### Racism in the math classroom: “Pythagorean theorem” and the two myths of “Euclid”

Tuesday, February 8th, 2022*The term “Pythagorean theorem” is #racist, based on a lie that (White) Greeks did a superior axiomatic math unknown in (Black) Egypt. Did they?*

*Was “Euclid” a White Greek male as imagined by Wikipedia or a Black Egyptian woman as in the cover of this book?*

Does racism affect math? Let us first understand how a racist *history* of math creeps into “standard” math terminology and hence into math *education*.

Everyone has heard of the “Pythagorean theorem”, for the term “Pythagorean theorem” is a stock part of K-12 math teaching today. For example, the official Indian class X math school text repeats this term “Pythagorean theorem” 32 times. But what exactly is the historical evidence for Pythagoras? What is the evidence for his connection to the theorem named after him? Do you know? I am talking of *evidence*, not stories. Anyone can tell stories, but no one seems to know any actual evidence for Pythagoras!

There is good reason to ask for hard evidence. The Egyptians built marvelous pyramids thousands of years before the purported date of Pythagoras. Surely they knew basic geometry? (Those pyramids were well-built hence are still standing!) So, why is credit given to Pythagoras?

### Greeks and racism

Egypt is in Africa, and Egyptians were Black, of African ancestry, as the features of the Sphinx corroborate. But, racist historians in the 19th century falsely appropriated the achievements of (Black) Egyptians to (White) Greeks, to fabricate a history of Greek achievements. This is known as Martin Bernal’s *Black Athena*[1] argument.

Afrocentrists understand that the term “Greek” is effectively a euphemism for “White”, as George James earlier thought. Bernal did not examine the “Pythagorean theorem” in detail, which task he left to me,[2] while agreeing with me that this fabricated history of Greek achievements goes back at least to the Crusades:[3] this Crusading history was later reused by racists to claim racist superiority.

During the Crusades (or pre-Crusades) a large Arabic library of Toledo was captured by Christians. The church tradition then was to burn “heretical” books, and the books written by its Muslim enemy were commonly presumed to be heretical. But the church realized that to win the Crusades it needed the knowledge in those Arabic books. So, instead of burning those books, the church decided to learn from them. But the church had a problem: how to explain the sudden U-turn, of learning from heretical books instead of burning them?

Therefore, the church resorted to a simple lie; it used its traditional method of false history,[4] to claim that all valuable knowledge in Arabic books was originally due to early Greeks. Why Greeks? Because the church had long regarded the *early* (pre-Christian) Greeks as its sole friends, being at war with everyone else (including later Greeks). Therefore, attribution to early Greeks was seen as a way to regard that knowledge in Arabic books as a “Christian inheritance”. Gullible Europeans, accustomed to blindly believing all sorts of things (such as virgin birth) from the church, easily swallowed this false history, and are still repeating tall tales of “Greek” achievements in math and science, without serious evidence, and often contrary to the evidence.

That is, (world) knowledge in Arabic books was wholesale appropriated to Greeks by the church. This knowledge was translated into Latin (starting around 1125 CE) and those texts (attributed to Greeks) became the basis for study in the first European universities, such as Oxford, Cambridge and Paris,[5] all set up by the church. Since then, the Western world has retained a deep and credulous faith in the purported achievements of the “Greeks” in science and math, no evidence needed, and all counter-evidence to be ignored.

Anyway, how did “Greeks as the sole friends of Christians” transform to “Greeks as White”? This involves a long story of how claims of *religious* “superiority” mutated to claims of *racist* “superiority” (or color prejudice),[6] and then further into colonial claims of civilizational “superiority” (“Greeks as West”). The key point is that all these claims of (Christian, White, Western) superiority are interlinked: the claim of one sort of “superiority” reinforces another, because it is the always same group of humans being talked about. After the end of apartheid in 1991, many people have stopped explicitly claiming that Whites are “superior”, but they can still claim (and do so all the time) that Greeks or the West did something superior.

**That is, claims of “Greek” achievements are an indirect way to promote claims of racist superiority in unexpected ways, and this happens in the math classroom.**

### Why “Pythagorean theorem”?

So, let as ask again, why is credit given to Pythagoras by using the terminology of “Pythagorean theorem”? As should be clear from the above prefatory remarks, the claim is that Pythagoras, a Greek, did something “superior”.

But, let us first understand the *opposite* point of view, of those who *defend* the term “Pythagorean theorem”. “Authoritative” Western historians assert that Egyptians, despite building pyramids, did not *understand *the “Pythagorean theorem”. For example, Gillings[7] quotes from T. L. Heath, “There seems to be no evidence that they [the Egyptians] knew that the triangle (3, 4, 5) is *right-angled*” (italics original).

