Indic knowledge courses

C. K. Raju

Indian Institute of Advanced Study

Rashtrapati Nivas, Shimla

Introduction

Will talk of 3 undergraduate-level courses

  • Calculus (and differential equations)
  • Statistics
  • History and philosophy of science (HPS)

Indic origins

  • Calculus is of Indian origin
  • as is probability and statistics
  • A corrected history AND philosophy of science needs to be taught
    • so students are aware of the historical Indian contributions to math and science.

Details of Indic origins: Background reading

Practical value today

  • WHY teach these Indic knowledge courses?
  • Just for sentimental value?
  • Because our ancestors did it?
  • A. NO. I teach them for their PRACTICAL value to students, TODAY.

So, what is wrong with existing courses on calculus and statistics?

  • They teach only mechanical application of formulae.
  • Students lack conceptual grasp of most elementary concepts such as
  • limit, derivative, integral, probability etc.
  • (will go into details shortly).

Ganita method

  • Teaching math (calculus and statistics) as ganita makes math easy and
  • enables students to solve harder problems not covered in usual courses
  • and to acquire a solid conceptual grasp.

A note

What is the practical value of the course on history and philosophy of science?

  • Western education taught us deshi = inferior and videshi = superior.
  • And hence taught us to imitate the West in every way.
  • As our school math texts still teach in class IX
  • that Indian geometry was inferior and we should imitate "superior" Western geometry.

Believed without checking

  • We accepted that claim of Western superiority
  • without ever checking it.
  • because teaching Western superiority at an early age indoctrinates us.

Consequence: our academic system 100% imitates the West

  • And we are frightened of not imitating the West
  • e.g. in mathematics,
  • without understanding that we taught most core mathematics to the West
  • We superstitiously fear that non-imitation may result in some terrible catastrophe.

The course on history and philosophy of science (HPS) teaches that

  • This claim of Western superiority is bogus
  • that it is based on a false history and bad philosophy of math.
  • This poorva paksha needs to be countered BEFORE we boast that we did this and that.
  • But we cannot contest it, because there is not a single university department or decent course in HPS in India.

But to understand how false history was used to trick and enslave us

  • we need to understand the tricks by which a false history was spread through the education system
  • including the trick of declaring a particular philosophy as (normatively) universal
  • as used in giving credit to "Pythagoras" for the so-called "Pythagorean theorem".

So the practical advantage of the course is that it will get rid of our sense of inferiority

  • it will give us the courage to innovate and get ahead
  • as we did before colonialism.
  • This is the right way to do it, instead of making wild claims of spacecraft in ancient India
  • which make us an object of ridicule.

Or false claims of "Vedic math"

Recap

  • Teaching math (calculus and statistics) using Indic methodology of ganita
  • makes math easy and enables students to solve harder problems not covered in existing courses.
  • Teaching history and philosophy of science exposes the false Western history of science
  • and the trick of using a bad philosophy.
  • and enables us to escape from the resulting sense of racial and cultural inferiority.

Calculus

  • Calculus originated in India with आर्यभट (5th c.)
  • Developed across a thousand years,
  • and was stolen and taken to Europe in 16th c.
  • for a solution of the European navigation problem

European navigational problem

  • was then the biggest scientific challenge in Europe from 15th to 18th c.
  • and many European governments offered large prizes for its solution
  • such as the prize offered by the British Board of Longitude set up
  • by an act of Parliament in 1711.

Note

भट is a dalit name (भट्ट the title of a Brahmin)

  • And Aryabhata's followers included the highest caste Namboodiri Brahmins from Kerala
  • such as नीलकंठ सोमसुत्वन who wrote a commentary on आर्यभटीय.
  • Caste system was quite different in ancient India.

Discovery doctrine

But we never study history of science, so are easily fooled

  • by false Western history repeated in math texts.
  • Will come back to history.
  • Let us move on to calculus.

