMAICS AND ASTRONOMY: SOME METHODOLOGICAL ISSUES
M.D.Srinivas
Centre for Policy Studies, Chennai
[email protected]
While there is a large corpus of modern scholarship on the history and achievements of the Indian
tradition of sciences, there has not been much discussion on the foundational methodology of
Indian sciences. Traditionally, such issues have been dealt with in the detailed bhashyas or
commentaries, which continued to be written till recent times and played a vital role in the
traditional scheme of learning. As regards Ganita or Indian Mathematics, it is in such commentaries that we find detailed
upapattis or “proofs” of the results and procedures, and some discussion of methodological and
philosophical issues. These texts also declare that the purpose of upapatti is mainly: (i) to remove
confusion and doubts regarding the validity and interpretation of mathematical results and
procedures; and, (ii) to obtain assent in the community of mathematicians. The notion of upapatti seems to be significantly different from the notion of ‘proof’ as understood
in the Greco-European tradition of mathematics. While the upapattis are presented in a sequence
proceeding systematically from known or already established results to finally arrive at the result to
be established, there is no attempt made to fit them into an axiomatic system. Nor do the Indian
mathematicians subscribe to the ideal of mathematics being a body of infallible eternal truths. Another significant feature of Indian mathematical tradition, which perhaps stems from the worldview of the Naiyayikas or Indian logicians, is that the method of indirect proof (known as tarka in
Indian logic), can be employed only for proving the non-existence of certain entities, but not for
proving the existence of an entity, which existence is not demonstrable (at least in principle) by
other (direct) means of verification. In this way, the Indian mathematical tradition may be seen as
adopting what is nowadays referred to as the ‘constructivist’ approach to the issue of mathematical
existence.
Further, in the Indian tradition, mathematical knowledge is not taken to be different in any
fundamental sense from that in natural sciences. In fact, the pramanas or the valid means for
acquiring and validating mathematical knowledge are the same as in other sciences, viz.
pratyaksha (perception), anumana (inference), sabda or agama (authentic text or tradition), though
it is noted that one could establish the validity of mathematical results without recourse to sabda or
agama.
To understand the methodology of Indian sciences, one has to start with the foundational works on
Indian logic and linguistics. The central concern of Nyayasastra or India logic is the study of
pramanas or means of acquiring valid knowledge. While a large part of this study is centred on
anumana or inference, as declared right at the outset by Vatsyayana in his celebrated commentary
on Nyayasutras of Gautama, anumana has to be pratyaksha-agama-asrita and not pratyakshaagama-viruddha. Anumana, as the name indicates, follows pratyaksha, and is not linked with any
special faculty of “reason”. Further, inference in Indian logic is both “deductive and “inductive’ – it
is verily the method of scientific inquiry.
Sabdasastra or the science of language was perhaps the earliest of Indian sciences to have been
rigorously systematised, and this systematisation became the paradigm example for all other
sciences. In his famous commentary Mahabhashya on Panini’s Astadhyayi, Patanjali explains that
the purpose of grammar is to give an exposition of all valid utterances. An obvious way to do this
is to enumerate all valid utterances individually. Since that is humanly impossible, one should
attempt to encapsulate larger and larger class of valid utterances by means of a set of general and
exceptional rules (utsarga and apavada-sutras). Patanjali further emphasises that the utterances
and their meanings are actually established in the world –one does not go to a Grammarian to make
utterances for him as one goes to a potter for pots. In thus characterising Paninian grammar, Patanjali expounds what is perhaps the basic
understanding of the Indian scientific effort: That truth resides in the real world with all its
diversity and complexity. For the linguist, what is ultimately true is the language as spoken by the
people in all their diverse expressions. Linguists make generalisations about the language as
spoken in the world. These generalisations are not the truth behind or above the reality of the
spoken language. These are not idealisations according to which reality is to be tailored. On the
other hand what is true is what is actually spoken in the real world, and some part of the truth
always escapes our idealisation of it. Many of these issues are further discussed by the great
philosopher Bhartrihari in his treatise Vakyapadiya. Texts of Indian astronomy often cite his dictum
that the procedures taught in sciences are only means (upaya) to easily accomplish desired
objectives in the world and they are not constrained or regulated in any other manner.
The tradition of astronomy in India goes back to the ancient texts of Vedangajyotisha which give
simple algorithms for fixing the elements of Indian calendar (Panchanga). The Vedangajyotisha
texts, as well as the later elaborate treatises on Indian astronomy, declare the raison d’être of the
science of astronomy to be the determination of time (as well as position and direction) by means
of the motion of the celestial bodies. Hence, it is the pragmatic concerns of calculating the
positions of the various planets and eclipses of the Sun and the Moon reasonably accurately, which
informed the efforts of the Indian Astronomers and in this they seem to have been eminently
successful at least from the time of Aryabhata). Thus, the Indian astronomers were in the business to calculate and to compute, not to form pictures
of the heavens as they ought to be. Indian astronomers do employ various theoretical models,
analytical as well as geometrical, which play a crucial role in their schemes of computation.
