• 논리연구
• /
• 제18권3호
• /
• pp.307-335
• /
• 2015

논증의 타당성에 대한 덤밋과 프라위츠의 증명론적 정의의 핵심사항 중의 하나는 열린 논증은 그 전제들에 대한 타당한 논증들을 그 결론에 대한 타당한 논증으로 전환하는 효과적인 방법이 있을 경우 타당하다는 조건이다. 그러나 그들의 정의에서 이 조건은 적절한 의미에서 결정 불가능한 전제들을 지니는 열린 논증들은 모두 사소하게 타당하게 된다는 부적절한 귀결을 지닌다. 필자는 프라위츠의 정의를 중심으로 증명론적 타당성 개념을 설명한 후, 이에 대한 사소성 문제를 제기하고 검토할 것이며, 이에 의거하여 프라위츠의 정의에 대한 수정안을 제시할 것이다.

• 한국수학교육학회지시리즈A:수학교육
• /
• 제24권2호
• /
• pp.105-116
• /
• 1986

For any thought and knowledge, its growth and development has close relation with the society where it is developed and grow. As Feuerbach says, the birth of spirit needs an existence of two human beings, i. e. the social background, as well as the birth of body does. But, at the educational viewpoint, the spread and the growth of such a thought or knowledge that influence favorably the development of a society must be also considered. We would discuss the goal and the function of mathematics education in relation with the prosperity of a technological civilization. But, the goal and the function are not unrelated with the spiritual culture which is basis of the technological civilization. Most societies of today can be called open democratic societies or societies which are at least standing such. The concept of rationality in such societies is a methodological principle which completes the democratic society. At the same time, it is asserted as an educational value concept which explains comprehensively the standpoint and the attitude of one who is educated in such a society. Especially, we can considered the cultivation of a mathematical thinking or a logical thinking in the goal of mathematics education as a concept which is included in such an educational value concept. The use of the concept of rationality depends on various viewpoints and criterions. We can analyze the concept of rationality at two aspects, one is the aspect of human behavior and the other is that of human belief or knowledge. Generally speaking, the rationality in human behavior means a problem solving power or a reasoning power as an instrument, i. e. the human economical cast of mind. But, the conceptual condition like this cannot include value concept. On the other hand, the rationality in human knowledge is related with the problem of rationality in human belief. For any statement which represents a certain sort of knowledge, its universal validity cannot be assured. The statements of value judgment which represent the philosophical knowledge cannot but relate to the argument on the rationality in human belief, because their finality do not easily turn out to be true or false. The positive statements in science also relate to the argument on the rationality in human belief, because there are no necessary relations between the proposition which states the all-pervasive rule and the proposition which is induced from the results of observation. Especially, the logical statement in logic or mathematics resolves itself into a question of the rationality in human belief after all, because all the logical proposition have their logical propriety in a certain deductive system which must start from some axioms, and the selection and construction of an axiomatic system cannot but depend on the belief of a man himself. Thus, we can conclude that a question of the rationality in knowledge or belief is a question of the rationality both in the content of belief or knowledge and in the process where one holds his own belief. And the rationality of both the content and the process is namely an deal form of a human ability and attitude in one's rational behavior. Considering the advancement of mathematical knowledge, we can say that mathematics is a good example which reflects such a human rationality, i. e. the human ability and attitude. By this property of mathematics itself, mathematics is deeply rooted as a good. subject which as needed in moulding the ability and attitude of a rational person who contributes to the development of the open democratic society he belongs to. But, it is needed to analyze the practicing and pursuing the rationality especially in mathematics education. Mathematics teacher must aim the rationality of process where the mathematical belief is maintained. In fact, there is no problem in the rationality of content as long the mathematics teacher does not draw mathematical conclusions without bases. But, in the mathematical activities he presents in his class, mathematics teacher must be able to show hem together with what even his own belief on the efficiency and propriety of mathematical activites can be altered and advanced by a new thinking or new experiences.