Pythagoras Is Still Around: A Transcendent Idea?

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How do you know that a²+b²=c² is true?

Well, you must provide a proof. Many people of all stripes (just click on that link) have provided proofs for the theorem. From schoolchildren to serious mathematicians. Even president Garfield provided one.

But a philosophical question also arises. How, when you begin think about it, do you know that the completion of a proof is a description of reality itself? Do you try and apply (in this case a geometrical proof) to known experimentation and observation? Do you come up with a series of deep arguments that try and explain why it might be so, giving up the math to become a philosopher?

One explanation is, much like a wheel in the sky, the understanding (and I’m using this term in the Kantian sense) always seeks to understand something from no place.  So once you understand an idea, you can’t claim it’s ultimate origin, and you can’t necessarily prove its truth either without asking another set of questions.  Kant led himself to extremely complex and deep arguments about the matter.

So, Pythagoras was one of the first ancient Greeks to imply that the way we come to know truth may NOT directly related to our experience. We must, instead, follow the math and build our knowledge upon it.  

…but is it a transcendent idea?  Does our thinking about math somehow transcend all of our experience and our senses…and if so…how can we know it gives us knowledge of the world beyond what we know?

As a side note, Bertrand Russell, incidentally, supposed much of Platonism to be based on the thought of Pythagoras, and that some mysticism of the Pythagorean followers, and some of his rather strange beliefs still frame our thinking…

Addition:  It should be noted that in trying to answer the question of whether or not mathematics is transcendent (knowledge not born of experience, or in his terminology, a priori synthetic knowledge), Kant redrew the limits of our knowledge in a very important way…but pretty much left the issue there…for now.