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:
- The square root of two is a number.
- The square root of two is not rational.
- 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".