Tag Archives: Ganita

Vedanga Jyotisha

The Great King Suchi of Magadha
His calendar was a royal mess
because the equinoxes precess
until he learned 'the lore of time' from Sage Lagadha.

Jyotihsastra

Jyotihsastra is the ancient Indian ‘science of light’ [2]. It includes within it the field of astronomy, which was known as Nakshatra Vidya (the science of the stars). Jyotihsastra is used for dik-desa-kala nirnaya (triprasna), i.e. to determine (direction, location, time) [4]. The Vedanga Jyotisha is an ancient text focused on Jyotisha, one of the six Vedangas. The Vedic texts, including the Upavedas and Vedangas, are harmoniously interlinked into an integrally united knowledge system. No one part of this system can be properly understood through an isolated study [1]. A key purpose of the Vedas is the performance of Yagnas correctly and on time. Time-keeping is the goal of Vedanga Jyotisha.

We resume our study of Ganitasastra at ICP through an inquiry into Jyotihsastra. This post is not an exhaustive restatement of facts. Instead, we try to understand the motivation and intuition behind the Ganita features of Vedanga Jyotisha (VJ). The Shulbasutras, which are part of the Kalpa Vedanga are also rich in Ganita, and will be discussed separately.

College students asked a professor 'Sir, what is time?' who replied "I can tell you what is the time, but I cannot tell you what is time"[4].

Vedanga Jyotisha

VJ is the earliest extant Indic work on time-keeping in the form of a handbook that is devoted to Kalavidhanasastra, the science of time-keeping. It provides the calculations associated with a lunisolar calendar derived from the Brahmanas and the Vedic Samhitas. VJ is not a self-contained treatise and any missing definitions, unstated assumptions, etc., are to be inferred from prior Indic sources and commentaries [1].

Vedanga Jyotisha has absolutely nothing to do with Phalita Jyotisha or Astrology [11]. 

The VJ was compiled around 1350 BCE (between 1150-1550 BCE) and is attributed in its verse to Lagadha, and key ideas in the VJ have been shown to belong to the Vedic texts and derived from earlier periods. VJ is in verse form while the other 5 Vedangas (Nirukta, Chandas, Kalpa, Vyakarana, & Siksha) are in Sutras indicating that it is the earliest of the six [1]. VJ was neither the first nor the last word in Indic time-keeping and astronomy as the Indians continued to make pioneering contributions to Ganita and Jyotihsastra over three millennia. These techniques enabled the Indics to produce a stable working calendar that could be employed for diverse purposes, and was sought after by the rest the world. The ancient Indic calendar traveled to China, and many other places [11].

VJ is available in the form of two ‘rescensions’ denoted as Rigveda Jyotisha (Arca Jyotisha, RVJ, 36 verses, earlier version) and Yajurveda Jyotisha (Yajusa Jyotisha, YVJ, 43 verses), which significantly overlap. Deciphering these rescensions turned out to be a challenging task. This effort started in the 1830s, culminating in the authoritative work of Prof. Kuppanna Sastry [1] in the 1980s who succeeded in meaningfully explaining all verses. Virtually every contemporary study of VJ cites his scholarship.

Time-keeping traditions of India

Vedic Cosmology — The Dharmic View of Time

We will devote considerable space discussing the unbroken traditions of astronomy and time-keeping that preceded Vedanga Jyotisha.

The Indic approach to discovery quite naturally arises from Rta, the cosmic order that is an expression of Satya, the ultimate reality. This cosmic order is experienced at every level from the microcosm to macrocosm. Time is sacred in this cosmology, and we have the kalachakra representing cyclic time, and it is intuitive that elapsed time can be tracked using precisely recurring rhythms of different durations that abound in nature. 18th century British scientist John Playfair who studied Hindu time-keeping in a manuscript obtained from Thailand, wrote an extensive treatise and was amazed by the Indic conception of cyclical nature [8]. He made several other important observations, which can be found within the cited references.

The second is the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom. 
- Physical Measurement Lab at NIST

Our solar system is quite flat, and hence the moon and most of the planets are located in a narrow region around the ecliptic, the apparent path of the sun in the sky during the day. This is really convenient for observation since it eliminates the need to focus on the innumerable luminaries that are away from the ecliptic. The moon’s path is within 5% of the ecliptic. The Ancient Indics kept time based on the periodicity of the (apparent) motion of the sun and the real motion of the moon. The study of planetary movements was not necessary to achieve this goal and does not concern VJ.

why isn't there a solar eclipse every New Moon night?

Careful observation was a critical component of Vedic astronomy and this became the hallmark of the Indic approach to discovery and obtaining valid knowledge in general, where all schools of dharma unanimously accept Pratyaksha Pramana [11]. From the perspective of accurate time-keeping required for Yagnas, kalavidhanasastra is a pratyaksha sastra [4], and is not deduced from a ‘black box’ math model.

By the early Rig Vedic period, one or more calendars were already in use for managing day to day activities. Time-keeping is critical for agricultural planning, e.g. to coordinate activities associated with the beginning and end of seasons, and continues to be important to the Indian economy [11].

The earth's equatorial plane is tilted at an angle of 23.5° with respect to the ecliptic plane. This results in varying seasons and daylight hours.

The Vedic people knew about the solstices and employed a six-season calendar which is special to India (it included a rainy seasonVarsha Rtu with months Nabha and Nabhasya). Obviously, the ability to accurately predict the arrival date of monsoons has always had significant economic value in India. The twelve tropical months along with their seasons in the Yajurveda are [2]:

Madhu, Maadhava in vasanta (spring),
Sukra, Suci in greeshma (summer),
Nabha, Nabhasya in varshaa (rainy),
Isa, Urja in sarada (autumn),
Saha, Sahasya in hemanta (winter), and
Tapa, Tapasya in sishira (freezing).

In 2004, agricultural operations were mistimed in India. Why? The monsoon was officially considered 'delayed' in the government calendar. In reality, it arrived on time per the traditional Indic calendar [11].

The trinity of adhidaiva, adhibhuta, and adhyatma are integrally united via Bandhus in the Vedic knowledge system [2, 9].  There exists a deep and ancient connection between Yagna (‘ the workshop where Bandhus are forged between the microcosm and macrocosm’ [9]) and time-keeping. Knowledge of the luminary phases was used to ensure that the monthly (Darshapuranamaasa) and seasonal (Chaturmasya) Yagnas were performed at the correct times [3]. The Atri family priests had the knowledge required to predict solar eclipses. By the time of the Yajurveda, the Hindus knew that a solar year was slightly more than 365 days. And importantly from a VJ perspective, a five year Yuga was already known, along with the need for two intercalary months to complete a Yuga [1].

pic source. By careful and patient daily observation of the sun at the same time in the sky, one can find out when the solstices occur (‘when the sun stands still’).

Prajapati as Time

Prajapati the creator is central to Vedic tradition. In his book ‘Being Different’, Rajiv Malhotra quotes the Rig Veda: “yajna is the very navel of the universe. It was Lord Prajapati who first fashioned yajna, and through it he wove into one fabric the warp and weft of the three worlds (Rig Veda I,164,33-35).” [9]. Prajapati creates and embodies a self-sustainable, self-correcting universe using the correspondence principle of bandhuta to achieve a balance between homogeneity and heterogeneity [9]. Prajapati is time, the very creator of the Vedas, signifying that the knowledge within the Vedas has no beginning or end [2]. He is Rta, the cosmic rhythm moving in a spiral, which indicates the Kalachakra, cyclical time [14].

Prajapati and Yagna are central to Vedanga Jyotisha, and receive the first respects in the starting verses of the VJ. The natural periodic events such as seasons, days, etc. are the five limbs of Prajapati, who personifies and presides over the five-year Yuga [1]. The separated faculties and limbs of Prajapati unite to form the infinite diversity of the universe, and the Yagna becomes a time-design to unite this multiplicity and continue the cosmic rhythm [14]. This five year Yuga is mirrored in Yagna through the constructed five-layered Agnicayana altar [6]. The Aahavaniya altar is built using 396 bricks that represent the days of the year: 360 to represent the Vedic ritual year and an additional 36 to represent the thirteenth (intercalary) month [3]. Many such bandhus arise through Yagnas [2]. The five-year Yuga is also a feature of Jain astronomy [6].

VJ states that those who correctly understand the effect of time on movements of the luminaries in the sky can fully grasp the impact of the Yagnas. One who truly understands the Vedas and Vedanga Jyotisha can experience transcendental bliss. These verses underline the integral unity [9] of the outer-material and inner-spiritual realms. We can see this dharmic concept re-asserted two thousand years later in the initial verses of Aryabhatiya, and more recently in Ramanujan’s approach as well.

The Indics were more than pattern seeking enthusiasts; they sought within patterns the deepest unity underlying nature’s diversity, and from this emerged the Yuga.  Yaga, Yoga, and Yuga (or the 3 Ys, with apologies to Modi ji) – all have a root meaning ‘to unite’. In [13], Prof Subhash Kak notes: “the ancient Indian calendar is an attempt to harmonize the motions of the Sun and the Moon…. Yoga may be seen as the harmonization of the motions of the inner planets of the body.  Patanjali’s Yoga Sutra speaks of how meditation on the Sun reveals the nature of the world-system and meditation on the Moon and the Polestar reveals the arrangement and the motion of the planets and the stars. Such assertions imply that turning inward can provide insights.”.

Nakshatras in Vedic Tradition

Since the most ancient time, Hindu astronomy adopted the sidereal system. This was done implicitly using Nakshatras (stars or asterisms) in the Vedic period, and explicitly in the VJ, as well shall see later [16]. The Vedics used 27 Nakshatras in the vicinity of the ecliptic to track the lunar passage where the moon takes 27.32 days to return to a fixed reference point (sidereal lunar month). To identify the Nakshatra location of the sun, a heliacal rising and setting of a Nakshatra seems to have been employed, i.e., a Nakshatra may be visible near the horizon just before sunrise or sunset.  Texts point to a multi-disciplinary approach to Jyotisha employing a Nakshatra Darsha (expert observer/astronomer) and Ganaka (a calculation expert). The term ‘Nakshatra Vidya’ is mentioned in the Chandogya Upanishad [2].

Mentions of Nakshatra observations in ancient texts are useful because they allow us to date these events using the earth’s precession rate. For example, Kuppanna Sastry quotes the Satapatha Brahman, mentioning that ‘the asterismal group Kritika never swerve from the east while others do’, which was also confirmed by the commentator Sayana. This yields a date of around 3000 BCE. Independent studies using modern astronomical simulation software and mathematical calculations (statistical best-fit models) indicate that the Nakshatras were closest to the path of the moon around 3000 BCE [10]. Subhash Kak has written extensively on the astronomical codes embedded within the Vedas [12].

It is clear that long before Vedanga Jyotisha, there was significant progress in time-keeping. It confirms the epistemological continuity in Indic sciences, including Astronomy and Ganita since the most ancient of times [5].

Epistemological continuity is also evident in other India's diverse traditions including art, music, dance, etc.

These prior developments are the foundation on which VJ’s calendar stands.  Let us see how VJ improves upon the prior work.

Vedanga Jyotisha’s Methods

VJ introduced an analytical time-tracking (deterministic) framework that works in tandem with astronomical observations of the real, uncertain world. Hence VJ’s Ganita calculates the timings of lunar and solar events, while also retaining and working in sync with the traditional pratyaksha sastra.  The Ganaka can make predictions, and the Nakshatra Darsa can visually confirm the degree of accuracy of these estimates, and corrections effected as needed. The diagram below illustrates how the VJ methodology can be useful in taking the science of time-keeping forward and provide increasingly accurate answers to triprasna.

Contemporary time-keeping adopts a similar approach. The atomic clock serves as an unnaturally perfect model for daily usage, but is corrected by nature. Without the latter, the model-based time would very slowly but surely drift away from reality.

The most recent leap second was added on December 31, 2016.

Nakshatra-sector Coordinate system (NCS)

Nakshatras (as stars or star groups) have been an integral part of Indic culture and some of them serve as exemplars. Dhruva (a northern pole star) and the Vashishta-Arundati (Mizar-Alcor) pair are good examples.

Prior to the VJ, the Nakshatras were used to denote visible stars or constellations (27 or 28 in number) dotting the moon’s path. Hence, it was limited by visibility.  Furthermore, these Nakshatras served as approximately fixed positions for time-keeping but were not truly invariant due to earth’s precession (‘precession of the equinoxes’). The designated pole star, for example, changes over time and cycles every 25,920 years (about a 1° shift every 71.6 years).

The Ancient Indics must have been aware of the impact of earth’s precession on the Nakshatra locations because, by the time of the VJ, the nakshatra-sectors were taken as 27 equal sections of the ecliptic (about 13.3° wide) rather than specific stars or asterisms in the background [3]. This change yields multiple benefits.

  • The NCS is an invariant and uniquely Indic coordinate system that comes with a clearly specified origin (zero-point) that gives us a fixed starting coordinate. It is unaffected by the earth’s precession. The NCS resembles the modern-day ecliptic coordinate system calculation of the celestial longitude (since the moon’s path is very close to the ecliptic, tracking longitude was practically sufficient) [3].
  • The NCS represents a virtualized analytical framework that allows the time-keepers to algorithmically enumerate the ecliptic sector locations of all the full and new moons in a Yuga, as well as the position of the sun. This was not possible in prior Vedic traditions since theirs was a purely physical coordinate system indentified by stars and asterisms along the moon’s path. This VJ system is free of visibility issues [3]. The VJ specifies a coordinate system using an ingenious ‘Jāvādi arrangement‘. Of course, pratyaksha continues to guide accurate time-keeping.
  • This NCS helps us carry out the VJ calculations unambiguously.

Yuga

The VJ Yuga is a time cycle of 5 years of 366 days each. A five year Yuga was already present in Vedic tradition. The Yuga is an integral unit of time-keeping in the Vedanga Jyotisha and all calculations are given based on this Yuga and the NCS.  VJ assumes 12 synodic months in a synodic year plus two intercalary months (adhimasa or adhikamasa) over a Yuga to harmonize the lunar and solar calendars, giving us a total of 62 synodic months in a Yuga.  The VJ specifically includes the adhimasa as synodic months #31 and #62 of the Yuga.

A VJ Yuga is completed when the sun and moon are observed to return to the pre-specified origin region of the NCS. This is the key definition in the VJ. Here is Sri Kuppanna Sastry’s description [1]:

In other words, the Yuga begins when the Sun and the Moon are observed together in the Sravistha Nakshatra sector of the ecliptic [3].

The Parameters of a Yuga (YVJ)

Tracking the movements of spherical objects rotating and revolving around other moving spherical objects can be tricky. Here is a ‘coin rolling on a coin’ puzzle, where the inner circle serves as a fixed frame of reference. If the inner circle also rotates, then the answer is relative to the chosen frame of reference.

How many rotations will the smaller coin make when rolling around the bigger one? (source: https://plus.maths.org)

Earth-Sun System (days and years)

Saavana durations represent the time of the (apparent) motion of the sun relative to the earth as the frame of reference. Each saavana year in VJ lasts 366 days, giving a total of 1830 civil days in a Yuga. In reality, this frame of reference is itself slowly revolving around the sun in the same direction, and therefore saavana calculations ignore the resultant additional earth rotation (one per year). Sidereal periods are calculated with respect to a fixed reference point (e.g. distant star). The sidereal year includes this ‘missing’ rotation, giving us 367*5 = 1835 sidereal days in a Yuga. VJ’s Nakshatra Darshas would’ve observed 1835 risings of a Nakshatra (an invariant ecliptic sector) in a Yuga.

Earth-Moon System (months and fortnights)

The moon is ‘tidally locked‘ to the earth. The actual time the moon takes to go round the earth (sidereal lunar month) is the time it takes to complete a full rotation around its own axis. So one side of the moon always faces us as if it never rotates, and we never get to see the mysterious far side of the moon (photographed for the first time in 1959).

"And if the dam breaks open many years too soon 
And if there is no room upon the hill 
And if your head explodes with dark forebodings too 
I'll see you on the dark side of the moon". - Pink Floyd.

Let us calculate the number of moon rises and the number of sidereal lunar months in a Yuga. This visible side of the moon will be partially or fully observable on all sidereal days except the new moon days, of which there are 62 (one per synodic month). This gives us 1835-62 = 1768 moonrises in a Yuga, and 1830/1768 saavana days per moon rise on average.

Similar to the earth-sun system, the earth-moon system also yields an extra rotation per year depending on the frame of reference. Due to the earth’s revolution, the moon takes a couple of days extra to complete the synodic month (~29.53 days) relative to the earth. There will be 62+5 = 67 sidereal lunar months in a Yuga.

Since there are 27 sectors of the ecliptic, the moon visits 67*27 = 1809 Nakshatra sectors in a Yuga. Therefore, the moon traverses one sector in 1830/1809 = (1 and 7/603) Saavana days. The sun apparently visits 27*5 = 135 Nakshatra sectors, spending 13 and 5/9 days in a sector.

We now examine some of the larger units used in the VJ to keep time.

Larger Time Units

Saavana day: measured from sunrise to sunrise. The VJ takes the civil year to be 366 days long. Each day is divided into 124 Bhaagas (day-parts). 31 parts make a pada.

Tithi: This is a fundamental unit of the VJ equal to (1/30) of a synodic month. Hence a lunar month lasts 30 tithis, and the VJ assumes 360 tithis or 12 synodic months in a year in harmony with Vedic tradition.  Thus, a Yuga has 1860 tithis and 1830 saavana days. From this, we can calculate the VJ mean value for a tithi = 1830/1860 or 61/62 of a day.  The duration of a tithi depends on the moon’s orbit and is a variable quantity (+/- 15% the mean value), with the tithi at sunrise representing a day’s tithi [3]. Sometimes, the same tithi can mark two successive sunrises or a tithi can be lost between two sunrises. Tithi was already used in prior traditions. In the Rig Veda, atithi is a guest – one who arrives without a tithi, i.e. without prior notice [15].

A tithi can go AWOL
It can be a really close call
All ye star-crossed suitors beware
Date your Nakshatras with care!

Given the diversity of India, its calendars are also diverse. Reckoning dates for dharmic events can be tricky even in 2017. This informative subtitled video asks ‘When is Ugadi?’. Yugadi ~ start of a new yuga (new year), Hevilambi, per current Hindu lunisolar calendars.

There are also variations such as Amanta/Amavasyat versus Purnimanta calendars depending on whether the start of a month is from a new moon or full moon.

Rtu (season): Its duration is 62 tithis long, and therefore a Yuga will have 30 rtus, and 6 rtus a year. An important and unique feature of the Indian calendar is the use of six seasons including the all-important rainy season, the most celebrated and joyous of all rtus. The monsoons are governed by the annual wind patterns influenced by the Coriolis force [11]. The first rtu of the Yuga is Sishira rtu (winter).

