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I am not sure if this is the right stackexchange for this question. However, I'm wondering about the following thing:

We know n+ow that there are rational and irrational numbers. Pythagoras however, believed that

Pythagorean conjecture: "all numbers are rational".

I am thinking about how this conjecture can be deductively proven to be false within the Pythagorean system. For that we need a definition of "number", so that we can say:

  1. The square root of two is a number.
  2. The square root of two is not rational.
  3. Therefore not all numbers are rational.

My question is essentially about how we can deductively show that (1.) is correct, based on the Pythagorean definition. That is why I am wondering: What is the definition of "number" for the Pythagoreans on the basis of which statement (1) is true.

Nevertheless when I look for a definition online, I only read vague statements like "numbers are the source of all things".

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Vihart from youtube has a video explaining this. The answer to your question is answered at 4:00 to 6:57. Here it is below:

https://www.youtube.com/watch?v=X1E7I7_r3Cw

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