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Mathematical Discourse From Logic: The Theory ofInquiry (1938) 180 The ability of any logical theory to account for the distinguishing logical characteristics of mathematical conceptions and relations is a stringent test of its claims. A theory such as the one presented in this treatise is especially bound to meet and pass this test. For it has the twofold task of doing justice to the formal character of the certification of mathematical propositions and of showing not merely the consistency of this formal character with the comprehensive pattern ofinquiry, but also that mathematical subject-matter is an outcome of intrinsic developments within that pattern. For reasons suggested in the closing sentence of the last chapter, the interpretation of the logical conditions of mathematical conceptions and relations must be such as to account for the form of discourse which is intrinsically free from the necessity of existential reference while at the same time it provides the possibility of indefinitely extensive existential reference -such as is exemplified in mathematical physics. I.TRANSFORMATION AS A FUNDAMENTAL CATEGORY The end ofinquiry (in the sense in which "end" means both end-in-view, or controlling intent, and terminating close) is institution ofa unified resolved situation. This end is accomplished by institution of subject-matters which are respectively material means and procedural means-factual data and conceptual meanings. These instrumental subject-matters are instituted by operations in which the existential material of a given problematic situation is experimentally modified in a given direction. Conceptual subject-matters,consisting of possibilities of solution, are at the same time so constructed as to direct the operations of experimental selection and ordering by which transformation of existential material toward the end of a resolved situation is effected. The conceptions that represent possibilities of solutions must, moreover, ifinquiry is controlled, be propositionally formulated; and these propositions must be developed in ordered series so as to yield a final general proposition capable of directing in operations definitely applicable to the material of the special prob- lem in hand. Otherwise, there is an inference so premature as to yield an ungrounded proposition . In short, ordered discourse is itself a series of transformations conducted according to rules of rigorous (or necessary) and fruitful substitution ofmeanings. Such transformation is possible only as a system of interrelated abstract characters is instituted. Common sense conceptions, for example, do not satisfy the conditions of systematic interrelation. Hence the change of content they undergo in science as they are modified to satisfy this condition. Transformation of conceptual contents , according to rules of method that satisfy determinate logical conditions, is thus involved both in conduct of discourse and in the formation of the conceptions that enter into it even when discourse is intended to have final existential application. The logical principle involved may be restated in the following ways: (1) The subjectmatter or content of discourse consists of possibilities . Hence the contents are non-existential even when instituted and ordered with reference to existential application. (2) As possibilities , they require formulation in symbols. Symbolization is not a convenience found to be practically indispensable in discourse, nor yet a mere external garb for ideas already complete in themselves. It is of the very essence of discourse as concerned with possibilities. In their functional capacity, however, symbols have the same logical status as existential data. For this reason they are themselves subject to transformations. Historically, the operations by which symbol-meanings are transformed were first borrowed from and closely allied to physical operations-as is indicated in the words still used to designate rational operations; in gross, in such words as deliberation, pondering, reflection, and more specifically in counting and calculation. As meanings were modified to satisfy the conditions imposed by membership in an interrelated system, operations were also modified to meet the requirements of the new conceptual material. Operations became as abstract as the materials to which they apply and hence of a character expressed, and capable only of expression, in a new order of symbols. In the chapters preceding the present one, Mathematical Discourse we have been concerned with the relation of meanings and propositions in discourse where discourse is conducted in reference to some final existential applicability: In discourse of this type application is suspended or held in abeyance but relationship to application is not eliminated in respect to the content of the conceptions. When, however, discourse is conducted exclUSively with reference to satisfaction of its own logical conditions, or, as we say; for its own sake, the subject-matter...

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