How colonial education changed our math teaching

(and what we can do about it today)

C. K. Raju

Indian Institute of Advanced Study

Math is not universal

  • The math we teach today in school and university is formal math.
  • also known as axiomatic math
  • differs from traditional (pre-colonial) गणित

"Raju non-equation"

  • गणित \(\neq\) formal math

  • How exactly does गणित differ from math?
  • Don't we have \(1+1=2\) in both?

  • \(1+1=2\) is too difficult!
  • Let us start with something easier.

Culture influences math

Definition of angle

  • The definition uses a term उभयनिष्ठ
  • for the common initial point of
  • an angle made by two straight lines (rays).

  • उभयनिष्ठ not in Hindi (or Sanskrit) dictionary.
  • Student must infer its meaning from उभय and निष्ठ (in Sanskrit): "loyal to both".

  • Use of such a complicated term suggests that the corresponding concept is absent in Hindi.
  • Is it? What is the Hindi word for angle?

  • Common answer कोण.
  • Wrong!

  • Word कोण not found in India before 18th c.
  • (Coined by Samrat Jagannath 1723 in Rekhaganit, a Sanskrit translation of "Euclid's" Elements from Farsi.)

  • So, did Indians have no concept of angle earlier?
  • They did.

E.g.

  • But the concept of angle in India was different.
  • चाप (chord) is the correct traditional term for "angle".
  • E.g., famous निहत्य चाप वर्गेण चापं from Yukitidipika, 441
  • Angle = the relative length of an arc.

  • If an angle is about two straight lines, why is a semi-circular protractor needed to measure it?
  • And, what instrument in a compass box (geometry box) can be used to measure angle in the sense of चाप?

  • None. The two concepts of angle require different instruments for their measurement.
  • The चाप definition requires a flexible string.
  • More on this in the second lecture.

  • Immediate point is this: math is NOT universal.
  • Let us move on to the more difficult 1+1=2.

Prohibition of empirical proof

  • Formal math focuses on proof.
  • That \(1+1=2\) not so important in formal math.
  • Important thing is why \(1+1=2\).

  • In formal math, the "why" requires a "proof".
  • A special kind of proof,
  • for formal math prohibits empirical proof.
  • (Empirical proof = proof based on evidence of senses.)

  • E.g. Formal math does NOT allow you to "prove" \(1+1=2\) by pointing.
  • Why not? Because you can see the fruits.
    • That is empirical and prohibited.

  • What is wrong with seeing something?
  • The empirical is fallible:
  • E.g. one may mistake a snake for a rope or rope for snake.

  • Mistakes happen, don't they?
  • But formal math claims to be infallible : the theorems of formal math are supposedly certain truths.

  • The empirical is fallible, and the example (rope/snake) is accepted in India (न्यायावलिः 3041, रज्जुसर्पन्यायः)
  • Nevertheless, ALL Indian systems (without exception) न्याय-वैशेषिक, सांख्य-योग, मीमांसा (incl. अद्वैत वेदांत), लोकायत, बौद्ध, जैन, etc.
  • accept empirical proof (प्रत्यक्ष प्रमाण) as the first means of proof.

  • Consequently, गणित too accepts empirical proof.
  • E.g. आर्यभट, गणित 13: साध्या जलेन समभूरधऊर्ध्वं लम्बकेनैव
  • "Determine level ground by water, verticality by a plumb"

A key difference

  • Anything you can see is empirical hence fallible and no part of formal math.
  • But गणित accepts empirical proof.

Declaring empirical proofs as inferior

  • declares ALL Indian philosophy as inferior
  • and also Indian गणित as inferior
  • as our 9th standard math text does.

  • Science too accepts empirical proof (experimental proof).
  • And must be accepted as "inferior" to formal math.

A question

  • Will science become "superior" if we reject empirical proof?
  • Or will formal math improve if we accept empirical proof?

Note 1: Indians did use reason

  • Acceptance of empirical often caricatured as rejection of reason.
  • Not true.

.

  • E.g., Aryabhata used both reason and facts (far off trees cannot be seen, horizon is round) to deduce that the earth is round.
  • like a kadamba flower (Gola 7)

  • गणित uses BOTH facts (empirical proofs) and reasoning exactly like science.

Note 2: conflating two types of reasoning

  • One word "reason" used for both
  • scientific or normal reasoning (reasoning with facts), and
  • axiomatic of formal reasoning (reasoning without facts)

  • Using one word, confounds issues and misleads as in our school text.
  • Don't conflate the two meaning of reason
  • axiomatic reasoning (reason MINUS facts) with scientific reasoning (reason PLUS facts).