The size of calculus texts

  • Typical early calculus texts (e.g., Thomas1, Stewart2) today have over 1300 pages in large pages (and small type).
  • Thomas = 1228 + 34 +80 + 14 + 6 + 6 + xvi (=1384) pages; size \(11 \times 8.5\) inches.
  • Stewart = 1168 + 134 + xxv pp. (= 1327) pages; size \(10 \times 8.5\) inches + CD).

3 years to learn calculus

  • At an average of 1 page per day a student will take over 3 years to read these texts!
  • And calculus is taught across 3 years: 11th, 12th, and beginning undergrad courses.
  • But, at the end what does the student learn?

The difficulty of limits

  • Surprisingly little!
  • Understanding a simple calculus statement \[\frac{d}{dx} \sin(x) = \cos(x),\]
  • needs a definition of \(\frac{d}{dx}\).
  • That requires limits.

But limits NOT defined

  • Indian NCERT class XI text says:

    First, we give an intuitive idea of derivative (without actually defining it). Then we give a naive definition of limit and study some algebra of limits3

The formal definition of limits

  • On present-day mathematics, the symbol \(\frac{d}{dx}\) is defined for a (real-valued) function (of one real variable) \(f\),
  • using another symbol \(\lim_{h \to 0}\).
  • \[\frac{df}{dx} = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}.\]

Definition of derivative (contd)

  • \(\lim_{h \to 0}\) is formally defined as follows. \[\lim_{x \to a} g(x) = l\] if and only if \(\forall\: \epsilon > 0, ~\exists\: \delta > 0\) such that \[0 < |x - a| < \epsilon\: \Rightarrow | g(x) - l | < \delta, \quad \forall x \in \mathbb R.\]

The missing element

  • The texts of Thomas and Stewart both have a section called "precise definition of limits".
  • But the definitions given are not precise.
  • They have the ritualistic \(\epsilon\)'s and \(\delta\)'s.
  • But are missing one key element: \(\mathbb R\).

The difficulty of defining \(\mathbb R\)

  • Formal reals \(\mathbb{R}\) often built using Dedekind cuts.
  • Set theory provides a model for formal natural numbers \(\mathbb N\),
  • which provide a model for Peano arithmetic.

The difficulty of defining \(\mathbb R\) (contd)

  • \(\mathbb N\)  can be extended to the integers \(\mathbb Z\).
  • This integral domain \(\mathbb Z\) can be embedded in a field of rationals \(\mathbb Q\).
  • Finally, \(\mathbb Q\) can be used to construct cuts.

Dedekind cuts

  • \(\alpha \subset \mathbb Q\) is called a cut if
    1. \(\alpha \neq \emptyset\), and \(\alpha \neq \mathbb Q\).
    2. \(p \in \alpha\) and \(q < p \implies q \in \alpha\)
    3. \(\not\exists \, m \in \alpha\) such that \(p \leq m\quad \forall p \in \alpha\)

Dedekind cuts (contd)

  • Addition (\(+\)), multiplication (\(\cdot\)), and order (\(<\)) among cuts defined in the obvious way.
  • May be readily shown that the cuts form an ordered field, viz., \(\mathbb R\).
  • Called "cuts" since Dedekind's intuitive idea originated from Elements 1.1.

Elements 1.1: The fish figure

  • With W, E as centers and WE as radius two arcs are drawn, and they intersect at N and S.
  • Used in India to construct a perpendicular bisector to the EW line and thus determine NS.

Rejection of empirical methods of proof

  • Elements, I.1 uses this figure to construct the equilateral triangle WNE on the given segment WE.
  • Empirically manifest that the two arcs intersect at a point.
  • This appeal to empirical methods of proof was rejected in the West.

Rejection of empirical proofs

  • \(\mathbb R\) required for formal proof which prohibits empirical proofs (accepted in ganita).

Historical note

  • If real numbers are essential for calculus
  • then since Dedekind invented real numbers in 1878
  • No one prior to that had any proper understanding of calculus.
  • So how did Newton and Leibniz invent calculus without understanding?

Epistemic test

  • Lack of understanding is proof that knowledge of calculus was stolen
  • How do you prove that a student copies in an exam?
  • If he lacks knowledge of what he claims to know
  • and cannot explain what has written in his exam paper.

