デデキントの数学観 : 大学教授資格取得講演における概念拡張の仕組み

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  • Dedekind's View on Mathematics : The Framework of Concept Extension in Habilitationsrede
  • デデキント ノ スウガクカン : ダイガク キョウジュ シカク シュトク コウエン ニ オケル ガイネン カクチョウ ノ シクミ

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抄録

Various extensions of concepts take place in the course of developments of mathematical theories. Dedekind in his Habilitationsrede makes a penetrating observation on the development of concepts in mathematics and in science generally. He claims that the creative development of science not only is an historical fact, but also originates in the inner necessity (innere Notwendigkeit) of a theory under concern. Namely, a new stage of a theory is geared to one direction or another by its past results as well as by its inner necessity. In order to speculate on the conditions according to which a concept extension can be viewed as appropriate, I will analyze this reference in detail. The significance of this reference lies in its attempt to clarify the relationship or the tangent point between the objects before and those after the concept extension. It is my view that one should regard an extension as homogeneous if the two families of objects share a same function. Dedekind discusses concept extensions with various examples. A study of these examples shows that they can be classified into five mathematically distinct scenes. Through a careful analysis of those scenes, I have been led to the three principles which constitute the framework that enables us to explain homogeneity of an extension common to all those scenes. Those principles are the following : necessity (The extension is necessitated by an inner demand of a mathematical theory.); soundness (The confinement of an extended concept to the original domain coincides with the original concept.); conservation (The laws characterizing the original concept are valid also in the extended theory.). All the cases which Dedekind treats abide by these principles. The fact that there are many significant examples of such extensions in modern mathematics should justify the use of this framework for an account of homogeneous extensions.

収録刊行物

  • 哲學研究

    哲學研究 596 24-45, 2013-10-10

    京都哲学会 (京都大学大学院文学研究科内)

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