"Euclid" must fall:

The "Pythagorean" "theorem" and the rant of racist and civilizational superiority

C. K. Raju


Some 30 years ago

  • a group of influential Indian intellectuals decided
  • we need to tell our own stories.
  • Can't remain as characters trapped in another's story.

My contribution was a new story of the calculus

But a new story not enough

10 years ago

5 years ago (2016): "Rhodes Must Fall" agitation

  • Visited Pretoria (UNISA) and spoke on my experiences with decolonising mathematics.
  • This talk frightened some White racists in South Africa, one of whom (a professor of math education) wrote
  • "Blacks and women are bad at math", and to "decolonise" math,
  • one must teach them to think like the dead white men (like "Euclid") who supposedly invented math.

My response that Europeans (including early Greeks) backward even in basic arithmetic until 16th c.

Why censorship? Why not a rebuttal?

  • Because, as described in Euclid and Jesus,
  • the academic issue of lack of evidence for Euclid was settled way back in 2002
  • when David Fowler the leading Western authority on history of Greek math admitted
  • "'What is known about Euclid?' Nothing."
  • Plenty of COUNTER-EVIDENCE that Euclid was a black woman.

Politics of saving "Euclid" myth

  • Three types of vested interests.
  • White racists: That Whites are "superior" since they created a "superior" kind of mathematics
  • Formal mathematicians: That formal math (= Western ethno-mathematics) is the "superior" way to teach math (as currently taught in our universities)
  • Church dogma: Euclid myth essential to post-Crusade church theology of reason.

A key point of today's talk

  • Above three vested interests are interconnected
  • because racism is about the sense of "superiority" (Christian/White/Western), not primarily about the color of the skin.
  • "Euclid" must fall, to destroy that sense of superiority.

A question

Apartheid, segregation, slavery, persisted for 5 centuries.

  • South Africa apartheid rule 1948–1994.
  • Segregation 1865–1968).
  • Slavery (of Blacks in Americas) 1501–1888.
  • Today racism still persists (e.g. "Rhodes must fall", Peter Floyd etc.)

Racism is cruel and ethically depraved

  • But, today, I want to discuss something else.
  • The belief that the color of the skin
  • makes one human being SUPERIOR to another
  • is an exceptionally FOOLISH belief.

Q. How did Westerners accept this EXCEPTIONALLY FOOLISH belief for so LONG?

  • Not only common people but top Western intellects (e.g. Kant and Locke, and numerous others) accepted this foolish belief.
  • Whole systems of ethics, law, and governance in the West were based on this foolish belief for centuries.
  • Why were racists collectively and individually so foolish?
  • Western scholars have never asked this question. Why not?

Merely asking the question militates against the White/Western sense of superiority

  • And that sense of superiority, NOT color of the skin, is at the core of racism.

A persistent and foolish belief is a superstition

  • In the West, the church is the key source of superstitions.
  • E.g. the Bible Defence of Slavery involved the superstition
  • that black color of the skin is due not to heat of the sun but
  • due to the curse of the Christian god (curse of Kam).

Why were Westerners so foolish and superstitious?

  • Question needs to be broadened.
  • Not appropriate to study slavery alone
  • Must study Genocide + Slavery
  • ("Age of Enlightenment" should correctly be called "Age of Genocide+Slavery"!)

Must include genocides of indigenes

  • And inappropriate to treat the genocide of "Red" Indians as separate from
  • the genocide of the Maya, the Inca, the Aztecs
  • (or the genocide of the Australian aborigine).
  • Principle of "Occam's razor" or "simplicity of causes".

Genocide related to slavery, even economically

  • Genocide (of indigenes) provided the free-of-cost land
  • Slavery provided the free-of-cost labor
  • (in Australia convicts were used instead).
  • Genocide + slavery the basis of Western wealth.

Q. The Europeans who engaged in genocide, did they lack all moral compunctions?

  • In the 15th-16th c., church morality still the dominant morality for Europeans.
  • However, the church thought it moral TO DEMAND genocide or enslavement of all non-Christians.
  • Bull Romanus Pontifex orders that all non-Christians should be killed or enslaved.
  • And papal bull Inter Caetera declared non-Christians lost the right to land on being "discovered" by Christians (as upheld by US Supreme court).