However, as the texts themselves emphasise, these theoretical frameworks or models are artefacts
which serve the purpose of achieving drig-ganitaikya or consonance between computations and
observations. Indian astronomers are also aware that their astronomical parameters and even
theoretical procedures could get out of tune with reality sooner or later, and the Indian texts
repeatedly emphasise the need for updating and revising the parameters and theoretical schemes so
that their computations conform to observations.
In their attempt to achieve concordance between their calculations and the observed planetary
motions, Indian astronomers were sometimes ready to accommodate inexplicable or even
seemingly contradictory procedures as component part of their models. In fact, while attempting to
resolve one such seeming contradiction in the traditional method of calculation of the latitudes of
the interior planets, the celebrated astronomer Nilakantha Somayaji came up with a far more
accurate formulation of the equation of centre and the motion of interior planets – than was
available till then in the Indian, Islamic or the Greco-European traditions of astronomy – in his
seminal work Tantrasangraha. In his other great works, Aryabhatiyabhashya and Jyotirmimamsa,
Nilakantha has highlighted the importance of preparing the practitioners of this science for the
onerous task of continuously observing the skies, continuously checking their computations against
observations and repeatedly re-adjusting their parameters and theoretical procedures so as to make
their calculations accord with reality.
This pragmatic and open-ended approach to scientific theorisation seems to be a characteristic
feature of all Indian sciences. It stands in marked contrast with the quest for absolutely true and
universal laws of nature that seems to have been the dominant ideal of the Greco-European
scientific tradition.
The Science and Signs of Consciousness: The Western Mind and the Eastern Self
Sangeetha Menon, Ph.D Professor & Head, Consciousness Studies Programme
National Institute of Advanced Studies
Indian Institute of Science Campus, Bangalore 560 012
Email: [email protected]; [email protected] Website: http://niasconsciousnesscentre.com/sm-cv.html
Today, on one side life and physical sciences are increasingly excited about identifying the neural
correlates of consciousness. And on the other, there is consensus that better integration at both
neural and psychological levels lead to an individual's psychological and physical health. Such a
scenario inspires us to review the functionally interconnected nature of brain, and the
interconnections between the brain and the self itself. In current trends in cognitive sciences the
discussion on body crosses the classical divide between the body and the self in terms of nature
and function. Embodiment theories have helped to bring in the importance of the role of
subjective experiences to understand cognition, and place the process of knowing in a cultural
and social context. There are clear conceptual distinctions between the self, and mind in the
Indian philosophical traditions. In this lecture, I would critique of the growing trend in cognitive
sciences, particularly in affective neurosciences, and approaches, to reduce the experiential self
to a non-entity. It is shown that though the apparent goal is to highlight the inner qualitative nature
of experience, what is happening in the background is a role reversal. The outer body becomes
the inner self. The inner self becomes the outer body. The nature and functions of the self are
founded on the body by theorising embodiment as an alternate to neural reductionism. One of
the negative consequences of embodiment theories is that age-old concepts of freewill,
character and moral choices become flimsy and fleeting in the process of embodying cognition.
Can we make sense of embodiment without a self that is foundational and yet capable of
creative self-reflection? Some of the concepts of self and self-reflection from the Kashmir Saivism
philosophy might help in bringing out the complexity involved in distinguishing the mental from that
which is of the nature of the subject and a core self.
Error Correcting Code-like chanting procedures in ancient India
by
R. L. Kashyap and M. R. Bell
Abstract
The integrity of the oral version of the Rig Veda Samhita, the ancient sacred book of the Hindus,
dated at least earlier to 2000 BCE, has been preserved by the use of special chanting procedures
called vikratis which resemble the modern error correcting codes developed in the last fifty years
for error free transmission of messages or strings of symbols over electronic media. This book has
more than ten thousand verses in a variety of metres.
We consider here one particular error
correcting code-like chanting called as Krama-mala Vikrati and show its connection to the modern
Linear Block Code developed on the basis of advanced mathematical concepts. We also mention
the other error detecting procedures (not correcting) for Rig Veda Samhita based on the use of
binary numbers dated 200 BCE or earlier.
The key idea behind the special chanting methods called as Vikratis and modern Error Correcting
Codes is same namely the use of encoder and decoder. The modern Error Correcting methods are
based on the advanced mathematical concepts such as group theory. The sages seem to have
arrived at similar results purely by intuition.
