While the Pythagoreans did come up with many things (eg the first rigorously documented scientific experiments) my understanding was that it is known for certain that they definitely were not responsible for Pythagoras’ theorem (or this rationality corolary you're talking about), and that the earliest formulation of it that is currently known is from Babylon where it's documented to do with sizing of farm plots[1] about a thousand years before Pythagoras. The proof of the irrationality of the square root of two was so terrifying to the Pythagoreans that the legend has it they threw the dude who produced it off a boat into the sea to drown because it was such a heresy (although I believe that is also known not to be true and the pythagoreans knew that root 2 was irrational).
In that sense it's like Euler's number (first documented by Napier), Lambert's W function (Invented by Euler to solve a family of equations Lambert couldn't solve), Lagrange's notation for calculus (used by Lagrange yes but first also invented by Euler) etc etc.
[1] https://www.researchgate.net/publication/222892801_Methods_a...
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.
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…
Like my teacher always says: "If at the end of the class, you haven't learned something new, come see me"
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.
But I tend to think that Pythagoras had an even more incredible life than the legends.
Clearly Pythagoras would have sick martial arts moves, if they’d ever make a movie about him.
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.
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 ??
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!
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
Imagine not knowing about irrational numbers. You assume all numbers are just integers and fractional ratios between integers. It would be weird (terrifying?) that something as simple as a right triangle would require a whole category of numbers you can't express.
The first people whose thinking was preserved until the present.
Which is still noteworthy, but a different thing.
If you are drawing a diagram for a building and you need a distance equal to the diagonal of a square, you set your compass to the two points and use that distance. No need to determine that it can't be represented by a comfortable multiple of the sides.
But before Pythagoras, it was still an open question if for any two reals A, B there might be a rational Q such that QA = B. Whereas we now know that for "most" reals there is no such Q, thanks to Pythagoras.