Pythagorean Theorem and Its Essence and Properties

Let us look at two propositions first:

1. All the Bachelors are not married.
2. In a plane right triangle, the sum of the squares of the two right-angle sides is equal to the square of the third side. (Pythagorean theorem)

Both of these two propositions are a priori truths. Kant distinguished the differences between a prior and a posterior from two aspects. The first is the justification aspect, the a priori knowledge can be proved without sensory experience, but the knowledge of a posteriori need to have a sensory experience. For example, in the first proposition, all the bachelors are not married. I can know that this sentence is correct only by figuring out the meaning of this sentence. I do not really have to observe specific bachelors in my life to see if they are married. If there is another proposition that Donald Trump is the president of the United States. In order to prove whether this argument is correct, I need sensory experience. So 'Donald Trump is the President of the United States', this proposition is a posteriori. Another aspect is the origin of the knowledge. 

Simply put, a representation is from the mind itself, it is a priori; if it comes from sensory experience, then it is a posteriori. The geometric theorem is about a priori of spatial relations. In order to prove that the Pythagorean theorem is correct, we do not need to resort to sensory experience. We do not have to draw a right-angled triangle on the paper, weigh three to four on the right-angle side and five on the hypotenuse to prove that the Pythagorean theorem is right. The knowledge of the Pythagorean theorem is not derived from the specific sensory experience, as is the knowledge of 'Donald Trump is the President of the United States.' More importantly, we know that the Pythagorean theorem is knowledge of necessity rather than contingent. In any possible world, a plane right triangle that does not conform to the Pythagorean theorem can not exist.

If space is things in themselves, that is, the object itself has spatial properties that exist independently of the observer (we). Then we as observers can only perceive the existence of space through sensory experience. The geometric theorem of knowledge about spatial relations is a priori, which exists independently of sensory experience. But if our perception of space depends on sensory experience, we will not be able to acquire the knowledge of a priori of geometric theorems. However, we do have the geometric knowledge of the Pythagorean theorem.

Therefore, space is not thing in itself. Things that are independent of the observer do not have the property of extending in space; this spatial property comes from our mind. Only when space exists in our mind, we can acquire a priori geometry knowledge without relying on sensory experience.