The integral

  • What about \(\int f(x) dx\)?
  • Integral taught as the anti-derivative, with an unsatisfying constant of integration. \[\text{if}~ \frac{d}{dx} f(x) = g(x) \quad \text{then}\quad \int g(x) dx = f(x) + c\]

Riemann integral

  • It is believed that some clarity can be brought about by teaching the Riemann integral obtained as a limit of sums. \[\int_a^b f(x) dx = \lim_{ \mu(P)\to 0} \sum_{i=i}^n {f(t_i) \Delta x_i}\]
  • Here the set \(P = \{ x_0, x_1, x_2, \dots x_n \}\) is a partition of the interval \([a, b]\), and \(t_i \in [x_i,\ x_{i-1}]\).

The integral (contd.)

  • Once more defining even the elementary Riemann integral requires a definition of the limits.
  • hence real numbers.

The difficulty in defining functions

  • What is the definition of \(\sin(x)\)?
  • Most students think \(\sin(x)\) relates to triangles.
  • Do not learn the correct definitions of \(\sin(x)\) etc.
  • since the definition of transcendental functions involve infinite series
  • and is today believed to require notions of uniform convergence.

What the student takes away

  • Calculus texts trick the student into a state of psychological satisfaction of having "understood" matters.
  • The trick is to make the concepts and rules seem intuitively plausible
  • by appealing to visual (geometric) intuition, or physical intuition etc.

What the student takes away (contd)

  • Thus, apart from a bunch of rules, the student carries away the following images: \[\begin{aligned} \text{function} &= \text{graph} \nonumber\\ \text{derivative} &= \text{slope of tangent to graph} \nonumber\\ \text{integral} &= \text{area under the curve.} \nonumber\end{aligned}\]

Belief that visual intuition may deceive

  • The student is unable to relate the images to the rules.
  • Ironically, the whole point of teaching limits and real numbers is the belief that such visual intuition may be deceptive.
  • Recall that Dedekind cuts were motivated by the doubt that the "fish figure" (Elements 1.1) is deceptive.

Need for set theory

  • Real numbers are not the end of the story
  • whatever way we get real numbers
  • as Dedekind cuts or equivalence classes of Cauchy sequences
  • set theory is needed for it.

The difficulty of set theory

  • The construction of \(\mathbb R\) requires set theory.
  • Students are TOLD about set theory but not TAUGHT even the definition of a set.
  • What the student typically learns about set theory is something as follows.

    "A set is a collection of objects"

    or

    "A set is a well-defined collection of objects"

Russell's paradox

  • With such a loose definition it is not possible to escape things like Russell's paradox.
  • Let \[R = \left\{x | x \notin x \right\}.\]
  • If \(R \in R\) then, by definition, \(R \notin R\) so we have a contradiction.
  • On the other hand if \(R \notin R\) then, again by the definition of \(R\), we must have \(R \in R\), which is again a contradiction.

Why Cantorian set theory was rejected

  • So either way we have a contradiction.
  • From a contradiction any nonsense can be deduced: \[A \wedge \neg A \Rightarrow B\]
  • Paradox is supposedly resolved by axiomatic set theory,
  • but even among professional mathematicians, few learn axiomatic set theory.
  • Most make do with naive set theory.4

Have you understood the difficulties?

  • Can go on like this, but if you don't know about formal reals and axiomatic set theory
  • you will not understand these problems or further problems like Banach-Tarski paradox
  • So, here is a simple challenge problem
  • to highlight the difficulties

Cape Town challenge

Prize of Rs 10 lakhs

  • If you can offer a valid answer to my Cape Town challenge by tomorrow.
  • Serious offer, hence some caveats.
  • Easy to put your name on someone else's written response
  • So, you will be expected to publicly explain what you claim to have written.

In Indian calculus all these problems disappear

  • Empirical proof accepted
  • No limits needed for derivatives only finite differences (e.g. खंड ज्या)
  • needed for Aryabhata's method of numerical solution of differential equations
  • My study of practical application at C-DAC showed that numerical solution of differential equations
  • needed and suffice for ALL major practical applications of calculus.

