Greek history for idiots: Greediots and Pythagoras.
1: No axiomatic proofs in Greek math
Recently, I presented my talk on “Pre-colonial appropriations of Indian ganita: epistemic issues”. This was at a round table at IIAS Shimla which replaced the now-postponed conference on Indology.
The key point of my talk was that much present-day school math is an inferior sort of math which Europeans appropriated from Indian ganita without fully understanding it, and then returned during colonial times by packaging it with a false history and declaring it superior. A philosophical comparison between ganita and math was done in earlier posts and publications.
This post focuses on the false history aspect, going back to the purported Greek origins of the “Pythagorean theorem”.
False Western claim
Egyptians built massive pyramids very accurately. One would assume that to achieve that marvellous feat of engineering they knew the so-called “Pythagorean theorem”.
But in his book Mathematics in the time of the pharaohs, Richard Gillings speaks of “pyramidiots”: people who claim various sorts of wonderful knowledge is built into the pyramids of Egypt. Gillings’ argues in an appendix (citing the Greek historian Heath) that “nothing in Egyptian mathematics suggests that Egyptians were acquainted with…[even] any special case of the Pythagorean theorem.” Heath adds, “there seems to be no evidence that they [Egyptians] knew [even] that the triangle (3, 4, 5) was right-angled”. The Egyptologist Clagett chips in, “there have been exaggerated claims that Egyptians had knowledge of the Pythagorean theorem which is, of course, a formal Euclidean theorem of the Elements.”
First, Gillings, Heath etc. are not honest enough to add that there is no evidence for Pythagoras. Nor is there any evidence for the claim that he proved any sort of theorem. So, one should rightfully say, “There have been persistent false claims about a Pythagoras having proved a theorem, though there is no evidence that there was any Pythagoras nor any evidence that he proved any theorem.”
Obviously, Western history of Greeks is of very inferior quality, since the tacit norm is that stories about Greeks need no evidence and must be accepted on mere faith in Western authority: it is only stories about others which require evidence!
That is why I use the term Greediots to describe people who fantasize about all sorts of scientific achievements by Greeks without any evidence, starting from the “Pythagorean theorem”: if they can believe in that they can believe anything on their blind faith.
Religious connection of geometry
A key point: not only is there nil evidence for the story of the “Pythagorean theorem”, it is CONTRARY to all available evidence.
The Pythagoreans were a religious cult: their interest was in the connection of geometry to the RELIGIOUS belief in the soul as described by Plato, in Meno, Phaedo, Republic, etc. Anyone can check in two seconds this connection of geometry to the soul by searching for the 2nd, 3rd, and 4th occurrence of “soul” in Meno, a primary source for Plato readily available online from the MIT repository. But for Greediots the story of a theorem is what is important: so they don’t and won’t check facts. (Is Plato evidence for Greek thought? If not, why has no one ever explained the grounds for rejecting Plato? And what are the other “reliable” sources, if any, for Greek history? )
Proclus, in his Commentary, explicitly asserts that this religious belief linking geometry to the soul was the sole concern of the Pythagoreans with geometry. But Greediots not only have no evidence for their beliefs, they ignore all the counter-evidence.
As Proclus further explains in his Commentary (on the book Elements today falsely attributed to an unknown “Euclid”) the book does geometry with exactly the same religious concerns. The subtle issue here is to understand Egyptian mystery geometry (and related Greek mathematics) as a sort of meditative discourse which drives the attention inwards and away from the external world.
All this is explained at great length in my book Euclid and Jesus: how and why the church changed mathematics and Christianity across two religious wars, Multiversity, 2012. See the webpage, or look inside. But Greediots will be Greediots they not only have no concern with facts they will not tolerate a counter-narrative or allow any space for it.
No axiomatic proofs in Elements
The interesting thing is how this “virgin-birth history” propagated by Greediots creates false “facts”. Clagett’s claim that “the Pythagorean theorem…is, of course, a formal Euclidean theorem of the Elements” is one such false “fact” which is widely believed.
The real fact is there is no axiomatic or formal proof of the “Pythagorean theorem” in the book Elements of “Euclid”. One has only to read the book; its very first proposition has an empirical proof not an axiomatic one. But just as most people do not read Plato, most people do not read the Elements. They just naively assume that even if the myth about its author as Euclid is false, the myth about the book must be correct. (Ha, Ha, they don’t know how thick are the layers of church lies!)
After centuries, some including Bertrand Russell finally understood the absence of axiomatic proofs in the Elements. What is shocking is for how many centuries Western scholars collectively failed to realize that even the first proposition of the Elements is contrary to the myth of axiomatic proofs in it.
Unfortunately, Russell referred to the absence of axiomatic proofs in the Elements as “errors” made by Euclid. That is, he foolishly assumed, merely on the strength of the “Euclid” myth, that such was “Euclid’s” intention. He should instead have inferred the intent of the author from an actual reading of the book. Thus, there is (1) no evidence for Euclid, (2) there is no evidence he wrote the book, (3) there are no axiomatic proofs in the books, but (4) Plato and Proclus’ commentary on the book do tell us about a different religious intention of geometry and (5) a different author of the book.
Further, Westerners with their faith-based history of Greeks expect others to be gullible enough to believe that this mythical Euclid, writing in a social context far removed from that of the Crusades, was so perfectly tuned to the politics of the Crusading church that the church used the book as a text for centuries. Obviously, any intention can be attributed to a mythical figure, as politically convenient, and the gullible flock will swallow whatever nonsense their priest says. The remedy is to actually read the book, and apply one’s mind to it, which ALL Western scholars hilariously failed to do for centuries.
Collective failure
That collective failure speaks very poorly for the quality of Western academics: we need to laugh at them, not follow them.
Specifically, the proof of the “Pythagorean theorem” (47th proposition) in “Euclid’s” Elements depends on the 4th proposition of the book, which is the side angle side or SAS proposition. However, in all manuscripts of the Elements, the SAS proposition is proved EMPIRICALLY, by putting one triangle on top of another and seeing that the two are equal. An empirical proof is NOT the same as an axiomatic proof. There are no axioms in the book from which SAS can be proved. In fact, there is no known way to prove SAS axiomatically except in the trivial way of assuming the theorem as an axiom, as is done today.
Therefore, today SAS is taught as a POSTULATE/AXIOM in our class IX school texts. This is proof of the absence of axiomatic proofs in the Elements. But Greediots don’t do their ninth standard math either: they are fixated on false myths, and worried only about how to save those lies.
To reiterate, the proof of the “Pythagorean theorem” in the book Elements depends on the proof of SAS. Therefore, there is no formal mathematical proof of the “Pythagorean theorem” in the book Elements of “Euclid”. That is, while the book Elements has axioms and proofs, it has no axiomatic proofs. In fact there is no formal proof of the “Pythagorean theorem” in any Greek works or before the 20th c. On the other hand, non-formal (semi-empirical) proofs are available everywhere, including in India.
Myth stronger than facts
But, even a century after the absence of axiomatic proofs in the Elements was publicly exposed, the myth proves stronger. There is no shortage of “eminent” scholars who keep repeating that false claim of formal or axiomatic proofs in the Elements. Clagett is not an isolated scholarly Greediot. Needham also believed that Greeks did something special in math, as I have pointed out.
It is high time that Western historians acknowledged the falsehood of both terms in the phrase “Pythagorean theorem”: no Pythagoras, and no formal theorem. It is even higher time that Indians understand that much Western history of science is deliberate fraud since Western historians refuse to remedy even utterly false claims about the “Pythagorean theorem”.
