Archive for the ‘History and Philosophy of Mathematics’ Category

वंशवादी इतिहास या विज्ञान और कला का इतिहास?

Saturday, November 12th, 2022

हाल ही में जयपुर डायलॉग में मेरा एक वक्तव्य हुआ.  विषय था “प्राचीन भारत– वंशवादी  इतिहास  या विज्ञान और कला का इतिहास?”. वक्तव्य हिंदी में था.

इसका ट्वीट  यहां है.

और विडियो इस लिंक पर देख सकते  हैं. Ancient India | Dr. CK Raju, Meenakshi Jain, Abhijit Chavda, Ved Veer Arya

जिन्हें पढ़ना ज्यादा पसंद  है वह मेरे  नोट्स देख लें.

इस वक्तव्य में वही पुरानी बात 1+1=2  का दोबारा जिक्र हुआ है,  और कुछ समझाया भी है  कि यह प्रत्यक्ष से क्यों साबित नहीं हो सकता है मैथमेटिक्स में, और क्यों इसे स्वयंसिद्ध मानना भी गलत है.

वैसे मैं समझता हूं की 1+1=2 का सबूत खामोशी से या गाली गलौज से भी हो सकता है,  जैसे कि पहले कई फोरम में हो चुका है.  गाली गलौज को उचित  ठहराया जाएगा  क्योंकि  यह संवेदनशील मामला है. अगर हमारे विश्वविद्यालय के व्याख्याता   इस मामले में अपना अज्ञान मान लें  तो नैतिकता की शायद यही मांग होगी कि उन्हें अपने पद से इस्तीफा देना चाहिए. लेकिन जिसे राजनीति मालूम है, और  जो ब्रिटिश हुकूमत  और चर्च  के गुप्त तरफदार हैं, उन्हें बाकी कुछ ज्ञान की क्या जरूरत है?

Jonathan Crabtree: a dangerous fraud

Monday, August 1st, 2022

Abstract: J. Crabtree is a fraud who knows neither math nor history, and would not be allowed to teach either subject to Australian school kids. I have called him a church mole, since he is a White man talking of “Bhartiya math” as a con trick to gain entry into Indian education. His object is to sow all kinds of confusion, while slyly defending key interests of false church history related to math. One such aspect of false church history is “Euclid” for EVIDENCE of whose existence I have offered large prizes. Crabtree does not have ANY evidence. Therefore, to save this church lie he pretends to “review” my Euclid book, 10 years after it was published! In typical church manner, he starts with a gross misrepresentation of my position, to launch an ad hominem attack (on my person) by telling some brazen lies which lies are proved false by the very subtitle of the book.

Declaration of personal interest. Crabtree  first wrote to PM Modi saying I should be given the Bharat Ratna, India’s highest award. When I didn’t rise to that bait, he now has many nasty things to say about me, as part of a filthy “review” of my book Euclid and Jesus.

Note. This blog is about Jonathon J. Crabtree an Australian based in Melbourne (NOT to be confused with Jonathon Crabtree an American based in Maryland).

Crabtree describes himself variously as a “private researcher”  and as a “K-8 math researcher” and “historian”, etc.

His educational qualifications? He attended B.Com in Economics at the University of Melbourne from 1980 to 1983. It is not clear whether he actually graduated even in that, since he says that he had an accident on his way to the University in 1983. His word that he is even a graduate is suspect: we need to see the (irrelevant) degree, since as shown below, he is a brazen liar.

In his ORCID entry, he has not indicated any employment record, since 1983 until now, which is a good 39 years of blank space. What was he doing that he is so ashamed to admit? Was he a priest? Certainly, he had NO academic employment ever, and certainly nothing to do with math or history, even as a school teacher.

But he says in his ORCID entry,”1983 to present | Elementary Mathematics Historian (Research) Employment” His “employment” apparently was directly with (the Christian) God: for he says (see above link) “I made it a promise to God that if I could walk again, that I would fix mathematics.  So that was 1983 and so since 1983 until now 37 years, I’ve been working on my personal goal of keeping my promise to God to fix mathematics.”

