§ 1. PHILOSOPHICAL CERTAINTY IS ALTOGETHER DIFFERENT IN NATURE FROM MATHEMATICAL CERTAINTY

One is certain if one knows that it is impossible that a cognition should be false. The degree of this certainty, taken objectively, depends upon the sufficiency in the characteristic marks of the necessity of a truth. But taken subjectively, the degree of certainty increases with the degree of intuition to be found in the cognition of this necessity. In both respects, mathematical certainty is of a different kind to philosophical certainty. I shall demonstrate this with the greatest possible clarity.

The human understanding, like any other force of nature, is governed by certain rules. Mistakes are made, not because the understanding combines concepts without rule, but because the characteristic mark which is not perceived in a thing is actually denied of it. One judges that that of which one is not conscious in a thing does not exist. Now, firstly, mathematics arrives at its concepts synthetically; it can say with certainty that what it did not intend to represent in the object by means of the definition is not contained in that object. For the concept of what has been defined only comes into existence by means of the definition; the concept has no other significance at all apart from that which is given to it by the definition. Compared with this, philosophy and particularly metaphysics are a great deal more uncertain in their definitions, should they venture to offer any. For the concept of that which is to be defined is given. Now, if one should fail to notice some characteristic mark or other, which nonetheless belongs to the adequate distinguishing of the concept in question, and if one judges that no such characteristic mark belongs to the complete concept, then the definition will be wrong and misleading. Numberless examples of such errors could be adduced, and for that very reason I refer only to the above example of touching.