Specifically consider the correct preservation of the chanting of one verse M, consisting of m
words. The Vikrati will generate another verse ‘M' based on the words of M having many more
words, typically 4 m or more. Let the verse O be the corrupted version of M, the version recited by
a chanter who makes unconscious errors. Applying mentally the decoder idea - reverse of the
encoder idea - to the corrupted output of O yields the correct verse M under appropriate conditions.
Alternatively another person who is familiar with the decoder and hears the output O can suggest
the correction. The sages were very much aware of the trade off between the redundancy in the
code quantified by the length of the vikrati version ‘M' versus the error.
Evolution of human understanding of Nature
Mayank Vahia, TIFR
Human beings have been experimenting with nature for the past two million years.
Human approach to science can be divided into four distinct stages namely ad hoc,
religious, pragmatic and axiomatic. Indian approach has been largely pragmatic.
Indian contribution itself came in three distinct time phases, namely early period till
about 1000 BC, the golden period from 1000 BC to 500 AD and the Siddhantic
period from 500 AD onwards. The approach of the Indian scientific community was
self-limited due to cultural constraints and hence did not produce the dramatic
scientific revolution similar to that of Europeans. Based on the insights developed
by Indian and other cultures, the Europeans, using an axiomatic approach,
developed new insights with spectacular results. This approach is also now
reaching its limits. In the light of this, we will discuss the future trends in science.
Theories of Shaabdabodha and Principles
of Information Coding: A computational
Perspective
Amba Kulkarni
Abstract
Indians have developed the theories of Shaabdabodha to decipher and
understand the information coded in language strings. Indian grammatical
tradition discusses four factors viz. Aakaanksha (expectancy), yogyataa
(mutual congruity), sannidhi (proximity) and taatparya (purport) as essential
factors in the process of verbal cognition (Shaabdabodha).
These four factors are useful for a computational linguist in understanding the
dynamics of information coding in a language string. They provide answers for
a) ‘where’ does a language string code information?, b) ’what’ kind of
information does it code?, and c) ’how’ is such information coded?
In this paper we unfold these factors and explain how do they help in
understanding the dynamics of information coding in a language string. Finally
we also discuss how these factors also suggest an upper limit on automatic
information extraction and the need for BIG DATA in automatic processing of
Natural languages.
Trea%sesonbuildingscience:MayamataandMaanasaara:
TheirApplica'ons
T.Satyamurthy
Ever since the man started dwelling in abodes either natural cavern or under the
shades of trees, to protect him and his progeny from heat and cold, he was very
careful to select a safe place bere< of malevolence forces. For such residen>al
placesheascertainedtranquilityeitherbyritualsorbyadheringsomeregula>onsof
theperiod.Thereareamplearchaeologicalmaterialstoshowthatsuchprac>ceof
holding some unwri@en rules was prevalent even in the earliest society. We find
ampleprehistoricpain>ngsspreadthroughouttheworldheraldingsuchprac>ces.
InIndiancontext,suchremainsareseeninrockartfromprehistorictoearlyhistoric
period.Theearliestsuchwelldesignedroyalandna>vearchitecturearefoundtobe
documentedinthe2ndBCEmuralpain>ngsofAjantaandinmanyreliefsinbuddhist
andbrahminicalcavearchitectureincludingthePallavaremainsinMamallapuram.
Such prac>ces or their experiments were well documented in oral literature
includingVedasandlaterintheformofclassicalwri>ngsandisbeingmatchedby
vernacularliterature.
Suchindependentworksareclassifiedunder“technicaltrea>ses”knownasShilpa
sastra and they are almost encyclopedias that concentrate on architecture both
dwellingsandplacesofworship.Theformerdealsmorewiththespacethatwould
becomfortableforthedaytodaylifeofhumanbeingsincludingthepetsthathelp
himinhislongjourney.Thela@erisagreatscienceencompassingmanyfacul>esof
knowledge like geology, astronomy, space science, civil engineering, architecture,
material science ,environment –discipline, biology and mathema>cs. Trigonometry
playedaprimeroleingivingfinalshapetothebuilding.
ThemasterofsuchProjectsnamedasSthap>wasrespectedbyallandconsidered
as the creator of world itself “Visvakarma”. He is qualified in all above subjects
besidessoundknowledgeonSastraandPuranas.Inshapingespeciallythedwelling
ofGodshisresponsibilityistobringdowntheenergyinmacrocosmtomicrocosm.
Theidolinstalledisnotameresymboloffaith,butthereflec>onoftheuniversal
force and that would vibrate with energy. His skill is to unearth such energy and
retainitforever.
AllsuchexperiencesarewelldocumentedinMaanasaara.Totally18suchworkson
architecture and 32 on art are enlisted there. Among them most of them are lost
and only five are now available in Sanskrit text. They are 1.Mayamatam
2.Visvakarmiyam, 3.Maanasaara 4.Manusaram 5.Kasyapam. All these works have
comprehensive formulae for architects and ar>sts to model their assignments.
