Now as concerns the argument, its conclusion is that in induction (causal inference) experience does not produce the idea of an effect from an impression of its cause by means of the understanding or reason, but by the imagination, by “a certain association and relation of perceptions.” The center of the argument is a dilemma: If inductive conclusions were produced by the understanding, inductive reasoning would be based upon the premise that nature is uniform; “that instances of which we have had no experience, must resemble those of which we have had experience, and that the course of nature continues always uniformly the same.” (Hume THN, 89) And were this premise to be established by reasoning, that reasoning would be either deductive or probabilistic (i.e. causal). The principle can't be proved deductively, for whatever can be proved deductively is a necessary truth, and the principle is not necessary; its antecedent is consistent with the denial of its consequent. Nor can the principle be proved by causal reasoning, for it is presupposed by all such reasoning and any such proof would be a petitio principii.
For it amounts to nothing more than the claim that, given any collection of mutually disjoint nonempty sets, it is possible to assemble a new set — a transversal or choice set — containing exactly one element from each member of the given collection.
But might there be some feature of the proof that compromises its ability to offer the kind of justification we are after? One aspect of the proof that suggests itself is its reliance on the Axiom of Choice....
Where a condition on conceptual coherence is consistency with ZFC, Zermelo-Fraenkel set theory plus the Axiom of Choice, a theory most mathematicians believe to be true and indispensable for the formal development of mathematics.
If this is the justification then it would render the whole proof implausible because as Penelope Maddy has correctly pointed out, mathematicians have "accepted" the axiom of choice largely because of pragmatic reasons (for ease of proving some theorems like Zorn's lemma). They haven't accepted it because of its philosophically sound obviousness (in fact, many of them and logicians and philosophers have skeptically questioned its truth much as the later have inductive reasoning). Given these pragmatic historical considerations, they seem to mirror the pragmatic considerations of why people accept inductive reasoning as generally true (how will we live without assuming natural regularity and uniformity?).