The biggest religious hate crime in history

  • As described by Las Casas (who accompanied Columbus in his second voyage 1493)
  • the subsequent genocide ("boil there you offspring of the devil" etc.)
  • was a religious hate crime (his initial estimate 10 million killed).
  • The criminals thought they were being "moral": only imitating what their Christian god would do in hell.

Considering genocide + slavery makes it clear

  • In the 15th and 16th c. genocide and slavery were justified using Christian RELIGION.
  • No mention of race anywhere.

So, we have a new answer to Theodore Allen's question

  • Q. What came first? Whiteness or slavery?
  • A. Sense of superiority based on church dogma
  • that Christians are "superior" and OUGHT to kill or enslave non-Christians.
  • since initial "moral" ground for enslaving Africans was that they were non-Christian.

This "moral" justification for slavery eventually ran into trouble

  • after the trans-Atlantic slave trade
  • when many Black Africans converted to Christianity.
  • It was then that the "curse of Kam" was highlighted to
  • declare Blacks as "inferior" Christians.

Why use the color of the skin?

  • Because church Inquisitors had earlier used the technique of quick visual cues
  • such as adherence to Moorish dress
  • to spot heretics or suspect Christian converts among Moors
  • and use that to subject them to physical tortures.

This Biblical justification ("curse of Kam") was thin

  • hence Immanuel Kant uses a different "secular" justification for his recommendation to whip Blacks
  • he echoed Hume that "Blacks are not creative".
  • Q. But would Kant/Hume have used this argument
  • if they believed George James?

Substance of George James' Stolen Legacy

  • (not literally that Aristotle plagiarized from the Great Library of Alexandria)
  • but that Greek philosophy is derived from Egyptian
  • due to TRANSLATIONS from Egyptian (and Persian) to Greek in Ptolemaic times
  • and/or later-day APPROPRIATIONS by motivated racist historians (e.g. Bernal, Black Athena) etc.

And/Or even simple accretion into later texts

Will simply summarize

  • From the 5th c., when the church first incorporated the myth of Christian superiority into its dogma
  • it also created a secular justification for the myth using false history (Orosius, History Against the Pagans).
  • This trick of systematically false history went ballistic during and after the Crusades
  • when a false history of SCIENCE was concocted.

Difficulty of winning the Crusades (they lasted 5 centuries) was

  • because Muslim Europe was scientifically FAR ahead of Christian Europe.
  • Hence, the church appropriated all scientific knowledge in Arabic texts
  • made it theologically correct by the easy trick of attributing its origin to early Greeks.
  • Why early Greeks? Because Eusebius declared them as the sole "friends of Christians". So, this made it a Christian inheritance.

The myth of "Euclid" is a Crusading concoction dating to this period (ca. 1125 CE)

  • Zero evidence for "Euclid" (no primary Greek text with his name in it)
  • "Euclid" (= Uclides= aql-i-des = rational geometry) a possible subtitle
  • Plenty of counter-evidence.
  • But "Euclid" was appropriated for reasons related to church dogma
  • (will return to that later).

The shift from religious superiority to racist superiority

  • saw a corresponding adjustment in the false history.
  • Greeks were now portrayed as Whites (as they still are)
  • rather than "friends of Christians".

Du Bois's dream

  • Need to expand the question beyond genocide+slavery
  • to include also colonialism.
  • Du Bois, in his Dark Princess, realized that the colonized were equally oppressed
  • and dreamt of freeing the oppressed by uniting them against the oppressor.

After the collapse of colonialism following World War II

  • and the collapse of the fake-dream of a unipolar world after the end of the Cold War,
  • given the rise of China,
  • Du Bois's dream is no longer a romance but now seems a realistic possibility.
  • False history a key element of both racist and colonial claims of superiority (e.g. Macaulay)

But racist history further mutated after colonialism.

