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Poststructuralism and Deconstruction: A Mathematical History
Abstract
Explaining his love of philosophy, Slavoj Žižek notes that he 'secretly thinks reality exists so that we can speculate about it'. This article takes the view that links between mathematics and continental philosophy are part of reality, the reality of philosophy and its history, and hence require speculation. Examples from the work of Jacques Derrida and Henri Poincaré are discussed.
... From Plato to Husserl and from Kant to Bertrand Russell, many philosophers have been influenced by their contemporary (and sometimes by their own) mathematical thoughts and initiatives, while mathematicians have always reflected on the philosophical movements of their period. For a good overview of this cross-fertilisation, see, e.g., [10]. Since the first half of the 20th century, well-known and popular(ised) concepts of the philosophical mainstream have been, among others, logicism (Frege, Russell), logical positivism (Vienna Circle) and formalism (Hilbert). ...
The traditional way of presenting mathematical knowledge is logical deduction, which
implies a monolithic structure with topics in a strict hierarchical relationship. Despite many recent developments and methodical inventions in mathematics education, many curricula are still close in spirit to this hierarchical structure. However, this organisation of mathematical ideas may not be the most conducive way for learning mathematics. In this paper, we suggest that flattening curricula by developing self-contained micro topics and by providing multiple entry points to knowledge by making the dependency graph of notions and subfields as sparse as possible could improve the effectiveness of teaching mathematics. We argue that a less strictly hierarchical schedule in mathematics education can decrease mathematics anxiety and can prevent students from ‘losing the thread’ somewhere in the process. This proposal implies a radical re-evaluation of standard teaching methods. As such, it parallels philosophical deconstruction. We provide two examples of how the micro topics can be implemented and consider some possible criticisms of the method. A full-scale and instantaneous change in curricula is neither feasible nor desirable. Here, we aim to change the prevalent attitude of educators by starting a conversation about the flat curriculum alternative.
... 288-291] , which is rather closer to the continental philosophy, and speaks also about the 'deconstruction' of the concept of mathematical 'proof'. For more about Derrida, and the relashionships of Poincaré, Brouwer and Weyl with post-structuralism and deconstruction see the very insightful paper by Tacić, [65], as well as his book [64]. The above list gives a very short description of the contemporary currents of the philosophy of mathematics, which are beyond those described for instance in Shapiro's book. ...
First a survey of some basic ingredients, like cognition and intelligence and split brain research is presented. Then, some proposals on which this paper is based are exposed. We proceed with an exposition of the main points in the development of the concepts of 'structure' and 'points', while efforts shall be made to delineate the concept of 'structure' and the Protean nature and the dialectics of the concept of 'point' in various characteristic cases. On the basis of the above the concept of 'level of reality' is examined in connection with the concept of 'completeness' and its importance in mathematics in general. Finally some conclusions for mathematics are drawn and a survey of the present state of the philosophy of mathematics is presented, pointing out also some omissions, that we usually find in the current literature.
... Conversely, mathematics had a profound influence on philosophy as far back as Pythagoras, through Plato, Spinoza, Kant, Frege, Wittgenstein, Russel, Husserl, Heidegger, and almost everybody, all the way to Derrida. Speaking of Derrida, Tasik has a very insightful paper [9] , soon to be expanded into a book, about Deconstruction and mathematics. In this lecture, I will not discuss philosophy per se, but will attempt to show how Derrida's seminal insights have the potential to revolutionize the practice of doing mathematics. ...
The inequality (DERRIDA+TURING)>(DERRIDA)+(TURING) will be illustrated by computerized deconstruction of Roger Apéry's miraculous proofs of irrationality.
Short abstract: The closely interrelated themes of affordance and agency in medias res are brought together in a case study of the development of expertise in archaeology by focusing on learning to identify (type) pottery, and on learning to excavate. In learning to type pottery, a novice is inculcated into the language-games of pottery. The formulation of typologies, meanwhile, shows how such language-games form, and how these language-games afford a semantic field that supports archaeologically mundane communications between archaeologists. The event of an excavation is used to focus on social dynamics seen from a perspective of agency in medias res and to demonstrate how wider social, economic and political influences intervene within archaeological discourse and practice to alter the agency of archaeologists in terms of their cognitive authority, and that of archaeology as discipline.
Brouwer and Weyl recognized that the intuitive continuum requires a mathematical analysis of a kind that set theory is not
able to provide. As an alternative, Brouwer introduced choice sequences. We first describe the features of the intuitive continuum
that prompted this development, focusing in particular on the flow of internal time as described in Husserl's phenomenology.
Then we look at choice sequences and their logic. Finally, we investigate the differences between Brouwer and Weyl, and argue
that Weyl's conception of choice sequences is defective on several counts.
- Jacques Derrida
Jacques Derrida, Positions (Chicago, Chicago University Press, 1981), p. 35.
English translation quoted from Arthur Miller
- See Henri Poincaré
See Henri Poincaré, "Les géometries non-euclidiPnnes," Rev. Génerale Sci. Pures Appl. 2
(1891), 769-74. English translation quoted from Arthur Miller, Imagery in Scientific Thought
(Cambridge, Mass., MIT Press, 1986), p. 19.
5.
Ibid.
English translation quoted from Manfred Frank, What is Neostructuralism?
- See Ferdinand De Saussure
See Ferdinand de Saussure, Cours de linguistique génerale, Edition Critique, ed. by Rudolf
Engler, Vol. 2 (Wiesbaden, Harrasowitz, 1967-74), p. 23. English translation quoted from
Manfred Frank, What is Neostructuralism? (Minneapolis, University of Minnesota Press,
1989), p. 426.
English translation quoted from van Heijenoort
- Hermann Weyl
Hermann Weyl, "Diskussionbemerkungen zu dem zweiten Hilbertschen Vortag über die
Grundlagen der Mathematik," Abhaundlungen aus dem Mathematischen Seminar der
Hamburgischen Universität 6 (1928), 86-88. English translation quoted from van Heijenoort,
J., From Frege to Gödel (Cambridge, Mass., Harvard University Press, 1967), p. 484.
English translation quoted from van Heijenoort, From Frege to Gödel
- David Hilbert
David Hilbert, "Die Grundlagen der Mathematik," Abhaundlungen aus dem Mathematischen
Seminar der Hamburgischen Universität 6 (1928), 65-85. English translation quoted from van
Heijenoort, From Frege to Gödel, p. 475.
Quantized Singularities in the Electromagnetic Field
- See Paul Dirac
See Paul Dirac, "Quantized Singularities in the Electromagnetic Field," Proc. Roy. Soc.
London A 133 (1931), 60-72.