This glibly overlooks the fact that the Pythagorean proposition is *better* restated using a *rectangle* and its diagonal, as Indians did[8] in the Manava sulba sutra. 10.10. (The diagonal divides the rectangle into two right-angled triangles.) Egyptians undeniably knew what a rectangle was and very probably understood the “Pythagorean” proposition in the same way, for a rectangle. But Gillings goes on to contemptuously describe as “pyramidiots”, those who believe things about pyramids without *textual* evidence. (Ancient Egyptians wrote on papyri, which crumbles easily.)

Such expressions of contempt, for Black Egyptians, are to be expected from historians of math, of the 19th and early 20th c., when segregation prevailed and racism was explicit, even if slavery had ended. That contempt is an indirect way to establish that Greeks did something superior (hence Whites/Westerners are superior) which is the primary concern of those making such derogatory statements. However, this violates a basic rule: history needs evidence. History without evidence is a devious way to promote prejudices.

Thus,* expressions of contempt for Black Egyptians may accomplish the unstated purpose of racist historians, but prove nothing about Pythagoras*. This tactic, of expressing contempt for Black Egyptians, however deviously hides the fact that there is actually no *evidence* for Pythagoras: whether he actually existed, whether he gave a proof of the theorem named after him, and, if so, what that proof was.

Since “Greek” is effectively a euphemism for White, the implicit racism of present-day mathematicians lies in their determination to hold fast to that racist term “Pythagorean theorem”, without evidence for Pythagoras or his connection to the “Pythagorean theorem”. Indeed, one could better turn around Gillings’ abuse and call “authoritative” Western historians “Greediots” for believing in the history of Greek achievements in math without serious evidence. In fact, there is ample *counter-evidence *from more reliable non-textual sources: early Greeks were extremely bad at math, even at basic arithmetic, and this is corroborated by the primitive calendar[9] of both Greeks and Romans, but we don’t go into that issue here.

### Jumping to the myth of “Euclid”

Since, the primary goal of false history is the racist one to assert Greek “superiority” (hence White “superiority”) this is done in other ways. Long after the abolition of segregation and apartheid, the Egyptologist Clagett[10] asserted in 1999,

“there have been exaggerated claims that Egyptians had knowledge of the Pythagorean theorem which is, of course, a formal Euclidean theorem of the Elements”.

The first thing to note here is the tactic of “myth jumping” (used especially by Greediots) to justify and “save” the terminology of the “Pythagorean theorem”. Racists are unwilling to honestly admit that there is no actual evidence for Pythagoras, which would lead to the collapse of centuries of fake claims of “superiority”. But the only “evidence” they can produce for the myth of the “Pythagorean theorem” is just another myth. That is, Greediots will just “jump” from the myths of Pythagoras to the myth of Euclid. This “myth jumping”, or passing off one myth as evidence for another, is just the tactic of telling a thousand lies to defend one lie and tire out and confuse the questioner. But let us persist in trying to understand those thousand lies.

Notice how Clagett’s assertion *complexifies* matters. To understand how such assertions of Greek “superiority” deviously reinforce claims of White superiority, we now need to understand several complex issues: (1) who was “Euclid”? (2) What is *actually* there in the book Elements of geometry which he purportedly wrote? And (3) what is a formal theorem?

Briefly,“Euclid” is mere myth: there is nil primary evidence for “Euclid”, and my decade-old prize[11] of around USD 3000, for *primary* evidence for Euclid stands unclaimed.

Nevertheless, and though “Euclid” was supposedly from Alexandria in Africa, where the default skin color ought to be black, Wikipedia depicts “Euclid” as a white-skinned male. This is done deviously by showing an image of “Euclid” as a Caucasian stereotype. This Wikipedia portrayal of “Euclid” is an easy way to understand how “Greek” is a euphemism for White.

Long ago, I objected to the similar Caucasian stereotype of “Euclid” in Indian school texts. It was changed to an image, not a stereotype, but still Caucasian! Despite centuries of depicting “Euclid” as white-skinned, there is no acknowledgment of the connection of fake “Greek” history to racism.

Instead, the common smart-alec response is: “how does the color of Euclid’s skin matter”? Now, if it *really* does not matter there should be no objection to depicting “Euclid” as Black. Will Wikipedia do it? No way! On the contrary, when I asserted to the contrary that Euclid was a black woman my article was censored *after* publication.[12] Wikipedia will censor it in another way: it will say that (as a non-White) I am not a “reliable source” — racism used to defend racism!

But that claim that Euclid was a Black woman was not an offhand claim; it was based on deep research as described in my book on Euclid and Jesus,[13] to which we will return in part 2 of this article.

### Jumping to the myth of axiomatic proofs in the “Euclid” book

When it is pointed out that there is no evidence for Euclid, myth jumpers will typically defend the myth (of Euclid) by saying “There is the book”. Indeed, there is a book, from about the 10th c. But according to this second “Euclid” myth about the *book*, it contains a special and “superior” way of doing geometry by proving “formal mathematical theorems”. Does it? It is easy to get lost in this barrage of ever more complex lies.

