We proceed upwards, however, from such questions as the common sense of mankind generally is competent to decide, to those in which special knowledge is required to give value to the judgment; and here we must distinguish between those departments of inquiry in which scientific conclusions are arrived at by a process of strict reasoning, and those in which they partake of the nature of common sense judgments.
Of the former class we have a typical example in mathematics, and in those exact sciences which make use of mathematics as their instrument of proof; but even in these, it is common sense which affords not only the basis, but the materials of the fabric. For while the axioms of geometry are self-evident truths which not only do not require proof, but are not capable of being proved in all their universality, every step of a demonstration is an assertion of which our acceptance depends on our incapability of conceiving either the contrary or anything else than the thing asserted. And thus the certain assurance of the proof felt by every person capable of understanding a mathematical demonstration, depends upon the conclusive self-evidence of each step of it. But we not unfrequently meet with individuals, not deficient in ordinary common sense, who cannot be brought to see this self-evidence; whilst, on the other hand, the advanced mathematician, when adventuring into new paths of inquiry, is able to take a great deal for granted as self-evident, which at an earlier stage of his researches would not have so presented itself to his mind. The deliverances of this acquired intuition can in most cases be readily justified by the reasoning process which they have anticipated. But the genius of a mathematician--that is, his special aptitude developed by special culture--will occasionally enable him to divine a truth, of which, though he may be able to prove it experientially, neither he nor any other can at the time furnish a logical demonstration. In this divining power we have clear evidence of the existence of a capacity which cannot be accounted for by the mere co-ordination of antecedent experiences, whether of the individual or of the race; and yet, as already shown, such co-ordination has furnished the stimulus to its development.