What was the outcome of working since 1983? His research? First article on “math” is in 2015, a good 32 years after his claimed his employment with his God began! What a fraud! If he did not cheat his God, by twiddling his thumbs for 32 years, he is cheating us by telling the above story to add 32 years of fake “experience”!

So, after 32 years, did he come up with something profound? No: only an alleged new reason why two negatives make a positive. This is in a magazine Vinculum published by the Mathematics Association of Victoria, which encourages first time writers: “Experienced and novice writers are encouraged to submit articles. Editors are able to support you in developing your ideas for an article, especially useful for first time writers.” Apart from another such article in the same magazine, he has some conference papers. That is the basis of his alleged “expertise” in math and history. In fact, like all frauds and cheats he claims to have some miraculous powers (like those of the silly Crusading “holy grail”) by calling himself “the Indiana Jones of math”.

The problem with this fraud: A White man saying “Bharatiya math” is an easy con-trick to get him entry with gullible Indians who do not check facts and believe he may be trying to do some good. He does not know any math, of course, but does not know any Sanskrit either.  What “Bharatiya math” will he do? He drops a few names, Aryabhata, Brahmagupta and so on, but seems thoroughly confused about negative numbers. (more…)

The church origins of (axiomatic) math

Friday, April 15th, 2022

Formal math is like the emperor’s new clothes, a metaphysical (=unreal) cloak over useful normal math. Formal math itself is neither useful not beautiful.
Axiomatic or formal math is like a metaphysical (=unreal, invisible) cloak over useful normal math. This metaphysics is neither useful for any applications of math, nor beautiful. Modern math is NOT “rigorous”; just resurrects the con-trickery of medieval church theology by injecting a metaphysics (fantasy) of infinity/eternity into math. It has the political value of giving Westerners control over mathematical knowledge.

The previous article[1] explained why the terminology of the “Pythagorean theorem” is racist. That terminology is defended by appealing to the two myths of “Euclid”, used to assert that White Greeks did “superior” geometry compared to Black Egyptians. But both myths of “Euclid” (the myth of the person “Euclid”, and the myth about the book “Euclid” that it has axiomatic proofs in it) are false.

It is a completely bogus claim that “Greeks used axiomatic proofs which originated with Euclid”. There are no axiomatic proofs in the actual “Euclid” book, whoever wrote it. This fact was publicly admitted by Bertrand Russell[2] and David Hilbert[3] over a century ago.

How then did axiomatic proofs actually originate? Who FIRST advocated axiomatic proofs? These questions are important, for axiomatic proofs are what are used in the math taught in universities and schools today. But Western mathematicians (and historians) never asked this question about the real origin of axiomatic proofs. All Westerners gullibly believed for centuries in the myth of axiomatic proofs in the “Euclid” book. Today they just carry on believing in the Greek-birth myth of axiomatic math (like the virgin-birth myth, underlying church ethics), using myth jumping, or the tactic of defending one myth by invoking another myth as evidence.

But if the “Euclid” book as the origin of axiomatic proofs is just a false creation myth, one has to ask this question about the real origins of axiomatic proof, and answer it.

Why the Crusading Church adopted reason

The fact is that the earliest recorded case of actually (not mythically) using the axiomatic method of proof is its use by the church. An influential priest of the Crusading church, Thomas Aquinas,[4] used axiomatic reasoning to prove his “angel theorem” that many angels can fit on a pin. That is, axiomatic proofs originated during the Crusades, shortly after the “Euclid” book arrived in Europe as a crusading trophy. This correlation was later confounded with a causal relation, that “axiomatic proofs originated with Euclid”!

