The Commens Dictionary

Quote from ‘Lowell Lectures. 1903. Lecture 3’

…a necessary reasoning is one which would follow under all circumstances, whether you are talking about the real world or the world of the Arabian Nights’ or what. And that precisely defines mathematical reasoning. It is true that a distinctly mathematical reasoning is one that is so intricate that we need some kind of diagram to follow it out. But something of the nature of a diagram, be it only an imaginary skeleton proposition, or even a mere noun with the ideas of its application and signification[,] is needed in all necessary reasoning.