Oct 13, 2012

Thompson’s Nurturing of Mathematical Intuition




By Marsigit
Yogyakarta State University

Gödel 1 explained our surprise at the emergence of paradoxes such as Peano's construction of space-filling curves, or Weierstrass's discovery of continuous but nowhere-differentiable functions, by accusing us of


carelessly mixing our pre-theoretic intuitions, with our more refined, analytic and topological ones; such a clash, between familiar geometry, say, and the set-theoretic reduction of point-sets, will undoubtedly arise at some stage; the paradoxical appearance can be explained by a lack of agreement between our intuitive geometrical concepts and the set-theoretical ones occurring in the theorem; therefore, he suggested that we must drive a wedge between our pre-formal and formal intuitions, in the hope of separating out errors coming from using the pre-theoretical intuition. However, Thompson, P.,1993, insisted that Gödel suggestion exercises in discrimination seems notoriously difficult to carry out, especially when it is tempting to refine intuitions of one generation; he claimed that far from being a once-and-for-all clarification of our logical optics, they have historically either turned out to be fallacies, or at best become the naivest intuitions of the next. On the other hand, Thompson 2 indicated that when Frege speaks that the truths of Euclidean geometry, as governing all that is spatially intuitable, looks as though, at last, we may have found a domain in which our intuitions are constrained and held within strict and well-defined bounds. He insisted that while the patterns we are trained to recognise are codified as schemas, the schemas we are most keen to apply are occasionally poorly-tuned, not suitable for the context, or totally in default when we project them into new situations; they may be indispensable as a heuristic, but the fact that they are so familiar often seduces us into the jaws of paradox.

Thompson, P.,1993, further claimed that if intuition in mathematics is properly characterized as a living growing element of our intellect, an intellectual versatility with our present concepts about abstract structures and the relations between these structures, we must recognize that its content is variable and subject to cultural forces in much the same way as any other cultural element. Thompson 3 insisted that even the symbols designed for the expression and development of mathematics have variable meanings, often representing different things in the 19th and 20th centuries, by virtue of the underlying evolution of mathematical thought; it must therefore remain an important strategy to aim to develop an increasingly versatile and expressive medium for the representation of familiar ideas. Further, Thompson argued that as working mathematicians, with increasingly abstract material, it seems that the ability to reason formally, which requires the explicit formulation of ideas, together with the ability to show ideas to be logically derivable from other and more generally accepted ideas, are great assets in broadening the scope and range of the schemas which become second nature to us, and are instrumental in extending the familiar territory of our intuition. Thompson 4 then summed up that during all but a vanishingly small proportion of the time spent in investigative mathematics, we seem to be somewhere between having no evidence at all for our conclusions, and actually knowing them; second, that during this time, intuition often comes to the forefront, both as a source of conjecture, and of epistemic support; third, that our intuitive judgments in these situations are often biased, but in a predictable manner. Ultimately, Thompson 5 concluded that although any satisfactory analysis of the role of intuition in mathematics should recognize it as a versatility in measuring up new situations, or even conjecturing them, using a rich repository of recurrent and strategically-important schemas or conceptual structures, painstakingly abstracted from sensory experience by the intellect, constrained by the languages available to us at the time, and influenced by the accumulated resources of our cultural and scientific heritage: what intuition does not do is constitute an insight gained by reason, through some remarkable clairvoyant power that is an insight, which, for Gödel, seemingly paved the way towards a crystal-clear apocalyptic vision of mathematics, or, for Descartes, paved the way into the ultimate structure of the human mind.

References:
1. Thompson, P.,1993, The Nature And Role Of Intuition In Mathematical Epistemology, University College, Oxford University, U.K,
2Ibid.
3Ibid.
4Ibid.
5Ibid.

3 comments:

  1. Heni Lilia Dewi
    16709251054
    PPs Pendidikan Matematika Kelas C 2016

    Intuisi memang memiliki peran yang sangat penting bagi pengetahuan matematika. Hal ini dipaparkan dan dikuatkan oleh pendapat para ahli, salah satunya yaitu Thompson. Dia mengatakan bahwa peran intuisi dalam matematika yang harus dipahami sebagai fleksibilitas dalam mengukur situasi baru, atau bahkan menebaknya, dengan menggunakan kumpulan skema berulang atau strategis yang penting secara strategis atau struktur konseptual, dengan susah payah diabstraksikan dari pengalaman sensorik oleh intelek, yang dibatasi oleh bahasa yang tersedia bagi kita pada saat itu, dan dipengaruhi oleh sumber daya akumulasi warisan budaya dan ilmiah kita.

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  2. Harumas Anom
    14301244013
    S1 Pendidikan Matematika I 2014
    Dari berbagai pendapat mengenai intuisi, salah satunya pendapat dari Thompson, berpendapat bahwa intuisi adalah apa yang melekat pada pikiran kita, sehingga ketika diberi stimulan suatu persoalan maka intuisi akan bekerja dengan secara otomatis. Intuisi juga harus dilatih agar tetap cekatan dalam respon, terutama dalam proses pembelajaran guru tidak boleh memberikan pengetahuan secara instan, namun harus memberi kesempatan kepada siswa untuk menggunakan intuisinya agar tetap terlatih dan mandiri.

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  3. Nama: Hendrawansyah
    NIM: 17701251030
    S2 PEP 2017 Kelas B

    Assalamualaikum wr wb

    Setiap orang memiliki intuisinya masing-masing.Intuisi dibangun dari hasil pengalaman dan pengetahuan.Intuisi mempunyai andil yang sangat penting dalam membangun dan mengembangkan kecerdasan seseorang di dalam matematika.Dalam postingan ini, menggambarkan bagaimana menjaga dan memlihara intuisi agar tidak pudar dalam diri seseorang.Sesuai dengan pendapatnya Thomson yang mengatakan bahwa sebgai ahli matematika yang selalu berhadapan dengan kehidupan abstrak nampaknya memerlukan pikiran secara formal untuk memperoleh rumusan gagasan sehingga diturunkan secara logis dari gagasan lain dan umumnya diterima.Ini merupaka aset yang paling penting dalam mencapai jangkauan dan memeperluas wilayah intuisi.

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