But why did Aquinas want to use reason? The context of the Crusades is critically important. Crusading spies like Adelard of Bath (who first brought the “Euclid” book from Muslim lands to Christian Europe) explained that Muslims accepted reason, unlike Christians who went by authority.[5] Indeed, there was a whole Islamic theology of reason, called the aql-i-kalam.[6]

This observation by the Crusading spy Adelard, that Muslims used reason, was very important for the real aim of the Crusades, which was to grab Muslim wealth, by trying to convert Muslims by the sword. But force failed against Muslims for centuries (after the first Crusade). The then military inadequacy of all the squabbling nations of Europe, combined under a religious flag, was amply manifest by the time of Aquinas. So, it was evident that Muslims could not be converted to Christianity by the sword, which method had earlier succeeded with pagans in Europe. This method (the sword) did not work even in the then Muslim parts of Europe (al Andalus, Spain etc.). These were the wealthiest parts of Europe then, hence a prime target of church greed.

Muslims obviously rejected the authority of the church. They also regarded the Bible as corrupted by the church, as even the scientist Isaac Newton did.[7] Force, church authority, and Bible all three were ineffective. But what else could work to convert Muslims? That is, the church had no option but to turn to reason to try to persuade Muslims to convert by reasoning. For this purpose, the church first erected a Christian theology of reason, parallel to the Islamic theology of reason. But how? The word “reason” occurs less than 100 times in the Bible, depending on the translation. So what were the church sources for this transformation? The church could hardly acknowledge that it was mimicking its religious enemy.

Therefore, the first step the church took was to appropriate reason and claim it as a Christian inheritance: this was achieved by its traditional secular method[8] of fooling opponents by concocting false history, since Orosius. False Crusading history simply attributed all “useful” knowledge in Arabic books (captured from Muslims) to early Greeks.[9] It was asserted that key Arabic books (incorporating world knowledge) were of Greek origin. The thought did not cross the minds of gullible Westerners that this tale of the sudden recovery of a long-lost heritage, during a religious war, might by a political stratagem.

Asserting the myth of the Greek origins of “reason”, in “Aristotle” and “Euclid”, enabled reason to be appropriated as a Christian inheritance, on the belief (since Eusebius) that the early Greeks were the sole “friends of Christians” (since they were before Christianity). So strong was the grip of the Crusading church on the minds of Westerners, that gullible Westerners are still sold on those Crusading myths of the Greek origins of reason. We have already examined, in the first part of the article, the falsehood of the myths about Euclid. (“Aristotle” will not concern us here.)

How and why the church invented axiomatic reasoning

However, even if reason came to be regarded as a Christian inheritance, it could still be dangerous to church dogmas. This danger was recognized, before reason was accepted into the church fold. Hence the Inquisition placed a number of reason-related books, translated from Arabic, on its notorious “Index”, and persecuted those engaging with these books.

This was the problem (resolving the danger of reason to church dogma) to which Aquinas and his schoolmen found a simple solution. The solution was to subtly twist reason to turn it into a church tool, so that “reason” could be used to support church dogma, and prove exactly what the church wanted: neither more nor less. How? By inventing axiomatic reasoning!

To understand how the church twisted reason, let us review the difference between normal reasoning and formal reasoning (or axiomatic reasoning). Normal reasoning begins with facts or observations. We see smoke, and reason that there must be fire. This is a good way of reasoning, and is very ancient;[10] its use in India is documented from long before any fictitious Greeks, such as Aristotle (of Toledo[11]) or “Euclid”. This is the kind of reasoning used also in science: for facts and observations are a key part of science.

However, Aquinas could not reason in this normal (or scientific) manner, for facts or observations are contrary to church dogmas about the nature of God [theology=God-knowledge]). Therefore, the church invented axiomatic reasoning, or reasoning which excludes facts. (more…)

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).


[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,

[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.

[4]“Euclid must fall-Part 1”, cited above,

[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’.

[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,

[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,; C. K. Raju, Mathematics, Decolonisation and Censorship, 2017,

[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),

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

How to make calculus easy

Friday, January 7th, 2022

Calculus is difficult because real numbers are wrongly believed essential to it. Reverting to the way calculus originated makes it easy.