There are some later works on construc>on of houses like Vastu vidya and
Manyshyalayacahndrika. There is no dearth for regional trea>ses like
Tasntrasamuchchyyaforkeralawoodenarchitecture.
Amongallofthem,Mayamataoccupiesafairlywelldefinedplaceandservesasthe
manualforthearchitectsofancientIndia.Maanasaaraisahandbookforthemto
haveaccuratemeasurementsinpropor>onandtheformeriscomprehensiveinall
aspectsofthebuildingscience.Permuta>onandcombina>onsprovidedthereare
morecomputedmanually.
MayamataisgenerallyplacedasthemedievalSanskritworkwithspecificprinciples
ofCholacountrybuildings.Nevertheless,thereferenceofMayanastheauthorand
master architect of Tamil Country is found men>oned in many Tamil Sangam
literatureofearlyhistoricperiod.Thetradi>onoffollowingtherulesformulatedby
Mayan in Tamil country con>nued for centuries and finally dra<ed in Sanskrit by
scholarsforthebenefitoftheen>reEngineeringcommunityofBharatavarsha.
It is considered as part of Saiva Agamic literature without the connec>on being
underlinedbyanypronouncedsectarianismanditsdra<ingmusthavebeendone
withvastexperiencegainedbytheauthors.Thearchitectureitdescribeshadpeak
of its maturity and almost the principles are followed in many edifices of Chola
period,thepossibilityofitsdatea<er10thCEcannotberuledout.Itaccommodates
allbuildingsconstructedforimmortalsandimmortalsandmaanasaaraprovidethe
computedmeasurementswithpermuta>onandcombina>ons.
Subjectsanalyzedandguidedinthetextincludevastu,VaastuandIconography.It
enumeratesthequalityofarchitects,orienta>onandlayouts,founda>ondeposits,
joineryandritualsforconsecra>onandrenova>onandassociatedrites.Thetemple
architecture enumerated here follows the op>on characterized by false storey,
whichproliferatedinPallavatemplesandwhichcametoitsfullzenithinflowering
colors in the chola period. Many Saivisam works like Kamigama is devoted with
architectureandliterallycrammedwithversesorevenen>repassageswhichareto
befoundintheMayamata.Mayamatathusinfluencedallotheralliedliteratureof
contemporaryandtheperiodthatfollowed.
MaanasaarahasgotnumerousparallelswithMayamatabutconsideredbyscholars
asthattheybelongslightlydifferentbranchesofsouthIndiaschoolofarchitecture.
The talacheda in describes definitely place it as the southern order text, but P.K
Acharyatheeditorandcommentatoroftheworkassignsanextremelyearlydateto
it to Gupta period. He considers it as the unique source of all presenta>ons of
architecture in purana and agama as well as specialized texts like Brhatsamhita or
Mayamata.
Itisallcomprehensiveworkandcompleteinallthetextsdealingwitharchitecture,
inthesensethatallspecifictypesmen>onedinothertextsaretobefoundinit.Itis
generallyagreedthattheyaretruerepresenta>onsofaphaseinthedevelopment
of the school which they represent. The originality of its teachings is the real
Document of a medieval technical text, well preserved in its original form. Co-
relatedstudiesofthearchaeologicalremainsbringoutmanyunknownfacetsabout
Indianarchitectureandarchitects.
Ayurgenomics:Canitbeapanaceafortheglobalchallengeinunravellingthegene%csof
polygenicdisorders?
ThelmaB.K
DepartmentofGene%cs
UniversityofDelhiSouthCampus
NewDelhi
Thecomple>onofthehumangenomeprojectwhichsequencedtheen>rehumangenome,atthe
beginning of this century, followed by impressive high throughput technologies have enabled
iden>fica>onofarangeofcommonandrare,largeandsmallvaria>onsthroughoutthegenome.
Despite this nearly complete knowledge, predic>on/preven>on of common complex disorders
such as type 2 diabetes, hypertension, cardiovascular diseases, Asthma, Parkinson’s disease,
Rheumatoid arthri>s etc which cons>tute ~60% of all human gene>c disorders has not been
possible. The major reason seems to be the clinical heterogeneity underlying such complex
phenotypes,warran>ngalternateparadigmsforinves>ga>on.
Ayurveda, the Indian tradi>onal system of medicine, on the other hand, uses the >me tested
deepclassifica>ontooltosubgroupallindividualsintothreepredominantprakri>typesofvata,
[email protected]>specificdiseasesuscep>bili>esarealsowelldocumentedinAyurveda.An
innova>ve alternate of ‘Ayurgenomics’, which combines these ayurveda doctrines with
contemporary genomic tools is hypothesized to provide the key to the black box of gene>cs of
complextraitsplaguinggenomeresearchers.Earlysuppor>vefindingsadop>ngthisapproachin
rheumatoidarthri>swillbepresented.
