## Indic knowledge courses

C. K. Raju

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.

### 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.

### 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.

### 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

• 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.

### 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.

### 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"

• 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

### 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 (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.

### 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.

### 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

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.