=129.= The object of pure Physic is the unfolding of the laws of the intelligible world; the object of pure Mathematic that of unfolding the laws of human intelligence.--SYLVESTER, J. J. _On a theorem, connected with Newton’s Rule, etc., Collected Mathematical Papers, Vol. 3, p. 424._ =130.= First of all, we ought to observe, that mathematical propositions, properly so called, are always judgments _a priori,_ and not empirical, because they carry along with them necessity, which can never be deduced from experience. If people should object to this, I am quite willing to confine my statements to pure mathematics, the very concept of which implies that it does not contain empirical, but only pure knowledge _a priori_.--KANT, IMMANUEL. _Critique of Pure Reason [Müller], (New York, 1900), p. 720._ =131.= Mathematics, the science of the ideal, becomes the means of investigating, understanding and making known the world of the real. The complex is expressed in terms of the simple. From one point of view mathematics may be defined as the science of successive substitutions of simpler concepts for more complex.... --WHITE, WILLIAM F. _A Scrap-book of Elementary Mathematics, (Chicago, 1908), p. 215._ =132.= The critical mathematician has abandoned the search for truth. He no longer flatters himself that his propositions are or can be known to him or to any other human being to be true; and he contents himself with aiming at the correct, or the consistent. The distinction is not annulled nor even blurred by the reflection that consistency contains immanently a kind of truth. He is not absolutely certain, but he believes profoundly that it is possible to find various sets of a few propositions each such that the propositions of each set are compatible, that the propositions of each such set imply other propositions, and that the latter can be deduced from the former with certainty. That is to say, he believes that there are systems of coherent or consistent propositions, and he regards it his business to discover such systems. Any such system is a branch of mathematics. --KEYSER, C. J. _Science, New Series, Vol. 35, p. 107._ =133.= [Mathematics is] the study of ideal constructions (often applicable to real problems), and the discovery thereby of relations between the parts of these constructions, before unknown.--PEIRCE, C. S. _Century Dictionary, Article “Mathematics.”_ =134.= Mathematics is that form of intelligence in which we bring the objects of the phenomenal world under the control of the conception of quantity. [Provisional definition.]--HOWISON, G. H.