False history-3: Inquisitional model (after 1452 CE)

  • Orosian model and Crusading model not the only two models of systematically false history due to the church
  • e.g. Byzantine Greek translation of Syriac work of Ibn Shatir available in Vatican library
  • attributed to Copernicus who merely translated it into Latin (with errors).
  • But big story of "Copernican revolution".

False history-4: "Christian discovery" model

Focus of this talk on philosophy, not history

Today would like to emphasize philosophy, not history

  • False history results in bad philosophy
  • as in Kant's "ethical" conclusion that Blacks (since "not creative"), should be whipped into silence.
  • To bring out how obnoxious is this conclusion, let us ask
  • Should Kant's followers be whipped into silence?

False myth of "Euclid", results in a bad philosophy of mathematics

  • This false myth not confined to WHO is the author of a particular book (Elements)
  • Amazingly, this false myth is also about the CONTENTS of that book.
  • On the myth, the book first gave axiomatic proofs in geometry.
  • But the book Elements does NOT have a single axiomatic proof.

Clarification about axiom

  • An axiom = postulate = assumption
  • NOT 'a self evident truth'. (No such thing.)

Definition of axiomatic proof

  • A sequence of propositions in which
  • each proposition is either an axiom
  • or is derived from preceding propositions by some 'rule of reasoning'
  • such as Modus Ponens (\(A,~ A \implies B, ~\therefore B\)).

Exclusion of facts, observations

  • An axiomatic proof EXCLUDES use of facts or observations, or anything empirical
  • That is, it EXCLUDES
  • "A is a fact, hence A" or
  • "I observe A, hence A".

Formal vs normal reasoning

  • Hence, axiomatic or FORMAL reasoning (reasoning MINUS facts)
  • differs from NORMAL (or scientific) reasoning used worldwide (reasoning PLUS facts).

E.g. NORMAL inference that earth is round

E.g. 2: NORMAL proof of "Pythagorean" proposition

With NORMAL reasoning ORDER of propositions NOT critically important

  • E.g. in the Yuktibhasa the "Pythagorean" proposition is the FIRST proposition
  • not the LAST (or second-last) proposition as in the Elements.
  • On the "Euclid" myth, Euclid's primary contribution was to the arrangement
  • or the ORDER of propositions in the Elements (so that the proof of each proposition depended only on the preceding).

750 years of European myth about math

  • This myth repeated in e.g. foolish Cambridge exam regulations (toward end of 19th c.)
  • for the Previous and Local examinations.
  • Why do I call this foolish?
  • Because, no one read the book. There are NO axiomatic proofs in the Elements!
  • (And the prescribed Cambridge text had all empirical proofs!)

Book Elements has NO axiomatic proofs.

  • Not of the first proposition (we OBSERVE the two arcs intersect, but no axiom for it)
  • Not of the last (or second-last, Pythagorean proposition)
  • whose proof depends on Prop. 4 (SAS)
  • proved empirically by SUPERPOSING one triangle on top of another and OBSERVING the two are equal.

Western authorities on axiomatic proof in "Euclid"

  • If you don't believe me, and haven't read or won't read the book,
  • like ALL Europeans failed to carefully read this prescribed text
  • or even its first proposition for some 750 years,
  • here is the "colonial proof" (what Western authorities have to say).

Dedekind, Russell, and Hilbert on "Euclid"

Hilbert's rewrite of "Euclid" completely changed the book

  • e.g. Hilbert's geometry is synthetic — no length measurement allowed.
  • Those interested in further mathematical details, please join my geometry workshop next month.
  • Today, my talk concerns racism and the related underlying political claim
  • of a civilizationally "superior" form of mathematics — axiomatic mathematics.

Based on a false claim of "Euclid" AND his purported axiomatic proofs

  • NEITHER of which exists,
  • but which false claims of "superiority" are still being peddled
  • by "reputed" Western historians such as Needham, Gillings, and Clagett.

This complete adherence to myth

  • and utter disregard of easily verifiable facts
  • publicly admitted over a century ago
  • ("no axiomatic proofs in the actual Elements")
  • entitles them to the title of Greediots!