Plant Systematics in Amarakosha
S Sundara Rajan
Poornapragna Samshodhana Mandira, Bangalore
Amarakosha authored by Amarasimha(Circa 5th CE) is a popular
Lexicon (Nighantu) and encompasses various aspects of societal life of
ancient India.Though technically known as Namalinganushasana
(dealing with names and genders) it has been the most popular
Nighantu in Sanskrit literature.The book is divided into three parts
(kandas) and a number of chapters (vargas) based subject wise.
The present paper deals with one of the Vargas of the second Kanda
namely 'Vanaushadhi varga’. It includes classification of plants, types of
forests, description of plant parts (Phytography), biodiversity and a list
of nearly 300 plants with their synonyms (paryaya pada).There is also a
mention of Parasites, Epiphytes etc indicating a deep understanding of
growth habit of plants. From the modern botanical standpoint the
Vanaushadhi varga classifies plants based on the reproductive
structures clearly antedating the basics of classification proposed by
modern scientists by more than a millennium later! Furthermore,
etymological analysis of synonyms provides basic data for identification
of plants.
Finally, the paper analyses a series of works dealing directly or
peripherally with plants like Arthasastra, Charakasamhita,
Vrkshayurveda,Amarakosha and Brhatsamhita and traces in detail how
there is a continuing tradition of using flowers and fruits in aid of
classification of plants in these works which spread over a millennium
starting from 3rd BCE to 5th CE. Specific evidence in support of these
facts mentioned above will be discussed.
Symbol, Number and Infinities: Alaukika Gaṇita of
Jaina
Navjyoti Singh, IIIT Hyderabad
Abstract: The Digamber Jaina tradition developed mathematics of infinities in
the founding text Ṣaṭkhaṇḍāgama and specially in its commentary Dhavalā
done at Shravanbelagola in Karnataka. I shall outline two of the Jaina proofs
with trans-finites quantities (ananta rāṣī-s) and point their novel use of
transfinite-induction in proofs. Further, I shall introduce Jaina concept of a
class of finite numbers (asaṁkhyāta rāṣī-s) which are beyond any numbernaming-schemes (such schemes capture only saṁkhyāta rāṣī-s or recursive
ordinals). Jaina-s make fundamental distinction between three classes of
numbers: (1) symbolic realm (less than the cardinality of śrutakevali) of finite
recursive ordinals or saṁkhyāta rāṣī-s; (2) experiential realm (less than the
cardinality of avadhijñānakevali) of finite non-recursive ordinals or
asaṁkhyāta rāṣī-s, and; (3) knowledge realm (less than the cardinality of
sarvajña or pūrṇakevali) of transfinite ordinals or ananta rāṣī-s. This
distinction has far reaching implication for the relation between symbol,
number and infinities and the idea of computability. I will reflect on the
possibility and implication of accepting smallest non-recursive ordinal being
finite, that is, Church-Kleene Ordinal being finite as the Jaina tradition
suggests. This will be supported by invoking Goodstein ordinal limitation
theorem and its unprovability in Peano arithmetic. I shall end with a
speculative note regarding consequence of Jaina Alaukika Gaṇita on the idea of
computability.
The Vaakya system of Astronomy M.S. Sriram
Prof. K.V. Sarma Research Foundation, Adyar, Chennai.
Abstract
In the Vaakya system of astronomy prevalent in south India, the true
longitudes of the Sun, the Moon and the planets can be found at regular
intervals, using vaakyas or mnemonics. These are based on the various
periodicities asociated with these celestial bodies.
The vaakyas for the Sun are the simplest. These are based on the facts that
the mean rate of motion of the Sun is nealy 1 degree per day, and that the
expression for the true longitude of the Sun has a simple form. Based on
these, we have vaakyas giving the longitude of the Sun on any day, and
the instants of entry into particular zodiacal signs and so on.
For the Moon, the vaakyas are based on the fact that its anomaly
completes very nearly 9 revolutions in 248 days. Correspondingly, there
are 248 Candravaakyas for the Moon, which give the longitudes of the
Moon at mean sunrise on 248 successive days, beginning with the day at
the mean sunrise of which the Moon's anomaly is zero.
For the planets, the vaakya system is necessarily more complicated, as the
planets orbit around the Sun, whereas we need to determine their positions
as vieewed from the Earth. There are elaborate tables of vaakyas for the
the longitudes of planets which involve their 'zodiacal anomaly', as well as
the 'solar anomaly'.