To understand this claim we first need to understand what is a formal mathematical theorem. Summarily, a proposition proved using the axiomatic method of proof is called a formal mathematical theorem. *The essence of the axiomatic method is NOT the use of reasoning but the exclusion of the empirical* (facts, observations). This involves reasoning beginning from assumptions, called axioms or postulates. **To reiterate, the novel aspect of the axiomatic method is the exclusion of facts (or observations)**.

Thus, the use of (scientific) proofs which use both reasoning AND facts (or observations) is very old, found in the Indian Nyaya sutra[14], and certainly predates anything the Greeks might have done. (Indeed, even the *objections* to deductive reasoning, by the Indian Lokayata,[15] predate the Buddha, hence any “Greek” text on reasoning.) But scientific proofs are different from axiomatic proofs because they begin with facts or observations, so let us return to axiomatic proofs.

However, Clagett’s claim is yet another act of myth jumping: jumping from the myth of the *person* Euclid to the *myth* about the “Euclid” *book — *that it has axiomatic proofs. It is hard to imagine that myths can be told about the *contents* of a book which is before one’s eyes. But the “Euclid” myth was erected by the Crusading church, which had great experience in the matter, through centuries of “reinterpreting” the Bible.

The fact is that there are, in reality, no axiomatic proofs in the book *Elements*, only the widespread *myth* of Euclid says there are. Greediots implicitly believe all sort of Crusading myth to be true. That is, apart from the myth of the “Pythagorean theorem”, and the myth of the person “Euclid”, this second “Euclid” myth (of axiomatic proofs in the “Euclid” book) is equally a false but deeply cherished aspect of Western tradition.

So deeply cherished that, towards the end of the 19th c., Cambridge University adopted this third myth as part of its *math exam regulations*. This was hilarious for the following reason. Empirical proofs, i.e., proofs based on facts or observations, are *prohibited* in the course of an axiomatic proof.[8] But the textbook specially got prepared by Cambridge University, for those new exam regulations,[9] was full of empirical proofs, starting from its proof of the very first proposition of the “Euclid” book!

This fact (of non-axiomatic proofs in the “Euclid” book) had already been noticed by Richard Dedekind a little earlier. The second “Euclid” myth of formal theorems (or axiomatic proofs) in the “Euclid” book was completely busted when Bertrand Russell explained that *there are no axiomatic proofs in the Euclid book*. Certainly, it has no axiomatic proof of the “Pythagorean theorem”. But that myth is still used to assert purported Greek “superiority” in geometry, so let us be absolutely clear that it is a myth.

In speaking of the contents of an old book we need to specify the manuscript, for different manuscripts may differ. The earliest Byzantine Greek manuscripts asserted that the “Euclid” book was actually written by someone else: Theon,[16] or based on his lectures. This was embarrassing because Theon came some 7 centuries *after* the supposed date of “Euclid”. (The church fought against his “paganism” and the library of Alexandria of which he was the last librarian was destroyed by a church mob as part of the church destruction of every last “pagan” temple in the Roman empire.)

But in the 19th c., a racist historian Heiberg solved this problem of divergence of facts from the “Euclid” myth: he “discovered” a manuscript of the “Euclid” book in the Vatican. That 19th c. manuscript (of unknown origin) is today declared the “authentic” and “original” source of “Euclid” from over 2000 years earlier(!).[17] The widespread acceptance of this claim shows the kind of “evidence” that Western historians have for their racist myths.

Anyway, even in this manuscript, the Pythagorean proposition is the second last proposition. Its proof depends upon the fourth proposition called the side-angle-side theorem or SAS, which states that two triangles with equal sides and equal included angle are equal. This proposition is proved *non-axiomatically, *in the “original Euclid”*. *It is proved empirically by picking up one triangle, moving it in space and putting it on top of the other triangle to *see* that the two triangle are equal. (Notice, also, how the myths pile on: the original term was “equal” NOT congruent; it related to *political* equity, which the church fought bitterly against, to assert that Christians are “superior”.) Anyway, contrary to the cherished myth, the “Euclid” book has NO axiomatic proofs from its first to its last proposition.

In fact, NO manuscript of the “Euclid” book (Arabic, Latin, Byzantine Greek) has an axiomatic proof of the “Pythagorean theorem”. Recognizing this, David Hilbert even wrote a whole book[18] to supply the missing axioms and the missing axiomatic proofs in the “Euclid” book, which the myth said were in the book. (Hilbert ‘s “rewrite” of the “Euclid” book badly mangled the original, because supplying axiomatic proofs was never the intent of the real author of the Elements. The actual book and intent both relate to Plato’s geometry — copied from Egyptian mystery geometry — used for mathesis or soul arousal,[19] as we will see in more detail in part 2 of this article.)

