Should we teach it with

normal or formal math?

C. K. Raju

Indian Institute of Advanced Study

Rashtrapati Nivas, Shimla

ckr@ckraju.net

- Padma Vibhushan, and US President's medal
- turned 100 a few days ago.
- He has an impish sense of humour
- like his joke about the "Rao-Cramer inequality in height" (Cramer was very tall.) 😜

**Econometrics**a stock example (but won't get into it).**History**: Kosambi's pioneering application of statistics (Z-test) to punch-marked coins.**Literature**: e.g. the case of Shakespeare (frequency table, other authors).**Indic studies**: Statistical analysis of Kautilya's Arthashastra suggests multiple authors (T. R. Trautmann 1971, Brill)

- BUT there is a problem.
- Most people in social science and humanities are mathematically illiterate.
- E.g. Romila Thapar confounded math and statistics
- when she wrote "Kosambi applied math to history".

- A women's group once called me to explain
- how to prove drug trials were faked?
- (They had a court-case against some drug).

- Use a simple chi-square test, to separate real from fake data.
- They asked in chorus "what is chi-square?"
- "If \(X\) has a normal distribution, \(X^2\) is distributed as \(\chi^2\)."
- "What is a normal distribution?"

- Oh, just use the exponential function: \(\frac{1}{\sqrt{2 \pi} \sigma} e^{-\frac{x^2}{2 \sigma^2}}\).
- "What is the exponential function?"
- At this stage I despaired. 😩

- applies also to science and engineering students.
- Last year I gave a talk on Indian calculus in Pune University.
- I asked "what is the derivative of \(e^x\)?".
- Whole hall shouts: \(e^x\).

- Only 2-3 hands go up: an infinite series.
- \(e^x = 1 + x + \frac{x^2}{2!}+ \ldots\).
- "How do you add an infinity of terms?"
- Whole hall is silent.

- Issue of infinite sums involves (formal) "real" numbers (as such infinite sums).
- E.g. \(\sqrt 2 = 1+\frac{4}{10} + \frac{1}{100} + \ldots\).
- In MIT I asked: what is a real number?
- Whole hall was silent. (Check the video.)

- Students of MIT were actually smart. They guessed there would be further questions ahead:
- about axiomatic set theory: a metaphysics of infinity
- on which the theory of "real" numbers depends.
- So, I tried this out in Univ. of Cape Town 2017.

- Prove 1+1=2 in formal REAL numbers (not integers or natural numbers) axiomatically,
- from FIRST PRINCIPLES (in the manner of the 378 page proof of 1+1=2 in cardinals by Russell and Whitehead)
- WITHOUT assuming any result from AXIOMATIC SET THEORY.

- Minati Panda is Professor of Mathematics Education at Zakir Hussain Centre.
- Or take Dhruv Raina who has a PhD in History and Philosophy of Math from some Western university.

- If either of them offers a valid answer to my Cape Town challenge by tomorrow.
- Not enough time?
- Take a week, but I reduce the prize to Rs 1 lakh.
- 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.

- Hang a notice outside your door:
- "Professor of mathematics education
- "but do not know why 1+1=2"

.

- I was taught the STARTING point of statistics is this
- Let (\(\Omega, B, P\)) be a standard Borel probability space.
- \(\Omega\) is a topological space
- \(B\) is the Borel \(\sigma\) -algebra
- \(P\) is a probability measure on it.

- you need to know topology
- measure theory
- and Lebesgue integral.

- Advanced topics will get into
- stochastic processes
- Wiener measure
- stochastic integral (Ito integral) etc.

- even for students of IIT
- or AI practitioners
- or theorists in finance.

- Statistics is useful in social science and humanities
- but, in its present form, too difficult for people in social science and humanities to understand.
- Is there a remedy?

- Because today we use formal math.
- Proof of 1+1=2 in normal math is kindergarten stuff
- because normal math accepts empirical proofs,
- Formal math prohibits the empirical.

- as Bertrand Russell showed long ago.
- Further, in formal math, 1 may mean different things.
- 1 as cardinal or integer \(\neq\) 1 as "real" number.
- Hence, answering my Cape Town challenge may take perhaps 1000 pages.

- not in a grocer’s shop, not in calculating rocket trajectories.
- (as I demonstrate by doing ballistics in my course on "Calculus without limits").
- I was a formal mathematician before I joined C-DAC and worked with the Indian space program (ISRO).

- One reason I turned away from formal math is that
- it adds huge difficulty but nil practical value.

- Facts are contrary to church dogmas.
- During Crusades the church adopted its theology of reason (from Islam, to persuade Muslims)
- For which theology the church invented reasoning WITHOUT facts (axiomatic reasoning) and
- glorified it as "superior".

