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

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)


Statistics is very useful even in social science and humanities.

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

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.


  • 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

  • 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)

"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.

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


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)


  • 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.


  • 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

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)

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

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

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

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.

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.

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.


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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