Bertrand Russell arguing with a grocer
Axiomatic proofs (like Russell’s 378 page proof of 1+1=2) add zilch to the practical value of math in a grocer’s shop, but they make math excessively difficult. The purported “superiority” of axiomatic proofs is a church superstition which was politically convenient to the Crusading church in developing its rational theology which used proofs based on reason minus facts (or observations).

Having stolen the calculus from India, Europeans eventually realized that they did not fully understand it. Hence, they invented “real” numbers, long after the purported “discovery” of calculus by Newton and Leibniz. Real numbers add to the difficulty but not to the practical value of calculus.

Real numbers are regarded as essential to define core calculus concepts such as derivative and integral, and to sum its infinite series. But they are not actually defined in the typical fat “Thomas’ calculus” texts, of over 1300 pages. Failure to define core concepts naturally makes calculus difficult.

Further, real numbers have nil practical value, since all current applications of the calculus, such as calculating rocket trajectories, are done numerically on computers which use floating point numbers and cannot use metaphysical (=unreal) real numbers.

Original Indian understanding of calculus

The solution to calculus difficulties, then, is not to cancel the calculus, but to make it easy, by cancelling real numbers. This can be easily achieved by reverting to the original Indian understanding (epistemology) with which calculus originated. That has three key components. (1) Aryabhata’s (5th c.) approach to calculus as concerning the numerical solution of difference/differential equations. (2) Using the “non-Archimedean” arithmetic of Brahmagupta’s (7th c.) polynomials (or unexpressed arithmetic), which algebra Europeans failed to grasp, but which first enabled Nilakantha (15th c.) to correctly sum infinite series. (This “cancels” real numbers, which are Archimedean.) (3) The philosophy of zeroism or sunyavada (which asserts that belief in exactitude is erroneous, since nothing endures exactly for even two instants).

This way of doing calculus makes it very easy. This has been pedagogically demonstrated with 8 groups of students across 5 universities in 3 countries. It also enables students to solve harder, real-life problems not covered in calculus courses at K-12 or beginning undergraduate level.

Church origins of axiomatic proofs

Axiomatic real numbers only provide the metaphysical (or fantasized) exact sum of an infinite series, such as that for the number pi(π). The axiomatic method was brazenly imputed to the “Euclid” book, when that book first arrived in Europe as a Crusading trophy. Hilariously, however, the “Euclid” book has no axiomatic proofs, and “real” numbers are typically used for an axiomatic proof of even its first proposition.

Actually, axiomatic proof, or a method of reasoning which dodges facts, was a church innovation. During the Crusades, the church had a political requirement for a Christian rational theology to counter Islamic rational theology. Hence, the church accepted reason, but cunningly prohibited facts which contradict so many of its dogmas. It declared reasoning (minus facts) from (metaphysical) assumptions a “superior” method of reasoning. The church brazenly read this axiomatic method into the “Euclid” book, to hide its real origins. Europeans, under church hegemony, gullibly believed that for centuries, though the “Euclid” book actually has no axiomatic proofs.

Today, it is a stock Western superstition that axiomatic proof is (epistemically) “superior”, since “infallible”, like the pope. Actually, axiomatic proofs are highly fallible.

The political advantage of making math difficult

But the West won’t abandon the axiomatic method, because the resulting difficulty of math makes most people ignorant of math. Ignorant people must trust authority, and trusted authorities are Western, as colonial education and Wikipedia both teach us. That is, the difficulty of math has the current political advantage (for the West) that it helps Western (or Western-approved) mathematicians to dominate mathematics, needed for science.

However, the difficulty of calculus also means fewer people available for technology development. Thus, the question today is whether, for the sake of its soft power, the West is willing to lose the technology race, and the hard power from its slender technological lead over others, as the California new math framework indicates.

How long can the false history and bad philosophy of math last?

The question is also how long that soft power will last, before the truth is exposed, in this age of information. And whether and for how long the colonised non-West will continue to blindly believe those false myths and superstitions (about the history and philosophy of math) used by the coloniser to teach the fundamental lesson of colonial education: “we are superior, imitate us”.