Note 3. Greeks never used axiomatic reasoning

  • Our school text says "Greeks alone used "reason" in math". (Which reason?)
  • This is false for two reasons.

  • The "Greeks alone" part is wrong; others, such as Indians too used reason in math. We saw that.
  • The "Greeks used reason" part is misleading because of the two meaning of reason.

  • Greeks gave axioms and proofs, separately BUT
  • no axiomatic proofs (i.e., proofs without using the empirical).

  • E.g. First proposition of the Elements gives an empirical proof.
  • One sees the two arcs intersecting.
  • But they may or may not have a point in common.

  • Also the 4th proposition (Side-angle-side theorem).
  • Original proof was by putting one triangle appropriately on top of the other to see that the two triangles are equal.
  • (But anything one can see is NOT formal math, therefore, today we teach SAS postulate.)

  • 4th proposition needed for the proof of the penultimate ("Pythagorean") theorem.
  • Hence ALL original Greek proofs of the Pythagorean theorem involve the empirical.

Definition of axiomatic proof

  • A sequence of sentences.
  • Each sentence must be either an axiom
  • or derived from preceding sentences by a rule of reasoning.

E.g.

  • A implies B
  • A
  • Therefore, B.

  • Simple?
  • Not quite.

  • Moreover, Russell only proved 1+1=2 for "integers" (actually "natural numbers").
  • In formal math, a separate proof is required for 1+1=2 in formal "real" numbers.

  • Cape Town challenge: Give a full axiomatic proof of 1+1=2 in "real" numbers (without assuming any axiomatic set theory).
  • May need a 1000 pages or more (no one did it).
  • Try this challenge on some IIT professors of math or anyone you think is knowledgeable in math.

Are axiomatic proofs infallible?

  • How exactly do you know that Russell's proof is valid?
  • That it does not contain 22 mistakes?

  • Only by reliance on Russell's authority.
  • (He got the Nobel prize!)
  • Authority is MORE fallible than empirical proof.
  • Therefore, deduction is MORE fallible than empirical proof.

Another example of fallibility of deduction

  • The game of chess is pure deduction.
  • You always lose to machine because even the top grandmaster always makes mistakes.

  • Belief in the infallibility of deduction is mere SUPERSTITION
  • like the belief in the infallibility of the pope.

Calculation vs proof

  • What did we gain from Russell's complicated proof?
  • गणित has proof but focuses on calculation because
  • all practical value comes from calculation.
  • In a grocer's shop what is important is THAT 1+1=2, not its non-empirical proof.

How did Russell do his groceries?

  • Columbus supposedly proved (empirically) that the earth is round
  • but did not know how to calculate the earth's radius
  • resulting in the huge European navigational problem until the 18th c.

Are axiomatic proofs superior?

  • Our school text (class IX) teaches that axiomatic proofs are superior.
  • (Hence we should do formal math not गणित.)
  • Is that correct?

What is an axiom?

  • An axiomatic proof begins with axioms not facts.
  • Today, axiom = postulate(= assumption)
  • as our class IX text admits.

  • But wrong assumptions lead to wrong conclusions.
  • Lokayata explained this using the story of the wolf's pug marks

E.g. in modern terms

  • Axiom 1. All animals have two horns.
  • Axiom 2. A rabbit is an animal.
  • Theorem. Therefore, a rabbit has two horns.
  • So mathematical theorems need not be even remotely valid knowledge.

  • How does one know axioms are valid?
  • Usually, no way to test the axioms of mathematics empirically.
  • That is, axioms of formal math are pure metaphysics (irrefutable in the sense of Karl Popper).

E.g.

  • Our class IX school text says geometric points are invisible (no size). (Remember: If you can see it, it is not formal math.)
  • Is that a con-trick like the Emperor's new clothes?

  • Hence, a line which consists of points is also invisible. (Class V text defines line as a breadth-less length.)
  • The text states an axiom: a unique (invisible) straight line connects any two points.

Note 1: Difference between metaphysics and abstraction

  • People wrongly say math is abstract.
  • The word "dog" is abstract: dogs come in many shapes, sizes, and colours.
  • But no child has a difficulty with this abstraction.

  • For we can point to several instances of dogs (that is how a child learns the meaning of dog).
  • We do not define the word dog, but children learn to discriminate between dogs and cats by seeing.
  • However, we cannot point to an invisible point. (It is metaphysics, not mere abstraction.)
  • (Definitions by pointing are prohibited in formal math.)

