Kyary, thank you for sharing this serious reflection.
I don’t think the concept ‘possible’ is licit except as a recognition of potentials of actuals. However, ‘potentials’ is applicable only to actuals that are concrete. So it is not applicable to vector spaces, for example, in pure mathematics. And it is not applicable to morphisms in the category of vector spaces, and so forth. I submit we must walk before we do geometry. The possibility-relations between purely mathematical structures and their transformations are analogues of the possibility-relations reflecting poteniality-relations among concretes, such as relations engaged in walking.
Kant, of course, was presuming Euclidean geometry (as I recall, he does not engage the link between that geometry and algebra from Descartes and Fermat giving us analytic geometry) as something that is instantiated in the physical concrete world. His reason for Euclidean space being most fundamentally mental rather than physical is that he rightly saw that the results we arrive at when in geometry class working the proofs in Euclid yield perfect, absolute universality of the resulting truths. And that is of a different character (he thought) than the sort of results we arrive at in empirical science. He was right on that and right in rejecting the doctrine of Wolff that the high necessity of the truths of geometry come from capacity of Euclidean geometry to be put into the form of a sequence of syllogisms (which was probably also the way Aristotle looked at it, but without the attempt of full-blown showing of it as in Wolff’s geometry text, from which Kant lectured). He was wrong in thinking we had no other cognitive access to features in the physical world to make into geometry that are not the methods of empirical science.
So to ask such questions as how is synthetic a priori knowledge possible is a legitimate use of ‘possible’, supposing your interlocutor accepts that the knowledge you are characterizing as synthetic a priori knowledge is rightly so characterized. Philip Kitcher in his book The Nature of Mathematical Knowledge is one of our contemporaries who rejects that characterization and would have to regard that particular possibility-question of Kant’s as fallacy of complex question.
That some spatial knowledge can be condition of much or all of our knowledge does not entail that that condition of knowledge cannot be physical.
I do not think we should join philosophers in their old arm-chairing of the evolution of consciousness or self-consciousness. We should rather overrule them or congratulate them with findings of modern developmental cognitive psychology***** and anthropology. For example, Michael Tomasello’s A Natural History of Human Thinking should be used to grade the arm-chair folk. One sorry blindness of philosophers from Descartes to Schelling is their failure to notice the fundamental social requirements for human consciousness.
I agree with your conclusion, Kyary, that Rand/Peikoff erred in thinking Kant closed the possibility of cognition of things as they are independently of consciousness on account of consciousness taking a specific form and having specific means of operation. I argue against this Rand/Peikoff conclusion in a paper currently at press. It should be published next April, and since there will be an online version of that journal, I’ll be linking it at this site when they publish.
I attach an excerpt from my fundamental paper “Existence, We” in which I oppose Rand’s approach to the idea of physical existence. Rand should have treated the concept “physical existence” in the way she had treated the developmental course of the concept “man”.
(Click on image for ease of reading.)
