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 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...