The documentatary evidence is fragmentary but there is significant evidence that a 5th-century BC Greek mathematician proved the incommensurability of the side of a square with its diagonal (it was apparently trivially known a century later since it appears in Plato's "Meno"). Whether that person was named Pythagoras or Hippasus or something else is really neither here nor there since he was pretty clearly part of the Pythagorean tradition that got associated with one name.

The point in any case is that incommensurability as a concept was not widely accepted at the beginning of the 5th century BC and was widely accepted at the end, and the name "Pythagoras" gets attached to the mathematicians who discovered that.

But like for that matter Plato's name wasn't "Plato"; that was a nickname his wrestling coach gave him.

Speaking of Plato’s wrestling coach…

Pythagoras was also a successful wrestling coach. In fact, he coached the most winningest Olympic athlete of all time: Milo of Croton. Milo won 5 consecutive Olympics over a 20 year period.

Pythagoras himself was thrown out of the boys Olympics at age 16 for being too effeminate (long hair), but then entered the men’s Olympics and won. Supposedly, he introduced some new kind of martial arts technique.

This is documented in Thibodeau, 2019 “The Chronology of the Early Greek Natural Philosophers.” Happy to share the refs. Ok, back to maths…

There is a PDF of your citation available at https://www.researchgate.net/publication/335965217_THE_CHRON...

I searched for "long hair" and found only this:

>The man with long hair at Samos’: They say there was a Samian boxer with long hair who went to Olympia and won after being mocked by his opponents for looking like a woman; he became proverbial. Eratosthenes says that Pythagoras of Samos won with long hair during the 48 th Olympiad; Duris represents this as Pythagoras being excluded, challenging the men, and beating many of them.

Like many stories concerning Pythagoras, I wonder if this was some local fable onto which his name later became plastered.

11. Eratosthenes, Olympic Victors 3rd century via Favorinus, Varied History, via Diogenes Laertius, Lives 8.47 “Eratosthenes (according to what Favorinus reports in book eight of his Varied History), said this man [sc. Pythagoras] was the first to box using technique, in the 48th Olympiad, letting his hair grow long and wearing a purple robe; after being excluded from the boys’ games and jeered at, he immediately joined the mens’, and won.” Olympiad 48: 588 to 584 BCE cf. Eusebius, Chronography, p. 93 Karst

“was the first to box using technique”

Clearly Pythagoras would have sick martial arts moves, if they’d ever make a movie about him.

According to wikipedia, at least, that's a different Pythagoras of Samos https://en.wikipedia.org/wiki/Pythagoras_(boxer)

The reference I gave gives a good reason why it isn’t a different Pythagoras… but, always hard to know for sure…

But I tend to think that Pythagoras had an even more incredible life than the legends.

Thank you so much!

Like my teacher always says: "If at the end of the class, you haven't learned something new, come see me"

I do find it amusing that most if not all of the famous classical philosophers would stand up just fine to the "post physique" meme.

Wouldn't 5 consecutive Olympics be over a 16 year period?

Slightly off topic, but one of the coolest (to me) archaeological discoveries is from ~1700 BC, showing a calculation of sqrt(2) to the equivalent of six decimal places: https://en.m.wikipedia.org/wiki/YBC_7289

Stigler's Law of Eponymy

Let me guess: Stigler didn't invent that.

Correct! He heard it from a friend and decided it needed his name on it, that way it is an example of the law itself.

I haven't read the paper you posted yet, but just to clear up a common confusion, Pythagorean triples are not the Pythagorean theorem.

The theorem is a logical statement about all right triangles, and it has a proof that the statement holds. Pythagorean triples are specific instantiations of the relation for some known triangles and probably would have served as evidence that the statement was even provable.

Historically we probably had triples long before we had a proof of the theorem, just like many of the theorems proved by Euclid were probably already known as rules of thumb.

Compare with an open problem today, like the Riemann hypothesis.

I mean, it's correct that Pythagorean triples are not the same thing as the Pythagorean theorem, but "Pythagoras" (or at least the Pythagorean school) proved things about modular arithmetic with the triples. The geometric interpretation was they were the measures of the sides of right triangles and that no other right triangles were "measurable" (that is, they could not be expressed as having 3 rational side lengths).

This is exactly right!

Oh, how I wish there was a book on the history of science and math. Like how they went from the Four Humours or Phlogiston and Spontaneous Generation and Luminferous Ether theories to what they had later. Like how scientists all thought the earth was 100 million years old for a couple centuries until the discovery of radioactivity.

I want a book that would speak about how people made fun of Ignaz Semmelweis for washing hands in hospitals, until Pasteur in France and John Snow in England showed evidence for the germ theory of disease. How people used leeches and bloodletting, and when / why they stopped. (And maybe anecdotes like How Washington Roebling building the Brooklyn Bridge died from a gangrene because he thought pouring water over his wound was enough, and his son finished it)

I want a book that would explain the experiments that led to the theories, like Michelson-Morley that challenged the Lumeniferous Ether model. Or how people first discovered X-rays and didn’t know what to make of them.

How, indeed, did people prove to others that atoms existed? I don’t mean Democritus’ theories 2500 years ago, I mean what made people convinced the world was made from atoms?

And then the experiments that led to the standard model, how was it developed? The word Quark, where it came from, the reactions of scientists to Quantum theory etc.

Our science comes shrinkwrapped, showing only the end result, not the history of thought and the places where (eg Andalusia in the 1100s or China in 20 AD). To me it is very interesting when people weren’t sure yet but got the right result based on circumstantial reasoning, didn’t have high-powered electron microscopes and still somehow realized about atoms and molecules (Gregor Mendel etc). Or how Darwin and Mendeleev developed theories genetics, why they so thoroughly discounted Lamarckian evolution and were they correct etc.

I want a book that discusses the history including the ERRONEOUS theories, the time frames, the experiments that challenged prevailing theories, the controversies, and what made people find the correct theories.

And maybe the people (eg Newton avoided women, and studied Hebrew to reconstruct Biblical history, Michael Faraday was a devout Christian etc.)

Is there such a book? Can you guys recommend ??

I may be falling into a trap here by answering this, but you might enjoy The Structure of Scientific Revolutions by Thomas Kuhn.

It is not that we do not teach this anywhere at all, but we certainly do not teach this in the introductory science classes, where the goal is to provide an accessible path to the already available "shrink wrapped" conceptual framework.

That is hard enough already. Showing the history of science properly is easily a hundred-fold more monumental of a task. It is a subject which is studied, and taught -- but for the specialists in the history and philosophy of science. There are many books dedicated both to cursory high level overviews of the history of science, and to some specific episodes in this history, going into nuances of what happened and how it happened.

For example, for the history of mathematics, one can consult "The History of Mathematics: An Introduction" https://www.amazon.com/History-Mathematics-Burton-Professor-...

Here is a much more specialized book, showing how much trouble relatively modern physicists had in correctly conceptualizing "heat" and "temperature": "The tragicomical history of thermodynamics, 1822–1854" https://scholar.google.com/scholar?q=history+of+thermodynami...

Beware! The history of evolution of ideas in science and technology is a vast, vast and an extremely fascinating field, with many dangerous rabbit holes!

Not a book but there's an informative history of physics channel "Kathy Loves Physics & History".

Alpha Beta Gamma: History of Radioactivity https://www.youtube.com/watch?v=-G_CtD2UnAc

First Law of Thermodynamics: History of the Concept of Energy https://www.youtube.com/watch?v=a9c7u-FM-Wc