Practice question · Multiple choice
Why was the discovery of incommensurable magnitudes so damaging to Pythagoreanism?
Hints
- The doctrine was that all things are number, meaning whole-number ratio.
- The diagonal of a unit square cannot be written as any such ratio.
Show the answer
A. It showed that some geometrical lengths cannot be expressed as any whole-number ratio
Why
The diagonal of a unit square is , and no ratio of whole numbers equals it. Since "all things are number" meant whole-number ratio, geometry itself contained a counterexample, produced by the school’s own methods.
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