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It's funny that everybody remembers Pythagoras for the right-triangle theorem, but that wasn't what made him important at the time. The right-triangle square eq

It's funny that everybody remembers Pythagoras for the right-triangle theorem, but that wasn't what made him important at the time. The right-triangle square equality had been known empirically for centuries. What he proved that was so completely earth-shaking is that for some right triangles A, B, C there is no rational Q for which QA = C. That was what was so important about his proof.

moioci 2y ago
That would seem to imply that for A =/= 0, C/A is irrational. This seems counterintuitive.
- asolove 2y ago
  What is counter-intuitive? In a triangle with sides A=1, B=1, then C=root(2), so C/A is irrational. That's what was so impactful about the discovery. 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.
  - Ekaros 2y ago
    For some reason that feels so weird that it would be that late "discovery"... Once you define a square(sides same length) the length of diagonal is one of the first questions. And this being very weird number is something I believe someone must have thought about long before that point of time.
    - asolove 2y ago
      These are more or less the first people to think about geometry rigorously as an abstract system. Anyone previous would have just pointed to the hypotenuse and said “it’s that length right there” and not asked a further question.
      - dontlikeyoueith 2y ago
        > more or less the first people to think about geometry rigorously as an abstract system. The first people whose thinking was preserved until the present. Which is still noteworthy, but a different thing.
    - bandrami 2y ago
      OK but at the time it was literally an open research question: given two reals A, B is there always a rational Q such that QA=B? Number theory as such was still in its infancy but I think it's impressive that this was exactly the right question to ask and they understood how important it was.
    - InitialLastName 2y ago
      A lot of early math was done using geometry tools rather than symbolic representation. 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.
  - moioci 2y ago
    My bad. I was thinking A, B, and C were integers.
- hinkley 2y ago
  The 3/4/5 triangle is rational. The unit right triangle is not. You’ve dropped the “some” from the parent.
- bandrami 2y ago
  For "most" right triangles, yes, C/A is irrational. In fact the triangles for which C/A is rational are vanishingly rare (though Pythagoras proved many important things about them[1]) 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. 1: https://en.wikipedia.org/wiki/Pythagorean_triple https://en.wikipedia.org/wiki/Pythagorean_triple
- deleted 2y ago
  [deleted]
seanhunter 2y ago
I'm not sure you've got that right. We know almost nothing about the historical Pythagoras and none of his actual writings survived. We only know of him through the Pythagorean brotherhood and other people (eg Plato) who he influenced. 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_and_traditions_of_Babylonian_mathematics_Plimpton_322_Pythagorean_Triples_and_the_Babylonian_Triangle_Parameter_Equations https://www.researchgate.net/publication/222892801_Methods_a...
- lordnacho 2y ago
  Stigler's Law of Eponymy
  - pfdietz 2y ago
    Let me guess: Stigler didn't invent that.
    - lordnacho 2y ago
      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.
- bandrami 2y ago
  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.
wolfi1 2y ago
I thought the Pythagoreans threw one of their members over board for proving the root of 2 to be irrational
- ants_everywhere 2y ago
  This is a myth, it's covered in https://en.wikipedia.org/wiki/Hippasus https://en.wikipedia.org/wiki/Hippasus
- bandrami 2y ago
  There's multiple stories about this; one of the better-attested is that for a time the brotherhood swore each other to secrecy (with threats of drowning) about it because it ran against the Parmenedian epistemology of the time.

moioci 2y ago

That would seem to imply that for A =/= 0, C/A is irrational. This seems counterintuitive.

asolove 2y ago
What is counter-intuitive? In a triangle with sides A=1, B=1, then C=root(2), so C/A is irrational. That's what was so impactful about the discovery. 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.
- Ekaros 2y ago
  For some reason that feels so weird that it would be that late "discovery"... Once you define a square(sides same length) the length of diagonal is one of the first questions. And this being very weird number is something I believe someone must have thought about long before that point of time.
  - asolove 2y ago
    These are more or less the first people to think about geometry rigorously as an abstract system. Anyone previous would have just pointed to the hypotenuse and said “it’s that length right there” and not asked a further question.
    - dontlikeyoueith 2y ago
      > more or less the first people to think about geometry rigorously as an abstract system. The first people whose thinking was preserved until the present. Which is still noteworthy, but a different thing.
  - bandrami 2y ago
    OK but at the time it was literally an open research question: given two reals A, B is there always a rational Q such that QA=B? Number theory as such was still in its infancy but I think it's impressive that this was exactly the right question to ask and they understood how important it was.
  - InitialLastName 2y ago
    A lot of early math was done using geometry tools rather than symbolic representation. 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.
- moioci 2y ago
  My bad. I was thinking A, B, and C were integers.
hinkley 2y ago
The 3/4/5 triangle is rational. The unit right triangle is not. You’ve dropped the “some” from the parent.
bandrami 2y ago
For "most" right triangles, yes, C/A is irrational. In fact the triangles for which C/A is rational are vanishingly rare (though Pythagoras proved many important things about them[1]) 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. 1: https://en.wikipedia.org/wiki/Pythagorean_triple https://en.wikipedia.org/wiki/Pythagorean_triple
deleted 2y ago
[deleted]

seanhunter 2y ago

I'm not sure you've got that right. We know almost nothing about the historical Pythagoras and none of his actual writings survived. We only know of him through the Pythagorean brotherhood and other people (eg Plato) who he influenced. 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_and_traditions_of_Babylonian_mathematics_Plimpton_322_Pythagorean_Triples_and_the_Babylonian_Triangle_Parameter_Equations https://www.researchgate.net/publication/222892801_Methods_a...

lordnacho 2y ago
Stigler's Law of Eponymy
- pfdietz 2y ago
  Let me guess: Stigler didn't invent that.
  - lordnacho 2y ago
    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.
bandrami 2y ago
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.

wolfi1 2y ago

I thought the Pythagoreans threw one of their members over board for proving the root of 2 to be irrational

ants_everywhere 2y ago
This is a myth, it's covered in https://en.wikipedia.org/wiki/Hippasus https://en.wikipedia.org/wiki/Hippasus
bandrami 2y ago
There's multiple stories about this; one of the better-attested is that for a time the brotherhood swore each other to secrecy (with threats of drowning) about it because it ran against the Parmenedian epistemology of the time.