=231.= [In mathematics] we behold the conscious logical activity of the human mind in its purest and most perfect form. Here we learn to realize the laborious nature of the process, the great care with which it must proceed, the accuracy which is necessary to determine the exact extent of the general propositions arrived at, the difficulty of forming and comprehending abstract concepts; but here we learn also to place confidence in the certainty, scope and fruitfulness of such intellectual activity.--HELMHOLTZ, H. _Ueber das Verhältniss der Naturwissenschaften zur Gesammtheit der Wissenschaft, Vorträge und Reden, Bd. 1 (1896), p. 176._ =232.= It is true that mathematics, owing to the fact that its whole content is built up by means of purely logical deduction from a small number of universally comprehended principles, has not unfittingly been designated as the science of the _self-evident_ [Selbstverständlichen]. Experience however, shows that for the majority of the cultured, even of scientists, mathematics remains the science of the _incomprehensible_ [Unverständlichen].--PRINGSHEIM, ALFRED. _Ueber Wert und angeblichen Unwert der Mathematik, Jahresbericht der Deutschen Mathematiker Vereinigung (1904), p. 357._ =233.= Mathematical reasoning is deductive in the sense that it is based upon definitions which, as far as the validity of the reasoning is concerned (apart from any existential import), needs only the test of self-consistency. Thus no external verification of definitions is required in mathematics, as long as it is considered merely as mathematics.--WHITEHEAD, A. N. _Universal Algebra (Cambridge, 1898), Preface, p. vi._ =234.= The mathematician pays not the least regard either to testimony or conjecture, but deduces everything by demonstrative reasoning, from his definitions and axioms. Indeed, whatever is built upon conjecture, is improperly called science; for conjecture may beget opinion, but cannot produce knowledge. --REID, THOMAS. _Essays on the Intellectual Powers of Man, Essay 1, chap. 3._