KANT'S PROOF OP THE PROPOSITION, ETC. 515 he introduces for the first time a statement to the effect that " self-evident propositions as to the relations of numbers are certainly synthetical," and produces as proof an examina- tion of his much-misunderstood instance of Arithmetical procedure, the addition of seven and five to make twelve. " The proposition, that 7 + 5 = 12, is not analytical." "For neither in the representation of seven, nor of five, nor of the composition of the two numbers, do I cogitate the number twelve." If we borrow from the later defence of this thesis in the Prolegomena, we find that Kant considers it absolutely necessary in numerical addition "to go beyond our con- ceptions and have recourse to an intuition ". We must represent before ourselves five fingers or five points, adding five to the number seven " by means of this material image, and by this process we at length see the number twelve arise," just as by representing to ourselves on paper the figure of Euclid's thirty-second proposition we can conclude that the sum of the interior angles of a triangle is two right angles. In neither case does Kant mean that the material image is necessary : it is only an empirical means of aiding us to reach the pure non-empirical intuition which lies be- hind. We should very much like to defend Kant's analogy between pure and empirical intuition, inasmuch as the many attacks directed against it in this connexion seem to be both ill-founded and inconclusive, but to do so would lead us far beyond our immediate task of exegetical exposition. And therefore we take Kant's argument to be quite sufficient of itself to warrant his conclusion, that " Arithmetical proposi- tions are invariably synthetical," and ask only how much Kant here claims to have proved. He expressly says that such propositions as 7 + 5 = 12 are not axioms but "numerical formulae," giving (in 1781) the ostensible reasons that we cannot have an infinite number of axioms and that axioms must be universal as in geometry. In 1783 they bear the simple name of "propositions" (Satze), and no mention is made of their non-axiomatic nature. They are treated as corresponding to the principles on which pure Geometry founds such principles as : " The straight line between two points is the shortest," but although Kant usually speaks of this example in language which implies that he himself regards it as an Axiom, it nevertheless ranks as a proved Proposition in Euclid, and diverse views of its real nature have always been common among mathematicians. For this reason it would be hazardous to attempt any elaborate comparison of these Arithmetical propositions with other instances of synthetic a priori propositions, but yet, when
