I don't know if it's true but we were always taught in the History of Maths course at Cambridge[1] that Pythagoras did not exist, or at least there was no evidence of a specific person called Pythagoras. Further if you followed the footnotes in most any research on the subject (especially claims that the Egyptians or Babylonians knew the theorem) you almost always end up at a loose end.
Maybe its because I'm on mobile, but your link brings me to a faculty page and nothing there suggests Pythagoras wasn't a person.
I even went through the syllabus of the most closely related of their past lectures (a quick history of ancient mathematics and mathematical ideas) and I quickly realized that I don't even know which book to read to try find anything related to Pythagoras' existence.
Is there a specific piece of text I could read that would verify your claim?
For what it's worth Wikipedia has him as a real person living 570 – c. 495 BC, son of Mnesarchus, a gem-engraver on the island of Samos, who travelled to Croton about 530 BC and set up some kind of school there.
It might be the mathematical equivalent of Winston Churchill... a person who definitely existed, but who didn’t say 90% of the famous bon mots attributed to him. It might have been that if you wanted your ideas to spread you had to attribute them to “Pythagoras.”
Or it was discovered by his followers and students (pythagoreans), or perhaps his name become associated with people who were mathematical philosophers at the time.
Yeah, it could be a case of syncretism: Pythagoras' school was perhaps a meeting of minds that took a bunch of disparate ideas/concepts from all over the world and finally nailed them down into more concrete shapes and merged similar but equivalent concepts into grander shared abstractions. Then things like "Pythagoras's school found" and "Pythagoreans believed" accidentally over the years get rubbed off to read "Pythagoras found" and "Pythagoras believed" in a "great man fallacy/conflation" between the school of thought and the nominal figurehead.
My understanding is that is possibly an incredibly common view on Greek history. For instance, I've heard that a lot of things attributed directly to Plato today seem better attributed to platonists in general and the wild maelstrom of activity in the forums.
Getting maybe further aside and into the weeds: Pythagoreanism was even a multiple-times revived religion in Greek culture, so that would give further reason to attribute things to Pythagoras as "religious texts" things from other scholars or things from later movements (such as neopythagoreanism, which I found in one rabbit hole deep dive has a lot of interesting but unproven ties to early Christian doctrine). (Similar to how so much scholarly work in Asian history got attributed to "The Buddha" before eventually some of them started attributing things maybe more accurately to just "a buddha".)
Prof Bursill-Hall had a fascinating theory that the early Christians were to blame for Pythagoras's elevation. The Pythagorean's believed, "all was number" and identified 2 and 3 with man and woman, 5 (= 2 + 3) with children, and 10 (= 2 + 3 + 5) with God(s).
The early church trying to enforce the slightly odd concept of the trinity found solace in a triangular number being equivalent to god.
As you say, proving this would be practically impossible.
So it works like SEO, if we mention the name of a celebrity then the site will get ranked higher in search results?
I guess they won't mind if we're thanking them. Thanks Larry Page for making the connection between academic citations and Google search results so we can all find things easily :)
Yes, but the point is the approach to proof (proving this for all cases) that represented a paradigm shift in mathematics, largely attributed to Thales.
I think this is the most important point modern readers need to make.
If we discover a proof earlier than Thales, regardless of whether it’s Greek, we might revise the discovery of trig. But this is about certain triples being used practically in Babylon (who recorded everything) without any recorded interest in the relationship between triples.
Because Greeks first recorded the important part —- the proof —- I think it’s factual and not at all offensive to refer to Ancient Greek culture as influential in that recorded innovation in geometry.
I’ve seen a movement to broadly give credit to other cultures for Pythagoras, but knowing about triples is just not enough to say that they promoted innovation leading to trig.
> this is about certain triples being used practically in Babylon (who recorded everything)
The difference isn't that Mesopotamia generated more records than other cultures. It's that Mesopotamian records become practically indestructible when burned.
(This didn't help much in Babylon itself, which didn't burn.)
With regard to Alexandria, if Euclid had access to all the resources of Alexandria, and Elements is the second-most translated and transcribed book (behind the Bible) then Euclid had the best known explanation of where the Greeks got the theorem (right or wrong) and also the historiographic influence to credit Pythagoras (also, of course, right or wrong). There might be a missing chapter of Euclid.
WRT Babylon, I don’t know how many tablets are left undeciphered. This article (or a comment) mentions the famous cosecant tablet. We have the tools to fill in the gaps for their language and math. I think extrapolating past that is at least as ahistorical as speculating about what was lost at Alexandria.
