• Journal for History of Mathematics
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• v.20 no.3
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• pp.105-126
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• 2007

The present paper considers how to teach mathematical concepts. In particular, we aim to a balanced, unified achievement for three elements of concept loaming such as concept understanding, computation and application through one's mathematical intuition. In order to do this, we classify concepts into three patterns, that is, intuitive concepts, logical concepts and formal concepts. Such classification is based on three kinds of philosophy of mathematics : intuitionism, logicism, fomalism. We provide a concrete, practical investigation with important nine concepts in theory of vectors from the viewpoint of three patterns of concepts. As a consequence, we suggest certain solutions for an effective concept learning in teaching theory of vectors.

• Journal for History of Mathematics
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• v.22 no.3
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• pp.1-30
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• 2009

Frege's logicism has been frequently regarded as a development in number theory which succeeded to the so called arithmetization of analysis in the late 19th century. But it is not easy for us to accept this opinion if we carefully examine his actual works on real analysis. So it has been often argued that his logicism was just a philosophical program which had not contact with any contemporary mathematical practices. In this paper I will show that these two opinions are all ill-founded ones which are due to the misunderstanding of the theoretical place of Frege's logicism in the context of contemporary mathematical practices. Firstly, I will carefully examine Cantorian definition of real numbers and Frege's critiques of it. On the basis of this, I will show that Frege's aim was to produce the purely logical definition of ratios of quantities. Secondly, I will consider the mathematical background of Frege's logicism. On the basis of this, I will show that his standpoint in real analysis was much subtler than what we used to expect. On the one hand, unlike Weierstrass and Cantor, Frege wanted to get such real analysis that could be universally applicable. On the other hand, unlike most mathematicians who insisted on the traditional conceptions, he would not depend upon any geometrical considerations in establishing real analysis. Thirdly, I will argue that Frege regarded these two aspects - the independence from geometry and the universal applicability - as those which characterized logic itself and, by logicism, arithmetic itself. And I will show that his conception of real numbers as ratios of quantities stemmed from his methodological maxim according to which the nature of numbers should be explained by the common roles they played in various contexts to which they applied, and that he thought that the universal applicability of numbers could not be adequately explicated without such an explanation.

• Research in Mathematical Education
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• v.7 no.1
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• pp.1-10
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• 2003

This research reviewed literatures on proof in mathematics education. Several views of proof can be classified (and identified) such as psychological approach (Platonism, empiricism), structural approach (logicism, formalism, intuitionism) and social approach (ontology, axiomatic systems). All these views of proof are valuable in mathematics education society. The concept of proof can be found in the form of analytic knowledge not of constructive knowledge. Human beings developed their knowledge in the sequence of constructive knowledge to analytic knowledge. Therefore, in mathematics education, the curriculum of mathematics should involve the process of cognitive knowledge development.

• Journal for History of Mathematics
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• v.12 no.1
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• pp.32-44
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• 1999

This paper is a sequel to [8]. Development of modern logic was initiated by Boole and Morgan. Boolean logic is one of their completed works. Cantor created the set theory along with cardinal and ordinal numbers. His theory on infinite sets brought about a remarkable development on modern mathematical theory, but generated many paradoxes (e.g. Russell Paradox) that in turn motivated mathematicians to solve them. Further, mathematicians attempted to construct sound foundations for Mathematics. As a result three important schools of thought were formed in relation to fundamentals of mathematics for the resolution of paradoxes of set theory, namely logicism developed by Russell and Whitehead, intuitionism lead by Brouwer and formalism contended by Hilbert and Bernays. In this paper, we examine the logic for intuitionism which is originated by Brouwer in 1908 and study Heyting algebra.

• Korean Journal of Logic
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• v.23 no.2
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• pp.117-159
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• 2020

In the early Wittgenstein's Tractatus, both philosophy of logic and that of mathematics belong to the most crucial subjects of it. What is the philosophical view of the early Wittgenstein in the Tractatus? Did he, for example, accept Frege and Russell's logicism or reject it? How did he stipulate the relation between logic and mathematics? How should we, for example, interpretate "Mathematics is a method of logic."(6.234) and "The Logic of the world which the proposition of logic show in the tautologies, mathematics shows in equations."(6.22)? Furthermore, How did he grasp the relation between mathematical equations and tautologies? In this paper, I will endeavor to answer these questions.

• Korean Journal of Logic
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• v.8 no.1
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• pp.69-88
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• 2005

Hume's principle says that the number of Fs is the same as the number of Gs iff there are just as many Fs as Gs. Frege seems to suggest at Grundlagen $\S64$ that (i) the content of the two sentences are the same, (ii) the left hand side sentence is a result of 'carving up the content' of the right hand side in a new way, (iii) 'the true order of things' are from the right to left rather than the other way round. We examine here if there is a room for arguing these three theses altogether within Frege's philosophy, and give a positive answer to it.