Justification in Mathematics and its Teaching

While we have the term "proving" for "deductive reasoning in mathematics", there is no consented term for the devising and evaluation of reasons leading to the acceptance or rejection of axioms and definitions. We suggest to speak about "justifying" in this case. The terms "proving" and "justifying" can be used in a mutually exclusive way in a given theoretical framework, because axioms or definitions are not proved, and for a theorem it is not sufficient "only" to be justified. Processes of justification are an essential element of mathematical practice and reasoning without which the activity of proving and the role of axioms cannot be understood. To state more precisely the problem treated in the present paper, we start from a statement by Felix Klein. We then discuss the complementary relation of justifying and proving and give two historical examples. In the online supplementary material, the question of how "justifying" could and should play a role in mathematics education in secondary school is discussed along two examples.