The VJ also specifies the duration of rtu using the NCS (4 and 1/2 Nakshatra sectors per Rtu). Knowing the start date of a rtu is also important because of the Chaturmasya rite that has to be performed.

Ayana (solstice): Ayanas divide the sidereal year into two halves. There are 10 ayanas in a Yuga.

Paksha: half a synodic month, equal to 15 tithis. The bright half is the Shukla Paksha and the dark half is the Krishna Paksha. A Yuga has 62 Shukla and 62 Krishna Pakshas.

Parva: The Yuga is divided into 124 Parvas. Therefore it is equal in duration to a Paksha. The Parva Raashi (R) is the accumulated heap of Parvas since the start of a Yuga and is quantified as follows [3]: R = 2(12(y-1)+m) + p + K,

where  y = current year of the Yuga, m = elapsed months in the current year,  p is the elapsed parvas in the current month, K is an conditional correction factor (2 per 60 elapsed parvas) to adjust for intercalation.

Visuva (Equinoxes): The day when the sun apparently starts to move south or north and they occur at the mid points of each of the 10 Ayanas in a Yuga.

The interval between two successive Visuvas will be 124/10 Parvas = 12 Parvas and 6 Tithis. Hence the time elapsed in a Yuga until the N-th Equinox can be obtained by multiplying this inter-Visuva number by (N-1) and simplifying.

Bhaamsa (Amsa): To track the position of the sun and moon, every ecliptic sector is also divided into 124 equal Bhaamsas, mirroring the Parva time division of the Yuga. Hence, there are 27*124 Bhaamsas that spans the 360°. The Bhaamsa after p parvas is the remainder obtained after dividing 11p/124 [3]. The VJ rescensions state an equivalent conditional and arithmetic rule that anyone can use, similar to the previous expression for parva raashi R.

Kalaa: A day is divided into 603 parts. This number is chosen so that the time taken by the moon to traverse one of the 27 nakshatra sectors (1 and 7/603 days) = 610 Kalaas, a whole number.

VJ gives many ingenious algorithms (abhiyukti) to keep track of the number of Parva, Bhaamsa, Paksha, etc. that have passed since the start of the Yuga. The interested reader is referred to Prof. Kuppanna Sastry’s work.

Intra-day Units of Time Keeping

Researchers point to 4 different kinds of times tracked by VJ [3] apart from the cosmic time. We point these out while listing the different time-keeping units.

Akshara (2 Maatraa) ~ 0.57 seconds. Time taken to pronounce a long vowel. This  time-unit is interesting and suggests the existence of a long and well-established oral tradition.

Kaastaa (5 Aksharas) ~ 1.15 seconds.

Kalaa revisited (124 Kaastaas) ~ 2 minutes, 23 seconds. Kalaa establishes a link between the rate of speech to the average rate of lunar motion.

Naadika (10.05 Kalaas) ~ 24 minutes. Mechanically-kept time using a water clock. Passages in the Vedas [7] suggest the use of a particular water clock of the ‘overflowing type’.

Muhurta (2 Naadikas) ~ 48 minutes. Solar time based on the Sun’s apparent motion. Amazingly, the ancient muhurta measure has been preserved and passed through several generations and is used in India the exact same way, to this day.

Length of local daylight time in Muhurtas = (12 + 2N/61), where N is the number of days after the winter solstice.  Since there are 183 days in an Ayana, the maximum increase is 6 Muhurtas. Using this, the ratio of the longest to shortest day is 18/12 = 3:2. This number depends on the latitude, and therefore helps us identify the source location of VJ.

Ahoratra (30 Muhurtas) ~ ‘day and night’, or 24 hours.

Bhaaga: The local time given in 124-th parts of a day starting from sunrise. Thus we see three divisions of ‘124’ in the VJ: Parvas in a Yuga, Bhaamsas in a Nakshatra sector, and Bhaagas of a day.

Note how a speech rate is linked to lunar time, then to mechanical time, and solar time. These physical temporal cycles of varying durations are ultimately united with the cyclical cosmic time through periodic Yagna performed at the right times.

Ganita

Algorithms

The VJ approach to specifying numerical constants is pretty elegant. The high-level parameters, which are fewer in number, are enumerated. For the myriad of lower level constants that proceed down to the intra-day level, it cleverly specifies algorithms based on a linear estimate (mean motion), using rules derived from modulo arithmetic. By specifying any three independent parameters of a Yuga, all other Yuga parameters can be calculated as derived values [1].  YVJ rescension’s second verse is famous for asserting the position of Ganita as the pinnacle of sciences [1].

VJ’s methods demonstrate ancient Indic abhiyukti. They do not provide a proof of correctness, but are to be validated by pramana. When a Ganaka’s analytically predicted quantity is in conflict with observation (pratyaksha), it is the model result that is discarded, and this also forces the model to improve.

Rule of Three: Linear Estimate

The VJ uses mean motion (average rate) as a first-order approximation within its calculations using the “rule of three“.

For example: Suppose we have a known average increment for a quantity ΔQ over a time period Δt, we can calculate an average rate of change = ΔQ/Δt. What will be the accumulated value of Q after T time units? A linear estimate will simply multiply this rate by time to obtain Q = (ΔQ/Δt)*T.  The VJ states the rule of three in verse, so that it can be used repeatedly as a subroutine: calculate an average rate and multiply the increment by this rate to generate the desired output.

Modular arithmetic

The VJ works with periodic quantities that get reset to 0 after reaching a maximum value. Doing calculations with such quantities requires expertise with modular arithmetic. 3000+ years before Gauss introduced formal modular arithmetic in 1801, the Hindus were actively applying modular arithmetic for calculating a variety of elapsed and remaining time values and the positions of the full and new moons over a Yuga.

Javadi Table

The Javadi arrangement is an important contribution from a Ganita and VJ calendar perspective. It represents a virtualized (independent of stars in the background) invariant ecliptic coordinate system with a zero point taken as the new moon near the Winter Solstice, which is tied to the start of a Yuga [3]. Javadi ~ Jau Adi, i.e., arrangement of Nakshatras starting from Jau (Ashvayujau) [1]. The position of the sun and new/full moon can be located unambiguously by the Javadi name of the Nakshatra sector and Bhaamsa within that sector. The table exhibits compact data organization and a circular ordering of the NCS data so that Sravistha represents Nakshatra sector 0 (or 27).

Simple Coordinates

From the Yuga parameters and the NCS, the ‘distance’ between successive full (new) moons can be calculated as follows:

The moon passes through 1809 Nakshatra sectors in a Yuga. There are 62 full moons and 124 pakshas in a Yuga. There, the distance between two full moons is 1809/62 = 29 and 22/124 Nakshatra sectors, and a paksha length is half this value (14 and 73/124). By partitioning a sector into 124 bhaamsas, we obtain a simple  (sector, bhaamsa) coordinate system using the original Vedic ordering of Nakshatras = (N_original, B) of new and full moons, where N_original and B are whole numbers.

Order-and-Chaos

The ganita properties of the full and new moon’s bhaamsas are interesting and we did not find much discussion on this, so we make an attempt here. A brief ganita description is in the appendix at the end of the post. Let us start from a new moon at bhaamsa B(0) = 0, and add 73 bhaamsas to obtain B(1) = 73 for the first full moon, and further 73 bhaamsas to get the bhaamsa B(2) of the second new moon, and so on.  We can observe the following patterns:

  1.  The full or new moon will be wandering around, visiting each and every bhaamsa number exactly once. A full or new moon will never be seen twice in the exact same location (bhaamsa) of the ecliptic within a Yuga. When it does so at bhaamsa 0 in the Sravistha sector, the Yuga is reborn (reminiscent of Kolam patterns).
  2. The 62 full moons of a Yuga occur at odd numbered bhaamsas, and new moons at even bhaamsas (if we start the Yuga at bhaamsa zero). At the full moons, the Sun’s coordinates will be 13.5 Nakshatras apart, i.e 13 sectors and 62 bhaamsas away.
  3. The nakshatra sector and bhaamsa are themselves linked, so if you specify just the bhaamsa, you can obtain the corresponding N_original value:
N_original = 5B mod 27 
N_original is remainder we get when we divide 5 times its bhaamsa by 27.

Of course, one can also calculate the N_original values directly as an independent check in case the input bhaamsa values are off. The VJ authors next transform the original Nakshatra sector list into the Javadi arrangement. It simplifies the required Ganita a bit (B instead of 5B).

Javadi Coordinates

The Nakshatra sector numbers can be transformed into a certain Javadi arrangement (N_original→N) using the following equation:

N = 11 N_original mod 27

Successive (original) Nakshatra sectors are 11 sectors apart in the Javadi arrangement. Conversely, successive sectors in the Javadi arrangement are 5 sectors apart in the original table. The Javadi arrangement starts from Ashvini and the final list is shown below [1]. The Sanskrit verse form of the Javadi representation is depicted at the top of this post.

Bhaamsa Generation Algorithm

This transformed N is related to B in this Javadi arrangement through a simpler modular equation compared to N_original. The (N, B) Javadi coordinates for all full and new moons of a Yuga can be iteratively generated (see appendix) and are shown in the plot below (X-axis = Javadi nakshatra sector indices, Y-axis = bhaamsa numbers). These coordinates would repeat every Yuga.

Javadi coordinates in terms of (Nakshatra-sector index, Bhaamsa) for all full and new moon in a VJ Yuga
Examples

The first full moon in a Yuga is at B = 73, which gives us a remainder N = 19 when divided by 27⇒ coordinates (19, 73). Therefore, the full moon occurs in the Magha sector per the Javadi table. The next full moon will be at bhaamsa B = 73+22 = 95. Applying N = 95 mod 27 ⇒ N = 14, i.e. Uttaraphalguni (14, 95).  Multiple full moons (2 or 3) can fall in the same Nakshatra sector, but always at different bhaamsas. For example, the next and only other full moon (38-th) in the Magha sector will occur when B = (73-54) = 19. The two full moons that occur in Magha are circled in light-blue in the above picture. Note that the 3 new moons along the Y-axis at Sravistha (X = 0) are at least 54 bhaamsas (about 5.8°) apart.

Ecliptical Coordinate System

The (N, B) from Javadi are equivalent to an ecliptic longitude. These results have been compared with those generated using the modern ecliptic coordinate system, and they are quite close [3]. Tracking the bhaamsas empirically is important and this can be done mechanically using a water clock. The Javadi table is deterministic and assumes fixed synodic month duration [3], so that every Yuga starts at coordinates (0, 0). This is not so in reality, and in the next section, we can see the maximum error that is possible. Since the origin is shifted, so will the calculation for every successive full moon. While the full moons may occur in the same Nakshatra sector, the bhaamsas will be off unless the origin-shift is accounted for. The Javadi table can be used as an approximate framework/guide for the Yagna calendar and supplemented with direct observation.

We have only discussed only a few of the high-level VJ calculations. For a detailed discussion, refer to [1].

Accuracy of some VJ calculations

Mean Tithi

VJ Value = 61/62 of a saavana day.

Modern estimate of an average synodic month ~ 29.5306 days

Modern value of tithi ~ (29.5306 * 12)/360  ~ 354.367/360

Absolute Error = |354.367/360 – 61/62| < 0.05%

A Yuga has 1860 tithis, so accumulated error ~ 0.896, or less than a tithi per Yuga [2].

Mean Moonrise Rate

VJ value = 1830/1768 ~ 24 Hours 50.4864 minutes [2], i.e., the moon rises about 50.4 minutes later every day. This agrees with the modern average moonrise value really well.

Start time of a Yuga

The new moon at the start of a new Yuga may not be exactly at bhaamsa 0 of the Sravistha sector. It has been shown that up to 46 bhaamsas error can accumulate over a period of 500 years [3]. Since the moon traverses a Nakshatra sector (124 bhaamsas) in 610/603 saavana days, using the rule of three, we find that the moon traverses 46 bhaamsas in 9 hours. This is less than the minimum gap (54 bhaamsas) between successive full or new moons in the same Nakshatra sector. The maximum possible cumulative error in the start time of a Yuga after 100 Yugas is 9 hours [3].

Yuga: Self-Organizing System

In general, the VJ seems to be relatively more accurate while calculating lunar periods compared to solar periods [2]. Over the next two millennium, the Hindu lunisolar calendars were significantly upgraded. The Ancient Indics were aware of the uncertainty in the true motions of the sun, earth, and moon, and the need for corrections. The Indian comfort with uncertainty [9] is perhaps reflected in the fact that the civil calendar was deliberately set up as a simple, convenient, and approximate framework for the astronomical (Yagna) calendar. The discrepancy between the arithmetic and astronomical calendar can be fixed using an intercalary day at the end of the Yuga [1]. They also synchronized the sidereal and tropical year using appropriate corrections. Beyond these basic corrections, the lunar-solar year gap can accumulate over Yugas. It has been discovered by researchers [1, 3, 6] that the properties of the VJ Yuga yields a self-correcting system that automatically cancels out these errors.

Lunisolar correction

Five tropical years at 1350 BCE = 5*365.1734 ~ 1825.9 days

Duration of a Yuga = 62 * 29.5306 ~1830.9 days

Difference ~ −5 days per Yuga or roughly one extra day per tropical year.

If this discrepancy is allowed to accumulate over 6 Yugas (sometimes 7), the total gap will be approximately a synodic month. A Nakshatra Darsha doesn’t even need to know the Ganita behind this. He/she simply sees the sun and moon together in the Sravistha sector to signal a new Yuga. The unnecessary intercalary month 61 is automatically skipped, which resets the accumulated error.

Some corrections were made by observation of the moon phase. At the new moon the moon rises and sets with the sun. If the moon rises just after sunrise, it indicates a time near new moon. Such observations enabled the Vedics to develop the rules required for an accurate timing of the Yagnas since certain Yagna performers would incur a penalty if they erred in the timing [1]. Thus Vedic Yagna is the creative driving force that inspires this self-correcting calendar. A self-harmonizing Yuga seems natural in Prajapathi’s self-organizing universe.

Date and Source of VJ

Date

Embedded within VJ’s verses is an astronomical date-stamp about Sravistha. If α-delphini is taken as the Yogatara (principal star) of Sravistha, then between 1550 BCE and 1150 BCE, the nakshatra Sravistha and the sun would have been close at the winter solstice, i.e., the Nakshatra rises and sets heliacally at the winter solstice, and this is not possible for dates outside this period [3]. If a certain other star other than α-delphini is chosen as the Yogatara, the date gets pushed back beyond 1800 BCE [7]. Kuppanna Sastry’s ganita calculations using the earth’s precession rate, and based on the observation of the VJ author that the winter solstice was at the start of the Sravistha segment, yields dates in the range [1150, 1400] BCE. Statistical analysis of the Nakshatra system shows that a maximal proportion (80%) of the Yogataras occupy their respective Nakshatra sectors in [1300 +/-300] BCE, indicating the finalization of the NCS during this period [3]. From [5], we find mention of a date of 1255 BCE when King Suchi of Magadha, a student of Lagadha [6] set forth VJ and dated it by including an astronomical note about the summer solstice. When combined with other independent considerations such as the visibility of the Saptha Rishi (Ursa Major) from Bharatvarsha, the timing of Yagnas in conjunction with seasons, full moon, and prescribed Nakshatras, we obtain a date range [1400 +/- 300] BCE for Vedanga Jyotisha [3].

Source

Multiple works show that the Nakshatra (star) system was most likely designed around 3000 BCE [2, 3, 10]. There is clear evidence of a continuous unbroken epistemology of time-keeping from the Rigveda Samhitas to the Vedanga Jyotisha.

Independent researchers have studied the 3:2 ratio of longest to shortest day, which is only possible around a certain latitude. This includes locations in far-northern India as well as other places. The calendar with a rainy season is also special to India. By also taking into account VJ’s date, several locations get eliminated from consideration, and Kashmir appears to be a likely location of the VJ author among the feasible candidates. This has been an independent conclusion reached by multiple scholars.

The Challenge of Vedanga Jyotisha

Kuppanna Sastry has listed three fundamental requirements for a scholar who wants to study and interpret Vendanga Jyotisha in its original Sanskrit verse [1]:

  • Sound scholarship in Sanskrit
  • Knowledge of Western Astronomy
  • Full understanding of the concepts and practices of Hindu Astronomy

Teamwork

Those who have been frustrated in this task have lacked one or more of the requirements stated above. It is not necessary for one person to have all three skillsets. We have a precedent from 3000 years ago, when Nakshatra Darshas and Ganakas combined their skills to take Indic science and technology forward. Today, traditional Vedic Pandits grounded in Sanskrit and Hindu cosmology, and STEM professionals can work as a team to overcome new challenges in many areas. The first and third requirements involve dharmic tradition, which requires shraddha and sadhana, something every team member must imbibe. The Swadeshi Indology initiative serves as an inspiring example in this regard.

Several luminaries have contributed their expertise toward explaining the time-keeping ideas of Jyotihsastra. This post summarizes the student notes compiled while learning from and exploring these truly enlightening works, which are listed in the references below.

'If you were in Darkness, what would you want more than anything else; what would it be that every instinct would call for? Light, darn you, light!' - Nightfall, Isaac Asimov.

References
  1. KV Sarma and Kuppanna Sastry. Vedanga Jyotisa of Lagadha In its Rk and Yajus Rescensions. With the Translation and Notes of Prof. T. S. Kuppanna Sastry. Critically edited by K. V. Sarma. Indian National Science Academy. 1985.
  2. Subhash Kak. Astronomy and its Role in Vedic Culture. Chapter 23 in Science and Civilization in India, Vol. 1. The Dawn of Indian Civilization, Part 1, edited by G.C. Pande. ICPR/Munshiram Manoharlal, Delhi, 2000.
  3. Prabhakar Gondhalekar. The Timekeepers of the Vedas: History of the Calendar of the Vedic Period (From Rgveda to Vedanga Jyotisa). Manohar Publishers. 2013.
  4. K. Ramasubramanian. Perspectives on Indian Astronomical Tradition. HH Dalai Lama Premises. Dharmasala. 2016.
  5. Kosla Vepa. The Origins of Astronomy, the Calendar, and Time. Lulu.com. 2011.
  6. Narahari Achar. Enigma of the Five Year Yuga of the Vedanga Jyotisa. Indian Journal of the History of Science (33). 1998.
  7. Narahari Achar. A Case for Revising the Date of Vedanga Jyotisa. Indian Journal of the History of Science (35). 2000.
  8. John Playfair. The Works of John Playfair (Vol. 3).. with a memoir of the author. Edinburgh, A. Constable & Co. 1822.
  9. Rajiv Malhotra. Being Different: India’s Challenge to Western Universalism. Harper Collins. 2011.
  10. Sudha Bhujle and MN Vahia. Possible Period of the Design of Nakshatras and Abhijit. Annals of the Bhandarkar Oriental Research Institute. 2006.
  11. C. K. Raju. The Cultural Foundations of Mathematics: The Nature of Mathematical Proof and the Transmission of the Calculus from India to Europe in the 16 c. CE.  Pearson Education. 2007.
  12. Subhash Kak. The Astronomical Code of the Rig Veda. Oklahoma State University, Stillwater. 2011.
  13. Subhash Kak. The Wishing Tree: Presence and Promise of India. iUniverse Inc. 2008.
  14. Kapila Vatsyayan. The Square and The Circle of The Indian Arts. Abhinav Publications. 1997.
  15. R. N. Iyengar. A Profile of Indian Astronomy before the Siddhāntic Period. ISERVE Conference, Hyderabad, India. 2007.
  16. Kuppanna Sastry. The Main Characteristics of Hindu Astronomy in the Period Corresponding to Pre-Copernican European Astronomy. Indian Journal of the History of Science (Vol 9). 1974.
Appendix & Acknowledgements
Acknowledgments: Thanks to N.r.i.pathi garu for encouraging me to write this post, and for his Baahubali-esque patience and valuable feedback.
Appendix

The bhaamsas of the full or new moon are generated using the recurrence relation:

B(k+1) = B(k)+73 mod 124.