In this talk , we sketch the essential features of the Vaakya system. Indic Mathematics and the Computational Thinking
Metaphor
K.Gopinath,IISc
TheIndicciviliza>onhasmanyinnova>onsanddeepinsightstoitscredit,asbefitsageographical
expansewellendowedwithnaturalresourcestosupportalargeandinquisi>vepopula>onforthe
last many millennia. Some of these ideas have influenced and diffused into other knowledge
systemsbutthereiss>llnoclarityonthedepthandextentofthese. Aninteres>ngperspec>ve
thatseemstohaveinformedagoodpartoftheIndictradi>onisthatofदृग्गिणतैक्यdrg.ganitaikya,
thedesirabilityofconcordancebetween“observed”andthe“computed”inareasofenquirythat
haveapredic>veaspect;duetothisaspectthecomputa>onaltradi>oninastronomyinIndiawas
verypronounced.Muchearlier,VrddhaGarga(pre700BCE?)saysदशर्नं गिणतं चैवयुगपद् योगसाधकम्
|(Observa>onandcalcula>onwhentheygotogetheriseffec>ve).
This la@er computa>onal aspect in the Indic tradi>on has been argued by Prof. Roddam
Narasimhaas“computa>onalposi>vism”andadis>nguishingfeaturetoooftheIndictradi>onin
S&T as a whole; this is in contrast to the “Greek” geometric tradi>on that emphasized axioms
(“obvioustruths”)andproofs.Furthermore,thisIndicperspec>vealsointeractsinteres>nglywith
an another perspec>ve, that of “computa>onal thinking”, as advocated by current computer
scienceresearchers.Wegiveabriefoutlineoftheseideas;therearethuscomputa>onalaspects
inareassuchaslanguage,logic,astronomy,mathema>csandarchitectureintheIndictradi>on.
ConceptNotefor
Seminar on
“Science and Technology in the Indic Tradition: Critical Perspectives and
Current Relevance”
at IISc Bangalore, Feb 4-5 2017
K.Gopinath
TheIndicciviliza>onhasmanyinnova>onsanddeepinsightstoitscredit,asbefitsageographical
expansewellendowedwithnaturalresourcestosupportalargeandinquisi>vepopula>onforthe
last many millennia. Some of these ideas have influenced and diffused into other knowledge
systemsbutthereiss>llnoclarityonthedepthandextentofthese.Thisseminaron“Scienceand
TechnologyintheIndicTradi>on:Cri>calPerspec>vesandCurrentRelevance”willexploresome
oftheseaspectsbyprovidinghighlevelperspec>vesonselectedareasofscienceandtechnology
byeminentresearchers.Moreconcretely,someofthetalksinthisseminarwillthrowlightonan
interes>ng perspec>ve that seems to have informed a good part of the Indic tradi>on: that of
दृग्गिणतैक्य drg.ganitaikya, the desirability of concordance between “observed” and the
“computed” in areas of enquiry that have a predic>ve aspect; due to this aspect the
computa>onal tradi>on in astronomy in India was very pronounced. This la@er computa>onal
aspect in the Indic tradi>on has been argued by Prof. Roddam Narasimha as “computa>onal
posi>vism” and a dis>nguishing feature too of the Indic tradi>on in S&T as a whole; this is in
contrasttothe“Greek”geometrictradi>onthatemphasizedaxioms(“obvioustruths”)andproofs.
Furthermore,thisIndicperspec>vealsointeractsinteres>nglywithananotherperspec>ve,that
of “computa>onal thinking”, as advocated by current computer science researchers. Below we
give a brief outline of these ideas. Some of the talks in the seminar will therefore be on
computa>onalaspectsinareassuchaslanguage,logic,astronomy,mathema>csandarchitecture.
Notethatweusetheterm“Indic”torestrictourdiscussiontocontribu>onsmadebythoseliving
in India as understood historically (so it includes, for example, those who lived in areas such as
current Afghanistan) but also exclude those working directly in the “European” tradi>on of
mathema>cs(foreg.SrinivasaRamanujan).Interes>ngly,eveninRamanujan'scase,hisworkwas
basedonhisprodigiouspowersofcalcula>ontoarriveatmathema>calstructures(forexample,
hisintui>ngtheamazingpar>>onfunc>on)withoutmuchemphasisonanaxioma>c/deduc>veor
proofcentredstyle,thela@erbeingspeciallyemphasisedin“European”mathema>cs.(Notethat
the issue is not so much that this style is not useful but that it may be inadequate in certain
contextsandcomputa>onalonesbe@er.)
BriefbackgroundonComputa>onalThinking
Twofundamentalaspectsofcomputerscience,asBhateandKak(“Panini’sGrammarandComputerScience,”AOBRI,
72, 1993, p.79-94) put it, are the crea>on of new compu>ng algorithms and machines that have powerful
computa>onalandcogni>veabili>es:thisincludesdevelopmentofnewtechniquesofrepresen>ngandmanipula>ng
knowledge, inference and deduc>on. Also, in a longer term perspec>ve, it is the development of techniques that
maketheelucida>onofthecomputa>onalstructureofnatureandthemindeasier.
