Analytic and synthetic judgments

The Prolegomena opens not with an answer but with a warning. In the Preamble (§§1–3) Kant announces that metaphysics has a “peculiarity” no other science shares, and that before we can ask whether it is possible we must fix “the peculiarity of all metaphysical cognition.” That housekeeping is one distinction — between two kinds of judgment. See Prolegomena to Any Future Metaphysics §2.

An analytic judgment is explicative. Its predicate is already contained — covertly — in the concept of the subject; the judgment merely draws it out. “All bodies are extended.” To be a body just is to occupy space, so “extended” adds nothing; it only unpacks what “body” already meant. Its distinctive mark: to deny it is a flat contradiction (“an unextended body” cancels itself).

A synthetic judgment is ampliative. Its predicate adds something not contained in the subject-concept, so it genuinely extends — amplifies — our knowledge. “All bodies are heavy.” Weight is nowhere inside the bare concept body; you had to reach beyond the concept to attach it. And its denial is no contradiction: an unheavy body is strange, perhaps false, but not self-cancelling.

Now the claim that startled Kant’s readers. All mathematical judgments are synthetic (§2). This sounds impossible — surely arithmetic is just definition, the very model of analytic truth? Kant says look closer at his favourite case: 7 + 5 = 12.

Take the concepts 7, 5, and the sum of them. Analyse them as hard as you like. All you find in “the sum of 7 and 5” is the thought of uniting two numbers into one — never which number results. The concept 12 is simply not thought there. To reach it you must step outside the concepts and count: exhibit the units (fingers, points, strokes) and run them together. That going-beyond is the synthetic ingredient. See Prolegomena to Any Future Metaphysics §2.

Geometry is the same. “A straight line is the shortest between two points” is synthetic: straight is a concept of quality (direction), while shortest is a concept of quantity, and no amount of unpacking “straight” yields “shortest.” You must construct the line in intuition to see it. So both branches of mathematics trade in synthetic judgments — informative, not merely definitional.

Why does this tiny distinction carry the whole book? Because it lets Kant relocate the problem. The empiricists — Hume above all — lacked a sharp analytic/synthetic cut. Hume’s “relations of ideas” quietly lumped genuine mathematical knowledge in with mere definitions, so he never saw the real puzzle: knowledge that is at once ampliative and certain in advance. Metaphysics wants exactly that kind of truth, not empty analytic unpacking. See An Enquiry Concerning Human Understanding IV.