The Problem about the Existence of an Irrational Number √2 in the Examples of the Proof by Contradiction in the Textbooks according to the Perspective of Intuitionistic Mathematics versus Classical Mathematics: ‘√2 is Not a Rational Number’ versus ‘√2 is an Irrational Number’

Intuitionistic mathematics philosophy, which distanced itself from classical mathematics, takes the view that mathematical objects are constructed by the human mind, like constructivism, which has been discussed in mathematics education. This paper aims to analyze the content of school mathematics and obtain educational implications based on a discussion of central issues in intuitionistic mathematics. Concerning the proof by contradiction, which can reveal the difference between intuitionistic mathematics and classical mathematics, among the expressions of the proposition presented as an example of the proof by contradiction in textbooks, in ‘√2 is not a rational number’, contrast to ‘√2 is an irrational number’, avoided a mention about its existence as an irrational number. With this background, we examined the issues of intuitionistic mathematical philosophy, focusing on the existence of mathematical objects, and based on this, we analyzed the textbook's related proofs by contradiction in terms of the existence of √2 as an irrational number. This analysis found that the perspective of intuitionistic mathematics permeates the expression and the proof which do not mention and avoid the existence of √2 as an irrational number. Based on the above, we discussed the reflection on the existence of mathematical objects in school mathematics and the educational value of mathematics from a humanistic aspect.