It may be doubted whether Pillon’s eulogy is altogether sound in its approval of the “reconciliation” of Hume and Kant, for such a reconciliation of opposites may well appear impossible. Renouvier himself faced this problem of the reconciliation of opposites when at an early age he inclined to follow the Hegelian philosophy, a doctrine which may very well be described as a “reconciliation of opposites,” _par excellence_. Dissatisfied, however, with such a scheme Renouvier came round to the Kantian standpoint and then passed beyond it to a position absolutely contrary to that of Hegel. This position is frankly that opposites cannot be reconciled, one or the other must be rejected. Renouvier thus made the law of contradiction the basis of his philosophy, as it is the basis of our principles of thought or logic. He rigorously applied this principle to that very interesting part of Kant’s work, the antinomies, which he held should never have been formulated. The reasons put forward for this statement were two: the principle of contradiction and the law of number. Renouvier did not believe in what mathematicians call an “infinite number.” He held it to be an absurd and contradictory notion, for to be a number at all it must be numerical and therefore not infinite. The application of this to the Kantian antinomies, as for example to the questions, “Is space infinite or finite? Had the world a beginning or not?” is interesting because it treats them as Alexander did the Gordian knot. The admission that space is infinite, or that the world had no beginning, involves the admission of an “infinite number,” a contradiction and an absurdity. Since, therefore, such a number is a pure fiction we _must_ logically conclude that space is finite,[29] that the world had a beginning and that the ascending series of causes has a first term, which admission involves freedom at the heart of things. [29] It is interesting to observe how the stress laid by Renouvier upon the finiteness of space and upon relativity has found expression in the scientific world by Einstein, long after it had been expressed philosophically. As Renouvier had treated the antinomies of Kant, so he makes short work of the Kantian conception of a world of noumena (_Dinge an sich_) of which we know nothing, but which is the foundation of the phenomena we know. Like Hume, he rejects all notion of substance, of which Kant’s noumenon is a survival from ancient times. The idea of substance he abhors as leading to pantheism and to fatalism, doctrines which Renouvier energetically opposes, to uphold man’s freedom and the dignity of human personality.