Based on my Kant 101 level knowledge of the subject, it's tempting to think:
- Kant's problem of accounting for synthetic knowledge is to explain how we are able to make certain (non-analytic) judgments in advance of experience, which experience then bears out.
- Kant's answer is that our minds organize experience in such a way that whatever input comes in from the noumena we will always represent a scenario in which these propositions hold true. So, for example, I can know in advance of experience that there are no round squares because my mind organizes experience in such a way that it couldn't represent a round square.
BUT if this understanding is right, then Kant's answer doesn't seem to do a very good job of addressing the problem he poses. For, the mere fact that my mind couldn't represent a scenario in which ~P does nothing to ensure (or even make it likely in any obvious way) that I would *realize* that I couldn't have an experience as if of P.
I mean, think about what we find attractive. It might be a psychological fact about me that no physically possible configuration of matter would strike me as constituting a person who is both tall and attractive. The algorithms that produce my feelings of attraction and the ones that detect tallness might be such that no possible sensory input could set off both. But none of this entail that I *know* that I am incapable of finding tall people attractive. Perhaps all I know, at any given time, is that I have not seen or imagined an attractive tall person *yet*.
Thus even if Kant's claims that mathematics, principles of causation and so forth are somehow an artifact of the way the human mind organizes experience were true, this (it seems) would not yet constitute an explanation of how we can manage to know about these subjects.
So the puzzle is: what should Kant's reply be and/or where is the interpretive failure in the argument above?