CHARACTER AND LIMITS OF BELIEF AND CERTAINTY. 915 
even meet with those, who claim for mathematics the honor of being the only branch of 
human knowledge that deserves the name of a science, and we may learn in history, that 
eminent philosophers have attempted to establish their systems on a mathematical basis,— _ 
systems of faith,—systems of religion,—systems of ethics,—systems of jurisprudence. We 
-may even ourselves, have sometimes regretted that subjects which seemed to us of the 
most vital importance, should be veiled in obscurity, and we must therefore rest our confi- 
dence in them, solely on the authority of others. We may have longed for,more light,— 
for a more confident belief,—for certainty, or at least an approximation to certainty. 
215. But notwithstanding this longing,—this earnest cry of all humanity for more posi- 
tive knowledge,—the world still believes that the relations of form and proportion, are the 
only ones that are susceptible of a rigorous and satisfactory analysis. Whatever its origin, 
this idea is certainly a mistaken one. 
216. Even in mathematics, all things are not proved. The very idea of proof is merely 
the idea of some new truth, deduced as a necessary inference from the relations existing 
between other truths which had been previously recognized. If we were obliged to prove 
everything, we could prove nothing, for we could never have a starting-point. We must 
therefore recognize certain truths as self-evident; we have already discovered that there 
are such truths, truths that form the substratum of all our knowledge, and of all our 
belief,—the axioms of science. These axioms are not confined to mathematics, neither 
have mathematical axioms any greater certainty than any other self-evident truths. The 
axioms of our own existence, and of the existence of something independent of ourselves, 
have no mathematical characteristics, and yet they are as indubitable as the axiom that 
the whole is greater than a part. Every necessary conviction of the mind, every proposi- 
tion that we receive unhesitatingly as true the moment we comprehend it, in other words, 
every self-evident truth, is incontrovertible, and every science that can be built upon such 
truths, and upon a correct perception of the relations that subsist between them, is a true 
science, and constitutes a portion of the absolute knowledge to which we are all capable 
of attaining. 
217. ‘There is, then, in the nature of things, no reason why we may not have metaphy- 
sical sciences, as well as physical sciences,—sciences of mind, of morals, and of law, as 
to an ull-sided activity 2? We are compelled to answer, No; for they do this only in relation to a knowledge of 
quantity, neglecting altogether that of quality. Further, is this mathematical evidence, is this coincidence of 
theory and practice actually found to hold in the other branches of our knowledye? The slightest survey of the 
sciences proves the very reverse; and teaches us that mathematics tend necessarily to induce that numb rigidity 
into our intellectual life, which pressing obstinately straight onward to the end in view, takes no heed or account 
of the means by which, in different subjects, it must be differently attained.’”” Von Weiller, Klumpp, Goethe, 
D’Alembert, Descartes, Du Hamel, Arnauld, and others, are quoted in further illustration of our author’s views. 