This is an example of a linear congruential generator (LCG) that is commonly used in computer simulation models. The sequence of bhaamsas visited by the full or new moon in a Yuga are pseudo-random numbers. Since 73 and 124 are relatively prime, this LCG is guaranteed to have a full period (124) that exactly spans a Yuga. The Hull-Dobell theorem (1962) proves the result for the general case. It is also easy to see that if B(k) is even, then B(k+1) will be odd, and vice versa. We can simply generate the bhaamsas to verify this.  The following algorithm generates the chronological sequence (N(k), B(k)) of all new and full moon positions of a Yuga in Javadi coordinates:

1. Initialize: k = 0, B(0) = 0.
2. N(k) = B(k) mod 27. If B(k) is even, it is a new moon, else full moon.
3. B(k+1) = B(k) + 73 mod 124.
4. if k=123 stop. Else, k=k+1; go to step 2.

An Indic Perspective to Mathematics — 3

mahaviracharya

(This is the concluding part of the sequel to ‘Introduction to Ganita’)

Part 1 (Introduction) 

Part 2 (Ganita – Math Encounters)

Part 3 (below): Ganita prevailed over Math in their encounters, but what did it really win? While Ganita’s results were absorbed into Mathematics, the underlying pramana and upapattis were rejected. We explain why this happened, and its implications.

Digestion Of Ganita, the Needham Question, and the Road Ahead
Nothing in life is to be feared, it is only to be understood. Now is the time to understand more, so that we may fear less - Marie Curie.
The Digestion of Ganita

It appears the ancient Babylonians had something in common with the Indians: they were pattern-seekers. As far as trying to understand how the world around us works, Richard Feynman rejected (Greek) Mathematics in favor of what he recognized as a Babylonian method, as discussed in this lecture below. Despite this endorsement, it is the Greek approach that drives Mathematics today, while the Babylonian culture can be found only in famous museums today. Why?

It would not be a problem for any civilization to view and benefit from imported knowledge by employing a native lens, without denigrating and destroying the external source tradition, and based on mutual respect. However, when knowledge from another culture is deliberately cannibalized and appropriated as a predator, it is a serious problem. It turns into a process known as’digestion‘. We now describe how Ganita was digested into Mathematics after their encounters.

The Digestion of Ganita into Mathematics

This process of digestion has been laid out by Rajiv Malhotra in [4]. We apply this description step-by-step to see how Ganita was digested into Mathematics.

Step 1.The less powerful culture is assimilated into the dominant one in such a way that: the dominant civilization dismembers the weaker one into parts from which it picks and chooses which pieces it wants to appropriate“.  During their encounter, all the important results of Ganita, starting from the place value system with zero, to algebra, trigonometry, algorithms, combinatorics, … to calculus were accepted by Europe to obtain real-life benefits. However, the underlying epistemology and approach of Ganita that has worked so well for 2000+ years, and could be used to generate such astonishing results in the future were amputated from Ganita. Only the results were retained within Mathematics.

Step 2.appropriated elements get mapped onto the language and social structures of the dominant civilization’s own history and paradigms, leaving little if any trace of the links to the source tradition“. The formal Math rooted in the Greek tradition was enhanced and expanded so that the Ganita results could be systematically re-derived and reinterpreted in a compatible manner. Later, the beneficial features of the native Encuvati system of pedagogy was appropriated into the British teaching approach, and ‘undesirable’ features were deleted to ensure compatibility with ‘Christian values’ [15]. Once this process was complete, the source tradition of Ganita was expendable.

Step 3.the civilization that was thus mined gets depleted of its cultural and social capital because the appropriated elements are modified to fit the dominant civilization’s own history, and these elements are shown to be disconnected from, and even in conflict with, the source civilization“.

A. The credit for a re-engineered calculus was given to Newton/Leibniz and not Madhava and the Kerala School. We are taught the Pythagoras theorem and proof, not Baudhayana’s result and validation procedure. Fibonacci numbers, not Gopala-Hemachandra series. The IEEE journals recognize Arab numerals, not Hindu numbers, and so on. The list is long. In almost all these cases, the standard reason is that the Indians had not proved their results using the formal system devised by the west, even though each of these results were generated first by Ganita and also satisfactorily validated within the source tradition, often centuries earlier. The Ganita tradition was erased from the history of Mathematics.

B. On the other hand, the following types of claims are created:

  • Vedanga Jyotisha was full of astrology and religious mumbo-jumbo
  • Ganita was some kind of elite “Vedic Mathematics”
  • Hindu tradition was backward, caste-ridden, superstitious and incapable of producing such advanced scientific results.

Whereas, the exact opposite is true.

  • Vedanga Jyotisha is the science of time-keeping, and “the entire Jyotisa does not have a single sentence relating to astrology or prophecy” [1], whereas the main goal of European calendar reform was to advance the cause of organized religion [1]
  • Ganita was pragmatic and accessible to ordinary Indians including vegetable vendors who taught the greatest Arab scholar of their time [14], while today’s formal Mathematics is indeed the preserve of an elite few [1].
  • Hinduphobia is rampant in the Humanities departments of Western universities, which is subsequently exported to Indian universities, even as the digestion of Hindu science and technology results continues unabated [16].

Step 4. The final result is catastrophic for the source civilization: “the depleted civilization enters the proverbial museum as yet another dead culture, ceasing to pose a threat to the dominant one. After being digested, what is left of a civilization is waste material to be removed and destroyed.”  A mathematical monoculture was imposed on India during the colonial era after uprooting the ‘beautiful tree‘, India’s indigenous decentralized education system whose Ganita curriculum was sensitive to local requirements. Few students and teachers in Indian schools and universities today are aware of the source Ganita tradition. Among those who recognize the word,  few realize it is not an Indian neologism for Mathematics. Is this not an instance of cultural genocide?

How can we protect and revive the authentic and practical Ganita tradition that was the head of all the Indian sciences? To do this, we must identify the nature of the civilizational ‘Poison Pills’ within Ganita.

Civilizational Poison Pills

Rajiv Malhotra introduced the idea of civilizational poison pills from an Indian perspective in ‘Indra’s Net’. [13]. “Poison pills are those elements or tenets that cannot be digested into the DNA of a predator, because consuming them would lead to the destruction of the predator’s constitution. If a predator absorbs such an element, it will mutate so profoundly that it will lose its original identity and qualities.”  We now try to identify the poison pill in Ganita that needs to be preserved.

Ganita’s Poison Pill

The Indians achieved a smart reduction in uncertainty in calculations to a contextually admissible level, instead of beating themselves up trying to attain complete certainty. Ganita and Vedic thought recognizes that human understanding of the cosmos is never fully complete. In [4], the Indian and western mindset is compared thus: “Indians indeed find it natural to engage in non-linear thinking, juxtaposing opposites and tackling complexities that cannot be reduced to simple concepts or terms. They may be said even to thrive on ambiguity, doubt, uncertainty, multitasking, and in the absence of centralized authority and normative codes. Westerners, by contrast, tend by and large to be fearful of unpredictable or decentralized situations. They regard these situations as problems to be fixed. As we shall see, there is in fact some scholarly evidence that demonstrates this view of Western attitudes.” For a mindset that revels in perfection, this element of uncertainty that was acceptable within Ganita is a poison pill. This anxiety was evident in all stakeholders in Europe during the Ganita-Math encounters.

Western Fear of Uncertainty

Practically every Western point of view from the ultra-secular, to the religious during the Ganita-Math encounters was in conflict with Ganita’s poison pill:

  • In the abacus-algorismus battle, Ganita’s idea of ‘one manifesting as many’ in its place value system and the way it managed non-representability was suspect, given the scope for ‘chaos’ and ‘fraud’.
  • For a reasoning mind like Descartes, measuring the ratio of curved to straight lines involved an irreducible uncertainty, an understanding of which was beyond the human mind. This gave rise to the term ‘irrational numbers’ [1]. Not surprisingly, he rejected the idea of infinitesimals too.
  • Philosopher Thomas Hobbes was no friend of the Jesuits. But he too found the absolute, perfect order found in Euclidean geometry was its most appealing aspect and reflected his own perspective. As noted in [12] “in their deep structure, the Jesuit papal kingdom and the Hobbesian commonwealth are strikingly similar. Both are hierarchical, absolutist states where the will of the ruler, whether Pope or Leviathan, is the law.”
  • The Jesuits, Protestants, Eastern Orthodoxy, Anglicans, and a vast majority of Christian sects may have disagreed on some theological points, but all subscribed to the history-centric truth claims of the Nicene Creed [4]. At least three aspects of Mathematics would’ve appealed to them:
    • Calendar and time-keeping helped preserve history centric dogma and reestablish the importance of clergy.
    • The top-down, hierarchical perfect Eucliean order.
    • Proving theorems without need for empirical demonstration. History-centric Christianity treats the body as a vessel of original sin. Embodied knowing is problematic for this mindset.
  • Pioneer Jesuat monk Cavalieri underwent an inner struggle [12] after ingesting this poison pill, and all but disowned his Ganita-based idea of ‘indivisibles’.
  • Scientists who championed the cause of the infinitesimals, and their successors could never come to grips either. The Tagore-Einstein conversation is a good example. As mentioned in [4] “Not even Einstein was able to reconcile himself to the uncertainty inherent in quantum mechanics, prompting him to remark: ‘God does not play dice with the universe.’ But Shiva and Parvati, the Hindu cosmic couple, do happily play dice. Indian philosophy is receptive to the uncertainty theories of physics.

See Article 

However this poison pill does not negatively impact the Indian mindset. Why? Our Ganita Post discusses in detail, but we briefly summarize here for the sake of completion.

Ganita’s Comfort in Dealing with Uncertainty

The Indians were comfortable working with contextually accurate estimates for non-quantities like √2 and π, recognizing that the result could be improved upon.  Hindu society has no central authority that could ban innovation or the exploration of the realms of uncertainty. Its decentralized structure produced independent thinkers and innovators in every era. Dharma systems have built-in safeguards against Hobbesian/Church absolutism. As Rajiv Malhotra explains in [4] “Chaos is entrenched in the Vedas, the Puranas and Hinduism in general for a reason: its role is to counterbalance and dilute any absolutist tendencies as well as provide creative dynamism through ambiguity and uncertainty.” Ganita inherits all these features, and must retain all these properties for best results.

The inevitability of uncertainty was no cause for panic. It even opened up a degree of freedom for (dharmic, ethical) optimization using Yukti.  This comfort with uncertainty is visible right through Ganita’s storied history from Paanini‘s Ashtadhyayi before the common era, to the Aryabhatiya in the 5th century C.E, within the calculus results of Madhava in the 14th century, to Ramanujan in the 20th century. This perspective placed the Indian creation of all its algorithms, interpolations, calculus, etc. on solid epistemological ground. Let’s look at the Aryabhatiya, as an example.

Aryabhata‘s R-sine difference table shown below required an algorithmic package that managed uncertainty every step of the way in a transparent manner: one method for estimating square-roots, another for interpolation, and yet another non-mechanical exception step to generate an optimal final estimate for each value in the table. The Kerala Ganita experts extended such prior work to infinite series, including their own innovative exception terms [1].

Source: Indian lecture series on Mathematics [14]

Western mathematicians who reviewed Ramanujan’s notes found that he often used the terms “nearly” and “very nearly”[10]. Ramanujan came up with clever, non-mechanical approximations for specific quantities like π. Some of his approximations eventually lead to exact results. His exact infinite series for π triggered the most dramatic leap in accuracy since Madhava [14]. Some examples of his approximations are shown below [10].

ramanujan-1

ramanujan-1

The Indian approach seeks balance between chaos and order [4] and represents a dharmic optimization under uncertainty.

Eliminating uncertainty and deleting Yukti, Upapatti, and Pramana from Ganita to digest it, drains it of key features that make it a powerful and reliable approach for solving real-life problems. Furthermore, lack of Pramana can lead to pseudo-science and fraud, as we will see shortly. Preserving these features within an Indian approach to Mathematics has the twin benefits of recovering pragmatism and making the subject understandable and usable by everyone. It protects against further digestion and denigration of the source tradition.

Finally, How can Ganita preserve this poison pill while continuing to retain its open architecture [13] and confidently exchange knowledge with other cultures?

The need of the hour is a thorough and systematic purva paksha of Mathematics and Modern Science, employing an Indian lens.

We don’t have to be a Manjul Bhargava to experience some differences between Ganita and Math.  We can simply try out the basic instruments employed within each subject.

Indian Rope vs Euclidean Geometry Box

One of C.K. Raju’s most important contributions is his cogent argument for a fundamental change in the way math is taught in Indian schools and colleges.

Source: fastudent.com

The rope is a key entity in Ganita and the Darshanas. A fundamental feature of the rope is its flexibility, reflecting the idea of ‘one manifesting as many’. The night-time confusion between a rope and a snake is an example that has been used Dharmic seekers to communicate the deep ideas about the nature of ultimate reality.

Source: Library of Congress

The knotted rope is a critical component of the ancient Indian navigational instrument known as the rapalagai  or kamal [1]. The ‘Sulba’ in the Sulba Sutras means ‘cord/string/rope’, and the rope served as a measuring tool since ancient times. Consequently, as C.K. Raju notes, the circumference can be the independent quantity measured quite naturally using a rope, with the straight line radius derived from this. A mathematical mind measures the straight line (Euclidean distance) first. A geometry box consists of an assortment of rigid straight-edged tools, and each one is used for a specific operation.

source: Indian Mathematics Lecture Series [14]
A knotted string can measured curved lines. When it is stretched taut between pins, it becomes a straight line, and with one of the pins freed, it behaves like a compass. This strings-and-pins set can be used to construct squares, rectangles, circles, etc, i.e., its flexibility reproduces the functionality of an expensive geometry box at a fraction of the cost. It unlocks the creativity of Ganita and is available even to the poorest student.

Indian Nyaya versus Aristotelian Logic

From the Indian point of view, two-valued (Aristotelian) logic can play a supporting role (e.g. like tarka [22]) but does not enable a person to attain a higher level of consciousness [4]. Note that such reductive logic is different from the holistic logic of Nyaya, which accepts multiple pramanas. In fact, no major school of Indian thought directly mentions deductive logic as pramana [22]. On the other hand, all major Indian schools of thought accept pratyaksha pramana, which in rejected by Mathematics [1]. Misusing two-valued logic (that has no place for uncertainty) as pramana negates Ganita’s poison pill.

Mathematics in India Today

The  current approach to teaching mathematics in India appears to be a stressful  and boring mixture of bits-and-pieces of Ganita mashed up with partially understood formal Mathematics imported from the west. This digested teaching approach has been successful in confounding multiple generations of Indian students. The modern rote/mechanical mode is a distortion of the original approach of recollective memory, which was a distinct mode of learning that cultivated the amazing computational (Ganita) abilities of the Indians [15].

Repeat after me:

“An acre is the area of a rectangle

whose length is one furlong

and whose width is one chain” – Pink Floyd, The Wall.

The 2016 Hindi movie ‘Nil Battey Sannata’ (~ 0/0) dramatizes this state of confusion. The movie claims that Math is a natural enemy of girls (“Ladkiyon ko Maths se purani dushmani hain“). While this may or may not be true,  the daughters of Lilavati  should not experience any difficulty with Ganita. For the great Shakuntala Devi, Ganita was a bandhu, not an enemy. The sophisticated Ganita within Kolam designs attests to the embodied learning capability within women. Let us also not forget the women engineers of ISRO who mastered the Ganita of rockets and spacecraft (yes, Ganita’s calculus without limits can do this well [1]).

ISRO staff celebrating ‘Mangalyaan’ success. credit: www.aniruddhafriend-samirsinh.com

The intrepid mother in the movie tells her daughter that “maths yaad karne ki nahi, samajne ki cheez hai“, while the maths-savvy classmate advises: “ek baar maths se dosti karke dekho, usse majhedaar aur kuch nahi“. A key scene in the movie shows everyday, familiar objects from real life being used to convey this ‘samaj’ – clearly a Ganita rather than an Euclidean solution to an Indian problem [15].

In formal math, even something as simple as a point (Bindu) gets hairy. (Euclid: A point is that which has no part, then graduate to this).  A blind import of western approaches into the Indian classroom without subjecting it to a thorough purva paksha,  is a folly not just restricted to Ganita, but one that been repeated in different areas of study including social sciences, economics, religion, art, etc. The net result is years of misery for most Indian students followed by a trip to the west to get it straight from the horse’s mouth. S. Gurumurthy has repeatedly noted the negative impact and the poor track record of such a reductive mathematics in solving practical problems in the Indian economic context.  We close with a discussion on contemporary mathematics and the way forward.

The Needham Question
"With the appearance on the scene of intensive studies of mathematics, science,  technology and medicine in the great non-European civilisations, debate is likely to sharpen, for the failure of China and India to give rise to distinctively modern science while being ahead of Europe for fourteen previous centuries is going to take some explaining” - Joseph Needham.

Many Indian scholars have attempted to answer this complex question. However, virtually all of these responses that try to provide social/religious explanations offer little insight due to a shallow understanding of dharma and Ganita traditions, and the inability to do a systematic Purva Paksha of the western approach using an Indian lens. We quickly summarize three perspectives below noting that we are only scratching the surface here.

