Throughout its long history, mathematics has involved the use ofsystems of written signs, most notably, diagrams in Euclidean geometry and formulae in the symbolic language of arithmetic and algebra in the mathematics of Descartes, Euler, and others. Such systems of signs, I argue, enable one to embody chains of mathematical reasoning. I then show that, properly understood, Frege’s Begriffsschrift or concept-script similarly enables one to write mathematical reasoning. Much as a demonstration in Euclid or in early modern algebra does, a (...) proof in Frege’s concept-script shows how it goes. (shrink)

In his Locke Lectures Brandom proposes to extend what he calls the project of analysis to encompass various relationships between meaning and use. As the traditional project of analysis sought to clarify various logical relations between vocabularies so Brandom’s extended project seeks to clarify various pragmatically mediated semantic relations between vocabularies. The point of the exercise in both cases is to achieve what Brandom thinks of as algebraic understanding. Because the pragmatist critique of the traditional project of analysis was precisely (...) to deny that such understanding is appropriate to the case of natural language, the very idea of an analytic pragmatism is called into question by that critique. My aim is to clarify the prospects for Brandom’s project, or at least something in the vicinity of that project, through a comparison of it with what I will suggest we can think of as Kant’s analytic pragmatism as developed by Peirce. (shrink)

Newton characterizes the reasoning of Principia Mathematica as geometrical. He emulates classical geometry by displaying, in diagrams, the objects of his reasoning and comparisons between them. Examination of Newton’s unpublished texts shows that Newton conceives geometry as the science of measurement. On this view, all measurement ultimately involves the literal juxtaposition—the putting-together in space—of the item to be measured with a measure, whose dimensions serve as the standard of reference, so that all quantity is ultimately related to spatial extension. I (...) use this conception of Newton’s project to explain the organization and proofs of the first theorems of mechanics to appear in the Principia. The placementof Kepler’s rule of areas as the first proposition, and the manner in which Newton proves it, appear natural on the supposition that Newton seeks a measure, in the sense of a moveable spatial quantity, of time. I argue that Newton proceeds in this way so that his reasoning can have the ostensive certainty of geometry. (shrink)

Françios Viète was a geometer in search of better techniques for astronomical calculation. Through his theorem on angular sections he found a use for higher-dimensional geometric magnitudes which allowed him to create an algebra for geometry. We show that unlike traditional numerical algebra, the knowns and unknowns in Viète’s logistice speciosa are the relative sizes of non-arithmetized magnitudes in which the “calculations” must respect dimension. Along with this foundational shift Viète adopted a radically new notation based in Greek geometric equalities. (...) His letters stand for values rather than types, and his given values are undetermined. Where previously algebra was founded in polynomials as aggregations, Viète became the first modern algebraist in working with polynomials built from operations, and the notations reflect these conceptions. Viète’s innovations are situated in the context of sixteenth-century practice, and we examine the interpretation of Jacob Klein, the only historian to have conducted a serious inquiry into the ontology of Viète’s “species”. (shrink)