- fraudulently attributes the origin of reasoning MINUS facts
- to some non-existent Greeks like "Euclid".
- We uncritically accepted all this on colonial education
- which was 100% church education when it came.

- accepts प्रत्यक्ष प्रमाण (pratyaksa pramana).
- But this is declared inferior, without debate
- though science too is based on the empirical.
- So, why is reason MINUS facts "superior"?

- cannot critically examine the Western philosophy of math
- They hope to earn money from the West
- hence seek Western patronage, and remain loyal to the West.

- So, for statistics, why use formal math which prohibits the empirical?
- In fact, normal math, which accepts the empirical is prima facie more suitable.

- statistics is fallible
- involves inexactitude (95% confidence level)

- accepts the EMPIRICAL (E.g. Ganita 13)
- relates to the real world (like statistics)
- accepts INEXACTITUDE (e.g. सविशेष in शुल्ब सूत्र, Baudhayana, 2.12, आसन्न in Aryabhata Ganita 10, etc.), like statistics
- Accepts FALLIBILITY (like science, and statistics) (failure of "Pythagorean" "theorem" on curved earth, Laghu Bhaskariya, 1.27)

- PROHIBITS THE EMPIRICAL,
- hence relates to a FANTASY world.
- Claims to be EXACT (in a fantasy world).
- Accepts church SUPERSTITION of INFALLIBILITY of deduction.

- History provides the answer.
- Probability and statistics (like calculus) originated in India as normal math
- were stolen by Europeans along with calculus in the 16th c.
- But Europeans made mistakes in understanding this imported knowledge

- applies to students who cheat in an exam
- but also to glorified figures in history, like Newton
- Those mistakes hard to understand, so I will give only simple examples.

- E.g. West repeatedly imported Indian arithmetic but failed to understand it
- E.g. Gerbert's abacus (10th c.). (Mistake: failed to understand efficiency of Indian arithmetic through algorithms.)
- Florentine law (1300 CE) against zero.
- Difficulty persisted until 16th c.

- Not only was the calculus stolen from India,
- but my thesis that the calculus was stolen
- was also stolen by Britishers!

- George Gheverghese Joseph ("Crest of the Peacock") serially plagiarised my work on transmission of Indian calculus to Europe,
- first case public in 2004, second case 2007, etc.
- Though Joseph stole and falsely claimed credit, he did not understand the math.
- and made many FOOLISH MISTAKES. (Will describe only one.)

- At that time I had not understood how Indians summed infinite series
- (Nilakantha undeniably did sum the INFINITE geometric series).
- But I knew that calculus on computers was done differently
- using floating point numbers, NOT real numbers. (Was teaching floats to computer science students in 2000.)

- I explained this, mentioning floating point numbers in my Hawai'i paper on Indian calculus (even used a C-program).

- Joseph and Almeida stole my point (without understanding) in a paper in
*Race and Class*2004. They wrote (p. 51)

"the Kerala mathematicians employed …floating point numbers to understand the notion of the infinitesimal and derive infinite series for certain targeted functions (p. 51)"

- 🤣 🤣 🤣. Complete NONSENSE.
- Floats are used in computers just because they have nothing to do with infinities and infinitesimals.

- 1. British and US law allows Christians to steal (land, knowledge) from non-Christians (Doctrine of Christian discovery)
- 2. He boasted that he could tell any lie, and Indians will believe him. They will never examine details, but go gaga if a Britisher praises Indians.

- The Indian press went gaga over Joseph in 2007 when he stole my WHOLE 1999 paper VERBATIM.
- PTI went by that upublised paper (mine, but now credited to Manchester Univ.)
- and refused to examine my Hawai'i paper or my 500+ page book on calculus and its transmission.
- ONLY the Hindustan Times published a retraction.

- No one else ever checked the facts.
- and Hyderabad university calls this ignoramus for a conference on math education.
- Because Western endorsement is our only test of truth.
- While colonised minds glorify similar "discovery" of calculus (without understanding) by Newton (sine series) or Leibniz (\(\pi\) series)
- and ape also the Western understanding of statistics.

- they failed to understand certain aspects of calculus (e.g. infinite series) and also probability
- Europeans modified calculus and probability to fit their SUPERSTITIONS about formal math
- This made the math very complicated (without adding practical value)

- So remedy is to decolonise math.
- Decolonisation is NOT blind rejection of everything Western (as often caricatured),
- it is a critical rejection
- of Western superstitions (arising from prolonged church hegemony).

- We allowed calculus to be stolen 500 years ago
- and are repeating the mistake by allowing the calculus transmission thesis to be stolen today.
- The solution is to understand and correct the mistakes made by those who steal (and hence make mistakes)
- for the issue is not mere glory, but better understanding.

