Statistics for social science and humanities:
Should we teach it with
normal or formal math?

C. K. Raju
Indian Institute of Advanced Study
Rashtrapati Nivas, Shimla
ckr@ckraju.net

First, my respects to one of our famous statisticians C. R. Rao

• 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.) 😜

C. R. Rao in IIT: Delhi (1976)

Introduction

Statistics is very useful even in social science and humanities.

• 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)

Many other applications (e.g. genetics etc.

• 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".

E.g. Gender studies

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

J. B. S. Haldane from my Institute had explained long ago

• 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. 😩

But ignorance of exponential function not limited to women's groups

• 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: \(e^x\).

"What is \(e^x\)?"

• 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.

Similar thing happened in MIT

• 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.

Cape Town challenge

• 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.

Let us try this in JNU

• 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.

Prize of Rs 10 lakhs

• 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.

Note

• 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.

Can't do it? No problem, just make a full disclosure.

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

.

Of course, issue is not limited to 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.

So, just to teach what is probability

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

This is just the beginning

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

Naturally this is too tough

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

Interim summary

• 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?

Yes, these difficulties all arise from the use of formal math.

Formal vs normal math

So, why is 1+1=2 so difficult?

• 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.

Prohibiting empirical makes even 1+1=2 very complicated

• 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.

This huge complexity adds NIL practical value,

• 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.

Why prohibit facts? The church gains!

• 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".

False church history (still in our school texts)

• 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.

In contrast EVERY philosophical system in India, without exception,

• 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"?

Our professors of mathematics who cannot explain why 1+1=2 axiomatically

• 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.

But statistics is about empirical data.

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

Secondly, statistics is about risk, uncertainty,

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

Normal math

• 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)

In contrast, formal math

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

How did the two incompatible things get combined? Formal math and statistics?

• 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

Epistemic test: those who copy make mistakes

• 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.

Simple example of failure to understand

• 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.

A second-order example of failure to understand

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

Serial plagiarism

• 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.)

My Hawai'i paper (1999/2001) was an early paper on Indian calculus and its theft.

• 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's gross mistake

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

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

When I first confronted Joseph with plagiarism (in 2000 Dec), he privately made two points

• 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 thief Joseph was right

• 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.

Result

• 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.

Key point: Since Europeans stole calculus and probability,

• 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)

Decolonisation

• 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).

But, history repeats when people repeat the same mistakes

• 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.

BUT

• 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.)

Indian origins of probability and statistics

General references, see articles in books

• "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.

Step 1: Probability and statistics arose in India as normal math or गणित.

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

अक्ष सूक्त (ऋग्वेद 10.34)

• 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..)

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 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.

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 found in many ancient Indian texts

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

Cochin-based Jesuits stole knowledge of math (calculus and probability) from India along with astronomy texts in the 16th c.

• 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 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.

Commentaries on Bhaskar's text such as Kriyakramkari

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

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.

Example of European failure to understand

• "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.)

In the case of calculus Europeans could not explain (until 20th c.) what an infinite sum was (even metaphysically).

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

Thus according to the "law" of large numbers (weak or strong)

• 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.

Subjective probabilities

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

However, quantum mechanics is probabilistic

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

Quantum logic corresponds to a quasi-truth functional logic

• 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

Karl Popper rightly argued that

• 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.

Short summary

• 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.

The remedy

Decolonised course on STATISTICS was developed in Malaysia in 2013.

• 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.

A decolonised course in calculus was developed

• 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.

My course on "Calculus without limits" teaches calculus that way.

• 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.

So what is the advantage?

• 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?

So, normal math works!

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

Conclusions

Decolonised statistics

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

The main thing colonial education teaches is to blindly trust and imitate the West

• 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.

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