The evidence of the pure and the applied doctrine of magnitudes. Comparison with the evidence for the proofs of God’s existence. Different methods of those proofs.

Abstract

Pure mathematics proves its doctrines merely in accordance with the laws of thinking, or a priori as one often calls it, without the help of experience and sensory knowledge. The power of its proofs rests upon the articulation of concepts. One analyzes the concept A, and finds a necessary connection between its characteristics and the concept of a predicate B. This yields an affirmative proposition [i.e., A is B; B is included in A]; the exclusion [of predicate B from the concept A] yields a negative proposition. Both, however, assert nothing further than the combination of concepts or ideal beings in accordance with the laws of what can be thought.