Ifwelookatcomputa>onalthinkinginearlyIndia,weseesomeverygoodexampleswrt:
a)Grammarianseg.Paanini,Kaatyaayana,Patanjali(Grammar~computa>on,nowestablishedincomputerscience).
b)Logicianseg.Gautama,Udayana,Gangesa,RaghunaathaShiromani(Logic~computa>on,alsonowestablishedin
computerscience)
c) Connec>ons with “cogni>ve science” or “inner sciences” like Yoga wrt mind sciences (psychology, neuroscience).
Thisisbecomingverypronouncedinthelastdecadeincomputerscienceandrelatedfields
In an influen>al paper (J. Wing, “Computa>onal Thinking,” CACM, 2006, 49 (3): 33), the two A’s of “Computa>onal
Thinking”aregivenas
a)Abstrac>on:abilitytothinkintermsofmul>plelayersofabstrac>onsimultaneouslyanddefiningtherela>onships
thebetweenlayers
b)Automa>on:abilityto“mechanize”theabstrac>onlayersandtheirrela>onships
Mechaniza>on is possible due to precise and exac>ng nota>ons, and models; note that both the (recursive) Indic
numbersystemandthePaaninangenera>vegrammarhavebeenthe1stsetofnon-trivialexampleswhichhadthis
propertyinthean>quity.Also,the“machine”canbehumanorcomputer,virtualorphysical.Forexample,Paanini’s
genera>vegrammarwassufficientlyinternalizedfortheSanskritlanguagethatanyliteratepersonhadtoknowitwell
enoughtouse/debatewiththeserulestodecideonthecorrectnessofsomeintricateques>onofwordforma>onor
seman>cs.
Computa>onalThinkingmorebroadlycanbeseenasAbstrac>on,Mechaniza>on,RecursionandBootstrapping;these
giveustheabilityandaudacitytoscale.
Interes>ngly,manyoftheseideasarepresentintheIndictradi>onandtheintentofthisseminaristobringtheseto
theno>ceoftheenquiringminds.
Forexample,Trivedi(VisualComputer,1989)discussestheuseofrecursivestructuresinbuildingtemplestodepictan
“evolving cosmos of growing complexity, which is self-replica>ng, self-genera>ng, self-similar and dynamic”.
Furthermore,“theproceduresarerecursiveandgeneratevisuallycomplexshapesfromsimpleini>alshapesthrough
successiveapplica>onofproduc>onrulesthataresimilartorulesforgenera>ngfractals.”Thetechniquesiden>fied
arefractaliza>on,self-similaritera>oninadecreasingscale,andrepe>>on,superimposi>on,andjuxtaposi>on.
BackgroundonComputa>onalTradi>oninIndicthinking
Let us briefly look at why a computa>onal approach for mathema>cs may be useful. There is a need for a healthy
dose of empiricism in complex domains of enquiry; this enabled, for example, the early Indian mathema>cians to
work on approxima>ons (infinite series) that the Western (“European”) mathema>cs could not develop or become
comfortable with except past 17th century. For example, consider some brilliant results that came out in the 14th
centuryCEsuchasMaadhava’sseriesforpi(misnamedlateras“Gregory”series),orfasterconvergentseriesforpiin
theKeralaschoolofmathema>csfromthe15thcenturyCEonwards.
Roddam Narasimha (“Axioma>sm and computa>onal posi>vism,” 37 (35) 2003, EPW) discusses "computa>onal
posi>vism"asadis>nguishingpropertyofIndianapproachtomathema>cs;itwasbasednoton,forexample,some
indefensiblemetaphysicsoftheGreeks(eg.“circlesareperfectshapes,allplanetsneedtobeexplainedasmovingin
circles, hence epicycles”) but on diverse models that each needed to evaluated for suitability. (Note that
Bhaakarachaarya’s Siddhaanta Shiromani men>ons दृग्गिणतैक्य drg.ganitaikya explicitly.) We summarize some of his
deepinsightshere.Thebasicposi>onhistoricallyhasmostlyeitherbeenadeduc>ve/“Logical”oneoftheGreeksor
the“Computa>onalPosi>vism”oftheIndics(notethatotherculturescertainlyhavehadsomecomponentofeither
butwewilltakeGreekandIndicasexemplars).Thela@eraxtude,o<enimplicitlyandsome>mesexplicitly,informed
the classical Indian mathema>cal approach to astronomy. Aryabha>iyam, for example, “provides short, effec>ve
methods of calcula>on rather than a basic model from which everything can be deduced”; essen>ally, it describes
algorithmic or computa>onal astronomy. This is opposite of Euclidean method of going from well stated axioms
throughaprocessofpurelylogicaldeduc>ontotheoremsorconclusions;evenwheretried,itwasusuallyana@empt
ataxioma>cdevelopmentwithadhocelementsinlargemeasure.A<erAryabha>iyam,aprofusionofdiverseideasin
mathema>csthenensuedsuchasthedevelopmentandfloweringoftrigonometryinIndiaandinnova>vesolu>onof
intricateintegerquadra>cequa>onsusingalgorithmictechniquessuchasChakravaala;thesefinallyfoundtheirway
toEuropethroughPersiaandArabia.Notethat(inthewordsofProf.MDSrinivas),“Indiantradi>on,unliketheGrecoEuropean tradi>on, does not envisage the possibility of arriving at absolute truth through the faculty of "reason".
