# Learning from the History of Mathematics

Mathematics is the historical product of our culture, and it goes without saying that learning about the history of this important aspect of our culture is of great educational value. However, I will not focus here on learning the history of mathematics, but on learning from the history of mathematics, not as a general cultural value but as an integral part of learning the present-day discipline of mathematics.

The use of history in the teaching of mathematics – and of science generally, for that matter – depends either implicitly or explicitly on some view of the rationale of mathematical development. We suppose that changes in mathematics were made for good reasons, which are simultaneously clarified and tested in the process. Therefore analytically compre¬hended accounts of historical developments are relevant means of conveying to learners the rationale of mathematical concepts and precepts. Such rational reconstructions are of limited use for understanding the so-called ‘real’ history of mathe¬matics, yet they are indispensable for learning from it. As has often been remarked, there is a tension between descriptive adequacy and educational demands. By itself, history just tells us of more or less distant and exotic practices to which we have no access other than through a philosophically com¬prehended account of their continuity with present mathematical practices. Abandoning any rational standpoint would land us in a relativism which would prevent us from distinguishing between genuine mathe¬matics and charlatanism, and would deprive us of any sense of continuity and direction. This may be alright as far as history per se is concerned, but it makes the relevance of the history of mathematics for our understanding of present-day mathematics highly questionable, especially to students. There is no learning from the past without some form of reconstruction.

As an example I will focus especially on the discovery of incommensurability by the Pythagoreans, of which Aristotle said that ‘they prove that the side and diagonal of a square are incommensurable by showing that if they were commensurable odd numbers would be equal to even numbers’. This statement is the basis of a reconstitution of the Pythagorean proof that is highly relevant educationally. Equally relevant for a proper understanding of the present-day discipline is Eudoxus’ solution to the problem thrown up by the said Pythagorean discovery, through a radical conceptual change (con¬solidated in Euclid’s Elements) from arithmetic to geometric concepts and the exact definition of proportionality in terms of identity and order relations. They are paradigmatic cases of cognitive activities shifting from the level of practical events and specific objects to the theoretical level of abstract and highly ideal concepts and imaginary suppos¬itions, which shifts could well have their counterparts in the desired cognitive development of the individual learner.