• Journal of Educational Research in Mathematics
• /
• v.27 no.4
• /
• pp.727-745
• /
• 2017

This study investigated the Aristotle's continuity and the historical development of continuity of function to explore the differences between the concepts of mathematics and students' thinking about continuity of functions. Aristotle, who sought the essence of continuity, characterized continuity as an 'indivisible unit as a whole.' Before the nineteenth century, mathematicians considered the continuity of functions based on space, and after the arithmetization of nineteenth century modern ${\epsilon}-{\delta}$ definition appeared. Some scholars thought the process was revolutionary. Students tended to think of the continuity of functions similar to that of Aristotle and mathematicians before the arithmetization, and it is inappropriate to regard students' conceptions simply as errors. This study on the continuity of functions examined that some conceptions which have been perceived as misconceptions of students could be viewed as paradigmatic thoughts rather than as errors.

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