## "Euclid" must fall:

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

C. K. Raju

## Background

### 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.

### 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.

### 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.

### 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

### 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)

### 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".

## 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.

• 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).

### 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).

### 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.

### 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.

## Conclusions

### 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

• 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 (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.