=1313.= It is commonly considered that mathematics owes its certainty to its reliance on the immutable principles of formal logic. This ... is only half the truth imperfectly expressed. The other half would be that the principles of formal logic owe such a degree of permanence as they have largely to the fact that they have been tempered by long and varied use by mathematicians. “A vicious circle!” you will perhaps say. I should rather describe it as an example of the process known by mathematicians as the method of successive approximation.--BÔCHER, MAXIME. _Bulletin of the American Mathematical Society, Vol. 11, p. 120._ =1314.= Whatever advantage can be attributed to logic in directing and strengthening the action of the understanding is found in a higher degree in mathematical study, with the immense added advantage of a determinate subject, distinctly circumscribed, admitting of the utmost precision, and free from the danger which is inherent in all abstract logic,--of leading to useless and puerile rules, or to vain ontological speculations. The positive method, being everywhere identical, is as much at home in the art of reasoning as anywhere else: and this is why no science, whether biology or any other, can offer any kind of reasoning, of which mathematics does not supply a simpler and purer counterpart. Thus, we are enabled to eliminate the only remaining portion of the old philosophy which could even appear to offer any real utility; the logical part, the value of which is irrevocably absorbed by mathematical science.--COMTE, A. _Positive Philosophy [Martineau], (London, 1875), Vol. 1, pp. 321-322._ =1315.= We know that mathematicians care no more for logic than logicians for mathematics. The two eyes of exact science are mathematics and logic: the mathematical sect puts out the logical eye, the logical sect puts out the mathematical eye; each believing that it can see better with one eye than with two. --DE MORGAN, A. _Quoted in F. Cajori: History of Mathematics (New York, 1897), p. 316._ =1316.= The progress of the art of rational discovery depends in a great part upon the art of characteristic (ars characteristica). The reason why people usually seek demonstrations only in numbers and lines and things represented by these is none other than that there are not, outside of numbers, convenient characters corresponding to the notions.--LEIBNITZ, G. W. _Philosophische Schriften [Gerhardt] Bd. 8, p. 198._