=1502.= All quantitative determinations are in the hands of mathematics, and it at once follows from this that all speculation which is heedless of mathematics, which does not enter into partnership with it, which does not seek its aid in distinguishing between the manifold modifications that must of necessity arise by a change of quantitative determinations, is either an empty play of thoughts, or at most a fruitless effort. In the field of speculation many things grow which do not start from mathematics nor give it any care, and I am far from asserting that all that thus grow are useless weeds, among them may be many noble plants, but without mathematics none will develop to complete maturity.--HERBART, J. F. _Werke (Kehrbach), (Langensalza, 1890), Bd. 5, p. 106._ =1503.= There are few things which we know, which are not capable of being reduc’d to a Mathematical Reasoning, and when they cannot, it’s a sign our knowledge of them is very small and confus’d; and where a mathematical reasoning can be had, it’s as great folly to make use of any other, as to grope for a thing in the dark, when you have a candle standing by you.--ARBUTHNOT. _Quoted in Todhunter’s History of the Theory of Probability (Cambridge and London, 1865), p. 51._ =1504.= Mathematical Analysis is ... the true rational basis of the whole system of our positive knowledge.--COMTE, A. _Positive Philosophy [Martineau], Bk. 1, chap. 1._ =1505.= It is only through Mathematics that we can thoroughly understand what true science is. Here alone we can find in the highest degree simplicity and severity of scientific law, and such abstraction as the human mind can attain. Any scientific education setting forth from any other point, is faulty in its basis.--COMTE, A. _Positive Philosophy [Martineau], Bk. 1, chap. 1._ =1506.= In the present state of our knowledge we must regard Mathematics less as a constituent part of natural philosophy than as having been, since the time of Descartes and Newton, the true basis of the whole of natural philosophy; though it is, exactly speaking, both the one and the other. To us it is of less use for the knowledge of which it consists, substantial and valuable as that knowledge is, than as being the most powerful instrument that the human mind can employ in the investigation of the laws of natural phenomena.--COMTE, A. _Positive Philosophy [Martineau], Introduction, chap. 2._ =1507.= The concept of mathematics is the concept of science in general.--NOVALIS. _Schriften (Berlin, 1901), Teil 2, p. 222._