A. Several centuries of foreign occupation

This occupation of India ranked among the worst and longest-running genocides in history and was characterized by violence that specifically the Indian intellectuals. Such a strategy is likely to have taken a heavy toll on Indian R&D output and institutions. When there was a sustained break from this violence, e.g., the time period of  the Vijayanagara empire,  we observe that Ganita, Ayurveda, astronomy, and other sciences achieved significant progress.

B. Civilizational inertia: complacency or weariness?

The sharpest debates in India occurred internally, between the various darshanas, which may have shifted the focus away from the study of external cultures entering India. There appears to be no evidence of a thorough study of the axiomatic approach from a native perspective. The Indians may have identified the lack of integral unity in the western approach and rejected it without any further examination of possible useful features.  CK Raju notes in [1] that it was only in the 18th century that India got the Elements translated from Persian into Sanskrit (by Jai Singh). This lack of a systematic Purva Paksha is not limited to Ganita alone but is also seen in many other areas, as pointed out by Rajiv Malhotra [16], suggesting an overly inward focus, careless disunity against an external threat, and a lack of strategic thinking.

C. The Unreasonable Effectiveness of Mathematics

Roddam Narasimha’s analysis examines a question complementary to Needham’s: what are the reasons for a sudden European resurgence after 1400+ years of backwardness in science and technology? He cites a key reason for their resurgence in the 17th century: the mathematization of science. Galileo is his study of the motion of falling bodies, used the calculus (via Cavalieri) to came up with the ‘law of the parabolic fall’. This is considered the first ever quantitative representation of motion using mathematical equations [12].  Scientists thereafter began to develop effective quantitative models relating different physical quantities like velocity, momentum, etc. using abstract models and calculus.Newton titled his famous scientific work as ‘Principia Mathematica‘. These mathematical models, however ‘wrong’ they may be, helped in new discoveries.

Indian Ganita experts too may not have anticipated this unreasonable effectiveness of mathematics when they rejected it for centuries. Narasimha summarizes this in [17]:  “Modern science seems to have acquired, perhaps by fortunate accident, the property that the great Buddhist philosopher Nagarjuna called prapakatva: i.e., it delivers what it promises; it may not be the Truth, but it is honest“.

The Road Ahead

Ganita, in the more recent interactions with modern science and math has made positive contributions, e.g., Satyendranath Bose and Narendra Karmarkar. The Bose-Einstein statistics comes out of counting exercise and is a significant contribution to Quantum Mechanics[17]. Karmarkar is famous for inventing the first practically effective algorithm for solving linear programs that is also theoretically efficient. Karmarkar’s proof of convergence demonstrates Yukti in gradually reducing the level of uncertainty in the solution quality in way that is both practically viable, and theoretically rigorous (a teeny bit of uncertainty remains in the end but it can be safely ignored).  Clearly, interacting with and exchanging ideas with other cultures can be beneficial, provided it is done with eyes wide open.  Scientists and applied mathematicians today employ a variety of different methods, including deduction, induction, inference, etc., along with empirical validation, etc., to come up with new findings and inventions.

Per Roddam Narasimha, the Indians paid a price for rejecting the axiomatic approach, but their stance was vindicated later by the 20th century developments in Quantum/Classical Mechanics and Logic [17].  Furthermore, modern science is being increasingly plagued by a variety of harmful ‘viruses’ that would not affect a ‘Ganita OS’.

Unreasonable expectations from Mathematics

The mathematization of science has succeeded, but only when the order it brings is honestly balanced by the reality check of an unpredictable nature.  The unbalanced mathematization of economics has resulted in a series of spectacular failures when applied in real life. Indian thinkers like S. Gurumurthy have studied these economic models in depth, and opted for a balanced Ganita-like method, bringing in empirical validation and Yukti to determine practical solutions anchored in Indian reality. Western social science, which mimics the axiomatic approach is degenerating into a self-serving pseudo-science that offers little insight. A sizable proportion of results published in modern scientific journals are not reproducibleThis highly cited 2005 article discusses the implications.  And then there is the issue of fraud that is peculiar to the western modeling approach based on Aristotelian logic.

Falling for Supermodels
Without+photoshop+_305e904f954ae7c6b82bd7893278408d
Source: funnyjunk.com

Supermodels sell an advertising pitch, not reality. Yet the temptation of falling for the perfection of abstract math models and ignoring the uncertainty of the real world can be too strong. As [17] notes: “The history of Western science is shot through with the idea of theories and models and of fraud. Ptolemy himself has been accused of fraud; so in more recent times have Galileo, Newton, Mendel, Millikan and a great variety of other less well-known figures. I believe the reason for this can be traced to faith in two-valued logic.” All models approximate reality. When this gap gets too wide, it makes sense to reject that model. However, it is tempting to reject reality in favor of a pet model or preferred hypothesis by cherry-picking data, fudging results, or tweaking the model in ‘creative’ ways to ‘make’ it work (e.g. some ‘AIT’ models in the Indian context).

Ganita does not suffer from this issue. Why? As noted in [17] that when “observation is the starting point and one has no great faith in any particular physical model, which was the prevailing norm of Indian scientific thought, the question of fraud does not arise. Indian scientists, even classical ones, do not appear to have accused each other of fraud. This could not have been mere politeness, as they did make charges of ignorance or even stupidity against each other (as Brahmagupta did on Aryabhata, for example). We could say that fraud is the besetting sin of a model-making scientific culture“.

Synthetic unity has its advantages and has revolutionized modern science, but progress based on Integral unity is more sustainable.

Some western scientists and mathematicians may have sensed this lack of Pramana. Poincare explored the role of intuition and inference in his candid 1905 essay [18]. We even get a hint of integral unity here. Albert Einstein was aware of the limitations of Math when he noted “As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.” Contemporary mathematician Terrence Tao recognizes that there is more to mathematics than just rigor and proof [19]. Thus, we see a limited move by Math toward the Ganita position while remaining firmly grounded in its native western tradition. Ganita can reciprocate in mutual respect, anchored in its own epistemology. We conclude with an informal discussion on emerging technologies.

Digestion by Machine: Math versus Ganita

Ganita is well-suited for this era of decentralized internet, analytics, big data, and digital computing which is algorithm driven. The emerging world of Artificial Intelligence is also very interesting. We touched upon AI citing an important observation of Subhash Kak [20] in our post on Ganita. As AI becomes highly sophisticated, it will be able to automate many human capabilities. It may eventually master the axiomatic approach and digest the Euclidean mathematician.

On the other hand the Indian approach to knowledge is rooted in the correspondence principle of Bandhu. Potential fallibility is acknowledged. Machines cannot replicate embodied knowing since they lack Bandhus, and they will not have the ability to attain a higher state of consciousness. For example, machines cannot chant mantras. Next, this ‘Euclidean’ robot will be able to master scriptures, and emulate all text-prescribed functionality of a cleric. It can function as a virtual holy establishment by delivering impeccable discourses. It will become an expert of theology by encoding history-centric truth claims as axioms and applying two-valued logic. However, it cannot become a Yogi.  Learning Ganita and internalizing the Dharmic worldview offers job security in the world of robots!  India can lead the way forward by carefully reintegrating useful features of modern science and math into its Vedic framework [21].

References:
  1. Cultural foundations of mathematics: the nature of mathematical proof and the transmission of the calculus from India to Europe in the 16th c. CE, C. K. Raju. Pearson Longman, 2007.
  2. Plato on Mathematics. MacTutor History of Mathematics archive. 2007.
  3. Plato’s Theory of Recollection. Uploaded by Lorenzo Colombani. Academia.edu. 2013.
  4. Being Different: An Indian Challenge to Western Universalism. Rajiv Malhotra. Harper Collins. 2011.
  5. Axiomatism and Computational Positivism: Two Mathematical Cultures in Pursuit of Exact Sciences. Roddam Narasimha. Reprinted from Economic and Political Weekly, 2003.
  6. Use and Misues of Logic. Donald Simanek. 1997.
  7. Computers, mathematics education, and the alternative epistemology of the calculus in the Yuktibhasa. C. K. Raju. 2001.
  8.  American Veda: From Emerson and the Beatles to Yoga and Meditation How Indian Spirituality Changed the West. Phil Goldberg. Random House LLC. 2010.
  9. Logic in Indian Thought. Subhash Kak.
  10. Ramanujan’s Notebooks. Bruce Berndt. Mathematics Magazine (51). 1978.
  11. C. K. Raju. Teaching mathematics with a different philosophy. Part 2: Calculus without Limits. 2013.
  12. Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World. Amir Alexander. Farrar, Straus and Giroux reprint / Scientific American. 2014.
  13. Indra’s Net: Defending Hinduism’s Philosophical Unity. Rajiv Malhotra. Harper Collins. 2011
  14. Mathematics in India – From Vedic Period to Modern Times: Video Lecture Series, by M. D. Srinivas. K. Ramasubramaniam, M. S. Sriram. 2013.
  15. Mathematics Education in India: Status and Outlook. Editors: R. Ramanujam, K. Subramaniam. Homi Bhabha Centre for Science Education, TIFR. 2012.
  16. The Battle For Sanskrit. Rajiv Malhotra. Harper Collins. 2016.
  17. Some thoughts on the Indian half of Needham question: Axioms, models and algorithms. Roddam Narasimha. Infinity Foundation. 2002.
  18. Intuition and Logic in Mathematics. English Translation of Essay by Henri Poincaré. 1905.
  19. The Pragnya Sutra: Aphorisms of Intuition. Subhash Kak. Baton Rouge, 2006.
  20. There’s more to mathematics than rigour and proofs. Terence Tao. 2009.
  21. Vedic Framework And Modern Science. Rajiv Malhotra. Swarajya Magazine. 2015.
  22. Epistemology and Language in Indian Astronomy and Mathematics. Roddam Narasimha. Journal of Indian Philosophy, 2007.
  23. The Math Page. Plane Geometry: An Adventure in Language and Logic based on
    Euclid’s Elements. Lawrence Spector, 2016.
  24. Continuity and Infinitesimals. Stanford Encyclopedia of Philosophy. 2005, substantive revision 2013.
  25. The Indian Origins of the Calculus and its Transmission to Europe Prior to Newton and Leibniz. Part II: Lessons for Mathematics Education. C. K. Raju, 2005.
  26. Why Write: Legos, Power, and Control.  F. D. Poston. Johns Hopkins School of Education.
  27. Indo-Portuguese Encounters: Journeys in Science, Technology, and Culture. Edited by Lokita Varadarajan. Indian National Science Academy. 2006.
  28. The Kerala School, European Mathematics and Navigation. D. P. Agarwal. Infinity Foundation Mandala website.
Acknowledgments: I'm deeply grateful to the ICP blogger and editor for their constructive comments, review, and feedback.

An Indic Perspective to Mathematics — 2

This is the Second in a Set of Posts as a follow up to our ‘Introduction to Ganita’.


AlgorithmGanita
Source: IIT Lecture Series on Indian Mathematics [14]
This Set of Posts on the Indic Perspective to Mathematics is the third installment of our continuing Series on Ganita.  Our first article in the Series celebrated Srinivasa Ramanujan. The Second provided an Introduction to Ganita. Emphases within quotes are ours.

Topic Outline

Part 1: (Introduction) 

In Part 1 of this Set of Posts on the Indic Perspective to Mathematics, we provided a background on the historical paradigms that drive the engines of Ganita and Western Mathematics respectively.

Part 2: (below) Ganita-Math Encounters. Ganita and Math came face-to-face when Indian Algorithms and Calculus traveled to Europe to help solve two critical problems: calculating with big numbers and managing the infinitely small. In a tense battle, Ganita’s balance of order and chaos prevails over the top-down Euclidean order backed by the church. We become aware of the massive contribution of the Vijayanagara empire to modern science.

Part 3: We adopt an Indic civilizational perspective of the Math-Ganita encounters. This gives rise to  interesting questions like ‘What was lost when Mathematics digested Ganita?’. We also look ahead, exploring the importance of Ganita and its Indian approach in a futuristic world.

Ganita-Mathematics Encounters
Experts have their expert fun
ex cathedra 
telling one 
just how nothing can be done. - Piet Hein.

In the Introduction to this Set of Posts, we studied the Greek origins of ‘Mathematics’. The abstract nature of Mathematics resulted in a drastically reduced practical output and Europe plunged into a 1000+ year dark era. During this period, Ganita contributions from Dharma thought systems helped keep math practically relevant in other parts of the world, right up to the 17-18th century CE. In particular, this injection of Ganita helped resolve two Math crises in Europe [1]. For the purpose of this post, we oversimplify and classify these problems as the ‘big’, and ‘small’ number’ crises. By helping resolve these crises, Ganita played a leading role in the birth and progress of modern science.

Big Number Crisis (Abacus vs Algorismus)

Here is example of a 10-digit Hindu number and its Roman numeral representation.

large numbers
credit: http://forbrains.co.uk/free_online_tools/convert_to_roman_numerals

There are several such websites that allow us to perform this conversion and three aspects stand out. First is the reference to ‘Arab numbers‘ in many sites. Second, is a maximum limit on the input. Third, ‘0’ or negative numbers are not valid input. The idea of ‘Arab numbers’ is of course, deep-rooted in the western STEM community to this day (IEEE journal publication guidelines still refer erroneously to ‘Arabic numerals’) since a large body of Ganita knowledge made it to the west via Arab translations of Sanskrit works. As can be gauged from the conversion tool, the Roman system is cumbersome for doing actual calculations. Its representation is additive in nature and there is no place value for zero, and the idea that placing a ‘0’ after a number would increase its value was befuddling. The west relied on the abacus / counting board, which was adequate for simple arithmetic calculations (the Indians did most of their routine arithmetic mentally). The introduction of ‘algorismus’ from India via Arab sources  around the 11th-12th century CE provided the merchants of Florence with an incredibly advanced way of quickly and accurately performing all kinds of numerical calculations [1].

Although traders found it to be practically useful, resistance to the alien method was stiff and it was several centuries (16th century) before the Hindu system gained unanimous acceptance. Well, almost. The British treasury preferred to place their money in the ‘secure’ hands of the abacus and held out until the 17th century [1].  By that time, the second math crisis in Europe was well underway.

Source: wikimedia.org.

Smiling Boetius‘ works with Hindu numerals to prevail over his opponent, Pythagoras, who is sadly stuck with a counting board abacus. This depiction of the victory of ‘algorismus’ is on the cover of Gregor Reisch’s Margarita Philosophica (1508) [1].

Aside from the suspicion of an Arab source in a crusading world, a technical reason for the distrust appears to be Ganita’s approximation techniques combined with the fear of zeroes being added to make sums bigger. To a mind accustomed to the perfection of Euclidean math, not even the tiniest quantity could be discarded. Such unacceptable imperfections could open the door to fraud and chaos [1]. The Indian approach, since the Sulba Sutras, recognized the non-representability of certain quantities (e.g. √2) and employed pragmatic and epistemologically secure approximation methods without anxiety, in order to reduce uncertainty (round-off error) to within an acceptable level [1]. ‘Algorismus’ was absorbed into European practice in order to resolve real-life calculations, but not the underlying pramana and empirical rationale (e.g. upapatti).  Why?

Small Number crisis (Infinitesimals and the Indian Origin of Calculus)

The Indian Background Story

Source: HaindavaKeralam| Zenith of Vijayanagara Empire
Brothers Harihara and Bukka, with the blessings of Rishi Vidyaranya, laid the foundation for one of the most important empires in Indian and world history in 1336 CE. In particular, the global scientific community owes the Vijayanagara empire a debt of gratitude.

While most regions of 14th century India reeled from the attack of fundamentalist invaders who had already destroyed India’s top universities and institutions, the Vijayanagara Empire became an oasis that protected and nurtured the Dharma. In particular, a school of Ganita was etablished in Kerala thanks to the prosperity and security enjoyed by the region during the Vijayanagara period, between the 14th and 16th century CE. An important member of this Ganita tradition was Madhava of Sangamagrama (~1350-1425 CE). This school produced a illustrious line of scholars who were the genuine adhyatmic and intellectual successors of Aryabhata, Bhaskara, and other great seekers. A major part of the foundation for modern science was laid by the Kerala school and the Ganita tradition they carried forward.

Recall that Aryabhata had already come up with finite difference equations for interpolation by 499 CE to generate fine-grained sine values. His practical approach essentially translates into Euler’s  18th century method for solving ordinary differential equations (ODEs). These results were subsequently improved upon by Brahmagupta (his second order interpolation result is known as ‘Stirling’s Interpolation Formula‘ today),  Bhaskara-2, and others [1]. Today, Indians are familiar with the phrase ‘Tatkal booking’ of train tickets. The ancient Ganita experts had developed algorithms  to calculate the Tatkalika gati of planets, their instantaneous velocity (an important quantity in Newtonian physics), as shown below.

Source: Lecture Series on Indian Mathematics [14]

We can observe a continual progress in India toward calculus, right from Aryabhata [1]. For all practical purposes, the Ganita school in Kerala during the Vijayanagara period can rightfully claim to be the developers of Calculus (from a formal mathematics perspective, western historians credit them for ‘pre calculus’). C.K. Raju has demonstrated the all-around practical viability of this epistemologically secure calculus without the use of ‘limits’ [11].

Madhava gave the world some beautiful and important results in infinite series by 1375 CE, centuries before Newton/Leibniz/Gregory/Taylor/McLaurin & Co.

madhava_collage
Source: Indian Mathematics, An Overview (https://youtu.be/p2WankcGP3Q)

In the derivation of these calculus results we can observe a smart management of the non-representability of infinitesimals based on order counting, along with a judiciously chosen exceptional / end-correction term (right side of the picture above). This is a really cool and important innovation that serves twin purposes, as explained by C. K. Raju below [1].

correction_term

There are many other novel ideas and instances of such Yukti within the Indian approach.  The interested reader can refer to [1] for a detailed description of the techniques employed.

It is worth comparing the meaningful Sanskrit non-translatable abhiyukti (expressing, or translating one’s Yukti in action) to its nearest English counterpart ‘algorithm’. The latter from the Latin ‘algorismus’, which in turn came from Al-khwarizmi who had translated Sanskrit texts of Ganita (see the picture at the top of this post). Jyesthadeva published the Ganita Yuktibhasa around 1530 CE in Malayalam, which provides the detailed mathematical rationale validating the Calculus results[1].

Why was Calculus Important to India?

Madhava’s infinite series with end-correction terms, allowed him to quickly calculate estimates for trigonometric values and π (pi) to very high levels of accuracy. For example, Madhava was able to calculate π to 11 decimal places, which represents both a quantitative, and methodological leap over prior brute-force type approaches (the next such dramatic leap was also due to Ganita, via Ramanujan) [14].  A natural follow-up question is: why were precise trigonometric values useful? Isn’t calculating π to many decimal places purely an academic exercise?  We summarize the reasons below, referring the interested reader to [1] for a detailed description.

