ABSTRACT

Another form of this tendency is shown by Kronecker, Borel, Poincare, and many other mathematicians, who refuse mere logical determination of a conception and require that it be actually described in a finite number of terms. These eminent mathematicians were anticipated by the empirical philosopher who would not pronounce that the " law of thought " that A is either in the place B or not is true until he had looked to make sure. This philosopher was of the same school as J. S. Mill and Buckle, who seem to have maintained implicitly not only that, in view of the fact that the breadth of a geometrical line depends upon the material out of which it is constructed, or upon which it is drawn, that there ought to be a paste-board geometry, a stone geometry, and so on ; 2 but also that the foundations of logic are inductive in their nature.3 " We cannot," says Mill,4 " conceive a round square, not merely because no such object has ever presented itself in our experience, for that would not be enough. Neither, for anything we know, are the two ideas in themselves incompatible. To conceive a body all black and yet white would only be to conceive two different sensations as produced in us simultaneously by the same object-a conception familiar to our experience —and we should probably be as well able to conceive a round square as a hard square, or a heavy square, if it were not that in our uniform experience, at the instant when a thing begins to be round, it ceases to be square, so that the beginning of the one impression is inseparably associated with the departure or cessation of the other. Thus our inability tq form a conception always arises from our being compelled to form another contradictory to i t / '