I think that there are several ideas that are both powerful and that people muddle when they talk about mathematics. Also the talk is often tinged with cultural pride.
The ideas are conceptualization, generalization, and rigor.
The Pythagorean triples are a conceptualization that for certain right triangles, the squares of the three sides obey a relationship.
The Pythagorean theorem is both 1. a generalization that that would be the case for all right triangles and 2. rigorously proved it. Furthermore, at least for the proof in The Elements, it was axiomatic rigor, proving this via a chain of logic starting from base concepts.
Now to the cultural pride aspect. Many cultures came up with concepts in mathematics. Many cultures generalized. Many cultures were rigorous.
The argument is/should really be that the Pythagorean theorem is on the far end of the rigorous, generalization axis. The better argument actually is that The Elements is.
My personal take is that the Pythagorean theorem isn't, at least in the long span of mathematics history, an exemplar of conceptualization.
And yes, there is some overlap and ambiguity with these ideas and you can argue about where the boundaries lie but I don't think we gain that much from that.
If people don't think that other cultures conceptualized, generalized, and were rigorous, just look at The Art of War, Bhagavad Gita, etc...
About conceptualization, my understanding is that ancient Greek followers of religious leader Pythagoras realized that this was not only a relationship between length and area, but a formula for irrational numbers. This might be a claim without evidence, and would not be the first time Western education has tried that.
Babylonians had no problem arriving at solutions to square roots, and a lot of their texts are yet undeciphered. If we find that Babylonians ruminated about irrational numbers, too, then we’re talking about a major revision in the history of math. Until then, to me, the concepts in the Greek treatments are more innovative than the practical usage in Babylon.
I think your take is supported if it turns out people have read way too much into the religion around Pythagoras. I think Pythagoras would be surprised to be famous for a proof when he’d rather be famous for musical scale or some weird ritualistic dance or something. In fact, as mathematicians we might be better served by later proofs, which is implied by your other points, I think.
Yes, irrational numbers were a great discovery (including how they were discovered) and the systematic treatment of math in The Elements was a better example of the advancement in approaches to knowledge and thinking. And yes, if the Pythagoreans actually thought that this was a formula for irrational numbers, that would indeed to quite impressive.
Also, rigor was romantacized. With the exception of The Elements, very little math till the last maybe 200 years had that level of rigor. Newton and Leibniz sure didn't.
This is one of my pet peeves, too, an obsession with referencing the first to try innovative math. Maybe Babylonian root finding is a simplified case of what we call Newton’s method. The minimum level of rigor for a concept to go from theorem to proof is both human and philosophical. We get very little recognition for the mathematicians putting in the hard work, and detrimental recognition for concepts that don’t exactly widen a field of math for innovation.
Recent movements to revise the recognition of Pythagoras are doing the right thing when they acknowledge the path dependence of unbroken written communication of ideas is often mistaken for awarding prestige to a particular institution, educational tradition, or culture.
The earliest known proof of Pythagorean theorem is due to Euclid. It is strange at all that any credit is attributed to Pythagoras, since Babylonians, Egyptians, Indians and Chinese all had discovered this [1]. And Pythagoras did not prove his theorem, to the best of our knowledge.
Also weird that (more generally) Euclid is only rarely mentioned by name, and more often nearly anonymously as the author of Elements. Yet, Pythagoras is named early and often.
From what I remember from my epistemology courses, it's due to historiographical reasons: there are a lot of cross reference of the existence of Pythagoras from other texts, so his historical context and existence is well anchored.
As for Euclid, apart from the Elements, there is pretty much no mention of him anywhere, which is even more troubling since the depth and reach of the Elements is massive and should have had an impact at the time. This led some people to think that Euclid may not be a person but a group of people.
This is what I remember from courses 15 years ago, I may be wrong or outdated on the subject ;)
This is close to what I was taught. I don’t know ancient Greek, so I wonder if that language draws a distinction between singular and plural in the way “the author” of Elements is structured.
Fwiw, I do recall that Pythagoras seems to have forbidden written records, so what survives are his followers’ notes, who apparently gave him singular credit where we might expect a collaboration.
> I don’t know ancient Greek, so I wonder if that language draws a distinction between singular and plural in the way “the author” of Elements is structured.
I don't know what you mean by "in the way 'the author' of Elements is structured", but yes, Greek draws a distinction between singular and plural. (And dual, though to a lesser degree.)