Integrals

  • No need of integrals (e.g. \(\int f(x) dx\) is solution of \(y' = f(x)\))
  • On this method, non elementary integrals (e.g. elliptic integrals) as easy as elementary integrals

Sine function

Sine function (contd)

  • Jiva changed to "jiba" in Arabic
  • (since Arabic has no v sound)
  • written without nukta-s as consonantal skeleton "jb".
  • misread as "jaib" by ignorant Toledo translators.

Infinite series

How was the infinite series summed?

  • INFINITE geometric series WAS first summed by Nilakantha (1501)
  • (Finite geometric series very old, found in Yajurveda 17.2, "Eye of Horus" fraction in ancient Egypt etc.)
  • How was it done WITHOUT real numbers and limits?

Brahmagupta's avyakta ganita

  • Brahmagupta's avyakta ganita of polynomials
  • was reproduced in al Khwarizmi's al jabr waal muqabala, later became Algebra.
  • This polynomial arithmetic is non-Archimedean
  • (means there are infinities and infinitesimals, but no unique limits)

Formula for finite sum is extended to infinite sum

  • and infinitesimals are discarded on the philosophy of zeroism.
  • Further details requires a full course.

Course on calculus without limits has been tried out

Course on calculus without limits (contd)

Advantages of the Indic calculus course

  • Conceptual clarity: no unreal "real" numbers, limits etc. which few understand
  • Teaches real life applications (e.g. ballistics with air resistance)
  • Teaches non-elementary integrals omitted from calculus courses (e.g. correct theory of simple pendulum using elliptic functions).
  • See e.g. tutorial sheet
  • Teaches that it is silly to memorise formulae by using MAXIMA (earlier MACYSMA).

Statistics

Indian origins of probability and statistics

General references, see articles in books

Probability and statistics arose in India as normal math or गणित.

  • Probability relates to game of dice.
  • The first account of the game of dice is in the RgVeda.

अक्ष सूक्त (ऋग्वेद 10.34)

Mahabharata (Sabha parva)

  • Shakuni wins the game by deceit
  • Hence, there was an idea of a "fair (or unbiased) game".

Mahabharata (Van parva 72)

  • Story of Nala and Damayanti
  • Their separation. Disguised Nala takes a job as a charioteer with Rituparna, king of Ayodhya.
  • Damayanti announces swayamvara (widow remarriage).

Counting the fruits on a tree by sampling

  • Nala and Rituparna dash to Vidarbha.
  • Stop on the way near a Vibhitaka tree (mentioned in the aksa sukta)
  • the five-sided fruit of which was used in game of dice.
  • Rituparna shows off his knowledge of ganita, by counting the 2095 fruits on the tree.

Sampling is the easy way to count

  • a large number of fruits in a tree
  • as Rituparna explains to Nala
  • who tries to check by cutting down the tree and counting.

The theory of permutations and combinations first found in many ancient Indian texts

When Cochin-based Jesuits stole knowledge of calculus they also stole probability from India

  • for the sake of their navigational problem.
  • Indian texts had verses giving trigonometric values which are very accurate
  • needed to determine latitude, longitude, and loxodromes
  • Accurate navigation was the biggest scientific challenge in Europe from 15th to 18th c. CE

Binomial coefficients and "Pascal's" triangle

  • Binomial coefficients found in Pingala's Chandahsutra
  • His commentator 10th c. Halayudha obtains binomial coefficients using 17th c. "Pascal's" triangle
  • called Khanda-meru by Bhaskar II.

Pascal was close to Fermat

  • whose 1657 challenge problem to European mathematicians remained unsolved until 18th c.
  • Was a SOLVED EXERCISE in Bhaskara's Beejaganita (87)
  • involving large numbers 226153980 and 1766319049.

Because they stole, Europeans failed to understand calculus and probability until 20th c.

  • they eventually turned it into formal math to fit their superstitions about math.
  • Most people do not do Lebesgue integral and find probability as a measure is very hard to understand
  • especially for management students who need it for marketing surveys etc.

