7 ms·

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.

sn41 5y ago
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. [1] https://en.wikipedia.org/wiki/Pythagorean_theorem#History https://en.wikipedia.org/wiki/Pythagorean_theorem#History
- ttyprintk 5y ago
  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.
  - ashtonkem 5y ago
    Euclid got Euclidean geometry, and that’s not nothing.
  - Galanwe 5y ago
    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 This is what I remember from courses 15 years ago, I may be wrong or outdated on the subject ;)
    - ttyprintk 5y ago
      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.
      - thaumasiotes 5y ago
        > 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.)
        ttyprintk 5y ago
        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.
        thaumasiotes 5y ago
        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".
  - ardit33 5y ago
    Euclidian plane.... term is used all the time. Euclidian Geometry as well. His ideas have a large part in Math and Geometry
  - thaumasiotes 5y ago
    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.)
- jules 5y ago
  How would one discover the Pythagorean theorem without having a proof?
  - tim333 5y ago
    You could draw a right angle triangle and notice that the squares match without being able to prove that.
    - jules 5y ago
      How would you notice that without proof? Seeing that two squares sum to the same area as a third isn't easy.
ttyprintk 5y ago
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.
- babesh 5y ago
  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...
  - babesh 5y ago
    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.
throw13579 5y ago
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] [1] https://en.wikipedia.org/wiki/Baudhayana_sutras#Pythagorean_theorem https://en.wikipedia.org/wiki/Baudhayana_sutras#Pythagorean_... [2] http://jwilson.coe.uga.edu/EMT668/EMT668.Student.Folders/HeadAngela/essay1/Pythagorean.html http://jwilson.coe.uga.edu/EMT668/EMT668.Student.Folders/Hea... [3] https://en.wikipedia.org/wiki/Srinivasa_Ramanujan https://en.wikipedia.org/wiki/Srinivasa_Ramanujan
- throw13579 5y ago
  Being downvoted. Let me correct myself, HN: all discoveries have been made by white people. These discoveries have been made in the last 6000 years, the only time that this universe has existed (according to the white people's good book).

sn41 5y ago

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. [1] https://en.wikipedia.org/wiki/Pythagorean_theorem#History https://en.wikipedia.org/wiki/Pythagorean_theorem#History

ttyprintk 5y ago
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.
- ashtonkem 5y ago
  Euclid got Euclidean geometry, and that’s not nothing.
- Galanwe 5y ago
  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 ;)
  - ttyprintk 5y ago
    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.
    - thaumasiotes 5y ago
      > 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.)
      - ttyprintk 5y ago
        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.
        thaumasiotes 5y ago
        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".
- ardit33 5y ago
  Euclidian plane.... term is used all the time. Euclidian Geometry as well. His ideas have a large part in Math and Geometry
- thaumasiotes 5y ago
  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.)
jules 5y ago
How would one discover the Pythagorean theorem without having a proof?
- tim333 5y ago
  You could draw a right angle triangle and notice that the squares match without being able to prove that.
  - jules 5y ago
    How would you notice that without proof? Seeing that two squares sum to the same area as a third isn't easy.

ttyprintk 5y ago

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.

babesh 5y ago
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...
- babesh 5y ago
  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.

throw13579 5y ago

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] [1] https://en.wikipedia.org/wiki/Baudhayana_sutras#Pythagorean_theorem https://en.wikipedia.org/wiki/Baudhayana_sutras#Pythagorean_... [2] http://jwilson.coe.uga.edu/EMT668/EMT668.Student.Folders/HeadAngela/essay1/Pythagorean.html http://jwilson.coe.uga.edu/EMT668/EMT668.Student.Folders/Hea... [3] https://en.wikipedia.org/wiki/Srinivasa_Ramanujan https://en.wikipedia.org/wiki/Srinivasa_Ramanujan

throw13579 5y ago
Being downvoted. Let me correct myself, HN: all discoveries have been made by white people. These discoveries have been made in the last 6000 years, the only time that this universe has existed (according to the white people's good book).