- If we can't even check or understand the crude mistakes made by Joseph et al., in last 20 years
- there is no hope Indians will ever understand or correct Newton's subtle mistake about calculus or
- Western mistake about "real" numbers, which are taught from class XI,
- or Western mistakes in understanding probability etc. (But, let me try.)

- "Probability in Ancient India",
*Handbook of Philosophy of Statistics*, Elsevier, 2012, pp. 1175-96. - "Probability",
*Encyclopedia of Non-Western science…*, Springer, 2016, pp. 3585–3589. -
*Cultural Foundations of Mathematics…transmission of the calculus from India to Europe in the 16th c. CE*, Pearson Longman, 2007.

- Statistics relates to game of dice.
- The first account of the game of dice is in the RgVeda.

- Translation
- (But when I pointed it out, in 2012, Witzel from Harvard LIED brazenly about it, "He said I used Wilson's translation", "talked of six-faced die".
- Shows you can't trust Western historians to be honest even about easily checked facts..)

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

- 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).

- Nala and Rituprana 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.

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

- from the time of the Jain Bhagwati sutra, Susruta, and Varahamihira etc.
- E.g. in the common school text ("Slate arithmetic") of Sridhara

- for the sake of their navigational problem.
- These 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

- Jesuits took Indian math texts to Europe where they "discovered" the math
- using the genocidal doctrine of Christian discovery
- that the first Christian to spot a piece of land or knowledge becomes its discoverer.

- 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.

- 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.

- were being translated and imported into Europe in connection with the European navigational problem.
- This was Christian discovery, NOT independent rediscovery.

- they eventually turned it into formal math to fit their superstitions about math.

- "What is probability?"
- If we toss a coin repeatedly, what we get is RELATIVE FREQUENCY of heads, say = \(\frac{\text{Number of heads}}{\text{Total number of coin tosses}}\).

- But relative frequency is NOT a fixed number.
- it keeps varying.
- (It is only near (asanna) \(\frac{1}{2}\) for an unbiased coin.)

- they invented formal (= unreal) "real" numbers and limits
- BUT THIS DOES NOT WORK TO DEFINE PROBABILITY.

- relative frequency converges to probability
- but only in a probabilistic sense (convergence in measure)
- (for a large number of trials the probability of large deviations become small)
- So, limits cannot be used to DEFINE probability without fallaciously begging the question.

- The other option is to regard "probability" as degree of subjective belief.
- In this case probability is defined over a boolean algebra (of propositions)

- "God does play dice"
- quantum probabilities are objective
- and quantum logic is NON-boolean
- (joint probability distributions do not exist)

- like Buddhist logic of चतुष्कोटी or Jain logic of स्यादवाद.
- as discussed in my books
*Time: Towards a Consistent Theory*, and*Eleven Pictures of Time*

- the frequentist interpretation of probability
- as also the subjective interpretation both fail.
- But his "propensities" are meaningless
- and his "solution" to the "problem of induction" is fallacious as I have explained.

- West has yet to understand "probability".
- How does statistics work then?
- Because the formal math of (\(\Omega, B, P\)) is NEVER used in practice.
- Especially not in social science and humanities.

- As per the stipulations of the Malaysian Qualifications Agency.
- Of course students need to know normal distribution, \(\chi^2\) etc.
- So, this requires as a pre-requisite a core course in decolonised calculus.

- Using the calculus as it developed in India
- starting from the 5th c. Aryabhata's numerical method ("Euler's" method of solving ordinary differential equations)
- and 7th c. Brahmagupta and his non-Archimedean arithmetic of polynomials which works better than "real" numbers
- plus the philosophy of zeroism (sunyavada) which accepts inexactitude.

- As later developed by the Aryabhata school in Kerala
- to sum infinite series.

- Taught for last 10 years in 3 countries.
- Tried out also with students of social science and humanities.
- CUTS, Sarnath, University Sains Malaysia (four groups),Ambedkar University, Delhi, CISSC, Iran, and SGT University, Delhi, NCR.

- Let me give a 25-year old example (also in my Hawai'i paper).
- In finance one encounters non-normal and fat-tailed distributions like the Levy distribution].
- This is a situation (stochastic differential equations driven by Levy motion) where formal mathematics fails, forcing computation.
- So, why not do things numerically from the beginning?

- even when formal math fails,
- as in this example.

- can help people in social science and humanities to acquire new insights.
- It is easy to learn.
- But

- The colonially educated are unable to think critically about the West.
- So, the suggested change probably cannot be implemented.
- People just have to keep suffering because they are voluntary slaves.