I confess I have not studied Kant sufficiently to say that his views differ, materially from mine, though I always thought they did until I read your interpretations of them. Perhaps I misunderstood the sense in which Kant used the term _a priori_. The term has been used in so many different senses that I prefer myself to drop it altogether. If it merely refers to priority in time there can be no practical doubt that, whether in the case of the human race or of an individual thinker, a large amount of sense-experience must have preceded even so simple an _a priori_ judgment as “twice two is four.” If the term merely refers to priority in logical validity it seems to me better to say that “such and such assertions are not dependent upon experience.” But Kant says of the assertion “7 + 5 = 12” that it is not only “_a priori_” but “synthetic”. By the latter term he means that its truth was _not_ deduced from definitions alone, and that the assertion therefore conveys real information. In this I believe he was wrong, and though he afterwards declares that “all knowledge _a priori_ is empty and cannot give information about things,” unless the true nature of _a priori_ knowledge is made more clear, people will inevitably continue to believe the contrary—and to believe moreover that Kant taught so. Any language which seems to imply that there is some dread necessity about mathematical truths—that they could not be otherwise if they would—is very misleading. Of course it is necessarily true that _if_ you have seven objects and add five more to them you will have in all twelve objects. But the whole objective difficulty is begged by the supposition. “Much virtue in if!” As I understand it the essence of the “laws” of pure mathematics is that they are verbal, that is they are only abbreviated expressions of the results of certain verbal processes. If the processes are repeated and the results similarly expressed, the results must always be the same. Our reason cannot “inform us about the form of existence” unless it is first given, as the _data_ or facts which correspond to the definitions of our symbolic arguments. It is only because our reasoning faculties are limited that symbolic arguments are necessary at all—that it is not evident to us at once that the conclusions of the most intricate mathematical calculations are given to us along with the _data_. Given the data, then in all possible worlds the conclusions must indeed follow, but only because they really are already _in_ the data which were given.