Agriculture and Trade were key contributors to an Indian economy that played a dominant role on the world stage from 0 CE (and earlier) through 1750 CE.

Agriculture
  1. Krishi was and is a dominant component of the Indian economy. It was (and still is) dependent on a successful rainy season, which means that accurately calculating the arrival time of monsoons is important. A couple of weeks ago, the Indian government announced a $60M supercomputer project to better predict monsoons.
  2. Vedanga Jyotisha is primarily a science of time keeping that has numerous applications and has been recognized by researchers as a key source of knowledge in the ancient world [1]. It enabled the Indians to maintain an accurate calendar. Thus, from a Krishi perspective, the Ganita of Jyotisha acted as a decision support system for planning and scheduling key agricultural activities.
  3. The Indian calendar date and time was calculated with respect to the prime meridian at Ujjain (long before Greenwich), which was then re-calibrated to obtain local times at locations all over Bharatvarsha that covered a vast area (ancient India was united by time too!). This local re-calibration:
    • ⇒ required the calculation of the local latitude and longitude (lat-long)
    • ⇒ which (in the Indian approach) used the size of Earth as input
    • ⇒ this required a value for π
    • ⇒ trigonometric values were also needed for lat-long calculations
    • Precise numerical values were required since tiny errors get magnified after multiplication by big numbers (in the order of the Earth’s radius). Thanks to the Ganita tradition, the Indians had access to good estimates that were continuously improved upon.
ujjainmeridian
Source: builtheritageconservation| The Ujjain Meridian
Overseas Trade

India has a culture of calculation and embodied knowing that goes back thousands of years. Many ordinary Indians even today take pride in their ability to think and calculate on their feet, or pull off some Jugaad without the aid of electronic devices. The pattern-seeking Indian nature is visible in their traditional approach to navigation, reflecting an ability to discover sufficient order even within an ocean of chaos. The metaphor of the Samudra Manthana truly comes alive here.

  1. India, thanks to its manufacturing and technological prowess, had established lucrative trading relationships as a net exporter with several countries, from ancient Rome to the far east. Much was this was done through open sea routes, and not just sailing close to the coast [1].
  2. Prior to the 11th century CE, accumulated navigational knowledge included seasonal wind patterns (‘wind lore’), nature of ocean currents (‘current lore’), etc., and the empirical wisdom of sea-craft. The ancient Tamizh seafarers made use of the Saptharishi mandalam (Ursa Majorin the southern hemisphere. This database of seafaring wisdom and best practices were preserved, improved upon, and transmitted from generation to generation via the oral traditions of the seafaring Jatis [27].
  3. Thus, the Indian sailors had already established a tradition of navigation and deep sea voyage without written charts (they rejected the method of dead reckoning‘ in order to stay alive). Their approach included an empirical understanding of ocean patterns, Ganita, and instrumentation like the rapalagai (kamal) for celestial observations. Tamizh navigators deciphered currents using a simple device known as mitappu palagai [27].
  4. Such historical data further debunks the theory that oral traditions were ‘pre-rational’ and the sole preserve of Vedic scholars. Hinduphobic Indologists like Sheldon Pollock are dismissive of such priceless oral traditions [16]. The western universal idea of history begins with written text and it is tough for this mindset to imagine open-sea navigation without written charts.
  5. Accurately determining the local lat-long using celestial observations (solar altitude at noon, pole star at night, etc.) was part of this approach.
  6. More reliable navigation in the open seas is possible if the 3L: latitude, longitude, and loxodrome can be accurately obtained for any given location. These were indeed calculated in multiple ways by the Indians using trigonometric values [1].
chola sea route pic
Source: Indo-Portuguese Encounters [27] | Chola Sea Route
Continual Progress in Calculating Accurate Trigonometric Values
  1. Aryabhata’s astounding publication of his R-sine difference table along with an interpolation method stepped away from the geometrical approach that was employed until then [1]. The Aryabhatiya was a prized intellectual property of its time. It significantly improved the accuracy of trigonometric values (given the sine value of an angle, one can use elementary identities to calculate all other trig values).
  2. Aryabhata’s work paved the way for Calculus. Over the next 1000 years, the Indians steadily improved upon prior estimates.
  3. Calculus was a natural outcome of this process of deriving ever more accurate trigonometric values. The Kerala school’s calculus extended the finite series based trigonometric results to a highly accurate infinite series based approach.

We refer the reader to this essay [28] by D.P. Agarwal for his summary of the Kerala School, European Mathematics, and Navigation. It is highly likely that this Ganita knowledge traveled to Europe via European missionaries in Kerala and played a key part in revolutionizing physics and mechanics via Newton’s Principia Mathematica and other works.  This story serves as background for the question: why did the idea of ‘infinitesimals’ which was a non-issue in the Ganita world, spark a crisis in Europe?

The European Background Story

Ancient Greek math hit a roadblock after encountering paradoxes tied to infinitely small quantities. Mathematics could not deal with the irritating uncertainty around infinitesimals and the problem of non-representability: For example, an infinite number of threads of minuscule but nonzero length, joined end-to-end should yield an infinitely long thread. On the other hand, combining even an infinite number of threads of ‘zero’ length would only yield zero. Aristotle believed that continuum could be divided endlessly and could not be made up of ‘indivisibles’.

A famous paradox (which used to be popular among those preparing for engineering school entrance exams in India) is that of Achilles and the Tortoise. Around 500 BCE,  Zeno of Elea came up with several such paradoxes that exposed the gaps in a seemingly perfect mathematics and two-valued logic. Unable to satisfactorily resolve such contradictions and deal with non-representability of certain quantities (a fundamental requirement for numerical calculations), Greek progress halted. The dark ages robbed the west of native expertise and appears to have hurt them in key areas including, but not limited to [1, 27]:

  • Astronomy, Navigation, Instrumentation
  • Calendrical Systems, Ship Building
  • Medicine and Botany

After more than a thousand years, between the 12th-16th century, we can observe the emergence of a new kind of Mathematics in Europe, which was fundamentally different in its epistemology from the Euclidean approach. This knowledge first arrived via Arab/Persian translations of Ganita works in Sanskrit, and later through Missionaries who had direct access to Ganita’s latest results in Sanskrit and local Indian languages. We kick off this discussion using the European calendar as a case study.

Trick question: What came after Thursday, October 4th, 1582 in Europe?

The answer is Friday, October 15th. The European (Julian) calendar was slow by about 11  minutes per year for about 1200 years across their dark age. Church and Biblical dogma reigned supreme from the time the Nicene creed was formalized in 325 CE. This dogma can be best understood as an instance of history-centrism [4], and a key to preserving the credibility of this ‘history’ of unique divine intervention is proper time keeping and dating of these events. This was a key motivation behind the European quest for a better calendar.

The Indians had maintained accurate calendars since ancient times thanks to Vedanga Jyotisha for use within multiple applications, and Buddhists even helped with calendars in China [1] (helping the Chinese is an old Indian habit…). The Roman Church realized in 1582 that their calendar was trailing the correct date by 11 whole days. This key project of calendar reform was taken up by Christopher Clavius (1538-1612 CE), a Jesuit priest. Thanks to his painstaking work, Pope Gregory was able to press the fast-forward button on the calendar (thereafter named after him), recommend a leap year correction, and the rest is history.

Milanese artist Camillo Rusconi’s sclupture, 18th century. Pope Gregory is on top of an urn depicting the 1582 promulgation of the Gregorian calendar. Source: http://vminko.org/ under GNU Free Documentation License 1.3.

C.K. Raju has uncovered the Indian source of this calendar bug fix [24, 1]: “Jesuits, like Matteo Ricci, who trained in mathematics and astronomy, under Clavius’ new syllabus [Ricci also visited Coimbra and learnt navigation], were sent to India. In a 1581 letter, Ricci explicitly acknowledged that he was trying to understand local methods of timekeeping from “an intelligent Brahmin or an honest Moor”, in the vicinity of Cochin, which was, then, the key centre for mathematics and astronomy, since the Vijaynagar empire had sheltered it from the continuous onslaughts of raiders from the north. Language was hardly a problem, for the Jesuits had established a substantial presence in India, had a college in Cochin, and had even started  printing presses in local languages, like Malayalam and Tamil by the 1570’s.“. The Jesuits have continued to exercise their influence on the Indian education system to this day. They also played a key role in the second Math-Ganita tussle.

Jesuit (Euclidean) Order versus Indian (Ganita) Chaos

The Jesuits are members of the Society of Jesus, an organization founded by St. Ignatius of Loyola (1491-1556) and rooted in Roman Catholicism. Per [12] “In the broadest sense, imposing order on chaos was the Society’s core mission, both in its internal arrangements and in its engagement with the world.

Sir Peter Paul’s ‘The Miracles of Saint Ignatius of Loyola’ (Source: wikimedia.org)

This painting of Ignatius of Loyola is richly symbolic. It depicts the victory of a perfect top-down hierarchical order over chaos. Loyola and his back-robed Jesuits are in the middle, watched over by angels at the top. Loyola is calmly performing an exorcism, expelling the chaotic evil spirits possessing the bodies of terrified people at the bottom of the picture. [12] provides an insightful description of this picture, noting the role of the black robed Jesuits of Loyola’s Society of Jesus: “They are Ignatius’s army, there to learn from their master, follow his directions, and ultimately take over his mission of turning chaos into order and bringing peace to the afflicted. For that was indeed the “miracle” of St. Ignatius and his followers. Like no one else, they managed to restore peace and order in a land torn apart by the challenge of the Reformation.“.

The Church gained immensely via this decisive mathematical triumph of calendar reform, and Clavius who played an instrumental role, realized the benefits obtainable by investing in mathematics. This was a time period characterized by fissures and dissent in Christianity, with several alternatives and reformations (e.g., led by Calvin) cropping up that challenged the exclusive authority of the catholic church. In this climate, Clavius felt that the top-down hierarchical perfection within Euclidean geometry would be a great fit for the Jesuit curriculum, and in sync with the primary goal of their founder St. Ignatius of Loyola.

As mentioned in [12] “It was clear to Clavius that Euclid’s method had succeeded in doing precisely what the Jesuits were struggling so hard to accomplish: imposing a true, eternal, and unchallengeable order upon a seemingly chaotic reality. Just as Ganita was recognized as the foremost of the sciences in India since ancient times, Euclidean Mathematics became a most important subject in Europe after the calendar reform. The Society of Jesus embraced Math and all was well for a while. The focus had shifted to other pressing topics. For example, navigational challenges had to be overcome in order to ‘discover‘ reliable sea routes to new lands.

The Indivisibles

Calculus created a rather sudden splash into Europe within 50 years of the calendar reform [1]. By that time, the calculus, which was rooted in Indian epistemology had already been developed and studied for two centuries.  Bonaventura Cavilieri (1598-1647), a Milanese Jesuat monk and a student of Galileo was an early adopter. While the Jesuits were more like a MNC, the Jesuats were a local group of Italian monks lower in the pecking order. However, Galileo’s endorsement boosted Cavalieri’s profile significantly. Cavalieri introduced the ‘method of indivisibles’, in which “planes and solids had an indeterminate number of indivisibles” and authored the book Geometria indivisibilibus (Geometry by Way of Indivisibles) in 1635 [12].

While the idea of indivisibles was embraced by the Galileans, the Jesuits were not as welcoming. Those who worked with infinitesimal quantities did so for its practical value in generating realistic new results and could not really establish any logical consistency needed to prove infallible theorems. Unlike Euclid’s Elements which used top-down deductive logic to prove specific theorems from axioms, the use of infinitesimals required the ground-up Ganita approach: to start from physical reality and work toward generalized results, which could lead to innovation and potentially unpredictable discoveries. Clearly, Yukti was not welcomed by the church whereas Galileo’s methods were more compatible with Ganita.

Galileo Galilei (1564 – 1642 CE)

Galileo had become a formidable opponent by that time. He had earlier discovered the moons around Jupiter, and as a prashasthi [16] to a rich grand duke who ruled Florence, named the moons after him and his family. In return, he was rewarded with benefits that included the post of ‘Chief Mathematician’ to the Duke in 1611, which also freed him up to pursue his work as an independent researcher. As [12] notes, “The Galileans also sought truth, but their approach was the reverse of that of the Jesuits: instead of imposing a unified order upon the world, they attempted to study the world as given, and to find the order within.” This started a conflict between the Galileans and the Jesuits.

For the church, the idea that matter could be broken down into infinitely small indivisible atoms was unacceptable. The archives of the Society of Jesus in Rome records for posterity the ruling of their leaders in 1632 on infinitesimals [12]:” Judgment on the Composition of the Continuum by Indivisibles”. …The permanent continuum can be constituted of only physical indivisibles or atomic corpuscles having mathematical parts identified with them. Therefore the said corpuscles can be actually distinguished from each other.” The church basically ducked the question of non-representability and banned the idea and the mathematical study of ‘indivisibles’.

Among the critics of these indivisibles was Thomas Hobbes, the philosopher author of the Leviathan, who deeply influenced Western thought. Hobbes was also an excellent mathematician and a devotee of the Euclidean approach. He was bitterly opposed in this battle of the infinitesimals by John Wallis of England, one of the founders of the Royal Society, the new science academy [12]. Wallis had little time for eternal proofs, and was firmly rooted in what we can unmistakably recognize as the pragmatic Ganita approach for solving real-life problems. Hobbes had tried in vain for several years to prove that he could ‘square the circle‘, and each attempt in this futile exercise was eagerly demolished by Wallis and exploited to the hilt in their public feud [12]. Eventually, Wallis’ team ‘won’ the contest (possibly in terms of cultural and scientific acceptance) and Newton came up with his famous work Principia Mathematica that relies heavily on calculus. Interested readers can refer to [12, 1] for a detailed discussion.

The Ghosts of Departed Quantities

It is worth noting some logical inconsistencies in the positions of both sides in this battle. The church was fighting to save their dogmatic belief in an infallible and orderly Euclidean math against a group injecting a practically useful but poorly-understood imported concept into this math. Every researcher seemed to have his own pet model showing how the math of the infinitely small worked.  In an important and devastating piece of satirical writing, Anglican church bishop Berkeley ridiculed the questionable fluxions of Newton, and Leibniz’s ‘infinitesimal change’ as “the ghosts of departed quantities”. CK Raju concludes (as do others) that this calculus was not on firm epistemological ground.

The European approach appeared to be mechanical and did not, for example, employ the end-correction terms that had helped keep Indian derivation transparent and anchored in a valid pramana [1]. Mathematicians could not accept, understand, or were unaware of the Ganita rationale behind the amazing calculus results derived by the Kerala School. For example, it is known that “Newton later became discontented with the undeniable presence of infinitesimals in his calculus, and dissatisfied with the dubious procedure of “neglecting” them” [24].  Mathematics was enhanced so that calculus was eventually placed on a firm formal foundation in the 20th century [1].

Transmission of Calculus from India to Europe

The etymology of ‘calculus‘ (17th century CE, Latin) relates to ‘reckoning’ and ‘accounting’. This focus is entirely empirical and on calculation, far away from the Euclidean world of theorems and proofs. On the other hand, it is directly corresponds in meaning, intent, and usage to Ganita. So far, research has uncovered three kinds of evidence linking Indian Calculus transmission to Europe: documentary, circumstantial, and epistemological. The interested reader is referred to [24, 1] for details. A primary, initial motivation for appropriating Ganita’s calculus results appears to be the practical problem of navigation: to obtain accurate trigonometric values required to calculate the 3L mentioned earlier [1].

A note in [24] on the circumstantial evidence is worth stating: “Unlike India, where the series expansions developed over a thousand-year period 499-1501 CE, they appear suddenly in fully developed form in a Europe still adjusting to grasp arithmetic and decimal fractions“. The 1400+ year discontinuity in the study of infinitesimals  in Europe was followed by a sudden upsurge in results in the 16th-17th century [12], right after Ganita’s documented achievements in Kerala and the establishment of European missions along the west coast. In fact, this was also a period when results from Ayurveda and Siddha began traveling to Europe giving birth to modern Botany, and similarly revolutionizing western medicine, health-care, and sanitation.

Epistemological Evidence

The epistemological evidence is fascinating to read [1]. A barrier in the western mindset as far as dealing with uncertainty manifests itself clearly in both the first and second math crises. As noted in [24]: “The European difficulty with zero did not concern merely the numeral zero, but related also to the process of discarding or zeroing a “non-representable” during the course of a calculation—similar to the process of rounding. Though the Indian method of summing the infinite series constituted valid pramana, it was not understood in Europe; the earlier difficulty with non-representables zeroed during a calculation reappeared in a new form. This was now seen as a new difficulty—the problem of discarding infinitesimals… In both cases of algorismus and calculus, Europeans were unable to reject the new mathematical techniques because of the tremendous practical value for calculations (required for commerce, navigation etc.), and unable also to accept them because they did not fit in the metaphysical frame of what Europeans then regarded as valid“.

Another instructive story (see page 3 of this essay), highlighting the outcome and unintentional humor caused by a borrow-copy-paste of Ganita without fully understanding its epistemology, is about how ‘sine’ and ‘cosine’ entered Europe. These mistranslated terms destroy the insight behind the original Sanskrit terms jya and kojya [1], baffling generations of Indian students studying Trigonometry.

To this day, neither organized religion and its theology, nor secular mathematicians, have been able to fully embrace the epistemology and validation procedure of Ganita. Why is this? And examining this question from the other direction, why did the Indians not take Euclidean math seriously for two thousand years? What is the future of Ganita? We study these civilizational perspectives in the third and concluding Post of this Set.

Selected References
  1. Cultural foundations of mathematics: the nature of mathematical proof and the transmission of the calculus from India to Europe in the 16th c. CE, C. K. Raju. Pearson Longman, 2007.
  2. Plato on Mathematics. MacTutor History of Mathematics archive. 2007.
  3. Plato’s Theory of Recollection. Uploaded by Lorenzo Colombani. Academia.edu. 2013.
  4. Being Different: An Indian Challenge to Western Universalism. Rajiv Malhotra. Harper Collins. 2011.
  5. Axiomatism and Computational Positivism: Two Mathematical Cultures in Pursuit of Exact Sciences. Roddam Narasimha. Reprinted from Economic and Political Weekly, 2003.
  6. Use and Misues of Logic. Donald Simanek. 1997.
  7. Computers, mathematics education, and the alternative epistemology of the calculus in the Yuktibhasa. C. K. Raju. 2001.
  8.  American Veda: From Emerson and the Beatles to Yoga and Meditation How Indian Spirituality Changed the West. Phil Goldberg. Random House LLC. 2010.
  9. Logic in Indian Thought. Subhash Kak.
  10. Ramanujan’s Notebooks. Bruce Berndt. Mathematics Magazine (51). 1978.
  11. C. K. Raju. Teaching mathematics with a different philosophy. Part 2: Calculus without Limits. 2013.
  12. Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World. Amir Alexander. Farrar, Straus and Giroux reprint / Scientific American. 2014.
  13. Indra’s Net: Defending Hinduism’s Philosophical Unity. Rajiv Malhotra. Harper Collins. 2011
  14. Mathematics in India – From Vedic Period to Modern Times: Video Lecture Series, by M. D. Srinivas. K. Ramasubramaniam, M. S. Sriram. 2013.
  15. Mathematics Education in India: Status and Outlook. Editors: R. Ramanujam, K. Subramaniam. Homi Bhabha Centre for Science Education, TIFR. 2012.
  16. The Battle For Sanskrit. Rajiv Malhotra. Harper Collins. 2016.

