Journal ArticleDOI
Geometry and generality in Frege's philosophy of arithmetic
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Some respects in which the philosophy of mathematics can fruitfully be informed by mathematical practice are developed through examining Frege's Grundlagen in its historical setting, through examining the relationship Frege envisions between arithmetic and geometry.Abstract:
This paper develops some respects in which the philosophy of mathematics can fruitfully be informed by mathematical practice, through examining Frege'sGrundlagen in its historical setting. The first sections of the paper are devoted to elaborating some aspects of nineteenth century mathematics which informed Frege's early work. (These events are of considerable philosophical significance even apart from the connection with Frege.) In the middle sections, some minor themes ofGrundlagen are developed: the relationship Frege envisions between arithmetic and geometry and the way in which the study of reasoning is to illuminate this. In the final section, it is argued that the sorts of issues Frege attempted to address concerning the character of mathematical reasoning are still in need of a satisfying answer.read more
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What does it mean to say that logic is formal
TL;DR: For instance, the authors argues that the notion of subsumption is not a concept or relation, but a restriction of functional application itself, since not all functions are concepts, and that subsumption does not correspond to any symbol in the Begriffsschrift, so logical relations that depend on it would be preserved under arbitrary category-preserving translations.
Journal ArticleDOI
Extending knowledge and `fruitful concepts': Fregean themes in the foundations of mathematics
TL;DR: In this paper, the notions de concepts productifs and de connaissance extensible developpees par Frege dans ses "Grundlagen" are discussed.
Journal ArticleDOI
A source book in classical analysis, edited by Garrett Birkhoff. Pp xii, 470. £12·50. 1973. SBN 674 82245 5 (Harvard University Press/Oxford University Press)
Book ChapterDOI
Proof Style and Understanding in Mathematics I: Visualization, Unification and Axiom Choice
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Recent Themes in the History of Early Analytic Philosophy
TL;DR: A survey of the emergence of early analytic philosophy as a subfield of the history of philosophy can be found in this article, where the importance of recent literature on Frege, Russell, and Wittgenstein is stressed, as is the widening interest in understanding the nineteenth-century scientific and Kantian backgrounds.
References
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Book
Introduction to higher order categorical logic
Joachim Lambek,Philip J. Scott +1 more
TL;DR: In this article, Cartesian closed categories and Calculus are used to represent Numerical functions in various categories and to describe the relation between categories. But they do not specify the topology of the categories.
Book
Mathematical Thought from Ancient to Modern Times
TL;DR: In this article, a comprehensive history of mathematical ideas and the careers of the men responsible for them is presented, focusing on geometry and trigonometry, and the role of mathematics in the medieval and early modern periods.
Book
Algebraic curves
TL;DR: A linear transformation with rational maps riemann sphere is presented in this article, where the presentation is kept as elementary as A linear transformations with rational map Riemann spheres the converse is where sense.