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Pythagoras didn't use modern algebraic notation, but I don't think it's correct to conclude from this that what he was doing wasn't algebra. He reasoned about r

Pythagoras didn't use modern algebraic notation, but I don't think it's correct to conclude from this that what he was doing wasn't algebra. He reasoned about relationships between abstract quantities in ways that we today would write down in algebraic notation. Does a change of notation mean that the reasoning itself is no longer algebra?

mathgenius 6y ago
The ancient Greeks did stuff by example, not using algebra. They said stuff like, "take 22 and multiply by 5..." with actual numbers involved, then at the end they say "and that's how it works in general". So it's up to the reader to generalize from the specific. Algebra works in the opposite direction, where the reader is obliged to specify from the generic. I happen to much prefer the ancient Greek way of doing mathematics, but it seems to have fallen out of fashion.
- tom_mellior 6y ago
  "The ancient Greeks" are certainly not a homogeneous group. Here's Euclid's proof of the Pythagorean theorem: https://mathcs.clarku.edu/~djoyce/java/elements/bookI/propI47.html https://mathcs.clarku.edu/~djoyce/java/elements/bookI/propI4... There are no concrete quantities in sight. I don't think that they were removed by the translator. Much more plausibly the original reasoning was symbolic in the way indicated here. Though again, possibly using different notation.
  - mathgenius 6y ago
    Sorry, I think I was referring specifically to the dude that did integer polynomial equations. His name starts with P i think.
    - tom_mellior 6y ago
      Ah, you might be thinking of https://en.wikipedia.org/wiki/Diophantine_equation https://en.wikipedia.org/wiki/Diophantine_equation. Sure enough, https://en.wikipedia.org/wiki/Diophantus https://en.wikipedia.org/wiki/Diophantus says: "It should be mentioned here that Diophantus never used general methods in his solutions. Hermann Hankel, renowned German mathematician made the following remark regarding Diophantus. “Our author (Diophantos) not the slightest trace of a general, comprehensive method is discernible; each problem calls for some special method which refuses to work even for the most closely related problems. For this reason it is difficult for the modern scholar to solve the 101st problem even after having studied 100 of Diophantos’s solutions”." I didn't know this, thanks.

mathgenius 6y ago

The ancient Greeks did stuff by example, not using algebra. They said stuff like, "take 22 and multiply by 5..." with actual numbers involved, then at the end they say "and that's how it works in general". So it's up to the reader to generalize from the specific. Algebra works in the opposite direction, where the reader is obliged to specify from the generic. I happen to much prefer the ancient Greek way of doing mathematics, but it seems to have fallen out of fashion.

tom_mellior 6y ago
"The ancient Greeks" are certainly not a homogeneous group. Here's Euclid's proof of the Pythagorean theorem: https://mathcs.clarku.edu/~djoyce/java/elements/bookI/propI47.html https://mathcs.clarku.edu/~djoyce/java/elements/bookI/propI4... There are no concrete quantities in sight. I don't think that they were removed by the translator. Much more plausibly the original reasoning was symbolic in the way indicated here. Though again, possibly using different notation.
- mathgenius 6y ago
  Sorry, I think I was referring specifically to the dude that did integer polynomial equations. His name starts with P i think.
  - tom_mellior 6y ago
    Ah, you might be thinking of https://en.wikipedia.org/wiki/Diophantine_equation https://en.wikipedia.org/wiki/Diophantine_equation. Sure enough, https://en.wikipedia.org/wiki/Diophantus https://en.wikipedia.org/wiki/Diophantus says: "It should be mentioned here that Diophantus never used general methods in his solutions. Hermann Hankel, renowned German mathematician made the following remark regarding Diophantus. “Our author (Diophantos) not the slightest trace of a general, comprehensive method is discernible; each problem calls for some special method which refuses to work even for the most closely related problems. For this reason it is difficult for the modern scholar to solve the 101st problem even after having studied 100 of Diophantos’s solutions”." I didn't know this, thanks.