(A full list of references will be published along with Part-3).

Acknowledgment: Big thanks to the ICP blogger and the editor for their constructive feedback, patience, and comments that helped shape and improve this post.

An Indic Perspective to Mathematics — 1

This is the first of a 3-part set of Posts that follows our ‘Introduction to Ganita’


baudhayanatheorem
Pythagorean or Baudhayana Theorem? (from Bhaskara’s Lilavati)
Topic Outline

This Post studies from an Indic perspective, the path taken by Mathematics from ancient Greece to reach its present form. We compare and contrast Math with Ganita (introduced in our previous post) and in this process, also gain a better appreciation for Ganita. In some places, oversimplifications are employed for ease of understanding, and to bring into focus certain latent aspects of the discourse. All emphases within quotes are ours.

For convenience, this Post has been divided into a set of three, to be published consecutively. The first part is presented today, but the entire set is previewed below:

Part 1: We study the origins and motivations of Math and the pivotal roles of Plato, Aristotle, and Euclid (via Elements) in shaping the initial course of Mathematics. We compare the Indian and Greek logic, noting the non-universality of logic. To each civilization and culture, their own: Pramana versus Proof. A fundamentally different understanding of the nature of ultimate reality guides the Math and Ganita approaches: The integral unity underlying Ganita versus a synthetic unity in which Math lives as a separately independent component.

Part 2: We observe and learn what happens when Ganita encountered Math. Sparks fly in a tussle between order and chaos when two sharply different approaches clash.

Part 3: We adopt an Indic civilizational perspective of the Math-Ganita encounters. This gives rise to  interesting questions like ‘What was lost when Mathematics digested Ganita?’. We also look ahead, exploring the importance of Ganita and its Indian approach in a futuristic world.

Part 1:Introduction
Dolores Umbridge: It is the view of the Ministry that a theoretical knowledge will be sufficient to get you through your examinations, which after all, is what school is all about.

Harry Potter: And how is theory supposed to prepare us for what's out there?

(Harry Potter and the Order of the Phoenix, by J. K. Rowling).

Mathematics is the ‘science of learning’ that originated in ancient Greece, and comes from the Greek root mathesiz, or learning [1]. Plato’s Republic (~375 BCE) mentions the five specific disciplines of mathematics as: Arithmetic, Astronomy, Plane and Solid Geometry, and Harmonics [2]. Plato founded the Academy in Athens and gave Western (Greek) philosophy to the world.  ‘Learning’ had a specific meaning in this philosophy. His ‘theory of recollection’ indicates that ‘mathesiz’ is all about a soul recollecting the knowledge it has forgotten. We cannot learn anything new, and only recall what we forgot [3]. His teacher was Socrates, and Aristotle was his famous pupil.  Plato took as ideal that which was perfect, unchanging, abstract, even spiritual, and regarded the phenomenal world riddled with uncertainty as inferior. He favored the rational over the empirical, and the goal of uplifting the soul as superior to the task of performing mundane calculations. For example, when it came to arithmetic, his views as the narrator in the Republic were pretty clear [2]:

I must add how charming the science of arithmetic is! and in how many ways it is a subtle and useful tool to achieve our purposes, if pursued in the spirit of a philosopher, and not of a shopkeeper!’

‘How do you mean?’, he asked.

‘I mean, as I was saying, that arithmetic has a very great and elevating effect, compelling the mind to reason about abstract number, and rebelling against the introduction of visible or tangible objects into the argument.”

Several elegant results came out this Greek approach which can be broadly viewed as a sequence of axiom/model followed by the use of deductive logic to prove an infallible theorem [5]. The exemplar for this approach is Elements, the treatise on geometry attributed to Euclid (~300 BCE), and this ancient work played a very powerful role in shaping the course of Mathematics. The impact of Euclidean geometry is visible to this day. However, progress in the realm of practical application and calculation was curtailed by the downgrading or even the elimination of the empirical.  While logic and deductive reasoning are indispensable in detecting inconsistencies in arguments and help in viewing existing ideas more clearly, scholars have recognized the limitations of logic when it comes to understanding the nature of ultimate reality:

  1. Logic can be misused when it is employed to find Truth. About Aristotle [6]: “it was, for him, a tool for finding truth, but it didn’t keep him from making the most profound errors of thought. Nearly every argument and conclusion he made about physical science was wrong and misguided. Any tool can be misused, and in these pre-scientific days logic was misused repeatedly“.
  2. Deductive reasoning can help us analyze existing ideas better and lead us to a different way of tackling a problem, but in itself cannot lead us to new knowledge.  “deduced conclusions are just restatements and repackaging of the content contained in the premises. The conclusions may look new to us, because we hadn’t thought through the logic, but they contain no more than the information contained in the premises. They are just cast in new form, a form that may seem to give us new insight and suggest new applications, but in fact no new information or truths are generated. This is especially noticeable in mathematics…“[6].

This Mathematics lived in an abstract infallible world divorced from reality.  One cannot also overemphasize the impact of Aristotle’s ‘law of the excluded middle’ on western thought – a law that leaves no room for uncertainty. The intellectual ideas of Greece were eventually digested [4] into Christianity via the so-called ‘Hellenic-Hebraic’ synthesis. This should come as no surprise given the motivation for the studying mathematics included ideas of absolute perfection and ‘uplifting of the soul’. Mathematics thus became intertwined with the theology of an organized religion. A comparative study of the Indian and the Greek approach bring out the sharp differences between the Ganita and the Mathesiz approaches. Ganita, the integral science of computing, is not the same as mathematics. Unlike the five categories of Mathematics laid out by Plato, Ganita is all pervasive.

via @Calvinn_Hobbes

In [4], Rajiv Malhotra comments on the influence of Aristotle on western thought: “The Law of the Excluded Middle dictates that the principle ‘P or not-P’ separates one thing from another in an absolute sense. All physical and logical entities are invariant units, mutually exclusive of each other. This is not just a pragmatic criterion for distinguishing one thing from another; it is the very nature of reality in both concrete and abstract realms. The law eliminates the possibility of things being mutually dependent, interrelated and interpenetrated. It is diametrically opposed to the intertwined and fluid relationships characteristic of integral unity…”.

There appears to have existed a state of tension between the fallible-and-real and the infallible-and-perfect domain in the western thought since the time of Plato, which manifests itself today as the anxiety-filled binary of ‘religion versus science’. Since this gap was never breached, only a synthetic unity was ever possible [4], and the resultant western approach is reductionist. The independent parts have to be subsequently synthesized to achieve unity. For example, we read in  [25] that “much of Western civilization is based on separating the parts. One date is separate from another, history separate from math which is separate from biology. It’s a world view we inherited from Newton and Descartes, so useful in many ways and disastrous in others. However, there has always been an alternative view of the universe as a single, totally interconnected system. You’ll find that in Eastern traditions.“. To this day, Mathematics and Science are treated and taught as two different school subjects. A key tussle here is between the ‘lower’ empirical world we can experience, and the ‘higher’ abstract-theoretical domain, with the latter being considered superior. This western view is even being taken as the universal approach to knowledge.

Western Universalism

Today, we can observe the promotion of the notion of a western universalism that traces its origin to the intellectual tradition of ancient Europe. For example, the choice of the logo for UNESCO, a world body, reflects a desire to preserve the memory of Parthenon in ancient Greece, which was damaged in wars eons ago. Key buildings in several prominent universities in the United States are designed to remind viewers of the glory of ancient Rome and Greece.

The UNESCO logo (Credit: wikimedia.org)

The belief in the dominance of Euclidean Mathematics is reflected in the argument between the ancient Greeks and Epicureans.

The Epicurean Ass

The Epicureans opposed the followers of Euclid who, from their perspective, appeared to be proving obvious results. For example, consider the following proposition in Elements as discussed in [23]:

Any two sides of a triangle are together greater than the remaining side.

In other words, a straight line is the shortest distance between two points!

If anyone wanted to ridicule mathematics for its insistence on the axiomatic method of orderly proof, this theorem offers a wide target. In fact, the Epicureans (those Athenian free-thinkers, who defined philosophy as the art of making life happy) did exactly that. They said that this theorem required no proof, and was known even to an ass. For if hay were placed at one vertex, they argued, and an ass at another, the poor dumb animal would not travel two sides of the triangle to get his food, but only the one side which separated them.”

C. K. Raju explains both sides of the argument [7]: “Proclus replied that the ass only knew that the theorem was true, he did not know why it was true. The Epicurean response to Proclus has, unfortunately, not been well documented. The Epicureans presumably objected that mathematics could not hope to explain why the theorem was true, since mathematics was ignorant of its own principles..” In the end, the Greek response cites the authority of Plato that mathematics “takes its principles from the highest sciences and, holding them without demonstration, demonstrates their consequences. [7].

Let us now introduce an Indic perspective.

In contrast with this Greek view, all Indian schools of thought accept empirical means of verification (e.g., pratyaksha pramana [1, 22]) while acknowledging the potential fallibility. All darshanas would reject any axiomatic approach that lacked valid pramana. The use of empirical rationale has existed in India since ancient times, including the Sulba Sutras (800 BCE or earlier) and is different from the axiom-theorem approach. C. K. Raju puts this in perspective: “Because no proof was stated it does not, of course, follow that the authors of the sulba sutras did not know why the result was true. But the method of proof that convinced them may well have  differed from the current definition of proof. Thus, it is incorrect to assert that the constructional methods used in the sulba-sutras implicitly lead to a proof in a formalistic sense. It is incorrect because the rationale for the formula for a right-angled triangle, from the constructional methods of the sulba-sutras right down to the 16th century Yuktibhasa, explicitly appeals to the empirical“. [7]

The Epicurean Ass argument has been kept alive in some form or the other to this day in a western worldview. From an Indian point of view, a Ganita expert like Srinivasa Ramanujan too was deemed a ‘wizard’ [14, Lecture 1] who did not know why his results were true, despite his point that he employed his own valid method, which produced so many astounding new and true results. He had to move from Kumbakonam to work in the U.K. to prove his results to the satisfaction of the formal math community in order to gain acceptance.

Indian Gurus, Yogis, Siddhas, and Tantriks who, through years of practice and sadhana, demonstrated amazing results in transcendental meditation, mind sciences, and medical sciences are sometimes labeled pre-rational Indian ‘mystics’ [4] as opposed to western ‘scientists’ who came up with sophisticated instrumentation that subsequently confirmed these results. Universities like Harvard periodically comes out with a research report ‘proving‘ prior findings in Yoga and Ayurveda from the Dharma traditions, which have been practically employed for centuries.

Public intellectuals like Rajiv Malhotra also ask: How often are these Hindu and Buddhist monks, who are the primary producers of this knowledge, credited as co-authors in the journal papers? This bias is propagated subtly by western scholars who study Hinduism. For example, Phil Goldberg who teaches at Loyola Marmount University, an institution rooted in the Jesuit Catholic tradition, compares ‘Indian philosophy and Western science’ in [8]. He also endorses the rejection of the ‘orange’ [saffron] robe of Dharma in favor of the authoritative western scientific garb of a ‘white lab coat’ in order to increase the credibility of Yoga and meditation techniques in the minds of westerners. Note the approach is one of extracting the benefits, and then rejecting/denigrating the Dharma source. Such biased attitudes have also helped feed an increasing Hinduphobia within western academia.

Two-valued logic is not universal. India had not one but several different schools of thought that also studied logic [22], including Nyaya and Navya Nyaya, as well the Buddhist Catuskoti, and Jaina Syadavada. In fact, the Buddhist understanding of integral unity as encapsulated in Nagarjuna’s brilliant arguments has been recognized as nothing short of a “death-blow to all synthetic unities that start with different essences and then look for unity” [4].

Indian Logic vs Greek Logic

There are several papers available that discuss the Indian approach to logic. For example see this work of Subhash Kak [9] and this discussion of Indian and Greek logic. In the popular textbook example for Indian syllogism versus that of Aristotelian logic, the first thing we notice are the ‘five steps’ in the Indian approach versus three in the Greek template [22]. The steps in the Indian rules of inference are not redundant and serve as a reality-check based on the correspondence principle of Bandhu [9], whereas the Greek argument is restricted to the infallible abstract domain. As Roddam Narasimha notes in [5] where he compares Greek Axiomatism and Indian Computational Positivism, the Indian distrust of deduction-based logic “appears to have been based on the conviction that the process of finding good axioms was a dubious enterprise. Note that logic in itself was not something that was shunned in India; without going into a detailed discussion of Indian systems of logic, it is enough to note here that time and again Indians use deductive logic to demonstrate inconsistencies or to refute the positions of an adversary in debate, rather than to derive what western cultures have long sought through that method – namely, certain truth.“.

The intellectual prowess of the ‘deductive logician’ has been promoted in popular western culture. For example, Sherlock Holmes is recognized foremost for his superb deductive reasoning, and is considered the most portrayed literary human character in history. However, an analysis of his stories show that Holmes relied a lot on anumana (inference) including the so-called abductive and inductive methods, and Conan Doyle did consider Holmes’ methods to be fallible, which resembles a Ganita approach to sleuthing!

Sherlock Holmes Portrait Paget.jpg
‘Sherlock Holmes’ By Sidney Paget (1860-1908) , Public Domain. Credit: Wiki Commons

CK Raju [1] calls out some flaws in the claim to universality of two-valued logic. First, the Hindu darshanas, Buddhist Catuskoti, and Jaina Syadavada offer solid alternatives from a different culture. These alternatives have always been compatible with the latest developments in science at every point in time, including Quantum Mechanics. We do not find any serious ‘religion vs science’ problem in India [4]. Even the materialist Charvaka school would reject this reductive logic for not accepting a Pratyaksha Pramana [1, 22]. Finally, it is tough to justify two-valued logic citing empirical evidence if its claim to dominance lies in its empiricism-free perfection [1].

A remaining argument in favor of a universality of two-valued logic and axiomatism is the endorsement by ‘higher authority’, representing a distorted version of Sabda pramana [22]. Indeed some proofs published in journals today are so abstract and technical that they can only be decoded by top formal mathematicians. The remainder of the global math community take it as truth based on the verbal authority of an elite few.

Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true - Bertrand Russell.
Vignette: Demotion of a Theorem

In middle school geometry, we learn about the congruence of triangles and come across the side-angle-side (SAS) postulate [23]:

“The fundamental condition for congruence is that two sides and the included angle of one triangle be equal to two sides and the included angle of the other.”

This result can be easily verified using empirical rationale (proof-by-superposition, as Euclid himself did), and would be perfectly acceptable in Ganita, but not in mathematics. This is because superposition involves moving one triangle and placing it on top of the other, which is considered a ‘fallible’ process. The SAS result is difficult to prove using logic alone and thus the SAS theorem was demoted to the status of an unproven postulate.

We conclude Part 1 by delineating a key, irreconcilable difference between Ganita and Mathematics. This difference also manifests in virtually every other field of study.

Summary: Fundamental Difference between Ganita & Mathematics

The ancient Indians recognized Nyaya (logic) and employed Tarka (reasoning) and even mastered it, but did not put it on a pedestal because of certain limitations. Results in Ganita, like all other Indian disciplines, are tied to a valid Pramana and rooted in reality, rather than an axiom-based proof operating in a separate abstract domain. The empirical approach can elevate the practitioner to a higher state of consciousness (The Bhagavad Gita recognizes it as a valid way to transcendental knowledge [4]).

Subhash Kak summarizes the Indian approach to acquiring knowledge based on bandhus [9]: “The universe is viewed as three regions of earth, space, and sky which in the human being are mirrored in the physical body, the breath, and mind. The processes in the sky, on earth, and within the mind are taken to be connected. The  universe is mirrored in the cognitive system, leading to the idea that introspection can yield knowledge“.  It is worth repeating what has been said before: In nature, the western civilization is intellectual, the Chinese civilization is philosophical, and the Indian civilization is spiritual (adhyatmic).

Ganita is rooted in an integral unity whereas Mathematics exists as a separately independent part of a synthetic unity.

This integral approach produced some of the most important contributions, from Hindu numerals, place value system with zero, to symbolic language for managing equations [5]  and calculus. On the other hand, the abstract nature of Mathematics resulted in a drastically reduced practical output while Europe drifted into a 1000+ year Dark Age. During this entire period, Ganita contributions from all Dharma thought systems proved to be crucial in keeping mathematics practically relevant in other parts of the world, up to the 17-18th century CE. We discuss these Ganita-Math encounters in the upcoming second part of this set of Posts.

Selected References
  1. Cultural foundations of mathematics: the nature of mathematical proof and the transmission of the calculus from India to Europe in the 16th c. CE, C. K. Raju. Pearson Longman, 2007.
  2. Plato on Mathematics. MacTutor History of Mathematics archive. 2007.
  3. Plato’s Theory of Recollection. Uploaded by Lorenzo Colombani. Academia.edu. 2013.
  4. Being Different: An Indian Challenge to Western Universalism. Rajiv Malhotra. Harper Collins. 2011.
  5. Axiomatism and Computational Positivism: Two Mathematical Cultures in Pursuit of Exact Sciences. Roddam Narasimha. Reprinted from Economic and Political Weekly, 2003.
  6. Use and Misues of Logic. Donald Simanek. 1997.
  7. Computers, mathematics education, and the alternative epistemology of the calculus in the Yuktibhasa. C. K. Raju. 2001.
  8.  American Veda: From Emerson and the Beatles to Yoga and Meditation How Indian Spirituality Changed the West. Phil Goldberg. Random House LLC. 2010.
  9. Logic in Indian Thought. Subhash Kak.

(The complete list of references will be published along with part 3).

Acknowledgments: I would like to thank the ICP bloggers for their constructive feedback and the editor for his incisive comments and ideas.

Introduction to Ganita

Lilavati1Bhaskara2
Lilavati by Bhaskaracharya 2

This introductory blog provides the background for an upcoming Ganita series here at ICP. All emphasis within quotes is by this author.