Despite indoctrination through globalised colonial education, the colonised just might wake up and cross-check that false history and discard that bad philosophy of math.

Why does California want to cancel the calculus?

Friday, January 7th, 2022

Because the calculus is difficult. Why? Because Europeans stole it, and hence, like cheats in an exam, they failed to fully understand it.

Matteo Ricci handwritten letter from 1581
Snippet of handwritten letter from Matteo Ricci (Cochin, 1581) just before Clavius’ 1582 Gregorian reform of the calendar, which itself was needed for European navigation (latitude determination in daytime). Ricci speaks of getting Indian calendrical knowledge from an “honest Moor or an intelligent Brahmin”. See, Cultural Foundations of Mathematics, Pearson Longman 2007. Europeans stole Indian calculus and its ultra-precise trigonometric values for their navigational needs: determining latitude, longitude, loxodromes.

Why does California want to “cancel” the calculus? Because the calculus is difficult. Why? (Read this article.) Because Europeans stole the calculus, and knowledge thieves fail to fully understand what they steal, just like students who cheat in an exam. See also this video of a talk at MIT and the abstract.

Astrology in university education – twenty years after

Saturday, September 11th, 2021

(Note: The following article speaks the truth, instead of taking sides. However, the Indian media being totally polarised, I could not publish the article in either English or Hindi. Hence, am now posting it on my blog, since I feel it is important for me to take a stand on the matter.)

Astrology is a superstition, but why are the colonised unwilling to admit that Johannes Kepler was a superstitious astrologer, who got his livelihood from astrology, and wrote in praise of astrology.  And what of Isaac Newton who superstitiously believed in Biblical creationism and apocalypse. His superstitions rubbed off into science and math as in the “eternal laws of nature”, not to mention his superstitions about the Indian calculus, all of which church superstition we happily teach in schools today.  But there is no outrage among the colonised who blindly accept all church superstitions in mathematics and science. That is exactly why it was the church which brought Western ethnoscience and Western ethnomath to the colonised in general and to India in particular. The real issue is about Western dominance, not science vs superstition.

The Indira Gandhi National Open University recently introduced a postgraduate course in astrology. A similar issue had arisen twenty years ago when the University Grants Commission (UGC) announced a scheme to open 16 university departments, to teach astrology across the country, in 2001. This was hugely opposed, and the late Kapila Vatsyayana organized a public debate, between scientists and astrologers, in the India International Centre, on the desirability of astrology in university education. The late Pushpa Bhargava, Raja Ramanna and I represented scientists. But the astrologers ran away from the debate, though I did later discuss this issue publicly with some other astrologers in other forums. The UGC eventually scrapped the scheme. However, some clarifications given 20 years ago are still relevant.

First, the term “jyotish”, which means time-keeping (through astronomy), is wrongly confounded with astrology (called “phalit jyotish”). The earlier UGC scheme was announced as pertaining to Vedic astrology. However, there is no mention of astrology in the Veda-s. Then, at the India International Centre, I had challenged the assembled scholars, in front of the international press, to show me a single sentence on astrology in the core text of Vedanga Jyotish.1 No one could do so, and some started asking for my copy of the Vedanga Jyotish, which they had obviously never seen before. The Vedanga Jyotishe is a manual of timekeeping, completely disjoint from astrology.

Indians persistently separated astronomy from astrology, which separation is not limited to the Vedanga Jyotish, last updated around -1500 CE. Thus, Nilakantha’s commentary on the Aryabhatiya is dated to +1500 CE.2 During this 3000 year period, there were numerous books written on astronomy in India. These included the Surya Siddhanta, the Aryabhatiya, the Laghu and Maha Bhaskariya of Bhaskar 1, the Brahmasphutasiddhanta of Brahmagupta, the Shishyadhivrddhida of Lalla, Vateshwar Siddhanta, and Gola, Tantrasangraha, Yuktidipika, etc. In none of these books do we find a single sentence related to astrology. The beginning of astrology in India is credited to the 6th c. Varahamihira, and his Brihat samhita, but even Varahamihira’s astronomy book Pancasiddhantika does not have a single sentence on astrology.

