Practice question · Sort into groups
Sort each claim by whether the Pythagoreans could maintain it after the discovery of incommensurables.
Groups: Still maintainable · Refuted by the discovery
- Every magnitude is expressible as a ratio of whole numbers
- Halving a string raises its pitch by an octave
- Mathematical structure underlies natural phenomena
- All things are number, in the sense of whole-number ratio
Hints
- The discovery concerns whether every magnitude has a whole-number ratio.
- The musical facts were never in question.
Show the answer
Still maintainable: Halving a string raises its pitch by an octave, Mathematical structure underlies natural phenomena
Refuted by the discovery: Every magnitude is expressible as a ratio of whole numbers, All things are number, in the sense of whole-number ratio
Why
The empirical discoveries survived; the metaphysical generalisation did not. Mathematical structure does underlie nature, but not every magnitude is a ratio of whole numbers, so the doctrine had to be weakened.
Practise Pythagoras and Early Pythagoreanism
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More questions on Pythagoras and Early Pythagoreanism
- A string of length 60 cm sounds a note. What length, in centimetres, sounds the octave above it?
- Why was the discovery of incommensurable magnitudes so damaging to Pythagoreanism?
- The Milesians identified the primary principle with a material substance, whereas the Pythagoreans identified…
- Arrange the steps of the classical proof that the square root of two is irrational.
- Which were features of the Pythagorean community?
- The Pythagoreans kept their mathematics strictly separate from their religious practice.
- Match each musical interval to its string-length ratio.