Losing one glove is certainly painful,

but nothing compared to the pain,

of losing one, throwing away the other,

and finding the first one again. 

—Piet Hein, Danish Mathematician

Introduction

Though often compared with the field of Mathematics, Ganita is best defined as the science and art of computation that originated in India. This is based on the definition offered by Ganesha Daivajna in his commentary (1540CE) on the classic Ganita treatise Lilavati of Bhaskara-2 [1]. Other descriptions of Ganita include ‘computing science’, ‘reckoning’, ‘science of counting’, ‘science of calculation’, etc. Although Ganita is related to Mathematics, they are not the same. The practice of Ganita cuts across multiple areas including Mathematics, Computing, Science, Logic, Analytics, etc. The term Mathematical Sciencesmay be closer to Ganita There is no exact English equivalent for the Sanskrit word Ganita, and it is better to use ‘Ganita’ as is.  

An ancient extant work of Ganita is the Sulvasutras (Sulbasutras), which are the oldest texts of geometry dating back to 800 BCE or earlier [1]. A verse in the Vedanga Jyotisha (1100 BCE or earlier) attests to the pride of place occupied by Ganita in ancient India.

Like the crest on the peacock’s head,
Like the gem in the cobra’s hood,
So stands Ganita*,
At the head of all the sciences.

The Ganita Culture of India

Indians are famous for their Ganita prowess. The greatest Persian scholar of his time, Ibn Sina (aka Avicenna, 10-11th CE) found that an Indian vegetable vendor’s calculating skills were superior to anything he knew [1].  European visitors during colonial times were astounded and remarked that “the natives of India are remarkable for the facility with which they acquire the mathematics; and indeed they excel in anything in which figures or numbers are concerned”.  The East India Company promised a reward of twenty pounds to its soldiers if they could learn arithmetic from the Indians [2]. It is well known that the word ‘algorithm’ comes from Algorismus, the latinization of ‘Al-Khwarizmi’, the person who translated several Sanskrit texts of Ganita (e.g. those of Brahmagupta) into Arabic. Thus, an algorithm implies the Indian method of computation, i.e., ‘Ganita’.  Much of Ganita and its methods made its way to Europe, first through Arab translators, and later through Jesuit priests stationed in India [3].

The Ganita curriculum in Indian schools prior to European colonization was functional, pragmatic, autonomous, and also guided by, and customized to local needs. The method of teaching Ganita in schools was ahead of its time. Researchers attribute this success to: “a culture of pedagogy grounded in a form of memory very different from the modern associations of memory with rote or mechanical mode. This could be characterized as recollective memory where memory practices constituted a distinct mode of learning and not merely aids to learning”[2]. 

To this day, the Ganita prowess, ability to recall, and the computing literacy of Indians is second to none. It is no coincidence that Indians excel in STEM disciplines. Understanding how Ganita works and what lies at its core is useful and interesting. Ganita can be a refreshing complement to the dull and dry Math taught in schools today. Training young minds to apply the Ganita approach to problem solving can offer them a competitive advantage. Manjul Bhargava, Fields Medal winner, is an example of a contemporary scholar who scaled the peak of Mathematics and is also well versed in Ganita and aware of its Indian tradition.

Above all, Ganita is a precious part of India’s heritage and culture. It is inevitable that India will regain its lost political and economic freedom. But this can and must be achieved without selling out or forgetting its traditions and indigenous knowledge systems like Ganita, Yoga, and Ayurveda. The trauma of a civilization that realizes that it squandered away its priceless cultural treasures will be unbearable.

The Scope of Ganita

This introductory video of an excellent IIT lecture series on Indian contributions to mathematics provides a good overview of Ganita. It was recognized that Ganita’s applications span the secular and sacred domains without any artificial distinction between the two. This integral nature of Ganita was embraced by all the great Buddhist, Jaina, and Hindu scientists and astronomers since ancient times.

pervasiveness ganithasarasangraha mahaviracharya 2
source: IIT lecture series on Ganita [1]
India’s Rishis and Ganita experts attributed their astonishing insights to a sacred source. The practice of Ganita offered a valid means of attaining this infinite, transcendental knowledge, and through this process, skilled practitioners also came up with ingenious practical solutions to a variety of problems.

Sacred Source of Ganita

Panini (BCE)

The Siva Sutras in Panini’s Ashtadhyayi, which one can consider as an early example of the Indian approach to science, were revealed to Panini (pronounced: Paanini) via the sacred sounds from Shiva’s Damru. In fact, some scholars consider the Indian approach to math and science to be the ‘Paninian approach’ [1]. Indian kids traditionally start their exam papers with a small notation above the top of the page as an invocation to Ganesha (e.g. Tamil kids draw a tiny ‘Pillayaar Chuzhi’). This is an ancient practice of a tradition that reveres wisdom and learning, and one that is worth preserving. From Panini to Ramanujan, we see a great line of Ganita scholars beginning their works with an explicit tribute to a divine deity and their sacred cosmology.

Aryabhata (499 CE)

In his Aryabhatiya, the great astronomer Aryabhata who’s statue today adorns UNESCO, begins by paying obeisance to Brahma who is recognized as “the god who is the one and the many” [5]. This is a pertinent point from a Ganita perspective which we shall see later. We learn the following from the commentaries on Aryabhatiya:

  • Bhaskara I : “It is said : ‘(Aryabhata) who exactly followed into the footsteps of (Vyasa) the son of Parasara, the ornament among men, who, by virtue of penance, acquired the knowledge of the subjects beyond the reach of the senses and the poetic eye capable of doing good to others’.” 
  • “Aryabhata’s devotion to Brahma was indeed of a high order. For, in his view, the end of learning was the attainment of the Supreme Brahman and this could be easily achieved by the study of astronomy”.
  • Aryabhata is obtaining new results by navigating through an existing ocean of knowledge: “Having taken a deep plunge into the entire ocean of the Aryabhata-sastra with the boat of intellect, I have acquired this jewel, the Karana-ratna, adorned by the rays of all the planets.

Nilakantha Somayaji (1444-1544 CE)

He was a great Ganita expert and astronomer from the Kerala School (who can be viewed as Aryabhata’s intellectual successors). Nilakantha was also recognized for his mastery of all six darshanas of Hinduism [6]. His great work Tantrasamgraha begins with an invocation to Vishnu. Commentators on this work note that the invocations recognize Vishnu as both the material and the efficient cause of the universe [7].

Srinivasa Ramanujan (1877 – 1920)

srinivasaramanujan

Ramanujan attributed his amazing results to Goddess Namagiri. His statements reveal a firm belief and appreciation of Hinduism and its understanding of ultimate reality. The source of his knowledge was beyond anything cognizable by ordinary senses. Thanks to his biography [8], there’s a lot of material describing, from a western perspective, Ramanujan’s amazing ability, and the following samples provide clues about his methods. Ramanujan is a role model for aspiring young Indian mathematicians and scientists, and this was acknowledged by the Nobel Laureate Astrophysicist, S. Chandrashekar.

  • Ramanujan’s belief in hidden forces and the powers of the supernatural
    was never, at least back in India, something about which he felt the need
    to apologize or keep quiet
  • Ramanujan “had grown up on the Indian gods and the relaxed fluidity of Hindu belief. In him, the natural and the supernatural, Jacobi and Namagiri, Number
    and God, found a common home, stood in something like an easy intimacy.
  • “…the mystical streak in him sat side by side, apparently at perfect ease, with raw mathematical ability may testify to a peculiar flexibility of mind, a special receptivity to loose conceptual linkages and tenuous associations.
  • his openness to supernatural influences hinted at a mind endowed with slippery, flexible, and elastic notions of cause and effect that left him receptive to what those equipped with more purely logical gifts could not see; that found union in what
    others saw as unrelated; that embraced before prematurely dismissing

Each of these independent Indian thinkers freely moves between the transactional and the sacred domains without anxiety. Their work was firmly anchored in Dharma, and serving this integrated unity. The deities invoked include the celestial Hindu trinity and the Devi. Dharma is not the same as religion [10], and this is not theology or missionary zeal working overtime to fudge mathematical models in order to make it compatible with religious scripture, prophecy, and God. Rather, Ganita’s findings arise from a seeker’s quest to learn the truth about the nature of ultimate reality. The Bhagavad Gita (verses 9.4, 3.40-41) recognizes the empirical to be rooted as well as culminating in the transcendental [13]. Ganita is a sacred and valid path to reach the transcendent, and the continuity in the views of four great scholars from different time periods in Indian history drives home this point. Given the importance attached to this sacred source by its foremost practitioners, it is more accurate to view Ganita as the integral science of computing.  Attempts to equate Ganita to a purely pragmatic and secular science or math is inaccurate and reductionist.

The creation stories in the Vedas lend themselves to a rich interpretation that trace out a fundamental Ganita template which was adopted by all these great practitioners. Toward this, we start with an algorithmic interpretation of Prajapati’s efforts to create a stable, self-organized universe [9].

Prajapati’s Algorithm

Prajapati employs an algorithm to create the cosmos. An iteration in this algorithm consists of an experimental trial, followed by an observation of the output data, which triggers a review and validation phase, followed by an adjustment of ‘design parameters’ and re-trial, if necessary. This process converges to Prajapati’s satisfaction within three iterations. However, no attempt is made to prove or claim with absolute certainty that among all possible universes, his is the most perfect and infallible. Since time is cyclical, such universes are dissolved and recreated with no beginning or end. The Rig Veda explicitly recognizes the inherent uncertainty associated with any answer to such questions [10].

  1. The first empirical trial produces a cosmos which is observed to be full of entities too similar in nature and they simply merge into each other, so that there is practically nothing to unite.  This is a homogeneous and ‘over-ordered’ universe where there is nothing left to know, and this system quickly becomes unmanageable. From a statistical perspective, there is little or no variance in this first universe.
  2. Prajapati increases variability in his second try but the output shifts to the other extreme. The world is now way too heterogeneous and there is no commonality between beings to relate to, and to unite. Nothing is certain and can be known, and chaos reigns.
  3. Learning from the first two attempts, Prajapati is able to achieve a good balance in his third version that overcomes prior problems, and the algorithm terminates with a stable universe.

How does Prajapati accomplish this task? In his book ‘Being Different’, Rajiv Malhotra says “Prajapati recognizes that all life should be situated between these opposing excesses of too much identity difference and too much homogeneity. Ultimately, he succeeds in producing just such a universe. He does so through the power of resemblance, known as ‘bandhuta’ or bandhu, which was discussed in Chapter 3. The Vedas abound in attempts at finding connections among the numerous planes of reality. This serves as a cardinal principle of all Vedic thought and moral discourse”.

Every entity created is unique, while also bearing some form of resemblance to each other.  Some resemblances may be more easily spotted, while others may be subtle and identifiable only after considerable effort. These Bandhus are the ‘conceptual linkages and tenuous associations’ revealed to Ramanujan after intense tapasya, and he is able to find “union in what others saw as unrelated” because the cognizable world is mirrored and mapped into the transcendental world, and vice versa via these Bandhus [11]. These strands of resemblances intertwine the elements of the universe into an integral unity, where every individual element’s identity is real but provisional, while always being rooted in the independent whole. There are no separately independent realities for individual elements and the methods of Ganita mirror Prajapati’s algorithm.

Bandhu
"The bandhus represent the laws that hold the universe together (Vishnu), paroksha is the dance of consciousness that is ever changing (Shiva), and Yajna is the process of
 change (Devi)" - Subhash Kak, Pragnya Sutra [PS, 12].

The idea of ‘resemblance’ is fundamental to the acquisition of knowledge that is required to make ‘risky’ and useful predictions about the future with a measure of confidence. This concept can be illustrated using the analogy of a modern business forecasting system. Suppose a company launches a brand-new product in the market and needs to know now how many units it is likely to sell in the next 6-12 months. Since no prior sales data about this product is available, no statistical method cannot be employed to directly calculate this number. To overcome this limitation, the new product A is mapped in terms of its selling attributes to that pre-existing product B which it resembles most. B’s data is borrowed to generate an initial sales forecast for A. Machine learning and AI techniques can be used to learn such recursive patterns, even deep ones, from unstructured data.

However, a machine has its limits. Computer Scientist and Sanskrit scholar Prof. Subhash Kak notes [12] “… knowledge emerges from a familiarization with its inner space and it may be seen to be a consequence of the bandhus (bonds) that exist between the outer and inner worlds. If there were no such bandhu, it would be impossible to make sense of the world. Machines only follow predefined rules and they don’t have bandhus, which is why they cannot be conscious. The bandhus are the ground that make awareness possible“.

Bandhus can be in the form of numbers, biological rhythms, sounds, lights, touch, etc.[11]. Or via Meghadutam? A study of the applications and motivation of ancient Indian geometry reveals the traditional Hindu approach of coexisting in harmony and synchronizing with nature by recognizing certain auspicious and sacred ratios and numbers (e.g. 108).  Two examples are stated below, which also bring to light the continuity and commonality in thought between the Harappan and Vedic time periods. We will discuss this in depth later in our series.

  1. The ratios and measurements used in Harappan architecture at Dholavira (2500 BCE or earlier)
  2. The dimensions and numbers of bricks used in Vedic fire altars.

Two of the three key notions of dharmic cosmology are recursion and paradox [11]. The former, via the principle of resemblance, injects a sense of order and certainty into our view of the transactional world, whereas the latter preserves the mystery and uncertainty about the true nature of ultimate reality. It is convenient in the Ganita context to understand this recursion using the Vedic metaphor of Indra’s net.

Self Organizing Patterns: Vedic Metaphor of Indra’s Net
"The Vedic deity Indra is said to have an infinite net consisting of a jewel in each node, arranged so that every jewel reflects all the other jewels; there is no separate self-existence of any jewel. Each is unique in its reflection of all others. Indra's Net symbolizes a universe with infinite dependencies and relations interwoven among all its members, none of which exists apart from but only in the context of this collective reality."  - Rajiv Malhotra, Indra's Net.

The links in this self-organized network are precisely the Bandhus. Since the ultimate reality is like Indra’s Net, Prajapati’s world allows order and information to emerge from what appears to be nothing but chaos and uncertainty (even soccer matches!). Such an Indra’s Net becomes a limitless source of useful ideas for Ganita. We provide three examples from Mathematics to illustrate this.

  • The ‘Rule of Five’ [14] states that: “There is a 93.75% chance that the median of a population is between the smallest and largest values in any random sample of five from that population.” Just five random samples are enough in nature, with no preconception about its probability distribution, to achieve a significant reduction in uncertainty – from being totally unsure, to knowing a lot about any group’s median behavior. This order has been hiding all along in plain sight.
  • The world around us is full of (approximately) normal distributions or bell curves, allowing a certain statistical order to emerge out of seemingly disorganized groups.
    source: usablestats.com

    Of course, not everything in nature is normally distributed. There are plenty of exceptions [14]. In [15], Lyon tries to understand how such normal distributions come about in nature. He argues that it is not because of the central limit theorem. He uses inference (which Indian logicians recognize as anumana) to understand how these patterns are generated in nature. By using the idea of ‘entropy’ to denote the degree of chaos (or disorder), we learn: “A further fact, which serves to ’explain’ why it is that this ’order generated out of chaos’ often has the appearance of a normal distribution, is that out of all distributions having the same variance the normal has maximum entropy (i.e. the minimum amount of information).” The balance between order and chaos in nature produces approximate bell curves, whose statistical properties can be gainfully employed to better understand this world. Sometimes, this Indra’s Net manifests itself as spectacular visuals.

  • In the brief video below, we can observe fireflies synchronizing. Thousands of fireflies light up at the same instant by simply doing their Dharma of flashing ‘strobes’ and sending out a visual signal, and in turn appropriately responding to incoming signals [16]. This was first noticed by western researchers in the jungles of Thailand. After the first sync-up, they remain synced. Self-organization is quite natural in the Vedic universe, and now we are beginning to see rigorous mathematical proofs reaffirming this reality. Inference and intuition was used by mathematicians in tandem with logical reasoning to understand the process of ‘sync’ and prove that synchronization is guaranteed in nature under certain conditions. Strogatz notes in [16] “The implication is that in a population of fireflies or brain cells, the oscillators have to be similar enough or nobody will synchronize at all.”  A certain balance between order and chaos is required for sync, and evidently, this is not uncommon in nature. After all, the dance of the universe is synced to the dance of Nataraja. Out of these self-organization principles emerge the beautiful equations and results of Ganita.

The Ganita of self-organization shows up prominently in Hinduism and in India. This decentralized ‘sync’ by insects could be quite naturally viewed by Hindus as a firefly Kumbh Mela. Pre-colonial India was largely decentralized. Self-organization reduces transactional costs and is environment friendly. Hinduism’s resilience and even a degree of ‘antifragility’ are due to built-in error-correcting mechanisms and the ability to constructively balance order and chaos [10]. The fidelity of Vedic chants has been orally preserved over several thousand years via embedded  layers of data redundancy that resemble ideas within modern methods of information transmission over a noisy communication channel. In the video below, Manjul Bhargava provides an example of Ganita in Sanskrit Kavya, which embeds an error-correcting code.

Several notoriously hard-to-solve mathematical problems (see example picture below) recognized in computational complexity theory are routinely managed in practice. Problem instances that actually manifest in nature appear to have certain data patterns and organization that allow them to be solved fairly quickly to the level of accuracy required by the practical application.

The Best ‘Bottleneck’ Traveling Salesman Route across USA (source: akira.ruc.dk)

Along with resemblances and patterns in nature comes paradox. Per Subhash Kak [11], “paradox is the recognition that the bandhu must lie outside of rational system, leading to the distinction between the “higher” science of consciousness and the “lower,” rational objective science“.  How does Ganita deal with paradox and uncertainty?

Ganita: At Ease With Uncertainty

A bit beyond perception’s reach

I sometimes believe I see

that Life is two locked boxes, each

containing the other’s key. 

—Piet Hein

(and in the words of Clint Eastwood, “If you want a guarantee, buy a toaster“).

In the Vedic period, there used to be enigmatic exchanges between scholars, known as Brahmodya, where a riddle about the nature of ultimate reality (Brahman, in Hinduism) was posed. The respondent remained silent if they could not decipher it, or countered with a deeper riddle if the hidden Bandhu was recognized [17]. (An entertaining version of this contest is the silent exchange via hand-gestures between Kalidasa and the scholar-princess Vidyottama). Dharma traditions recognize that our understanding of reality is likely to be incomplete. For example, Rajiv Malhotra notes in [10]: “There is equivalence in the relationship between sunya (emptiness) and purna (completeness or integral wholeness), the paradox being that the void has within it the whole“. With new knowledge and its associated benefits invariably comes uncertainty and ‘side effects’. There is no ‘free lunch’. Consequently, man-made algorithms are not infallible and dharma systems explicitly factor this in.

