Decolonising mathematics, socialism and cybernetics

C. K. Raju

Indian Institute of Advanced Study

Rashtrapati Nivas, Shimla

Introduction: 3 key questions

• (1) what is decolonization of mathematics?
• (2) how is that relevant to socialism?
• (3) what does that have to do with technology: cybernetics or artificial intelligence?

What is colonial mathematics?

• People wrongly believe "math is universal".
• Common question: what is "colonial" about 1+1 = 2?
• Have answered this question numerous times, and I will repeat the answer here.
• Key question in colonial mathematics is not that 1+1=2, by WHY 1+1=2.

WHY is 1+1=2?

• Most people will repeat what they were taught as children
• they will say 1 orange and 1 orange make 2 oranges
• But this involves an empirical process: you see the oranges.
• This is OK in "normal math", but NOT OK in formal math (= colonial math)

Divorce from the empirical (in Western ethno-math)

• Empirical processes are NOT allowed in formal mathematics (= Western ethno-mathematics)
• the math brought by colonial education.
• Why? Formal math demands an AXIOMATIC proof.

What is an axiomatic proof?

• An axiomatic proof is a sequence of statements in which
• each statement is either an axiom (= postulate = assumption) or
• is derived from preceding statements by means of some rule of reasoning,
• (such as modus ponens: A, A implies B, therefore B)

Axiomatic proof excludes facts

• At no stage in an axiomatic proof can one introduce a fact by saying
• "I observe this therefore it is true".
• Hence an axiomatic proof is divorced from the empirical.

Russell's axiomatic proof of 1+1=2

• No mention of apples or oranges here.
• Science requires calculus
• and calculus supposed requires "real" numbers, which are taught from class IX onward.
• Axiomatic proof of 1+1 = 2 in "real" numbers is even more difficult.

Cape Town challenge:

• a prize of ₹1 million for an axiomatic proof of 1+1=2 in "real" numbers
• offered to the faculty of one of our prestigious universities JNU.
• Similar denial of knowledge of "real" numbers at MIT.
• Requires knowledge of axiomatic set theory.

Metaphysics not mere abstraction

• "Real" number 1 does NOT correspond to anything real.
• Word "dog" is an abstraction, for dogs may come in many sizes shapes and colors.
• But one can point to physical instances of dogs.
• Children easily understanding the abstraction "dog", without defining "dog"
• but find metaphysics difficult (where they cannot acquire knowledge their own such as the difference between a dog and a cat)

Axioms of formal math are metaphysics not facts.

• E.g. Hilbert's axiom: "there is a unique straight line through two geometric points"
• is metaphysics, NOT a fact.

To dodge the fact, school texts teach a geometric point has no size.

• If so, that would make a geometric point invisible.
• An invisible point is metaphysics because it does not exist in reality.
• One cannot point to an invisible point.
• Metaphysics of set theory (needed to define "real" number 1) is a metaphysics of infinity.

No practical value

• An axiomatic proof (or metaphysical reasoning without facts) does NOT add any practical value to any application of mathematics
• in a grocery shop
• (You lived all your life without knowing or feeling the need for the axiomatic proof of 1+1=2.)
• All practical value comes from normal mathematics (you must relate numbers to objects in a grocery shop).

"It works" superstition (a high-tech counter-example)

• The colonised mind is terrified of challenging the master. "It works", they say. But what works?
• To calculate rocket trajectories one must solve differential equations.
• Differential equations require the calculus.
• The calculus requires "real" numbers (according to present day university teaching).

Rocket trajectories (continued)

• But, NASA or ESA calculates rocket trajectories using computers.
• Computers canNOT represent unreal "real" numbers:
• they work with floating-point numbers
• quite different from real numbers (e.g. associative "law" for addition fails).

Formal math does NOT work: Other examples

• "Pythagorean theorem" NOT exactly true anywhere in the world.
• not for triangles on curved earth, not for triangles in curved space.
• Calculus of variations: cycloid NOT brachistochrone with resistance.
• And if approximations are OK, why go to all the complexities of formal math?

