Careful what you ask for. I don’t have a problem with the Prolegomena because I don’t consider it the relevant text for the current discussion.
300 years after the fact, all there is, is opinion. My opinion is, most everybody, in concentrating on this or that, overlooks transcendental philosophy as a whole.
I can explain til I’m blue inna face, but there remains a serious problem: there’s no need for mathematical judgements or their synthetic
a priori classification, when I’ve known all about them since I was knee-high to a grasshopper, thanks to my 1st grade teacher. It’s extremely difficult to comprehend the reason for them when rote instruction has removed the consciousness of their applicability. That being said….
1)….the human being has not evolved out of the condition he was in 300 years ago: he still perceives and he still thinks, from which it follows the tenets of transcendental philosophy still hold;
2)….from 1), regardless of current opinion concerning the system prescribed by transcendental philosophy with respect to human cognition, each part of the system remains fully dependent on all the others;
3)…..from 2, mathematics being synthetic judgements
a priori is merely an example of what they are, where they reside in the system, and what they do for the system, but rely on something else for sufficient proof of their possibility.
It makes no difference if synthetic judgements
a priori are accepted or not; within the theory they are required, which just means to reject that part is to reject the whole. Which is fine, things do move on, after all.
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In the case of the conception of
a priori itself, Kant did not mean it with respect to time as such, but with respect to placement in the system as a whole. The systemic procedure in a nutshell, for knowledge of things, is perception through to experience. Kant allows
a priori to be pure or impure, but stipulates….probably for the sake of his editors…when he writes the word, he means the pure version, always, without exception. The pure/impure signifies whether or not the subject under consideration is empirical, subject being the propositional form thereof, indicating what he’s talking about at the time: impure means,
e.g., the subject conception is represented by a real thing, while pure, on the other hand, means, “….not such as is independent of this or that kind of experience, but such as is absolutely so of all experience….”.
Now, given the only two possible ways for the human cognitive system to work, either from perception of things, which is all the empirical side, or, from mere thinking of things, which is all the rational side, it follows that “independent from experience” makes explicit the term is restricted in its use to the rational side alone.
So,
a priori means within, or restricted to, any internal systemic function in which there is nothing having to do with empirical predication. To then say
a priori, as it relates to time is before experience, is not quite right, insofar as pure thought absent empirical conditions, is already that for which there never will be any experience anyway, so before experience or before the time of experience, in such case, is superfluous.
It is the entire point of transcendental philosophy, is to combat Hume’s reluctance to pursue pure rational thought as the ground of knowledge. In order to be successful, Kant had to demonstrate those conditions under which ALL knowledge stems, and that from the very condition Hume’s resolution was to “….consign it to the flames…”.
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To answer your question when do we know 7 + 5 = 12, we
know it when we represent it to ourselves by empirical example. Yet beforehand, we know
a priori there is nothing contained in the conception “7”, or in the conception “5”, from which we are given the conception “12”.
Because that is known with apodeictic certainty….
(when all you have is boards over there and nails over there)
….yet the are mathematical statements we know with equal certainty from experience….
(yet there’s houses everywhere you look)
….it remains to be undetermined how to get from one to the other….UNLESS….the cognitive part of the system as a whole, and in particular the part which reasons, does something with the two given conceptions…
(hammer the nails into the boards is the way to build a house; synthesis the “7” and the “5” in understanding is the way to judge the relation of two given conceptions having nothing to do with each other)
Full stop. You hammer all day long, you still don’t have a house; you synthesis the conceptions, you still don’t have the conception “12”. Now we see synthetic judgements
a priori are only representations of a very specific cognitive function, a synthesis done without anything whatsoever to do with experience, and of which we are not the least conscious. It is all an act of reason, which is that systemic faculty not so much involved in knowledge itself, but provides the principles by which it becomes possible. At this point we don’t care about the 12, just as we don’t care the house isn’t done yet. All we want is proof for a way to get the house built, and proof of a way to get to whatever the relation of 7 and 5 gives us.
We think nothing of combining 7 and 5. We don’t think anything of the combining of them. But we stop dead in our cognitive tracks, when the very same synthesis is just as necessary but for which immediate mental manipulation is impossible. The rote mechanism of mere instruction doesn’t work for a vast majority of us, when the synthesis is of, like, numbers containing many digits, or of a different form of synthesis altogether,
i.e, calculus. The principle is the same, though, for all of them.
And all that, is only half the story….