Decolonise math
=
Eliminate
the myth, fraud, and superstition
in formal math

Introduction

Racist claims of superiority

• The George Floyd (or Wisconsin) case recently sparked anti-racism protests.
• Racism persists because of the persistent Western/White claims of racial and religious "superiority".

Related claim: "superiority" of Western math, critically important for colonialism

• Colonialism changed the education system globally,
• The problem of decolonisation of math arises from the RELATED claim of "superior" Western math,
• supposedly first done by a Greek/White "Euclid" and "Pythagoras" etc.
• which the whole world should mimic.

Prejudices in math not understood

• Unlike racist prejudices,
• most don't understand the related prejudices in math.
• Tried to explain the connection in two recent newspaper articles on racism in English and Hindi.

In fact, many people laugh at the thought of decolonising math

• they ask: what is there to decolonise in 1+1=2? Ha, ha, ha! 😜
• They have in mind something like this.

Complexity 1: empirical disallowed

• This is an EMPIRICAL proof of 1+1=2: accepted in normal math.
• The proof depends on the senses: you can SEE the oranges in question. But
• Empirical proofs are PROHIBITED in Western/colonial/formal math.

You may recall this prohibition of the empirical

• from your school geometry lessons.
• Indian class VI (K-6) geometry text explains a "point".
• "Sharpen the tip of your pencil and make a dot on a piece of paper"
• "the almost invisible dot will give you an idea of a point."

"Of course, a real point must be invisible"

• That is, if you can see it, it is wrong!.
• That is "prohibition of the empirical".

Invisible location?

• The school text also says "a point determines a location".
• A doubt: But if it is invisible how do you know its location?

I explained this in a debate on decolonisation of math in South Africa

• (and laughed).
• White racists got enraged that I laughed at them
• A racist news reporter tried to rebut me

Proof by authority

• Alas, he could not explain HOW to locate an invisible point!
• So, he just cited the authority of an unknown mathematician
• "mathematicians routinely handle invisible points"! 😀

I hope you are getting the drift

• Prohibiting the empirical, forces reliance on authority.
• But WHICH authority?

"Superior" authority

• "White/Western authority is superior".
• so say defenders of Western/colonial math, like Wikipedia
• Believe the Western math "experts" who say, the king is wearing beautiful clothes!

That explains the connection between decolonisation and anti-racist protests

• Both reject claims of Western "superiority"
• which originate from religious prejudices.

A key part of the struggle to decolonise math is to RESTORE the simplicity

• of the empirical proof of 1+1=2.
• and explain why this does NOT negatively affect ANY practical application of math.
• (If prohibition of empirical results in something "superior", why not also prohibit experiments in science?)

"Superior" Western/colonial/formal math demands AXIOMATIC proof of 1+1=2.

• How?
• "Use Peano's axioms" to prove 1+1=2 axiomatically.
• So said the "mathematician"
• after the debate on decolonisation of math and science at the University of Cape Town in 2017.

BUT, he was WRONG!

• In Western/colonial/formal math "1" can mean several different things

Complexity 2: What exactly is 1?

• 1 (natural number)
• \(\neq\) 1 (integer)
• \(\neq\) 1 (rational number)
• \(\neq\) 1 ("real" number)

These different notions of "1" are similar but NOT identical.

• Cape Town "mathematician" WRONGLY assumed that by "1" I meant the natural number 1.
• Actually, I required axiomatic proof of 1+1=2 in "real" numbers.
• Can't use Peano's axioms with real numbers.

Why "real" numbers?

• Because science needs calculus which supposedly needs "real" numbers
• on the current text-book understanding of calculus.

Now we have a problem.

• In 2015 I gave a talk at MIT.
• I asked about "real" numbers and no one present was willing to define them. (Check the video.)
• If so many smart people are not quite sure what a "real" number is,
• how can they prove 1+1=2 in "real" numbers?

Those students of MIT and Harvard were actually smart

• they at least knew what they did not know.
• (My definition of "expert": one who knows what he does NOT know.)
• The problem arises with those people who think the notion of 1 as a formal real number is easy.

Formal real numbers are typically defined using

• Dedekind cuts
• or equivalence classes of Cauchy sequences.
• Both require extensive use of set theory.

Complexity 3: set theory (metaphysics of infinity)

• The above equivalence class is a hugely infinite set (each of whose elements is itself an infinite set)
• Today set theory is taught practically with mother's milk as Venn diagrams!

Set theory (continued)

• But (metaphysics of) such huge infinite sets invites paradoxes,
• e.g. Russell's paradox.
• As in Cantor's set theory (or naive set theory).
• So, one may end up talking complete nonsense.

Therefore, to prove 1+1=2 (in reals) one at least needs to know AXIOMATIC set theory

• But most people don't know it.
• Most can't even DEFINE a set in axiomatic set theory
• like a head of the math department in IIT, who said a set is a
• "collection of objects". 😄

Formal math forces most people to proceed on faith: on mere TRUST in the validity of set theory (required even for 1+1=2)

• To negate this reliance on faith, I demand that ALL the required results must be derived from first principles
• in the manner in which Bertrand Russell proved 1+1=2 in cardinals in 378 pages.
• Here is the formal challenge.

