24 THOMAS WHITTAKER : valid for thought whatever our experience may be ; and, on the other side, we cannot by means of it anticipate experience to the smallest extent. For real availability, it is absolutely- dependent on there being an order of which by itself it contains no assertion. In passing from Formal to Material Logic, we come first to the general principles of mathematical knowledge. Since Kant's investigation of these, it is allowed that they are " synthetic " and not merely " analytic ". That is to say, there are involved in mathematical demonstration proposi- tions which are neither an affair of hypothetical definition nor can be educed from definitions by means of the formal laws of thought. To take Kant's own examples. The geo- metrical axiom that "two straight lines cannot enclose a space " is not a truth that can be evolved by mere comparison of the concepts of the straight line and of space. Similarly with an arithmetical proposition such as 7 + 5 = 12 : no mere comparison of the concepts of the separate numbers can give the resulting number. In both cases, what is required is a construction in intuition or in the corresponding imagination, a process of mental drawing, or of numbering things or events in time. And the peculiarity of mathematical prin- ciples is that, upon such construction, recognition of the necessary truth of the proposition is the outcome of a single act of comparison. Thus they are not generalisations from experience. This last position of Kant has been contested from the experiential side. What remains incontestable is that, be- sides the principles of Formal Logic, mathematical science requires first principles peculiar to itself. The positions of Locke, of Leibniz, and of Hume in the Inquiry, are abandoned on this point. Kant's view as regards the peculiarity of mathematical reasoning, it may be observed, had been in part anticipated in the Platonic school. Plato himself had marked off Mathematics from what he called Dialectic which was at once Metaphysics and Logic on the one side, and from such an adumbration of Physics as was then pos- sible on the other. Aristotle divided Metaphysics proper from Logic ; and by Plato's successors, with the aid of the later Peripatetics, something was done to make clearer the precise character to be ascribed to mathematical truth. An intermediate position was assigned to it between laws valid for pure thinking, which are prior, and " laws of nature " emerging from observation or experiment, which are pos- terior. These distinctions were to some extent obscured in the early modern period, but may now be considered as
