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Viewed properly, formalism is not a single viewpoint concerning the nature of mathematics. Rather, it is a family of related viewpoints sharing a common framework—a framework that has five key elements. Among these is its revision of the traditional classification of the mathematical sciences. From ancient times onward, the dominant view of mathematics was that it was divided into different sciences. Principal among these were a science of magnitude (geometry) and a science of multitude (arithmetic). Traditionally, this division of mathematics was augmented by an ordering of the two parts in terms of their relative basicness and which was to be taken as the more paradigmatically mathematical. Here it was geometry that was given the priority. The formalist outlook typically rejected this traditional ordering of the mathematical sciences. Indeed, from the latter half of the nineteenth century onward, it typically reversed it. This reversal is the first component of the formalist framework.

*Keywords: *
formalism;
mathematical sciences;
science of magnitude;
science of multitude;
geometry;
formalist framework

*Article.*
*39057 words.*

*Subjects: *
Philosophy of Mathematics and Logic
;
Philosophy of Language

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