It is well known that Smritis have to be updated periodically while always serving  the unchanging Shruti.  Similarly, Ganita practitioners come up with increasingly better Siddhantas that progressively improve our understanding of natural phenomena. What is also important to remember is that the Indian approach to any field, including Ganita, is one of shraddha that is grounded in the sacred. We can be transformed by this experience and attain higher levels of consciousness that bring us ever closer to experiencing the ultimate reality.  This view is apparent in Aryabhatiya [5]: “the end of learning was the attainment of the Supreme Brahman and this could be easily achieved by the study of astronomy. In the closing stanza of the Dasagitikasutra, he says: “Knowing this Dasagitikasutra, the motion of the Earth and the planets, on the celestial sphere, one attains the Supreme Brahman after piercing through the orbits of the planets and the stars“.

The Integral versus the Synthetic Approach

We briefly compare two alternative approaches to dealing with paradox and uncertainty:

  1. Integral approach
    • Recognize reality with all its inherent diversity as is, as the ideal, and treat knowledge acquisition as a systematic process of reducing uncertainty.
    • Inference and intuition is useful in gaining new knowledge, and ingenuity is prized in such a tradition. Such knowledge is fallible, and new and improved methods are continually developed to reduce error to an acceptable level. Pragmatism rules, and the layman is familiar with the Ganita required for his/her own profession [2].
    • The validity of a method is demonstrated via Upapattis [1] that are rooted in reality.  From a logic perspective, the validity of knowledge is tied to the specific Pramanas it relies on, which may not be universally acceptable.
  2. Synthetic approach [10, 3]
    • Reject chaos as undesirable and consider ‘perfect order’ to be the ideal, and reality as subservient to this ideal ‘model’.
    • This binary mindset prefers to view reality as a bunch of separately independent systems where knowledge acquisition is preferably beyond doubt and free of empiricism.
    • Every new result is proven conclusively and universally using logic, starting from a minimal number of ‘self-evident’ axioms.

This distinction does not automatically imply that Ganita (example of integral approach) and modern science/math (largely synthetic approach) are in a state of irreconcilable conflict. As Roddam Narasimha notes [24]: “Modern science seems to have acquired, perhaps by fortunate accident, the property that the great Buddhist philosopher Nagarjuna called prapakatva: i.e., it delivers what it promises; it may not be the Truth, but it is honest“. What is undeniable and supported by fact is that by the 16th century CE, Ganita results had already laid the foundations for many crucial developments in modern science and mathematics [3].  Ramanujan is an example of an Indian who practiced the integral approach, and found a way to work constructively with western mathematicians so that his results could benefit the world.

It is interesting to see how Mathematician Hardy and Ramanujan reacted to each other’s approach as noted in [8].

  1. When Hardy asked for proof, we excerpt Ramanujan’s response: “…. I dilate on this simply to convince you that you will not be able to follow my methods of proof if I indicate the lines on which I proceed in a single letter. You may ask how you can accept results based upon wrong premises. What I tell you is this: Verify the results I give and if they agree with your results, got by treading on the groove in which the present day mathematicians move, you should at least grant that there may be some truths in my fundamental basis.
  2. Professor Hardy’s description of Ramanujan’s approach: “It was his insight into algebraic formulae, transformations of infinite series and so forth, that was most amazing. On this side most certainly I have never met his equal, and I can compare him only with Euler or Jacobi. He worked far more than the majority of modern mathematicians, by induction from numerical examples; all his congruence properties of partitions, for example, were discovered in this way. But with his memory, his patience and his power of calculation, he combined a power of generalisation, a feeling for form, a capacity for rapid modification of his hypothesis, that were often really startling, and made him, in his own peculiar field, without a rival in his day.”.  Prof. Hardy was careful not to tamper with Ramanujan’s mysterious ability, which was rooted in Ganita.
  3. On Ramanujan’s approach to the partition problem: “… the uncanny accuracy of their results attested to the power of the approximating technique they had used to get them ..So subtle and inspired were the approximations it permitted that it went beyond approximation to promise exactitude. …. Selberg, in fact, argues that Hardy’s insistence on certain methods of classical analysis actually impeded their efforts; and that lacking faith in Ramanujan’s intuition he discouraged a search for the kind of exact solution Rademacher produced twenty years later.”

Commentators have often termed this integral Indian approach as ‘Paninian’. We try to better understand what they mean by that.

The Real is the Ideal, and the Perfect is its Approximation

If the west has Euclid as the pioneer and exemplar for mathematics, India follows Panini. In [18], Dr. J. J. Bajaj explains this statement by using the commentary of Patanjali on Panini’s work: “In providing this characterisation of the science of grammar Patanjali laid his finger on perhaps the most essential feature of the Indian scientific effort. Science in India seems to start with the assumption that truth resides in the real world with all its diversity and complexity… As Patanjali emphasises, valid utterances are not manufactured by the Linguist, but are already established by the practice in the world. Nobody goes to a linguist asking for valid utterances, the way one goes to a potter asking for pots. Linguist do make generalisations about the language as spoken in the world. But these generalisations are not the truth behind or above the reality. These are not the idealisation according to which reality is to be tailored. On the other hand what is ideal is the real, and some part of the real always escapes our idealisation of it. There are always exceptions. It is the business of the scientist to formulate these generalisations, but also at the same time to be always attuned to the reality, to always to conscious of the exceptional nature of each specific instance. This attitude, as we shall have occasion to see, seems to permeate all Indian science and makes it an exercise quite different from the scientific enterprise of the West.

This discussion tells us that Panini’s is an integral approach rooted in the ultimate reality. On the other hand, the synthetic approach mentioned is popular in the west. Advances in modern science have been attributed to this approach. However, this approach can allow false assumptions to creep if the reality-check step is missing. There is an interesting story about the US Air Force set in the 1950s when they discovered that many of their pilots were losing control of their planes and crashing at an alarming rate.

source: thestar.com

The investigation eventually narrowed down the cause to the design of the cockpit, which was precisely engineered to a precise standard in the 1920s for the average American pilot. The USAF theory was that the average pilot had gotten bigger in the prior three decades and so the cockpit dimensions need to be re-sized upward. More than 4000 pilots were measured across 140 dimensions to compute a new standardized design. During this process, an analyst who was sifting through this data discovered that the total number of pilots who were average or near-average across these dimensions was exactly zero.  The ideal pilot simply did not exist.

A similar survey was conducted a few years earlier to find a lady in Cleveland who would closely match the ideal normal figure (‘Norma’). Among the nearly 4000 contestants, there was not one lady in the survey who matched Norma’s perfect vital statistics. Assuming that reality will conform to a non-existent ideal model is a recipe for disaster. USAF quickly realized that it was far better to design and periodically update designs based on the observed reality by explicitly taking uncertainty into account. This is exactly what the USAF did thereafter and switched their cockpit design philosophy to ‘individual fit’. It was a pragmatic response to an important problem that was jeopardizing pilot safety and costing millions of dollars. From an Indian perspective, the USAF chose the Ganita approach. Every US military branch embraced this idea soon after. A similar revolution is ongoing in healthcare, with allopathic medicine representing the synthetic alternative, and Ayurveda being the integral method. This integral approach to computing produced amazing results such as the decimal place value system and Algebra.

Integral Unity of Indian Place Value System

The Indian decimal place value system that is now used all over the world is startlingly simple and elegant. It arises from the sacred idea of ‘the One that manifests as many’ that exists in all Dharma thought systems (and Aryabhatiya paid obeisance to). Just like Panini was able to encode the infinite possibilities of pre-existing and all future utterances using a small number of rules, the Indian place value system too can represent all previously used and yet-to-be-used numbers in the universe using just a few symbols and rules.  Every digit in an N-digit number is denoted by its symbol that has a provisional reality, and through an established place value, it acquires a manifested form that unites into the whole number. As shown in the picture below,  some two thousand years ago, Rishis explain that the same symbol ‘1’ can realize different values, e.g. in the unit, tens, and hundreds place just as a lady can be a daughter, sister, mother, etc.

decimal pv system analogy to a lady3
source: Module 1 of IIT lecture series [1]
Algebra and Sanskrit

The place value system is essentially algebraic in nature. Bijaganita (Algebra via Arabic Al-Jabr) is a natural extension of this idea that arose independently in India (early algebraic results can be found in the Sulvasutras[1]). Here a single symbol like ‘x’ represents an unmanifest quantity that can potentially take one of many values. It eventually takes a fixed numerical value that is feasible to the equations representing the reality which it is part of. In [10], we find this algebraic concept mirrored in Sanskrit [10]: “When a word with a contextually determined meaning is reduced to only one of its many meanings, it is akin to assigning a specific constant value to an algebraic variable, thereby eliminating its usefulness as a variable.”

These context-sensitive meanings in Sanskrit, and the Contextual and Universal Dharma ethics are other well-known concepts that resemble this idea. For example, the word ‘Lingam’ which means symbol or icon has multiple contextual meanings [10]. The idea of equations and the introduction of a symbolic processing language to manage such equations also existed in India. The Bakshali manuscript provides evidence of this [6]. Aside from the decimal system, there were also the Katapayadi, Bhutasamkhya, and the Aryabhata notation that encoded numerical data in exquisite sacred verse [1]. Here are some bewitching examples.

This Paninian approach naturally motivates the generation of permutations and combinations while are fundamental to the idea of mathematical ‘probability and chance’, and finds application in Sanskrit Kavya [3]. The infinite-series results achieved by Madhava of the Kerala school long before McLaurin/Taylor/Leibniz, etc. also resemble this generating principle. The game of chess (Chaturanga), and snakes and ladders (Moksha Pata, Vaikuntapalli), etc. also have a similar structure and not surprisingly, originated in India. German Sanskritist Paul Thieme noted that a civilization that produced Paninian grammar could easily have produced also the game of chess, which it did [4].  The potential chaos that can arise from permitting multiplicity is ingeniously managed via the guiding principles of Dharma to produce harmonious order. All these discussions raise an obvious question – why are Indians so ‘tuned in’ to this integral approach?

Forest Civilization’s Pattern Seekers and Algorizers

The multiplicity of numbers, cascading permutations, infinitesimals running amok, and the never ending decimals of irrationals seemingly paralyzed the binary mindset at one point in time. On the other hand, this chaotic prospect caused little anxiety among the ancient Indians who were grounded in Dharmic view where such diversity are but natural forms of the One. In general, the practice of Ganita is appealing to those who seek recurring patterns and inter-connections in nature.

India is a forest civilization [10]. A significant portion of the narrative in two of its major works of Itihasa, the Ramayana and the Mahabharata, occur in the forest, which is a complex ecosystem where the inter-dependency of its members is Omnipresent.  It is but natural that the Indians are attracted to the infinitely repeating patterns that abound in nature and draw inspiration and inferences from them. On the left is a picture of the tessellations drawn by ancient Indians on rocks [19] several thousands of years ago (possibly upper Paleolithic period. This may remind some of Kolams). On the right is a visualization of the theta function [20] that Ramanujan may have studied while coming up with his equations for the ‘mock theta’ functions that he made famous.

tessellations in ancient Indian rock art theta functions

Based on an intuition and deep contemplation about certain connections and resemblances observed in nature, a Ganita expert comes up with a sequence of calculations. Scientist Roddam Narasimha describes this Indian approach [24] as that of pattern seekers and algorisers and that the Indian astronomer (like Aryabhata) can “discern patterns in planetary motion and make computations, and proceeds to devise clever algorithms to carry out such calculations“. He describes the Indian approach via the following sequence: observation →  algorithm → validated conclusion. Several Sanskrit keywords are used within this approach, for which no exact English equivalent word exists. We briefly summarize these keywords based on the discussion in [6].

Key Ganita Non-Translatables

Pramana – correct cognition, a means of acquiring valid knowledge. Pratyaksha and Anumana are two important Pramanas in Ganita.

Anumana – inference, the key reasoning component in Indian logic. This is not the same as deduction, but is a derived conclusion from the observation of patterns.

Pariksha – careful comprehensive observation. e.g. yantra pariksha:  observation using instruments. An extension of Pratyaksha (direct observation and perception), the oldest and most universal Pramana among darshanas.

Drg-Ganita –  ‘seeing and computing’. This is an important method introduced by Parameshwara of the Kerala school, which looks for agreement between what was computed and what is observed.

Siddhanta – a validated conclusion, or a validated algorithmic package. What happens when there is a clash between Siddhantas? In [6], “Nilakantha recommends that under such  conditions more observations need to be taken with instruments and compared with calculation, and that the numerical parameters should be changed (or the algorithms tuned) so as to improve agreement. In other words a new siddhanta has to be created. Siddhantas are thus human creations, and the best at any time may not remain
so for long—it is valid only for some finite periods of time.”

Yukti – skilful and ingenious practice. Ganita gives the pride of place to Yukti, sometimes overruling the primacy of the Agamas. Verse (2.5) from the Bhagavad Gita says “yogah: karmasu kausalam“, yoga is skill in action [6].  It appears that the Ganita tradition had little time for ‘pure’ theorists who lacked the Yukti or intent to deliver realizable results.

Anveshana – ‘wild goose chase’. In general usage, this word has a positive connotation but in the context of Ganita it represents a futile exploration.

Upapatti – a rigorous validation of results to the satisfaction of peer experts. Yukti is employed to constructively demonstrate how a result can be correctly reproduced by anyone else. This is not the same as the synthetic notion of abstract proof [1]. An important book in this regard is the Ganita Yukti Bhasa of Jyeshtadeva hailing from the Kerala school. It is a myth that Indian mathematicians provided no proof of their results. One has to read the accompanying commentaries on the results stated in Sutra form in order to understand all aspects of a Ganita result, including the validation step. The tradition of providing Upapattis is an old and well established one [22].

We conclude this introductory post by excerpting some passages from an essay on Mathematics by Henry Poincaré. In this essay, we get to read his independent views on the nature of reality. He also provides a balanced discussion of the pros and cons of different approaches that can be employed to generate new results. It is worth comparing his views with the ancient Indian perspective. This discussion also sets the stage for the next blog in this series.

Poincaré on ‘what is reality?’

We excerpt a couple of paragraphs from a 1905 essay [21] by the great French mathematician Henri Poincaré. From a Dharma and Ganita perspective, Poincare alludes to the integral unity of reality rather than a synthetic ‘artificial assemblage’. He also talks about the need for a ‘direct sense’ of the internal unity of a piece of reasoning in order to possess the ‘entire reality’. He also uses the principle of resemblance to explain his ideas.

The physiologists tell us that organisms are formed of cells; the chemists add that cells themselves are formed of atoms. Does this mean that these atoms or these cells constitute reality, or rather the sole reality? The way in which these cells are arranged and from which results the unity of the individual, is not it also a reality much more interesting than that of the isolated elements…?

Well, there is something analogous to this in mathematics. The logician cuts up, so to speak, each demonstration into a very great number of elementary operations; when we have examined these operations one after the other and ascertained that each is correct, are we to think we have grasped the real meaning of the demonstration? …. Evidently not; we shall not yet possess the entire reality; that I know not what which makes the unity of the demonstration will completely elude us.

“…often a very uncommon penetration is necessary for their discovery. The analysts, not to let these hidden analogies escape them, that is, in order to be inventors, must, without the aid of the senses and imagination, have a direct sense of what constitutes the unity of a piece of reasoning, of what makes, so to speak, its soul and inmost life. When one talked with M Hermite, he never evoked a sensuous image, and yet you soon perceived that the most abstract entities were for him like living beings. He did not see them, but he perceived that they are not an artificial assemblage, and that they have some principle of internal unity.


What we don’t know about India’s Ganita heritage is much more than what we currently know. Only a minuscule fraction of primary source texts of Ganita have been studied and interpreted so far. We have to thank researchers like the late K. V. Sarma for their tireless work in this regard.

"Our youth are hungry for a sensible knowledge of our past, but are denied an opportunity to acquire it by a marvellous educational system that shuns history in science curricula, and by the paucity of attractive but reliable accounts of the fascinating history of Indic ideas. Our academies, universities, museums and other institutions need to make such a project a national mission. Anything less would be irrational blindness to a unique legacy." - Roddam Narasimha [23].
Acknowledgment: I thank the ICP editor and bloggers for their constructive feedback and corrections.
* Indic epistemology traditionally places Ganita under Jyotisha. The original quote in Vedanga Jyotisha refers to Jyotisha in its enlarged meaning, hence the popular direct translation today of that word as referring to 'Math', used above as Ganita.

References
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  3. Cultural Foundations of Mathematics: The Nature of Mathematical Proof and the Transmission of the Calculus from India to Europe in the 16th c. CE., C. K. Raju, Pearson India. 2009.
  4. Vilasamanimanjari: a 19th century chess manual in Sanskrit. Shrinivas Tilak. 2011.
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    Kripa Shankar Shukla, in collaboration with K. V. Sarma. 1976.
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  11. Art and Cosmology in India. Subhash Kak. Patanjali Lecture given at Center for Indic Studies, University of Massachusetts, 2006.
  12. The Pragnya Sutra: Aphorisms of Intuition. Subhash Kak. Baton Rouge, 2006.
  13. Indra’s Net: Defending Hinduism’s Philosophical Unity. Rajiv Malhotra. Harper Collins. 2011.
  14. How to Measure Anything: Finding the Value of Intangibles in Business. 3rd Edition. Douglas W. Hubbard. Wiley. 2014.
  15. Why are Normal Distributions Normal? Aidan Lyon. British Journal of the Philosophy of Science. 2013.
  16. Sync: How Order Emerges from Chaos In the Universe, Nature, and Daily Life. Steven H. Strogatz,Hachette Books. 2012.
  17. Abhinavagupta’s Conception of Humor, svabhinava.org.  Sunthar Visuvalingam.
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  19. Computing Science in Ancient India. T. R. N. Rao and Subhash Kak. Center for Advanced Computer Studies. University of SW Louisiana. 1998.
  20. Ramanujan’s Mock Modular Forms: Indian Mathematician’s Dream Conjecture Finally Proven. Huffington Post Science 2012.
  21. Intuition and Logic in Mathematics. English Translation of Essay by Henri Poincaré. 1905.
  22. Ganita Yukti Bhasa: Rationales in Mathematical Astronomy of Jyeshtadeva. Vol 1. Malalayalam text critically edited with English translation by K. V. Sarma. 2008
  23. The ‘historic’ storm at the Mumbai Science Congress. Roddam Narasimha. Guest Editorial, Current Science, Vol 108 (4), 2015.
  24. Some thoughts on the Indian half of Needham question: Axioms, models and algorithms. Roddam Narasimha. 2002.
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