Kant, having observed that the geometers of his day could not prove their theorems by unaided argument, but required an appeal to the figure, invented a theory of mathematical reasoning according to which the inference is never strictly logical, but always requires the support of what is called "intuition." The whole trend of modern mathematics, with its increased pursuit of rigour, has been against this Kantian theory. The things in the mathematics of Kant's day which cannot be proved, cannot be known—for example, the axiom of parallels. What can be known, in mathematics and by mathematical methods, is what can be deduced from pure logic. What else is to belong to human knowledge must be ascertained otherwise—empirically, through the senses or through experience in some form, but not a priori. The positive grounds for this thesis are to be found in Principia Mathematica, passim; a controversial defence of it is given in the Principles of Mathematics. We cannot here do more than refer the reader to those works, since the subject is too vast for hasty treatment. Meanwhile, we shall assume that all mathematics is deductive, and proceed to inquire as to what is involved in deduction. In deduction, we have one or more propositions called premisses, from which we infer a proposition called the conclusion. For our purposes, it will be convenient, when there are originally several premisses, to amalgamate them into a single proposition, so as to be able to speak of the premiss as well as of the conclusion. Thus we may regard deduction as a process by which we pass from knowledge of a certain proposition, the premiss, to knowledge of a certain other proposition, the conclusion. But we shall not regard such a process as logical deduction unless it is correct, i.e. unless there is such a relation between premiss and conclusion that we have a right to believe the conclusion [Pg 145] if we know the premiss to be true. It is this relation that is chiefly of interest in the logical theory of deduction.