This chapter gives a valuable account of the work of German philosopher Immanuel Kant (1724–1804). Kant developed a critical philosophy, which assumed that knowledge exists and sought to understand the character of that knowledge and how it is possible at all. The answers he found developed into a vast and detailed metaphysics. Russell gives him the principal credit of having conceived a priori knowledge that was not "purely analytic" and for elevating the essential discipline of epistemology to a 'first philosophy.'

The traditional concept of the a priori was necessarily analytic, which meant that the predicate was always implicit in the subject. An example is the statement: "A bald man is a man." The idea of a man is clearly implied by the phrase "bald man," producing an obvious and almost trivial assertion. Before Kant, the law of contradiction, that "nothing can both be and not be," was sufficient to prove the truth of all a priori statements; a bald man cannot be bald and not bald without contradiction.

The work of English philosopher David Hume (1711–76) was an important precursor to Kant's work. Hume revealed that many cases of analytic knowledge, the most conspicuous case being that of cause and effect, were in fact synthetic. Hume argued against the traditional rationalist belief that effect could be logically deduced from cause, and he concluded that nothing could be a priori known about the relation between cause and effect. In answer to Hume's skeptical thought, Kant (whose thought derived from rationalist origins) alleged that there were many other propositions besides cause and effect that failed as analytic truths. He held that arithmetic and geometric propositions were also synthetic in that "no analysis of the subject will reveal the predicate." His notable illustration was the statement "7 + five = 12." In themselves, neither seven nor five contain the idea of twelve, and they must be put together first. Such observations allowed him to make a distinction between the a priori and analytic. His description took the form: "All pure mathematics, though a priori, is synthetic." Kant then investigated how such knowledge, a priori and synthetic, was possible.

Other schools of philosophy have attempted to answer this question. The empiricists depended on experience to arrive at pure mathematical knowledge, through the repetition of instances. We already admitted that this answer is flawed because it is possible to grasp the proposition that "two and two make four," through reflection on one instance. Kant's solution poses a sophisticated metaphysical picture, which Russell sketches before responding.

Russell's account of the Kantian system of reason first separates our experience into two elements: the element owing to the physical object and the element owing to our own nature. The next structural feature of Kant's philosophy is the distribution of these elements. Russell writes: "(Kant) considers that the crude material given in sensation—the colour, hardness, etc.—is due to the object, and that what we supply is the arrangement in space and time, and all the relations between sense-data which result from comparison or from considering one as the cause of the other." Kant supports this view because he believes that we have a priori knowledge of "space and time and causality and comparison," but not of "the actual crude material of sensation."

For Kant, the physical object, the thing in itself, is unknowable. What we can know is the "phenomenon," the object in our experience. The phenomenon is a product of the thing in itself and us. Thus, as it comes into our experience, the phenomenon will acquire characteristics which "conform to our a priori knowledge." Further thus, this knowledge cannot be valid outside of our experience. Despite having knowledge that it is a priori in quality, we cannot extend it outside our experience to a thing in itself. Russell reads Kant's conclusion as an attempt to reconcile the rationalist picture with the empiricist one.