However the colonially educated are deluded that jyotish means astrology. The same colonial education also impacts nationalists. Hence, they repeatedly return to the claim that astrology was an important aspect of Indian tradition since Vedic times. Twenty years ago, Pushpa Bhargava had challenged the teaching of astrology in the Madras High Court. In response, the UGC had said that astrology was an important aspect of ancient Indian tradition, a claim happily accepted by the judge (Kalifulla J.) Nobody asked for evidence that astrology was a significant part of Indian tradition, and nobody offered it.

To the contrary, the Buddha explained3 that common people praise him because he does not earn a livelihood by the unethical means of predicting uncertain future events, such as predicting the victory or defeat of kings in a war, or predicting good or bad rainfall. This was not any specifically Buddhist ethics, since it was the common people (then pre-Buddhist Hindus) who praised the Buddha thus.

In contrast, the West traditionally believed in prophecy. Herodotus4 begins his History with the story of Croesus, from Lydia (Turkey), who first made Ionian Greeks his vassals. Before fighting the Persians, Croesus checked the outcome with the Oracle of Delphi. “A great empire will fall” was the prophecy. Unsure about which Empire would fall, Croesus again sent an emissary to ask how long his own rule would last. “Until a mule rules the Medes (Persia)”. Croesus thought that hardly likely and battled Cyrus the Great and lost. The prophecy was then explained that Cyrus  was the mule since he was of mixed descent. Of course, had the outcome been different, there would have been no need for an explanation. This illustrates how foretelling the future was traditionally based on subtle con-tricks.

Prophets were given a very high religious status in the West. Hence, during the Crusades, the church tried to put down Muslims by the criticism that Paigambar Muhammad made no prophecy. Unfortunately, the strange response of Muslims to this critique has been to translate Paigambar (meaning messenger) as Prophet!

Traditional Western superstitions did not magically disappear with the advent of science. Johannes Kepler, famous for his “laws” of planetary motion, wrote on the fundamentals of astrology.5 Before he grabbed the high church position of Astronomer Royal to the Holy Roman Empire, Kepler was a practising astrologer, and he wrote that providing astrology as a means of livelihood to astronomers was proof the of pre-established harmony created by God!

Even Isaac Newton superstitiously believed in Biblical creation, some 6000 years ago. He explicitly used it to deny the antiquity of Egypt.6 He also believed the Bible correctly foretold the future apocalypse of the world at the “seventh trumpet”.7 Indeed, belief in prophecy, or the belief that the future can be foretold, persists in the scientific belief in the mechanistic evolution of the world according to some “eternal laws of nature”. This belief in “eternal laws of nature” is a Christian dogma first propounded by Thomas Aquinas.8 This dogma, is not, for example, acceptable in Buddhism,9 or Islam,10 or Hinduism.11

But, both Newton and Kepler believed in this dogma, and we teach it in our schools today.12 This dogma asserts that the future is determined and predictable by the knowledgeable, like prophets and Laplace’s demon. (On Karl Popper’s formulation, Laplace’s demon is a super-scientist, who knows all the laws of nature, a super-observer, and a super-computer, who can hence calculate the future.13) Of course, no one knows how the “laws of nature” or equations of physics (supposedly) causally determine human actions, any more than anyone knows how planets determine human actions. So, the difference between the demon and astrology is a matter of technique, not of principle.