Cape Town challenge

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

Nobody has met the challenge yet

• But my guess is that the proof will take at least a 1000 pages .

Moral 1: colonialism made math difficult

• Colonialism brought in formal math
• which makes even 1+1=2 enormously difficult,
• beyond even most professional mathematicians.

Moral 2: adds no practical value to math

• The difficulty of formal math does NOT add an iota of practical value,
• not in a grocer's shop,
• not in rocket technology.

Thus, though "real" numbers are deemed essential to calculus

• rocket trajectories are calculated using computers
• which CANNOT represent real numbers
• but use the finite set of floating point numbers instead.
• See this Hawai'i paper of 2001, available from http://ckraju.net/papers/Hawaii.pdf.

Moral 3: Prohibiting the empirical forces reliance on authority

• of a particular Western metaphysics of infinity
• (related to church dogmas of eternity)
• declared "superior" on the strength of Western authority.

(Reasoning without facts was a technique invented and glorified by the church in Christian rational theology, modified from Islamic rational theology.)

This is the situation that decolonisation of math seeks to correct. It seeks to

• make math easy,
• enhance its practical value, and
• reject dependence on Western authority to decide what is right and wrong in math.

The justification

It is claimed that

• Claim: axiomatic/deductive proofs are epistemically superior,
• they are certain or infallible (or less fallible) compared to empirical proofs.
• Is this true?
• Let us first see how the case for epistemic superiority is established.

Claim 1: Greeks did this "superior" mathematics.

• Egyptians built the pyramids accurately
• But they did NOT know "Pythagorean theorem" (Heath, Gillings, Needham, Clagget etc.)
• WHERE did Greeks do it? Related theorem found in "Euclid's" Elements,
• used as a church textbook since about the 13th c. CE.

Fact: Claim of Greek origins is FRAUD.

• Proofs found also in other cultures. (E.g. of Indian proof, see Hawaii paper.)
• (that proof involves the empirical)
• BUT no AXIOMATIC proofs in "Euclid's" Elements.
• Not even the first proposition in it is proved axiomatically. Certainly no axiomatic proof of the Pythagorean proposition.

This fact admitted since 20th c.

• Formal math started with David Hilbert's Foundations of Geometry, 1898
• which rewrote "Euclid" to provide axiomatic proofs missing in it.
• (Clear admission of absence of such proofs.)
• Also did great violence to the original: e.g. length measurement NOT allowed in Hilbert's geometry.

Myth: "Euclid" erred but intended pure deductive proofs

• How do you know the intentions? (and how were they so agreeable to the church that the book was adopted as a church textbook?)
• Read the book.
• Contrary to the claim, it has diagrams.
• Plato says (Phaedo) taking a person to a diagram shows that geometry arouses the soul and its innate knowledge (mathesis).

So the intention of the author was to use geometry to arouse the soul (and make it recollect its previous lives)

• using a notion of soul later cursed by the church.
• The commentator Proclus confirms this: that "mathematics leads to the blessed life".
• But the whole Western case for Greek origins of formal math is built on
• a single manifestly forged remark in the book by Proclus, but CONTRARY to everything else that Proclus says.

Do we know something else about Euclid to judge his intentions?

• No. No evidence for any Euclid.
• My "Euclid" challenge of USD 3000 to provide primary evidence for Euclid.
• standing for ten years.
• Eliminating "Euclid" would expose the church origins of the philosophy of formal mathematics. (Reason MINUS facts.)

Superstition: why are axiomatic proofs superior?

• Empirical proofs are fallible. Yes.
• But ALL systems of Indian philosophy (without exception) nevertheless accept the empirical as the FIRST means of proof
• as does science.
• Such "experimental errors" can be easily corrected inductively by repeating the observation/experiment.

Pure superstition that axiomatic proofs are INFALLIBLE.

• Lots of students flunk in math because they turn in erroneous proofs. So which deductive proofs are infallible?
• A long complex proof (such as Russell's proof of 1+1=2) may have errors. How can you be CERTAIN it is error free?
• Many erroneous proofs of e.g. Riemann Hypothesis published. E.g. by Kosambi.

• ALL Western scholars believed for SEVEN centuries that Euclid was a model of deductive proofs. But they were wrong even about the first proposition.

Chess is a game of pure deduction

• But EVERY human being almost ALWAYS makes an error, hence loses to a computer.
• Mind is far more fallible than the senses.
• Errors in deductive proofs can only be checked inductively or by authority
• Hence, deduction MORE fallible than empirical proofs or induction.

Summary. (Many more arguments can be given.)

• Formal math NOT epistemically superior,
• Claim of "superiority" based on fraud, myth, and church superstition.
• Decolonisation of math seeks to correct this
• by eliminating the fraud, myth and superstition in math (which creeps into science).

Western response

• Colonialism involved changing the education system of the colonised
• based on a key premise of the Western superiority in math and science.
• Decolonisation of math destroys that premise.

Hence the Western scholars have only responded politically, not philosophically.

• and refused to engage with either the arguments
• or the evidence.