Hypocrisy: Practical value comes from normal math not formal math

• That is whether groceries or rockets or AI
• what one does is normal math
• what is talked about (or taught) is formal math
• (But I teach decolonised calculus withOUT real numbers".)

No aesthetic value

• There is a Western myth that math is beautiful.
• Plato put mathematics on par with music as a way to arouse the soul.
• Egyptian mystery geometry, which Plato had in mind, did have aesthetics.
• However, formal math has no connection to soul or its arousal.

The church curse on the pagan "soul"

• By the sixth century the Christian Emperor Justinian had cursed Plato's "pagan" notion of soul.
• Justinian also shut down all schools of philosophy in mathematics in the Roman Empire (Christendom).
• Earlier, the mathematician Hypatia, was raped and brutally killed in a church, for advocating this notion of soul.

Formal mathematics supposedly based on "Euclid"

• "Euclid" a text used by the church, which cursed Plato's notion of soul.
• Hence, math no longer connects to the soul, or to soul arousal.
• This soul-less-ness of formal mathematics makes it ugly, like Russell's proof of 1+1=2
• Most of you avoided mathematics just because you were repelled by it.

Ugliness of formal math

• But, most of you probably like music.
• Thus, a clear disconnect has developed between mathematics and music, since Plato.
• I too left formal mathematics for the same reason: because it is ugly metaphysics of nil practical value.

Political value

• Q. So, if formal math has no practical value and no aesthetic value, what value does it have?
• A. It had great political value for the crusading church.

The church theology of reason

• During the Crusades, the church wanted to and did adopt reason
• To contest the Islamic theology of reason (aql-i-kalam)

But, the church could not adopt normal reason (= reason + facts)

• Since facts went against church dogmas
• Therefore, the church made a striking innovation
• It invented formal reasoning or metaphysical reasoning minus facts.
• This allowed Aquinas to "reason" about angels (no facts about angels, only fantasy).

Masking church innovation

• But, this church innovation of metaphysical reasoning
• is hidden behind several layers of masks,

Epistemic value?

• Today, the key Western claim is that an axiomatic proof has "superior" epistemic value.
• Decolonisation of math: central proposition is that
• this claim that prohibiting facts results in superior epistemic value is complete bunkum.
• Further, it is a church superstition (hence so widely believed).

Colonial claim of epistemic superiority

• Colonialism used this claim of superior epistemic value
• to claim civilizational superiority:
• only the West "understood" this superior epistemic value
• of metaphysical reasoning and metaphysical math.

The claim of civilizational superiority

• Just as racism involved a bogus claim of racist superiority:
• that Whites are superior to non-Whites,
• So also colonialism involved a bogus claim of civilizational superiority:
• that the West is superior to the non-West (especially in math and science)

Relation of racism to colonialism

• The two claims of racist/civilizational superiority are organically linked:
• supported by the same false history
• initially concocted by the church during the Crusades and the "Age of discovery".

False history of science

• During the Crusades the origin of all scientific knowledge in captured Arabic texts
• was indiscriminately attributed to early Greeks.
• During the so-called "age of discovery" not only were all native lands appropriated in three continents
• But also indigenous knowledge was appropriated and claimed to have been discovered by Christians

False history results in bad philosophy

• Our concern today is not only with correction of credits
• But also that this appropriation of knowledge resulted in the promotion of a BAD philosophy
• As in the bad philosophy of geometry actually due to the church, but attributed to "Euclid"
• Or in the bad philosophy of the calculus attributed to Newton.

Opposing colonialism

• Anti-racism seeks to uproot the claim of racist superiority,
• and involved toppling racist icons ("Rhodes must fall").
• So also decolonization seeks to uproot the claim of civilizational superiority,
• and involves toppling civilizational icons such as Euclid (and Newton etc.).

"Euclid" must fall

• No evidence for existence of "Euclid".
• Counter evidence: a black woman from Africa wrote the book Elements in the +4th c. CE,
• not a white male from Greece in the -3rd c. CE.