However, there is no rejec>on of the validity of the method of "logical reasoning" anywhere in the Indian
mathema>cal /astronomical literature (with logic being given equal importance as in काणादं पािणनीयं च
सवर्शास्त्रोपकारकम् |). Indianmathema>csmaynotbeformulatedasadeduc>vesystembasedonaxioms.However,
theuseofwellacceptedgeneralprinciplestoformulateimportantresultssuchastheBaudhaayana(“Pythagoras”)
theorem and systema>c use of reasoning (aided by diagrams etc if necessary) to demonstrate newer results from
already demonstrated results is seen all over in Bhaaskaraachaarya's works or Yuk>bhaasha. Similarly, while
observa>onsareveryimportantinIndianastronomy,logicalreasoningbasedonmodelsofceles>almo>onarealsoof
importance. Only the theore>cal models have not been raised to the pedestal of presen>ng the "true picture of
mo>onoftheheavens".”
Posi>vismpositsthatfactsaretheonlypossibleobjectsofknowledgeandsciencetheonlyvalidknowledge;thereis
no need for metaphysics. “Logical” posi>vism of the famous Vienna Circle (scien>sts, mathema>cians and
philosophers)infirsthalfof20thc.hadacentraltenetofverifiability:“astatementthatcannotbe'verified'washeld
tobemeaningless”.Thereareinthisperspec>veonlytwotypesofmeaningfulstatements:thenecessarytruthsof
logic,mathema>csandlanguage,andempiricalproposi>onsabouttherestoftheworld,withWi@gensteinposi>ng
thatproposi>onsoflogicandmathema>csaretautologies!However,GodelandPopperinthe1930’sdemolishedthis
schoolofthoughtcomprehensivelyintheirownways.
Computa>onal Posi>vism, as argued by Roddam Narasimha, is that computa>on and observa>on, when in
agreement,cons>tutetheonlyformofvalidknowledge;models,logic,metaphysicsetc.areeithersecondaryornot
relevant.Modelsmaynotbeunique(inthesensethatdifferentmodelsmayyieldverysimilarresultsinadomainof
interest)andlogictautology(Wi@genstein)!
ThebestexampleinIndiaisthatoftheKeralaschoolofmathema>cs,agroupofastronomersandmathema>cians
who, over a period of some three centuries, produced some very innova>ve and powerful mathema>cs applied to
astronomy.Thebasicgoalisthatofदृग्गिणतैक्यdrg.ganitaikya,theiden>tyoftheseenandthecomputed.Thereisan
effort to find “best” algorithms or computa>onal procedures that made the best predic>ons as determined by
comparisonwithobserva>onas,overaperiodof>me,discrepanciesbetweencomputa>onandobserva>ontendto
increase. Niilakantha (1444-1545 CE) says explicitly that “the best mathema>cians have to sit together and decide
how the algorithms have to be modified or revised to bring computa>on back into agreement with observa>ons”.
Also,“theindica>onthatashaastrahasbecomeslatha[inadequateorweak]isthatonenolongerhasdrgganitaikya
(concordancebetweentheoryandobserva>on)”(formoredetails,seeMDSrinivas,“IndianApproachToScience:The
Case Of Jyothisastra,” 2011). Furthermore, “one should search for a siddhaanta that does not show discord with
actualobserva>ons(atthepresent>me).Suchaccordancewithobserva>onhastobeascertainedby(astronomical)
observersduring>mesofeclipsesetc.Whensiddhaantasshowdiscord,i.e.whenanearlysiddhaantaisindiscord,
observa>onsshouldbemadeofrevolu>onsetc.(whichwouldgiveresultswhichaccordwithactualobserva>on)and
anewsiddhaantaenunciated.”Hencethisperspec>veisthatofdevelopingacomputa>onalmodelwithexperimental
backingandalsoclosetohowmodernscienceviewsmodelsandobserva>ons.