  • If the axioms of formal math are metaphysics, so are the theorems.
  • Mathematical theorems are at best, fallible relative, metaphysical truths (relative to both axioms and logic).
  • They are true only in a fantasy world, not the real world.

  • At best, one may choose the axioms so that the fantasy world of formal math somehow "approximates" the real world.
  • Does Harry Potter's world "approximate" the real world?
  • Will come back to that later.

The church intervention in math

  • Q. Who declared metaphysical knowledge to be superior to knowledge of the real world?
  • A. The church, which invented metaphysical reasoning.
  • Why?

  • During the Crusades the church accepted reason
  • because of its greed to convert Muslims
    • who could not be converted by force
    • rejected the Bible as corrupted
    • But accepted reason (aql-i-kalam).

  • However, church could not also accept facts (reason + facts = science)
  • for facts are usually contrary to church dogmas (e.g. virgin birth).

Angels don't exist

  • hence axioms and theorems about angels are irrefutable. But
  • irrefutable \(\neq\) infallible (merely metaphysics, unreal, fantasy) (Popper)

Colonial education came as church education

  • Macaulay said the colonised needed colonial education for SCIENCE.
  • However, the fact is that colonial education came through the CHURCH.

  • Not only mission schools, but also the big universities like Paris, Oxford and Cambridge
  • were set up by the church.
  • Our universities still imitate those universities.

  • The church set up these higher education institutions for ITS benefit
  • (to produce insular missionary minds).
  • But a nation of billion people is still deluded that the church education system was for OUR benefit (as stated in the latest NEP)

  • Main teaching of colonial education is the church propaganda:
  • "you are inferior, the West is superior, imitate it"
  • as we do.
  • Our class IX school text says, "Indian गणित is inferior, formal math is superior, imitate it."

  • As we saw formal math makes the simplest things like 1+1=2 very difficult.
  • Difficulty of maths is an added advantage for colonialism:
  • it keeps most people ignorant of math, hence dependent on authority.

  • Metaphysics is another way to force acceptance of authority.
  • No one can have any direct knowledge of metaphysics, like invisible points.

  • The ignorant are forced to depend on others they trust.
  • The other church trick is to teach you "trust the West", and "mistrust the non-West".
  • This way, you can be easily misguided, as Wikipedia does
  • and no one can guide you back on the correct path.

Hiding the church connection to math

  • Myths of Greeks, such as Euclid, are used to hide the church connection to math
  • The fact is that the church used the text on geometry by Euclid to teach "reasoning" to its priests.

Religious vs secular

  • Western math was tied to religious beliefs since Plato.
  • Word "mathematics" derives from mathesis meaning learning
  • or arousal of the soul to make it recollect its eternal knowledge from past lives. (Plato, Meno, 4th occurrence of "soul".)

  • Proclus explained this happens by sympathetic magic,
  • because the eternal truths of math arouse the eternal soul.

  • This superstition ("math has eternal truths" persisted with the church.
  • the major cultural influence on Europe.

  • Aquinas said: God rules the world with eternal laws of nature
  • naturally written in the language of eternal truth: mathematics.

  • Led to related superstition that math is exact
  • (Exactitude possible only in a fantasy world of metaphysics.)
  • Will talk more about it in next lecture.

What can we do about it?

  • Colonial education captures the mind, and makes change difficult.
  • E.g. most of you will say "We are ignorant of math, what can we do?"

  • That is exactly the church trick: to make you ignorant, hence impotent.
  • But the issue concerns you, your children or grand-children, so you must defeat this trick.
  • Here are some simple things you can do.

  • E.g. ignorance should not stop you from asking questions.
  • Though colonial education teaches you that asking questions is wrong.

So, be critical

  • Superstitions disappear if you are sceptical and ask questions.
  • E.g. Pile on the pressure on NCERT to provide primary evidence for false myths like Euclid in our texts.

Understand the church connection of colonial education

  • E.g. Ask why we are teaching inferior religious math instead of secular and practical गणित (just as we use an inferior religious church calendar).
  • Demand a PUBLIC review of colonial education brought by the church.

Aim to change at least one thing

  • Fight TINA. Show that things become better if done differently: that गणित is better than formal math.
  • There are concrete pedagogically demonstrated alternatives

  • such as "Calculus without Limits" (at university level)
  • and Rajju Ganit at school level (about which more in the next lecture).