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?

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:

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

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

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

"Euclid" must fall

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 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

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.