Recent foundational approaches to Infinitesimal Analysis are essentially algebraic or computational, whereas the first approaches to such problems were geometrical. From this perspective, we may recall the seventeenth-century investigations of the “inverse tangent problem.” Suggested solutions to this problem involved certain machines, intended as both theoretical and actual instruments, which could construct transcendental curves through so-called tractional motion. The main idea of this work is to further develop tractional motion to investigate if and how, at a very first analysis, these ideal machines (like the ancient straightedge and compass) can constitute the basis of a purely geometrical and finitistic axiomatic foundation (like Euclid’s planar geometry) for a class of differential problems. In particular, after a brief historical introduction, a model of such machines (i.e., the suggested components) is presented. Then, we introduce some preliminary results about generable functions, an example of a “tractional” planar machine embodying the complex exponential function, and, finally, a didactic proposal for this kind of artifact.

milici, p. (2014). A Geometrical Constructive Approach to Infinitesimal Analysis: Epistemological Potential and Boundaries of Tractional Motion. In From Logic to Practice (pp. 3-21). Springer [10.1007/978-3-319-10434-8_1].

A Geometrical Constructive Approach to Infinitesimal Analysis: Epistemological Potential and Boundaries of Tractional Motion

MILICI, Pietro
2014-01-01

Abstract

Recent foundational approaches to Infinitesimal Analysis are essentially algebraic or computational, whereas the first approaches to such problems were geometrical. From this perspective, we may recall the seventeenth-century investigations of the “inverse tangent problem.” Suggested solutions to this problem involved certain machines, intended as both theoretical and actual instruments, which could construct transcendental curves through so-called tractional motion. The main idea of this work is to further develop tractional motion to investigate if and how, at a very first analysis, these ideal machines (like the ancient straightedge and compass) can constitute the basis of a purely geometrical and finitistic axiomatic foundation (like Euclid’s planar geometry) for a class of differential problems. In particular, after a brief historical introduction, a model of such machines (i.e., the suggested components) is presented. Then, we introduce some preliminary results about generable functions, an example of a “tractional” planar machine embodying the complex exponential function, and, finally, a didactic proposal for this kind of artifact.
http://link.springer.com/chapter/10.1007/978-3-319-10434-8_1
milici, p. (2014). A Geometrical Constructive Approach to Infinitesimal Analysis: Epistemological Potential and Boundaries of Tractional Motion. In From Logic to Practice (pp. 3-21). Springer [10.1007/978-3-319-10434-8_1].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10447/105087
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