Phaneroscopy, Semeiotic, Logic
Part II. Mathematical Reasoning
Thus, many a pupil is mystified, and never discovers what mathematics aims to do. However, putting aside the cases of men who on account of bad teaching suppose they have no mathematical talent, when they may really be rather above the average in this respect; there remain over others who, with the best of teachings, never could be enabled to comprehend the pons asinorum and yet may be profound lawyers, naturalists, or historians.
This is a most singular psychological phenomenon; because there is no element of mathematical reasoning that is not found in all reasoning whatever, exceptions only in logical analysis and in the formation of conjectures. If the incapacity of high judicial and other intellects were limited for the majority of mathematical reasonings, we might attribute it to the difficulty of grasping the very intricate relations to which mathematical propositions usually refers. But since this incapacity extends to the pons asinorum and other propositions which present no such intricacy, we have to look elsewhere for the secret of it. Let us see, then, precisely what logical operations are involved in a very simple piece of mathematical reasoning. Suppose the question is how many rays, or unlimited straight lines, can at most each cut four given rays that cannot all be cut by each of an indefinitely great multitude of rays. For the sake of brevity, we will suppose that, of the four given rays, which we will distinguish as A, B, C, and D,...
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