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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.

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    Weren't the reals technically "discovered" (via the subset of the reals consisting in irrationals) as soon as the square root of 2 was known to be irrational? And how could prehistoric humans have discovered imaginary numbers without knowing the natural (counting) numbers, since the imaginary unit is mediated by the even root function on -1, which itself hearkens back to 1? The even root function itself hyperoperationally presupposes the number 3 for a negative operator (3 being the operator's index in the hyperoperator sequence). A priori or not, mathematics has a lot of integrity. Commented Apr 24, 2022 at 6:40
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    Kant does say that all knowledge begins in experience, but his deeper definition of apriority is "proactive" (see the later Groundwork sections) instead of "passive/reactive," like the senses. So counting, for example, is not something that happens to us, but something that we do. Commented Apr 24, 2022 at 15:00
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    From the start of the Intro. to the first Critique: "That all our knowledge begins with experience there can be no doubt. For how is it possible that the faculty of cognition should be awakened into exercise otherwise than by means of objects which affect our senses, and partly of themselves produce representations, partly rouse our powers of understanding into activity ... to convert the raw material of our sensuous impressions into a knowledge of objects, which is called experience? In respect of time, therefore, no knowledge of ours is antecedent to experience, but begins with it." Commented Apr 24, 2022 at 16:32
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    The Groundwork section is called "Of the Interest attaching to the Ideas of Morality." Commented Apr 24, 2022 at 16:34
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    Yes. Thank you.
    – ahron
    Commented Apr 26, 2022 at 3:37

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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.

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  • If we start with pure understanding / awareness and naught else - there is little in the way of abstract "knowledge" that this understanding is capable of. Because "knowledge" is referred in the context of "knowledge of/about ____" - hence knowledge needs an object. And objects arise only via experience. Like you said - to group objects, one needs to know groups and objects. But once one has known it, the knowledge can be applied as a priori for subsequent applications. So the first time it is a posteriori knowledge and then it is a priori for new applications. Is there a distinction for this?
    – ahron
    Commented Apr 24, 2022 at 9:54
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    Peripathetic axiom: "Nothing is in the intellect that was not first in the senses". Leibniz answer: "Except the mind itself".
    – RodolfoAP
    Commented Apr 24, 2022 at 16:45
  • What according to the perpatetic school/axiom, would constitute knowledge a priori? Are there any comments of Kant on the Aristotlean school of thought?
    – ahron
    Commented Apr 26, 2022 at 3:36

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