Greediots prize goes to Lefkowitz

  • Recall that Cheikh Anta Diop pointed out that
  • the that the Ahmes ("Rhind") papyrus (e.g. problem 41), etc.
  • has the right formula for the volume of a cylindrical granary
  • \[V = h \left(d - \frac{d}{9} \right)^2 = \frac{256}{81}r^2 h \approx 3.16 \cdot r^2 h \approx \pi r^2 h. \]

Recall also that the Sphere and Cylinder is attributed to Archimedes

  • but on accretive secondary sources from 16th c. with no clear connection to Archimedes.
  • But Lefkowitz asserts in Not out of Africa (p. 153)
  • "Archimedes determined that the volume of the cylinder was \(\frac{3}{2}\) the area of the sphere"
  • (Comparing volumes and areas! 🤣, 🤣 and based on 16th c. texts.) But let us get back to the story.

Dedekind, Hilbert, Russell rejected the MYTH of axiomatic proofs in the Elements

  • But they still accepted the SUPERSTITION that axiomatic proofs are "superior"
  • (they all tried to provide such axiomatic proofs).

Are axiomatic/formal proofs "superior"?

  • "superior" for what purpose? to whom?

"Euclid" (=Uclides=aql-i-des) and aql-i-kalam

  • During the Crusades the church suddenly switched to Christian theology of reason
  • of Aquinas and the schoolmen.
  • Why?
  • Crusades aimed to convert Muslims by force
  • the way "pagans" in Europe were earlier converted to Christianity by force.

But, militarily, Crusades (after the first) were a failure for centuries.

  • Further, Muslims could not be converted using the Bible which they rejected as corrupted.
  • But Muslims then accepted reason
  • as in Islamic rational theology (aql-i-kalam)
  • advocated by Averroes (Ibn Rushd)

Hence, church too decided to adopt reason

  • We have seen how: by giving reason a Greek ancestry (since no mention of reason in Bible)
  • But why? Isn't reason against church superstition?
  • NO. Facts go against church dogma, not reason alone.
  • Hence, the church accepted reason MINUS facts (formal reason).
  • Starting point of reasoning would be an axiom or assumption NOT a fact.

E.g. Aquinas' theorem in Summa Theologica

  • many angels can fit on a pin.
  • Starting from the AXIOM that angels occupy no space
  • since no FACTS about angels.
  • Anything convenient can be authoritatively assumed to "prove" whatever one wants.

Formal reasoning greatly suited the church and was hence glorified

  • Formal reasoning was read into Euclid (and that hid this tricky church invention).
  • This was a brazen misreading, since NO formal proofs in Elements.
  • But the book is full of figures which are "not rigorous" (Dedekind) and "logically worthless" (Russell)

Diagrams in the book relate to mathesis

  • but (Proclus) diagrams related to Plato's idea of mathematics as mathesis (e.g. Meno, Phaedo).
  • Shows that the Elements was about mathesis
  • (Egyptian mystery geometry) NOT irrefragable proof politically needed by the church.

Theorem-proving mathematics declared superior

  • because it served the Crusading church's political purposes,
  • but formally proved theorems not PRACTICALLY superior.
  • Unfortunately, many people have the "it works" superstition that formal math "works".

What "works" (and has always worked) is universal NORMAL math NOT Western FORMAL math

  • E.g. Egyptians built pyramids before formal math.
  • You purchase your every-day groceries without knowing the formal proof of 1+1=2
  • ab initio proof in real numbers (my Cape Town challenge, not met by anyone yet).
  • For practical purposes (such as calculation of rocket trajectories today) what is used is still normal math of inexact calculation.

Formal math needs "reals", normal math uses floats

  • E.g. "real" numbers deemed essential for formal math of calculus (as taught in universities)
  • But trajectory calculations for rockets done using computers, which uses floating point numbers, NOT real numbers.
  • [Calculus made simpler and better using Indian "non-Archimedean" arithmetic
  • but math axioms for university teaching laid down solely by Western authorities (even in China).]
  • Trick is to plant a superstition and make you obedient to and dependent on Western authority.

Formal theorems fail in real life because axioms/assumptions fail.