No axiomatic proofs in the Elements

• The Elements is a book on Egyptian mystery geometry (mentioned by Plato).
• It does NOT have any axiomatic proof of ANY proposition
• from its first proposition to its penultimate proposition ("Pythagorean" proposition).

Western gullibility

• But, from 1125 CE to the 1900 CE all Western scholars read the book, and
• foolishly believed the church propaganda that this is a book about axiomatic proofs (which suited the church)
• Only in 1878 Dedekind ATTEMPTED "real" numbers to provide an axiomatic proof of its first proposition.

Russell and Hilbert on Euclid

• Russell called the actual proofs in "Euclid's" book "a tissue of nonsense".
• Hilbert wrote a whole book Foundations of Geometry to provide the axiomatic proofs missing in the actual Elements.
• Propagandist claims of "Euclid's" non-existent axiomatic proofs still persist
• Despite this exposure (e.g. in India in class IX math school text).

Continuing propaganda of civilizational superiority

• Needham too went by the myth even AFTER exposure of "Euclid" by Russell and Hilbert.
• He wrongly asserted that the Chinese used only ("inferior") empirical methods of proof UNLIKE "Euclid".
• Likewise, Claggett wrongly asserts that the Pythagorean theorem is formally (=axiomatically) proved in Euclid's Elements.
• He hence concludes that Egyptian geometry was inferior!

Complete disregard of facts

• Thus, top Western scholars continue the propaganda of Western civilizational superiority in mathematics
• completely disregarding all facts.
• This is an extreme form of post-modernism (long before modernity!).
• Western history, like formal math, is "superior" since divorced from facts! 😜

Is deductive proof superior?

• Today we teach formal mathematics in the belief that "deductive proofs are infallible",
• hence superior to fallible empirical proofs.
• But, deductive proofs ARE fallible.
• Students COMMONLY err in deductive proofs, hence flunk in mathematics.

Since deductive proofs are potentially fallible, to ensure validity

• One must either repeatedly re-check the proof (induction)
• or blindly rely on authority (usual process).
• Hence, deductive proofs are MORE fallible than empirical proofs.

Chess is a game of pure deduction

• An error-free game must end in a draw.
• But, every Grand Master makes a mistake EVERY time.
• Hence, always loses to the computer.
• So much for the foolish superstition that "deduction is infallible".

Formal mathematics forces reliance on authority

• For, it is metaphysics, where axioms are decided by appeal to authority not facts.
• "Authority"= Western authority (calculus is taught with real numbers not non-Archimedean arithmetic as I recommend).
• This subordination to Western authority greatly suits colonialism.

Interim summary

• Colonial mathematics advocates axiomatic proofs
• on the false myth that "Euclid" gave (or intended) such proofs,
• and the superstition that such proofs are "superior" since infallible.
• Decolonisation of mathematics ("Euclid must fall") seeks to pull down this claim of superiority,
• AND the superstition about infallibility of deduction.

Axiomatic proofs just assume two valued logic

• But logic is not culturally universal
• (e.g. Buddhist logic and Jaina logic are not two-valued like Nyaya logic = "Aristotelean" logic).
• Nor is logic empirically certain (e. g. quantum logic is quasi-truth-functional like Buddhist logic)

Proofs/theorems vary with logic

• So if logic itself is decided by cultural dictatorship,
• or empirically,
• deductive proofs can never be more certain than empirical proofs.

Demand: Replace formal by normal math

• In a word, decolonisation seeks to reject formal/church math
• (and its myths and superstitions),
• and replace it with normal math.

Colonial education

• Macaulay sold colonial education (in India) using the myth of civilizational superiority.
• He spoke of the "immeasurable superiority" of the West in science
• (and no one checked the false history he used).
• Therefore, he argued the colonized needed Western education for science.

Education as counter-revolution

• The same Macaulay in a speech to British Parliament (18 April 1847)
• shortly before the Communist Manifesto sought to exorcise the sceptre of revolt then haunting Europe. He said
• Education is the best and cheapest means to stop revolts.
• (Because Western education then was 100% church education, which taught subordination to authority.)