To reiterate, the “Pythagorean theorem” is NOT proved axiomatically or formally in (any manuscript of) the “Euclid” book (before Hilbert rewrote it to force the book to fit the myth about it). The claim of a “superior” Greek axiomatic method in math is false (though widely believed).

Therefore, the terminology of the “Pythagorean theorem” and the claim of a “superior” Greek mathematics are bogus and without a valid historical basis, and, indeed, *contrary* to known evidence. That racist terminology remains a key part of math education.

### Summary and Conclusions

Since there is no evidence for “Pythagoras”, the terminology of the “Pythagorean theorem” is defended by “myth jumping” successively to each of the myths of the person “Euclid” and the myth of the “Euclid” book, that it has axiomatic proofs. But both those myths are false; there is ample counter-evidence against both myths. As such, the claim that (White) Greeks did math in a way superior to what (Black) Egyptians did is sheer prejudice. Hence, the related terminology of the “Pythagorean” theorem, a stock part of K-12 math education, is also racist.

(To be continued).

### References

[1]Martin Bernal, *Black Athena: The Afroasiatic Roots of Classical Civilization.*, vol. 1: The fabrication of ancient Greece (London: Free Association Books, 1987).

[2]C. K. Raju, ‘Black Thoughts Matter: Decolonized Math, Academic Censorship, and the “Pythagorean” Proposition’, *Journal of Black Studies* 48, no. 3 (2017): 256–78, https://doi.org/10.1177%2F0021934716688311.

[3]C. K. Raju, *Is Science Western in Origin?*, Dissenting Knowledges Pamphlet Series (Multiversity, 2009); C. K. Raju, ‘“Euclid” Must Fall: The “Pythagorean” “Theorem” and The Rant Of Racist and Civilizational Superiority — Part 1’, *Arụmarụka: Journal of Conversational Thinking* 1, no. 1 (2021): 127–55. https://cspafrica.org/wp-content/uploads/2021/10/6-AJCT-CK-Raju-Part-1.pdf.

[4]“Euclid must fall-Part 1”, cited above, http://ckraju.net/papers/AJCT-Euclid-must-fall-Part-1.pdf#page=26.

[5]trans Dana C. Munro, *Translations and Reprints from the Original Sources of European History, №3, The Medieval Student*, vol. II: №3 (Philadelphia: University of Pennsylvania Press, 1897).

[6]Raju, ‘“Euclid” Must Fall: The “Pythagorean” “Theorem” and The Rant Of Racist and Civilizational Superiority — Part 1’. http://ckraju.net/papers/AJCT-Euclid-must-fall-Part-1.pdf#page=25.

[7]Richard J. Gillings, *Mathematics in the Time of the Pharaohs* (New York: Dover, 1972).

[8]S. N. Sen and A. K. Bag, *The Śulbasūtras* (Delhi: Indian National Science Academy, 1983).

[9]*A Tale of Two Calendars*, 2015, https://www.youtube.com/watch?v=MvpuC7Dg4e0.

[10]Marshall Clagett, *Ancient Egyptian Science: A Source Book*, vol. 3. Ancient Egyptian mathematics (Philadelphia: American Philosophical Soceity, 1999).

[11] or see the related presentation.

[12]For more details see C. K. Raju, ‘To Decolonise Math Stand up to Its False History and Bad Philosophy’, in *Rhodes Must Fall: The Struggle to Decolonise the Racist Heart of Empire* (London: Zed Books, 2018), 265–70; also, https://thewire.in/history/to-decolonise-maths-stand-up-to-its-false-history; C. K. Raju, *Mathematics, Decolonisation and Censorship*, 2017, https://kafila.online/2017/06/25/mathematics-and-censorship-c-k-raju/.

[13]C. K. Raju, *Euclid and Jesus: How and Why the Church Changed Mathematics and Christianity across Two Religious Wars* (Penang: Multiversity and Citizens International, 2012).

[14]Satish Chandra Vidyabhushana, *The Nyaya Sutras of Gotama* (Allahabad: Pāninī Office, 1913).

[15]Haribhadra Suri, ed., षटदर्शन समुच्चय, 5th ed. (Bharatiya Jnanapeeth, 2000), 452 Karika 81, Commentary 559.

[16]T. L. Heath, *A History of Greek Mathematics* (New York: Dover, 1981).

[17]T. L. Heath, *The Thirteen Books of Euclid’s Elements* (New York: Dover Publications, 1956).

[18]David Hilbert, *The Foundations of Geometry* (The Open Court Publishing Co., La Salle, 1950), http://ckraju.net/geometry/Hilbert-Foundations-of-Geometry.pdf.

[19]Raju, *Euclid and Jesus: How and Why the Church Changed Mathematics and Christianity across Two Religious Wars*.