Indians understood long ago

  • that "Pythagorean" theorem fails for navigation because of sphericity of earth
  • as explicitly stated by Bhaskar 1 (7th c.) Mahabhaskariya II-5 (trans.)
  • But calling the "Pythagorean" theorem an "approximation" is an inadequate defence
  • "Approximation" worthless without error bounds (not available with the "exact" theorem)
  • and which error bound require accurate knowledge of radius of the earth (unknown to Europeans until late 17th c.)

"Pythagorean" CALCULATION (not theorem) needed for practical applications

  • Difference? "Pythagorean" CALCULATION (needed even for European navigation) requires knowledge of square ROOTS
  • unknown to early Greeks, but known to early Egypt, Iraqis, and Indians.
  • Note that calculation of \(\sqrt 2\) involves infinite series hence forever INEXACT.
  • Practical value of math is all based on normal math of inexact CALCULATION, not theorem-proving.

Is theorem-proving even EPISTEMICALLY superior?

  • Reason given for rejecting empirical proofs is that they are fallible. True.
  • But the related claim that "deduction is infallible" is a mere superstition.
  • As any math teacher knows students err often and submit many wrong proofs.
  • Authorities too have erred and published many wrong proofs (e.g. of Fermat's last theorem) etc.

Deduction far MORE fallible than empirical proof

  • because human mind errs more easily than human senses.
  • E.g. game of chess is an exercise in pure deduction
  • if neither side makes an error, it must end in a draw.
  • But, EVERY human being, makes an error in deduction almost EVERY time, hence loses to a computer.

Even a validly deduced theorem may be invalid knowledge

  • (because assumed axioms are not true in the real world).
  • We have seen the example of the Pythagorean theorem.
  • Another example is the Banach Tarski theorem of set theory based on the axiom of choice.
  • In fact, any nonsense proposition whatsoever can be proved as a formal theorem by selecting the axioms appropriately.

Contingent vs necessary truth

  • Church said "logic binds god" who can create the facts of his choice, but not an illogical world.
  • Therefore, it is believed that facts are "contingent" truth (true in some (Wittgensteinian) worlds),
  • while the relation of axioms to theorems is a "necessary" truth (true in all worlds).
  • This parochially assumes 2-valued logic used in formal math. But apart from axioms, logic too can vary.

Alas, logic is not unique, not certain


Church role in genocide, slavery and racism

  • Genocide and slavery were first mandated by the church (papal bulls Romanus Pontifex and Inter-Caetera),
  • accepted also by Protestants and the US Supreme court till today.
  • The underlying dogma of extreme Christian superiority (from 4th c. CE)
  • was justified using false history (5th c.), and a false history of science (12th c., onward)

Action points

  • Hold the church responsible for genocide + slavery.
  • Join the indigenes to condemn the church and
  • demand that the church recant publicly,
  • and ban it from giving moral advice for at least another 500 years.

Action points (history of science)

  • Recognize that false history (of Greeks, and science) is a centuries-old church con-trick
  • it is the basis of false claims of superiority used to support racism.
  • To overthrow it INSIST on real primary sources for history.
  • Learn to distrust Western authority and Wikipedia.

Action points (contd)

  • Use non-textual evidence to cross-check bloated claims based on corrupted texts.
  • Start your own courses in History and Philosophy of Science in every country.


  • Euclid is an important symbol of the false history of math concocted during the Crusades.
  • Total falsehood. Euclid did NOT invent axiomatic proof. NO axiomatic proofs in the book attributed to him.
  • Axiomatic reasoning (reasoning from assumptions instead of facts) an invention of Crusading church
  • very convenient for politics of Christian rational theology
  • NOT "superior" in any sense. Formal math has no practical value, no special epistemic value.

Euclid must fall

  • Destroying symbols important but not enough.

Action points (math)

Action points (reminder)

  • Colonial education began as church education and massively spreads superstitions
  • so all education needs to be decolonised.
  • But math and history and philosophy of science first priority because
  • false history and bad philosophy of science are used to promote racism
  • and colonial math wrongly believed to be "universal" and not in need of change.