From pure constructivism to imaginative constructivism – An analysis of Intuitionist Mathematics from the viewpoint of design theory

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    • Published: 08-19-2015 03:47 am
    • "Abstract: The paper considers one of the main constructivist mathematical theories - the Intuitionism. Brouwer, the father of Intuitionist Mathematics, describes mathematics as the study of mental mathematical constructions realized by a creative subject. This perspective on mathematics bears many similarities with more conventional design processes. This is not surprising considering the fact that both fields’ common foundational concern is defining new objects with novel properties. The paper uses C-K design theory as an analytical framework to interpret notions, results and philosophy of Brouwer. The mathematical reasoning process Brouwer explains is identified as a conceptive reasoning process, in the sense of C-K theory. From a design standpoint, the authenticity of Intuitionism is the introduction of incomplete objects and their expansion into the heart of mathematics by means of lawless sequences and free choices. Nevertheless, even though Intuitionist Mathematics authorize, and even welcome, design concepts and the reasoning process of the creative subject implicitly makes use of them, a concept space C and its impact on learning and reasoning is not considered. As such, Intuitionism remains a purely Constructivist approach as opposed to an Imaginative Constructivism it approximates. This, result combined with many parallels between the C-K theory and Intuitionist Mathematics, opens up interesting research perspectives both for Design theory and Philosophy of Mathematics."

      http://cgs-mines-paristech.fr/wp-content/uploads/2012/02/Kazakci.Hatchuel.Brouwer.v17.pdf

      I think both the constructivists, subjectivists and intuistic schools of mathematics relate to Abstract in NTJ and empirical, logicism, platonisism mathematics relate to Concrete in NTJ.
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