I mean the structure of the way later writers refer to Euclid. Apparently, from Wikipedia, "ὁ στοιχειώτης" but I hope an improved discussion/potential rabbithole occurs if we know whether that unambiguously translates to one male author.
Yes, it unambiguously translates to one male. That question is easy. A literal translation would be something like "the Elements guy".
But it doesn't mean the author was one male. Lots of texts have attributed authorship. The Homeric Hymns are attributed to "Homer". The Gospel of Luke is attributed to "Luke".
Pythagoras isn't named so much in his aspect as a mathematician. He's named because he is the center of a major Greek mystery cult. (The other major cult being centered around Orpheus.)
The Hindu mathematicians Baudhāyana certainly discovered the theorem before Pythagoras [1] and Bhāskarāchārya independenly proved the theorem [2] (though the latter seems it was after Pythagoras).
It is also possible to discover mathematical truths without providing a formal proof. Indian mathematician Srinivasa Ramanujan "independently compiled nearly 3,900 results (mostly identities and equations). Many were completely novel; his original and highly unconventional results, such as the Ramanujan prime, the Ramanujan theta function, partition formulae and mock theta functions, have opened entire new areas of work and inspired a vast amount of further research. Of his thousands of results, all but a dozen or two have now been proven correct." [3]
The mathematical statement versus rigorous proof was also a cultural difference: "Their collaboration was a clash of different cultures, beliefs, and working styles. In the previous few decades the foundations of mathematics had come into question and the need for mathematically rigorous proofs recognised. Hardy was an atheist and an apostle of proof and mathematical rigour, whereas Ramanujan was a deeply religious man who relied very strongly on his intuition and insights." [3]
To the highly-material Western mind, Hindus can be a little weird sometimes: "A deeply religious Hindu, Ramanujan credited his substantial mathematical capacities to divinity, and said the mathematical knowledge he displayed was revealed to him by his family goddess Namagiri Thayar. He once said, "An equation for me has no meaning unless it expresses a thought of God."" [3]
So Pythagoras likely didn't even formulate the theorem - he was more into numerology and only cared about sum of squares triplets, not the results about right triangles.
On the other hand, we have strong evidence that the Babylonians understood and applied the "Pythagorean" theorem long before Pythagoras lived.
Additionally, the first recorded explicit statement of the theorem is actually by the Indian writer Baudhayan, over a century before Pythagoras.
The oldest extant proof of the theorem comes from Euclid. The proof commonly attributed to Pythagoras actually first shows up in the Chinese Zhoubi Suanjing, several centuries after Pythagoras.
Quite simply, "Pythagorean theorem" is a major misnomer.
Here's a reason: sharing, communication, and popularization are even more important than the underlying idea. The Babylonians can fight the Chinese, Egyptians, and 5 other claimants for the historical footnote of who figured it out first. I don't really care. If the Pythagoreans evangelized the concept at a critical juncture in the history of the discourse that eventually became my education, I think that's a fine reason to leave their name on the theorem.
Of course, the Pythagorean theorem is a good springboard for discussing how this works, how attribution is often murky or flat out wrong, how evidence of various bits of math popped up far earlier in history than their official invention, yet remained sterile and ultimately went nowhere because they weren't shared, popularized, and expanded upon. Or it happened but stopped at cultural boundaries and didn't get to us by that path, so we give credit to the person who made that happen. In any case, receiving the idea is what we care about, not originating the idea. See: Euler did it first, Gauss did it first, ancient Indians did it first, ancient Chinese did it first, etc, etc.
My understanding is that some schools in India teach it as the "Pythagorean theorem" and others as the "Baudhayan theorem" - the choice of which term to use is definitely a colonial issue.
Agreed with ummonk. This is not a pre-colonial issue. This is an issue of colonial-era politics. The dating of many Indian texts was performed by European indologists who were funded by then-colonial governments in a then-colonized India with a view to advance the biblical worldview that the universe is about 6000 years old [1], which any reasonable thinker now knows is pure hogwash [2]. So Indian texts are, in fact, probably a lot older. I believe that a now-decolonized India needs to pay closer attention to its history.
During a heated argument, a person I know argued that humanity advanced as fast as it did during the 19th century after aliens invented(!) electricity and gave it to us.
An uncomfortably large percentage of our population doesn’t understand the theorem, no matter what it’s called. But sure, let’s quibble about the name.
