What are some good examples of a priori knowledge that must exist independent of experience and transcend it? How can we be certain that such is indeed a priori?
The example Kant mentions in the Critique of Pure Reason is that Mathematical knowledge transcends experience, and also talks of the trustworthiness of mathematical knowledge.
Here, I disagree. Mathematical knowledge is not truly a priori.
Were it truly a priori, the discovery of mathematical tenets could have gone differently. Caveman without the knowledge of counting might have come up with Real and Imaginary numbers.
But the discovery of rational numbers came at a time when the Integers were well known and found to be inadequate for certain human activity (dividing larger quantities into portions). The discovery of reals came after the rationals were known closely enough to find their inadequacies.
So it is experience that prompted the discovery of more knowledge which he wants to consider a priori.
Further, it cannot be entirely trustworthy either.
Were it entirely trustworthy, what was once established as mathematical fact ought to remain that way. But since mathematics is a work in progress with new discoveries, sometimes contradicting old ones, this claim is not valid.
And how can knowledge acquired by the analysis of objects and concepts (which stem from experience) be considered truly a priori? Because it cannot obviously be proven to be a priori, because the objects of experience cannot ever be truly removed from the understanding and the logic and rationale it applies. Since the understanding of every individual has already been exposed to and tainted by experience.