The colonially educated believe Indians were especially superstitious. But the experimental method was used in India, long before Bacon,14 and many traditional Indian astronomers spoke out against superstition. For example, it is said that Indians believed that Rahu  and Ketu are the cause of eclipses. This myth is undoubtedly found in the Purana-s. However, the eighth century Lalla titled the 20th chapter of his Sisyadhivrdhida15 as the “Correction of mythical knowledge”. Here he gives several arguments why demons such as Rahu, cannot be the cause of an eclipse. In the 26th sloka he says “In a solar eclipse, people in different parts (of the earth) see different portions of the Sun eclipsed. Some do not see (the eclipse) at all. Knowing this, who can maintain that an eclipse is caused by Rahu?” Further, Lalla (20:22) asks why eclipses occur only near the full moon or new-moon. In contrast, the Bible (Luke 23:44-45) states the superstition that God caused a solar eclipse at noon on the crucifixion of Jesus, which is impossible, because Easter, or the supposed date of resurrection of Jesus, is linked to the full moon when a solar eclipse is impossible. Before the 19th c., which Western astronomer rejected this Biblical assertion as a superstition?

The conclusion is that scientific thinking is a much older part of Indian tradition than astrology which was probably imported in the 6th c., and true nationalists ought to encourage that older tradition. On the other hand, church superstitions still flourish in science (and math) and the tail-wagging colonised who believe science is a matter of Western approval, not critical thinking, need to understand that.

1K. V. Sarma, ed., Vedanga Jyotisa of Lagadh, trans. & notes T. S. Kupanna Sastri (New Delhi: INSA, 1985).

2K. Sambasiva Sastri, ed., Aryabhatiya of Aryabhatacarya with the Bhasya of Nılakanthasomasutvan (University of Kerala, Trivandrum, 1930).

3Maurice Walshe, trans., Digha Nikaya: Long Discourses of the Buddha (Boston: Wisdom Publications, 1995), 68–72 Brahmajala sutta, section on Mahashila.

4Herodotus, The History, trans. G. C. Macaulay, n.d.,

5J. Bruce Brackenridge and Mary Ann Rossi, ‘Johannes Kepler’s on the More Certain Fundamentals of Astrology Prague 1601’, Proceedings of the American Philosophical Society 123, no. 2 (1979): 85–116.

6Isaac Newton, “Chronology of Ancient Kingdoms amended”,

7C. K. Raju, The Eleven Pictures of Time: The Physics, Philosophy and Politics of Time Beliefs (Sage, 2003) chp. 4, ‘Newton’s time’.

8Thomas Aquinas, Sumnma Theologica, First part of the second part, 91,1, n.d.,

9C. K. Raju, ‘Buddhism and Science’, 2016,, a conversation with the Dalai Lama.

10C. K. Raju, ‘Islam and Science’, keynote address, in Islam and Multiculturalism: Islam, Modern Science, and Technology, ed. Asia-Europe Institute University of Malaya and Japan Organization for Islamic Area Studies Waseda University, 2013, 1–14,

11Minutes of a meeting in the University Sains Malaysia, 2011, to discuss whether the belief in “laws of nature” should be part of a course in the philosophy of science.

12See, e.g., the 2021-22 NCERT text on science for class IX, chp. 10, p. 133.

13For Laplace’s demon see C. K. Raju, Time: Towards a Consistent Theory (Springer, 1994).

15For a general account, see “Indians against superstition”, extract from “Proofs and Refutations in Mathematics and Physics: an Indian Perspective”, in History of Science and Philosophy of Science, ed., P. K. Sengupta, Pearson Longman, 2012. For the original source see Lalla, ‘शिष्यधीवृद्धिद’, ed. Bina Chatterjee (Delhi: Indian National Science Academy, 1981).

“Euclid” must fall: racism, the church, and the axiomatic method (collected resources)

Friday, July 2nd, 2021

(Keynote Tübingen/Pretoria 13 May 2021. Related articles now online.)




Part 1: Racist prejudice and the false history of “Greek” achievements in math and science

Abstract.To eliminate racist prejudices, it is necessary to identify the root cause(s) of racism. American slavery preceded racism, and was closely associated with genocide. Accordingly, we seek the unique cause of the unique event of genocide + slavery. This was initially justified by religious prejudice, rather than colour prejudice. This religious justification was weakened when many Blacks converted to Christianity, after the trans-Atlantic slave trade. The curse of Kam, using quick visual cues to characterize Blacks as inferior Christians, was inadequate. Hence, the church fell back on an ancient trick of using false history as secular justification for Christian superiority. This trick had resulted in a false history of science during the Crusades when scientific knowledge in translated Arabic texts was indiscriminately attributed to the early Greeks, without evidence. This false history enabled belief in religious superiority to mutate into a secular belief in White superiority. After colonialism, and the Aryan race conjecture, the belief in White superiority further mutated into a belief in Western civilizational superiority, openly propagated today by colonial education. Hence, to eliminate racist prejudice, it is necessary to engage simultaneously with the allied prejudices about Christian/White/Western superiority, based on the same false history of science.

