Kant's epistemology in a nutshell
Kant (1724-1804): universal truths possible because of the universal operations of the intelligent (human?) mind.
1. Distinction between synthetic (new) versus analytic (tautological) knowledge, and a priori versus a posteriori knowledge.
2. In his Prolegomena (p. 260) Kant writes that it was Hume who "interrupted my dogmatic slumber." According to Kant, Hume sees mathematics as analytic a priori knowledge, and science as synthetic a posteriori knowledge -- thus no certainty in new knowledge, and no possibility for metaphysics. [for critique as misrepresentation of Hume click here]
3. Kant maintains that geometry discovers the universal laws of space, and algebra discovers the universal laws of time. Space and time are "pure intuitions" by which perception can take place, so they are a priori and universal.
4. Our mental abilities only give us knowledge of appearances ("phenomena") and not things-in-themselves ("nounema") -- much like Plato's cave.
5. Nounema --> space and time --> judgments of perception (subjective)
--> concepts --> judgments of experience (objective)
(Note that this is not a process in time!)
"The objective validity of the judgment of experience signifies nothing else than its necessary universal validity" -- unlike Plato's cave, appearances are governed by universal laws; the discovery of these laws is the process of science.
6. Like Space and Time, the Concepts ("pure concepts of the understanding," aka "Categories") are universal to all conscious entities. Concepts include things like Unity, from which we derive measurement, and Plurality, from which we derive magnitude; thus making objective science possible. One of his "concepts" is that of cause, thus solving (from Kant's POV) Hume's obstacle to the possibility of science.
7. Just as understanding requires these categories, reason requires ideas--by which he means ideals: the idea of the complete subject (psychological), the idea of complete conditions (cosmological), and the idea of complete complex of all that is possible (theological). We can never achieve these, nor stop trying. For example, Kant illustrates the cosmological idea with "antinomies" -- thesis/antithesis pairs that both appear to be true, eg "There are in the world causes through freedom" and "There is no freedom, but all is Nature." Kant claims the split between nounema (where freedom may lie) and phenomena (where necessity rules) solves such dilemmas.