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Why do they always gotta throw the Pythagoreans under the bus? “ Two and a half millennia ago, the Pythagoreans held as a core belief that every number is the

Why do they always gotta throw the Pythagoreans under the bus? “ Two and a half millennia ago, the Pythagoreans held as a core belief that every number is the ratio of two whole numbers. They were shocked when a member of their school proved that the square root of 2 is not. Legend has it that as punishment, the offender was drowned.” Not only is this story ahistorical, it is obviously wrong if you have developed the Pythagorean theorem.

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DoctorOetker 2y ago
A person who is born rich can proclaim it is easy to be rich. A person who has been educated with intellectual richess, for example having been shown the proof of irrationality of sqrt(2), can similarily think this observation is obvious. The Pythagoreans were a semi-secretive cult. It is not because you know a theorem that you automatically know all future proofs that apply this theorem as a step. https://en.wikipedia.org/wiki/Hippasus https://en.wikipedia.org/wiki/Hippasus We don't know if it happened or didn't happen.
- dr_dshiv 2y ago
  Oh stop. If you have the theorem how would you not test it with sides = 1. Of course we know it didn’t happen. The ancient stories of Hippasus don’t have anything to do with this libel. As is conveniently mentioned in the Wikipedia article you posted. The Pythagoreans were absolutely incredible — and yet this is the only story people throw around. It’s just laziness.
  - AlotOfReading 2y ago
    They probably didn't have the modern form we use where plugging in different values like 1 is a natural and obvious thing to do. Regardless, they were a weird religious cult. They could have just regarded numbers that didn't produce rational numbers as unnatural and not something that was going to occur in the actual functioning of the world.
    - thaumasiotes 2y ago
      > They probably didn't have the modern form we use where plugging in different values like 1 is a natural and obvious thing to do. No, but the form they used was that they thought about lines in a plane that had no particular scale other than whatever you might assign to them. "Plugging in 1" for the sides of a right triangle in that model just means that you look at an isosceles right triangle. It will always be obvious.
  - DoctorOetker 2y ago
    I am certainly open to the idea that > Of course we know it didn’t happen. The ancient stories of Hippasus don’t have anything to do with this libel. As is conveniently mentioned in the Wikipedia article you posted. I reread it BEFORE posting my initial comment. Can you point me to where ON THE WIKIPEDIA PAGE this story was conclusively debunked?
    - dr_dshiv 2y ago
      It’s not about conclusively debunking. There is no evidence for it at all! They say he drowned for his impiety at publishing the dodecahedron (stated first by Iamblichus, 700 years later). “The discovery of irrational numbers is said to have been shocking to the Pythagoreans, and Hippasus is supposed to have drowned at sea, apparently as a punishment from the gods for divulging this and crediting it to himself instead of Pythagoras which was the norm in Pythagorean society. However, the few ancient sources who describe this story either do not mention Hippasus by name (e.g. Pappus)[4] or alternatively tell that Hippasus drowned because he revealed how to construct a dodecahedron inside a sphere.[5] The discovery of irrationality is not specifically ascribed to Hippasus by any ancient writer”
      - DoctorOetker 2y ago