Statistics for AI

  • AI based heavily on statistics
  • Use without understanding can result in major calamities tomorrow.

Statistics for finance

  • Many science and engineering students shift to finance a lucrative line
  • This requires solid grounding in statistics, as explained in video of my recent JNU talk
  • Failure to understand can result in a huge crash like the sub-prime crisis.

History and philosophy of science

  • On my analysis, Macaulay imposed Western/colonial education using a false history of science
  • which we never checked.
  • We need to check it.

International HPS workshop

The course was popular

Course was localised to India

  • As per the original recommendation of the workshop
  • the course was to be localised to suit different countries.
  • Accordingly, when I taught it in SGT University in 2017
  • it was localised, as is clear from the final question paper.

The need to check history using facts from primary sources

  • One aspect that the course emphasizes is that
  • a whole lot of false history has been systematically created by the church/Western historians
  • Because establishing your inferiority is the first step to conversion.
  • So, those people who accept that everything indigenous is inferior are already half-converted.

The remedy

  • How to check that false history?
  • Today, most people google which takes them to Wikipedia.
  • But Wikipedia is an UNRELIABLE tertiary source.
  • Even standard texts are secondary sources which may be unreliable as the course teaches.

Case of Euclid

The need for philosophy

  • A second important issue to understand is
  • the trick by which a bad philosophy is used to change history.
  • Look at Q. 2b in the question paper.

Is "Pythagorean theorem" found in the sulba sutra?

  • It is often claimed that it is.
  • This ignores the poorva paksha or counter argument in our school text that Indians Egptians etc. had a statement,
  • but no deductive proof.

Indians did have proof

  • While Indians did have a proof as well
  • it involved empirical methods (but also used reasoning)
  • There is nothing wrong with proofs which use the empirical
  • accepted as the first means of proof by all Indian systems of philosophy
  • and by science.

Does prohibiting empirical make a proof superior?

  • All of our current math and stat teaching is based on that belief
  • which derives from a church superstition that proofs based on axioms and prohibiting facts are infallible.

Why math works

Indian "Pythagorean" calculation,

  • instead of \[d^2 = a^2 + b^2\]
  • the Manava sulba sutra$ has \[d = \sqrt {a^2 + b^2}\]
  • The two forms are NOT the same,
  • since calculation of square ROOTS was unknown in the West until the 12th c.

There are many other such differences that it is important to understand.

  • E.g. most ganita (arithmetic, algebra, "trigonometry", calculus, probability and statics) first went from India to Europe.
  • In geometry, we need to switch back to Rajju Ganita.
  • Cannot be all explained in a lecture: requires a full course.

Conclusions

  • Teaching math (calculus and statistics) as ganita makes math easy and
  • enables students to solve harder problems not covered in usual courses
  • and to acquire a solid conceptual grasp.
  • Teaching HPS courses is essential to end our
    • half conversion through a sense of inferiority through FALSE history and BAD philosophy
    • and resulting mental slavery to the West.

Final word: West is not best

  • You must fight; no one else will free you from mental slavery
  • The fight for freedom (swaraj) did NOT end 73 years ago.
  • This fight requires knowledge, not arms, so demand knowledge, not merely certificates
  • Accept knowledge from all sources, but choose what is best, not just what is West.

References (to texts)

  • G. B. Thomas, Maurice D. Weir, Joel Hass, Frank R. Giordano, Thomas' Calculus, Dorling Kindersley, 11th ed., 2008
  • James Stewart, Calculus: early Transcendentals, Thomson books, 5th ed, 2007.
  • J. V. Narlikar et al. Mathematics: Textbook for Class XI, NCERT, New Delhi, 2006, chp. 13 "Limits and Derivatives", p. 281.

Refereces to texts (contd.)

  • e.g. D. V. Widder, Advanced Calculus, 2nd ed., Prentice Hall, New Delhi, 1999.
  • e.g. W. Rudin, Principles of Mathematical Analysis, McGraw Hill, New York, 1964.
  • e.g. P. R. Halmos, Naive Set Theory, East-West Press, New Delhi, 1972.