Abstract
To answer the question, "what is the nature of mathematical knowledge?" is one of the important tasks in mathematics education. The purpose of this article is to point out the following as features of mathematical knowledge, by using the examples of Aristotle's syllogism and its Boolean algebraic expressions. (a) It is not always possible to separate the content of some mathematical knowledge from its representations. Modification of the representation of mathematical knowledge necessarily leads to change of the content to some degree. Therefore, we are not always able to present teaching materials in an easier way by modifying their representations. (b) Modification of the representation of mathematical knowledge facilitates the generation of new ideas and the development of mathematical knowledge. This is also important from a perspective of mathematics education, because the sequential system or hierarchy of learning mathematics reflects this feature. Therefore in developing teaching materials in mathematics classes, we should consider not only the merit of a modification from abstract representations of mathematical knowledge to concrete representations of them, but also the demerits, such as losing generality or multiplicity of use with which abstract representations could be endowed.