        You seem to misunderstand the Wikipedia page. You took initially took issue with the following snippet from the HN article: > “ Two and a half millennia ago, the Pythagoreans held as a core belief that every number is the ratio of two whole numbers. They were shocked when a member of their school proved that the square root of 2 is not. Legend has it that as punishment, the offender was drowned.” So I reference a Wikipedia page stating ancient / quasicontemporary sources describe such an event, but those ancient sources didn't attribute it to Hippasus by name. In the centuries after the supposed event authors start attributing this to Hippasus, which is under historical contention, since we don't find this attribution in writings from the same era as the event. The HN article (just check your own snippet) describe the person as "a member of their school", which is in line with the hermeneutic interpretation of ancient texts (who said what and when). Perhaps some day newfound libraries will be uncovered, or burnt scrolls deciphered, possibly finding contemporary (in the same lifetime) attributions of this discovery of the irrationality of sqrt(2) to Hippapus. Basically: oldest sources already describe the events without naming the "offender" that was killed. Currently the oldest sources attributing this "offense" to Hippapus are lifetimes separated from the supposed events, casting doubt on the identity of the victim.
  - wat10000 2y ago
    So you test it with sides = 1. Result: hypotenuse is some number n where n*n = 2. So far so good. How does this lead you to the obvious conclusion that n is irrational? (I’m familiar with the standard proof that it is, but that’s not something that just naturally falls out of this.)
    - dr_dshiv 2y ago
      Because you can’t make it with a fraction.
      - DoctorOetker 2y ago
        Ah yes, thanks for reminding us that the Pythagoras proved the square root of 2 was irrational by ... circular reasoning. /s
      - wat10000 2y ago
        Stating the definition of irrationality is not a proof. What do you see as the obvious thing that goes from n*n = 2 to “you can’t make n with a fraction”?
Sniffnoy 2y ago
How is this story obviously wrong if you've developed the Pythagorean theorem? The Pythagorean theorem has nothing to do with rationality. If you think the irrationality of sqrt(2) follows easily from the Pythagorean theorem, then by all means, please demonstrate! (These days the irrationality of sqrt(2) is obvious due to unique prime factorization, but the ancient Greeks didn't have that concept!)
- dr_dshiv 2y ago
  It’s because you can’t create square root of two with a fraction. Pythagoreans were all about fractions (such as their musical tuning).
  - Sniffnoy 2y ago
    That's not an argument, that's a restatement of the problem; that's what being irrational means. The problem is to prove that you can't create sqrt(2) with a fraction. How, exactly, would this statement have been obvious to the Pythagoreans? Can you give me an argument for this that they would have found obvious? One that uses the Pythagorean theorem even, perhaps, since you brought that up? Remember, no using concepts they wouldn't have had like prime factorization!
dhosek 2y ago
So many circular arguments. The Pythagorean theorem tells you that √2 exists, but not that it’s irrational. As for the story, it’s apocryphal, not ahistorical, but even so, it was too good of a story not to tell my students when I taught Math for Liberal Arts Majors (the version I’d heard was that the proof was presented while the Pythagoreans were on a boat and they were so offended by the idea that √2 is irrational, they threw the guy off the boat. I would guess that of all the things I said in lectures for that class, this is the one that my students would be most likely to remember).