Full article at:

Part 2: The practical failure (and political success) of the axiomatic method (or the church understanding of reason) in math

Abstract. Previously we saw that racist prejudice is supported by false history. The false history of the Greek origins of mathematics is reinforced by a bad philosophy of mathematics. There is no evidence for the existence of Euclid. The “Euclid” book does not contain a single axiomatic proof, as was exposed over a century ago. Such was never the intention of the actual author. The book was brazenly reinterpreted, since axiomatic proof was a church political requirement, and used in church rational theology adopted during the Crusades, as a counter to Islamic rational theology. Deductive proofs are MORE fallible than inductive or empirical proofs. Even a validly proved mathematical theorem, such as the “Pythagorean” theorem (based on Hilbert’s axioms, say), is invalid knowledge in the real world. There is no concept of approximate truth in formal mathematics. Nevertheless, the myth of “superior” axiomatic proofs in the “Euclid” book continues to be reiterated by Western historians, and colonial education teaches axiomatic mathematics. Actually, superior practical value comes from the two “Pythagorean” calculations well known to Indian/Egyptian tradition, but unknown to Greeks. The advantage of related decolonized courses in mathematics has been pedagogically demonstrated. But understanding and political will is needed to change colonial/church education.

Full article at:

Alternative to current school teaching of Christian chauvinist “Euclid’s” geometry

“Euclidean’ geometry vs Rajju ganita” Bengaluru, 5, 6 June 2021)

Details of workshop:

Prior reading list:

Presentations: Day 1, Day 2 (space bar moves to the next slide)

Videos: Day 1, Day 2 (2:45:55, and 2:55:59)

School text for class IX:

See also, Euclid and Jesus: How and why the church changed mathematics and Christianity across two religious wars, Multiversity, Penang, 2012.

“Euclidean” Geometry vs Rajju Ganita

Tuesday, June 8th, 2021

All the lies and obscurities of “Euclidean” geometry, as taught in the NCERT class VI, IX, and X school texts, stand exposed in these presentations and videos of the workshop held on 5 and 6 June 2021.

Presentations: Day 1, Day 2 (space bar moves to the next slide)

Videos: Day 1, Day 2 (2:45:55, and 2:55:59)

Both “Euclid” and the related philosophy of “superior” axiomatic proof are just a church fraud concocted during the religious fanaticism of the Crusades, to provide support for Christian theology of “reason” set up by Aquinas and the schoolmen.

Sad that Indians have been fooled for 2 centuries, and refuse to think. This incapacity is a design objective of colonial/church education. Sad that the top formal mathematician in the country and abroad are running away from public debate in the manner of fraudulent astrologers: they have no concern for truth or for the future of children.

Rajju Ganita or traditional Indian/African string geometry is so much better.

Rajju ganita cover

Rajju Ganita workshop Bengaluru, 5-6 June 2021

Friday, May 21st, 2021


brings you




Prof. C.K RAJU

A 2-day workshop to discuss:

(1) The falsehoods and obscurities in “Euclidean” geometry, as currently taught
(2) Why Indian śulba (string) geometry is a better way to teach geometry.

DATE: 5th and 6th June, 11AM to 1PM

Who can attend: Math Teachers, Home schoolers, anyone curious and interested in a new perspective on Ganita

Prerequisites : Participants are expected to have read the geometry sections in NCERT math texts (available online) from class VI to class IX. They should also attempt the home assignment at this link. For a further reading list see here.

Registration : Click Here