Examples of "a priori knowledge" in Kant

Question

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.

Answer 1

This is not a direct answer to the main question -you just need to Google for examples of a priori-, but, instead, an answer to your rejection of the possibility of it. This is how one day I came to discover the greatness of Kant and the huge error in reality that he's trying to solve. I think this will answer all your doubts.

TLDR. Instead of denying the possibility of a priori knowledge, that is, pure metaphysical knowledge, just consider the impossibility of having abstract ideas without some abstract framework of knowledge to start from. You cannot know what a point is without having knowledge of what space and time are (transcendental aesthetic a priori). You cannot group objects without subjectively creating the concept of a group (transcendental analytic a priori). Etc. If you came with answers for those, ask what was a priori needed for that. And if you come to a circular logic, you are wrong. Your history was not circular. It started somehow.

Long answer:

If you disagree with Kant, you just agree with empiricism (or rationalism). But it is just a matter of simple logic to conclude that both extremes are simply not possible. Specifically, the lack of a priori Knowledge.

You already know that Kant goal was to raise metaphysics to the level of a science. But in order for Kant to propose how to do it, Kant needed to define what is metaphysical knowledge. And his answer was that metaphysical knowledge was pure knowledge, that is, synthetic a priori. Hence the title of the book, Critique of Pure Reason. Such answer is consistent.

But if there is no a priori knowledge, then, there is no metaphysical knowledge, and that is simply impossible. How come, then, aesthetic knowledge is possible? Moral knowledge? Logical (and in consequence, mathematical) knowledge? Without metaphysical knowledge, science (the realm of empirical knowledge, and therefore, of empirical truth) would be the final truth, but we know that it is not the final truth, there are evidently infinite considerations that prove that science requires of metaphysical knowledge in order for it to be possible.

And more: the very thing you are doing, a philosophical question, would not be necessary. Because if there's no metaphysical knowledge, then there is no philosophy; in such case, philosophy would be reduced to pure traditional science. So, your question would not be necessary: it would be answered by science.

Moreover, quantum mechanics, relativity and even darwinisn evolution came to impose the inevitable and mandatory role of the subject -that is, of metaphysical knowledge-, in the process of knowledge development: science needs to have a formal understanding of metaphysical knowledge. Without metaphysical knowledge (which is, subjective knowledge), there is no objective knowledge. Instead of dismissing metaphysical knowledge, quantum mechanics shows the complete opposite extreme: we need to understand (and not only understand, but scientifically describe) the subject in order for the object to be understood. But for now, nobody took real efforts to raise metaphysics to the level of a science, except Kant.