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DoctorOetker 2y ago

A person who is born rich can proclaim it is easy to be rich. A person who has been educated with intellectual richess, for example having been shown the proof of irrationality of sqrt(2), can similarily think this observation is obvious. The Pythagoreans were a semi-secretive cult. It is not because you know a theorem that you automatically know all future proofs that apply this theorem as a step. https://en.wikipedia.org/wiki/Hippasus https://en.wikipedia.org/wiki/Hippasus We don't know if it happened or didn't happen.

dr_dshiv 2y ago
Oh stop. If you have the theorem how would you not test it with sides = 1. Of course we know it didn’t happen. The ancient stories of Hippasus don’t have anything to do with this libel. As is conveniently mentioned in the Wikipedia article you posted. The Pythagoreans were absolutely incredible — and yet this is the only story people throw around. It’s just laziness.
- AlotOfReading 2y ago
  They probably didn't have the modern form we use where plugging in different values like 1 is a natural and obvious thing to do. Regardless, they were a weird religious cult. They could have just regarded numbers that didn't produce rational numbers as unnatural and not something that was going to occur in the actual functioning of the world.
  - thaumasiotes 2y ago
    > They probably didn't have the modern form we use where plugging in different values like 1 is a natural and obvious thing to do. No, but the form they used was that they thought about lines in a plane that had no particular scale other than whatever you might assign to them. "Plugging in 1" for the sides of a right triangle in that model just means that you look at an isosceles right triangle. It will always be obvious.
- DoctorOetker 2y ago
  I am certainly open to the idea that > Of course we know it didn’t happen. The ancient stories of Hippasus don’t have anything to do with this libel. As is conveniently mentioned in the Wikipedia article you posted. I reread it BEFORE posting my initial comment. Can you point me to where ON THE WIKIPEDIA PAGE this story was conclusively debunked?
  - dr_dshiv 2y ago
    It’s not about conclusively debunking. There is no evidence for it at all! They say he drowned for his impiety at publishing the dodecahedron (stated first by Iamblichus, 700 years later). “The discovery of irrational numbers is said to have been shocking to the Pythagoreans, and Hippasus is supposed to have drowned at sea, apparently as a punishment from the gods for divulging this and crediting it to himself instead of Pythagoras which was the norm in Pythagorean society. However, the few ancient sources who describe this story either do not mention Hippasus by name (e.g. Pappus)[4] or alternatively tell that Hippasus drowned because he revealed how to construct a dodecahedron inside a sphere.[5] The discovery of irrationality is not specifically ascribed to Hippasus by any ancient writer”
    - DoctorOetker 2y ago
      You seem to misunderstand the Wikipedia page. You took initially took issue with the following snippet from the HN article: > “ Two and a half millennia ago, the Pythagoreans held as a core belief that every number is the ratio of two whole numbers. They were shocked when a member of their school proved that the square root of 2 is not. Legend has it that as punishment, the offender was drowned.” So I reference a Wikipedia page stating ancient / quasicontemporary sources describe such an event, but those ancient sources didn't attribute it to Hippasus by name. In the centuries after the supposed event authors start attributing this to Hippasus, which is under historical contention, since we don't find this attribution in writings from the same era as the event. The HN article (just check your own snippet) describe the person as "a member of their school", which is in line with the hermeneutic interpretation of ancient texts (who said what and when). Perhaps some day newfound libraries will be uncovered, or burnt scrolls deciphered, possibly finding contemporary (in the same lifetime) attributions of this discovery of the irrationality of sqrt(2) to Hippapus. Basically: oldest sources already describe the events without naming the "offender" that was killed. Currently the oldest sources attributing this "offense" to Hippapus are lifetimes separated from the supposed events, casting doubt on the identity of the victim.
- wat10000 2y ago
  So you test it with sides = 1. Result: hypotenuse is some number n where n*n = 2. So far so good. How does this lead you to the obvious conclusion that n is irrational? (I’m familiar with the standard proof that it is, but that’s not something that just naturally falls out of this.)
  - dr_dshiv 2y ago
    Because you can’t make it with a fraction.
    - DoctorOetker 2y ago
      Ah yes, thanks for reminding us that the Pythagoras proved the square root of 2 was irrational by ... circular reasoning. /s
    - wat10000 2y ago
      Stating the definition of irrationality is not a proof. What do you see as the obvious thing that goes from n*n = 2 to “you can’t make n with a fraction”?

Sniffnoy 2y ago

How is this story obviously wrong if you've developed the Pythagorean theorem? The Pythagorean theorem has nothing to do with rationality. If you think the irrationality of sqrt(2) follows easily from the Pythagorean theorem, then by all means, please demonstrate! (These days the irrationality of sqrt(2) is obvious due to unique prime factorization, but the ancient Greeks didn't have that concept!)

dr_dshiv 2y ago
It’s because you can’t create square root of two with a fraction. Pythagoreans were all about fractions (such as their musical tuning).
- Sniffnoy 2y ago
  That's not an argument, that's a restatement of the problem; that's what being irrational means. The problem is to prove that you can't create sqrt(2) with a fraction. How, exactly, would this statement have been obvious to the Pythagoreans? Can you give me an argument for this that they would have found obvious? One that uses the Pythagorean theorem even, perhaps, since you brought that up? Remember, no using concepts they wouldn't have had like prime factorization!

dhosek 2y ago

So many circular arguments. The Pythagorean theorem tells you that √2 exists, but not that it’s irrational. As for the story, it’s apocryphal, not ahistorical, but even so, it was too good of a story not to tell my students when I taught Math for Liberal Arts Majors (the version I’d heard was that the proof was presented while the Pythagoreans were on a boat and they were so offended by the idea that √2 is irrational, they threw the guy off the boat. I would guess that of all the things I said in lectures for that class, this is the one that my students would be most likely to remember).