Logical implication is often an enemy of dignity and eloquence. De Morgan[54] relates "a tradition of a Cambridge professor who was once asked in a mathematical discussion, 'I suppose you will admit that the whole is greater than its part?' and who answered, 'Not I, until I see what use you are going to make of it.'" And the care displayed by cautious mathematicians like Poincaré, Schoenflies, Borel, Hobson, and Baire in abstaining from pushing their arguments to their logical conclusions is probably founded on the unconscious--but no less well-grounded--fear of appearing ridiculous if they dealt with such extreme cases as "the series of all ordinal numbers."[55] They are, probably, as unconscious of implication as Gibbon, when he remarked that he always had a copy of Horace in his pocket, and often in his hand, was of the necessary implication of these propositions that his hand was sometimes in his pocket. FOOTNOTES: [51] _Md._, N. S., vol. iii., 1894, pp. 436-8. Cf. the discussions by W. E. Johnson (_ibid._, p. 583) and Russell (_P. M._, p. 18, note, and _Md._, N. S., vol. xiv., 1905, pp. 400-1). [52] The inhabitants of "Erewhon" punished invalids more severely than criminals. In modern times, one frequently hears the statement that crime is a disease; and if so, it is surely false that criminals ought to be punished. [53] _Irrelevant_ in a popular sense; one would not say, speaking loosely, that the fact that Brutus killed Cæsar implies that the sea is salt; and yet this conclusion is implied both by the above premiss, and the premiss that Cæsar killed Brutus. Cf. on such questions Venn, _S. L._, 2nd ed., pp. 240-4. [54] _F. L._, p. 264. [55] Cf. Chapters XXIX and XXXVII. CHAPTER XX DIGNITY We have seen, at the end of the preceding chapter, that logical implication is often an enemy of dignity. The subject of dignity is not usually considered in treatises on logic, but, as we have remarked, many mathematicians implicitly or explicitly seem to fear either that the dignity of mathematics will be impaired if she follows out conclusions logically, or that only an act of faith can save us from the belief that, if we followed out conclusions logically, we should find out something alarming about the past, present, or future of mathematics. Thus it seems necessary to inquire rather more closely into the nature of dignity, with a view to the discovery of whether it is, as is commonly supposed, a merit in life and logic.