Kant approaches the matter as he does because he is responding to Hume, and one of Hume’s initial challenges is about the origin of “ideas.” While the Problem of First Principles is about the justification of propositions, Hume’s Empiricist approach goes back to asking about the legitimacy of the very concepts, of which the propositions are constituted, in the first place…
Thus, Kant begins, like Hume, asking about the legitimacy of concepts. However, the traditional Problem has already insensibly been brought up; for in his critique of the concept of cause and effect, Hume did question the principle of causality, a proposition, and the way in which he expressed the defect of such a principle uncovered a point to Kant, which he dealt with back in the Introduction to the Critique…
In An Enquiry Concerning Human Understanding, Hume made a distinction about how subject and predicate could be related:
All the objects of human reason or enquiry may naturally be divided into two kinds, to wit, Relations of Ideas, and Matters of Fact. Of the first kind are the sciences of Geometry, Algebra, and Arithmetic; and in short, every affirmation which is either intuitively or demonstratively certain [note: these are Locke’s categories]. That the square of the hypothenuse is equal to the square of the two sides, is a proposition which expresses a relation between these figures. That three times five is equal to the half of thirty, expresses a relation between these numbers. Propositions of this kind are discoverable by the mere operation of thought, without dependence on what is anywhere existent in the universe. Though there never were a circle or triangle in nature, the truths demonstrated by Euclid would for ever retain their certainty and evidence.
Matters of fact, which are the second objects of human reason, are not ascertained in the same manner; nor is our evidence of their truth, however great, of a like nature with the foregoing. The contrary of every matter of fact is still possible; because it can never imply a contradiction, and is conceived by the mind with the same facility and distinctness, as if ever so conformable to reality. That the sun will not rise to-morrow is no less intelligible a proposition, and implies no more contradiction than the affirmation, that it will rise. We should in vain, therefore, attempt to demonstrate its falsehood. Were it demonstratively false, it would imply a contradiction, and could never be distinctly conceived by the mind. [Enquiries, Selby-Bigge edition, Oxford, 1902, 1972, pp.25-26]
Both paragraphs warrant quoting in full. The first now would seem properly more a matter of embarrassment than anything else. Whatever Hume expected from intuition or demonstration, it would be hard to find a mathematician today who would agree that “the truths demonstrated by Euclid would for ever retain their certainty and evidence.” If Hume’s fame rests on this point, there would be little to recommend it. The second paragraph, however, redeems the impression by giving us a logical criterion to distinguish between truths that are “relations of ideas” and those that are “matters of fact”: A matter of fact can be denied without contradiction…
For, indeed, outside of an axiomatized logic itself, the First Principles of Demonstration will be synthetic. However Kant can explain the truth of non-empirical synthetic propositions, i.e. those that are a priori instead of a posteriori, that will be his answer to the Problem of First Principles. Yet Hume himself is often poorly understood. While it is common to say that Hume denied the existence of synthetic a priori propositions, there is some question about whether he actually does. He says that the relationship of cause and effect is not discovered or known by any reasonings a priori, but that is not the same thing. A synthetic a priori proposition is not known from any reasonings. In fact, Hume does not see that the relationship of cause and effect is discovered or known from anything, since it is not justified by experience, in which there is no necessary connection between cause and effect, and there is in fact nothing in the cause to even suggest the effect, much less that the effect must follow. Hume’s famous explanation was a psychological one, that we become accustomed to the association of certain events (“causes”) with others (“effects”); but this, obviously, carries no weight whatsoever about the nature of things, which is what makes Hume, very properly, a Skeptic.
At the same time, Hume had no doubts whatsoever of the necessity of cause and effect.