Socialism

• Contrary to Marx's prediction, revolutions took place in poor and least educated countries, where the productive forces were least developed
• E.g. Russia was the poorest country in Europe then.
• After the revolution these countries came under tremendous military and economic pressure.
• Their only way forward was rapid economic and military development.

The irony

• That required a large scientifically trained work force.
• For this, post-revolutionary societies accepted Western/colonial education,
• which Macaulay had characterized as the best counter-revolutionary measure!

Specifically, for math teaching

• they accepted formal mathematics,
• with its metaphysics of infinity intertwined with the church metaphysics of eternity,
• calculus done with formal real numbers, limits etc.
• Statistics with Kolmogorov probability as measure.

They should have reflected on Marx's own experience with calculus

• Marx came even before Dedekind, Russell and Hilbert,
• before the absence of axiomatic proofs in "Euclid's" book was publicly exposed,
• before real numbers and limits.

Marx rejected Newton's confused calculus

• Then there were Newton's "fluxions".
• Based on Newton's incredibly confused idea that time itself flows
• (an idea rejected as confused by the 8th c. Indian philosopher Sri Harsa, for commonsense reasons today known as "McTaggart's paradox").
• Marx rightly rejected "the calculus of Newton and Leibniz" as "mystical" meaning confused.

Practical applications of Newtonian physics

• to planetary motion, and ballistics,
• were all achieved LONG BEFORE "real" numbers and limits,
• again demonstrating the complete irrelevance of formal math (real numbers, limits) to practical applications.

Using Western/church education for "science" subverted values

• Because Western education came along with church myths and superstitions,
• as in the accompanying false myths of math ("Euclid" and his purported axiomatic proofs)
• or the superstition that axiomatic proofs are epistemically superior
• (hence "trust the West" for axioms).

Mental subjugation

• These myths and superstitions also ensured Western authoritative control ("West knows best")
• over scientists in post-revolutionary societies
• subjugating them mentally,
• exactly as the colonized were mentally subjugated to the colonizer.

Socialism is not consistent with colonial values

• of even one powerful individual
• as the collapse of Soviet Union after Gorbachev demonstrated.
• China surely survived and prospered.
• But did socialism?

The Chinese cultural revolution

• Unlike Stalin who liquidated the kulaks and other rich sections of the pre-revolutionary society,
• Mao tried to co-opt them, but failed,
• resulting in his cultural revolution.

Acceptance of rhetoric of civilizational superiority by Mao and Deng

• Both Mao with his "for Olds",
• and Deng with his four "new-s" (four modernizations),
• accepted the propaganda of civilizational superiority, and
• both failed to understand the counterrevolutionary danger to values,
• posed by the import of Western/church education (for science).

Xi Jinping has understood the importance of older values

• but all failed to understand that, for example,
• older philosophy may be relevant to "modern" mathematics, science and technology.

I use zeroism (Buddhist sunyavad) to teach calculus without limits (decolonised calculus)

• which makes calculus easy,
• as also the statistics needed for machine learning.
• (Most people just use Google TensorFlow for machine learning, without understanding the underlying statistics.)
• Easier math will result in better and more robust algorithms. (Failure of AI programs could be catastrophic tomorrow.)

But AI also involves hardware

• And tomorrow's hardware may be quantum computers,
• based on quantum logic,
• different from 2-valued logic assumed in formal math,
• but very similar to the Buddhist logic of catuskoti (allows A and not-A).

Today science is still influenced by church dogmas

• Such as Aquinas's dogma of "eternal laws of nature".
• Already busted by quantum mechanics ("God plays dice"),
• but still no room for human creativity.

Future quantum computers will be android

• Android in the sense of part living + part machine.
• Would be impossible to control, unlike software
• or with old church techniques,
• of using myths and superstitions to control humans.

Conclusions

• For socialism to work,
• math and science education must be decolonised,
• by eliminating church myths and superstitions from current math and science.