Seven plus five is twelve.

You did not have to check. You have never counted a set of seven things and a set of five things and found the total to be eleven, and no future observation is going to unsettle you. So the claim is a priori — its justification owes nothing to experience.

Kant's harder claim is that it is also synthetic: the concept twelve is not sitting inside the concepts seven, five and plus waiting to be unpacked. Something had to be put together to get there.

If both are true, there is a kind of knowledge that is neither empirical nor merely definitional — and that possibility is what the whole Critique of Pure Reason exists to investigate.

The four boxes

The category only means anything against the three it is distinguished from, and laying them out is the fastest route into the subject.

Two distinctions, crossed. Analytic or synthetic: does the predicate unpack what is already in the subject concept, or add to it? A priori or a posteriori: does the justification require experience?

Analytic a priori. All bachelors are unmarried. True by the meanings, knowable from the armchair. Uncontroversial.

Synthetic a posteriori. Snow is white. Adds to the concept, and you had to look. Uncontroversial.

Analytic a posteriori. Empty, and necessarily so: if a claim is true by meaning alone, no experience is needed to establish it.

Synthetic a priori. Kant's category. Adds to the concept, and no experience is needed. He thought mathematics was full of them, that the principles of physics included them — every event has a cause — and that metaphysics had been trying and failing to produce them for two thousand years.

Why it looked impossible

The two traditions Kant inherited had the space carved up between them and neither left room for this.

Rationalists held that reason alone reaches substantive truths, and could not say how thinking about concepts delivers information about a world outside them. Empiricists held that all substantive knowledge comes from experience, and had to treat mathematics as either trivial or uncertain — neither of which describes how mathematics actually behaves.

Kant's answer is that both assume knowledge conforms to objects. If instead the objects of experience conform to the structure of the mind receiving them, then you can know something in advance about any object you could possibly encounter — not because you have met them all, but because you supplied the frame they must arrive in.

That answer is transcendental idealism, and it is worked out there. The point to carry here is that the two doctrines stand or fall together: the synthetic a priori is the thing to be explained, and transcendental idealism is the explanation.

Quine's attack

The twentieth century's decisive objection does not deny that synthetic a priori knowledge exists. It denies that the question is well formed.

W. V. O. Quine's Two Dogmas of Empiricism (1951) goes after the analytic/synthetic distinction itself. To call a statement true by meaning, you need an account of sameness of meaning; every account on offer turns out to presuppose the notion it was supposed to explain. And in practice, no statement faces experience alone — a recalcitrant observation can always be accommodated by adjusting something else, so no sentence is immune to revision and none is purely definitional.

Pull the distinction and the four boxes go with it. For much of the later twentieth century the synthetic a priori was treated as a pre-Quinean relic.

That has softened. Kripke showed in Naming and Necessity (1980) that the related distinctions had been run together for centuries — necessity is metaphysical, a priority is epistemic, and there are necessary truths known only through experience, such as water being H₂O. Once those come apart, Kant's question stops looking confused even to people who reject his answer.

And the original puzzle survives all of it. Mathematics is not obviously empirical, and it is not obviously trivial either. Any account of knowledge still has to say what it is.