I believe sites like these do a disservice to the ancient knowledge of Hindu people. There were many important discoveries, some unknown to the West for centuries, and still not having enough exposure (when you read math books, they start with Greeks, then maybe they gloss a bit over al-Khwarizmi and then jump right into the 17th century to all big names, as if nothing was happening for over a thousand years - well, a lot was going on, just not in Europe).
However, some people exaggerate in the other direction, I'm not sure if this is just a Hindu thing or a more global phenomenon, but there is a limit of what you can explain using Vedas, at least as far as science is concerned. (Metaphysics is something different, but this a completely different discussion.)
I agree. As an Indian it is just sad that there are people that do this. The effect of such articles is actually the loss of faith in the real findings of those ancient people caused by exaggerated claims presented in these articles.
There are so many actual great things and findings written in those texts, but it seems like people are either in the ship of "Everything is in Vedas" or "All this isn't worth researching". The lack of interest in researching these ancient texts caused by these 2 extreme sides is disheartening.
Per this source, in their own words Fibonacci credits Indian mathematician for his series. There is a great deal of disservice done to Eastern sources in the history of Science & Mathematics which is mostly because modern versions of these mostly skipped these places.
No doubt about it. There are A LOT of things that aren't credited correctly in the modern world, but these articles that I referenced aren't helping the situation at all. On the contrary, they're worsening the state all these topics are in with their exaggerated claims which make people question the integrity of the real work itself.
There should be thorough studies about Indology (at least in far greater number than there are right now) without all these click-baity articles associated with it.
The wiki article you link states 1) no direct evidence of a link or that their ideas were known beyond Kerala exists and 2) they might also have worked off ideas NY Islamic scholars. Really not sure what the specific claim or evidence you have is to give credit for 'many ideas' to this group.
Here's some evidence:
"The Baudhāyana Sulba Sūtra states the rule referred to today in most of the world as the Pythagorean Theorem."
"The Baudhāyana sūtras are a group of Vedic Sanskrit texts which cover dharma, daily ritual, mathematics, etc. They belong to the Taittiriya branch of the Krishna Yajurveda school and are among the earliest texts of the genre, perhaps compiled in the 8th to 6th centuries BCE." [1]
Note that the "perhaps compiled in the 8th to 6th centuries BCE", while still older than Pythagoras, comes from dating of Indian texts by European indologists. These indologists were funded by then-colonial governments in a then-colonized India with a view to advance the biblical worldview that the universe is about 6000 years old [2], which any reasonable thinker now knows is pure hogwash [3]. So Indian texts are, in fact, probably a lot older. I believe that a now-decolonized India needs to pay closer attention to its history.
Peter kingsley talks about this in both reality and a story that will pierce where you dive deep into the hidden connection between Pythagoreans and mystical traditions leading back to the mountains of Tibet.
The teachings of Falan Dafa also discusses this claiming that the earth and human civilizations has gone through cycles of destruction and renewal with the last period occuring in the time of the ice age where a small surviving group of mystics in Tibetan mountains managed to survive which matches up with the flood stories across cultures and recent scientific discoveries outlined by people like Graham Handcock.
I don't think you need big cycles of destruction and renewal like those guys claims.
Perfect example was what happened in Europe when the western roman empire disintegrated.
The whole knowledge transfer chain from master to student got disrupted.
So much so that the institutional knowledge base got eroded and people forgot how to do/maintain products.
Maybe because you a cluster of smaller empires no one could finance those prestige projects, without those big projects
the knowledge transfer chain will also be disrupted.
This me just thinking up loud and accepting the shit I learned 10~15 years ago and haven't used in a long time have simple been forgotten.
Human memory is extremely fragile, so the invention of paper must have been super important to keep those knowledge transfer chain going over the millennia.
There is a huge swath of Indian comments here, claiming vaguely that all these are Indian ideas without any evidence to back this up. Would be curious to see some actual evidence.
Not to forget that things can also be invented/discovered more than once independently.
Pythagoras had visited India as reported by
Diogenes Laertiu. Pythagoras theorem is actually Baudhayana Sutra and was widely used by Brahmins in India to build vedic hawan kunds.
Tomorrow, the newly formed "Estate of Pythagoras" corporation will begin selling shares in its upcoming lawsuits against "everybody who has ever used math" for IP infringement as appropriate in their many jurisdictions.
No one will really bet big on it until Disney takes them over.
The great man theory almost always fails. The life of Pythagoras is largely myth. This theorem was most likely known and proven well before him. This is old news.
