Philosophical Origins in Mathematics ?

Since the turn of the 19 th and 20 th centuries when systematic research on antique Greek mathematics began, the general tone of historiographical work in the field has been dominated by the so-called internalist attitude. H.G. Zeuthen, P. Tannery and others initiated a research tradition whose goal is to reconstruct the mathematical development of the minus 5 th and 4 th century Greece by providing a technical and conceptual analysis of the available source texts, notably of Euclid's Elements. Such a reconstruction relies on today's conception of mathematical knowledge, inasmuch as it proceeds by decontextualising antique texts and separating their 'mathematical content' from the 'form of expression', the latter regarded as incidental and irrelevant. According to this view, the history of mathematics appears as a development of interrelated mathematical concepts and propositions, and the growth of mathematical knowledge lies in our increasing understanding of the network of these conceptual relations. Thus our better understanding of mathematics can, and should, shed light on the actual content of ancient mathematical texts by disregarding the 'clumsy' and 'cumbersome' language of expression and reformulating the original problems in the more suitable mathematical language of our modern age. Notwithstanding some seriously problematic characteristics of this attitude to which I shall partly return, the internalist historiographic tradition developed an insightful and coherent interpretation of early Greek mathematics such as had never been available before. One of its key concepts is 'geometric algebra', which expresses the view that many of the mathematical problems solved in Euclid are genuinely algebraic problems dressed up in a 'geometric garb'. There are two reasons why the Greeks used a geometrical language to solve algebraic problems: (i) the supposedly scandalous discovery that some 'magnitudes' (quantities) cannot be measured with the same unit, i.e. they are 'incommensurable' and cannot be written as the ratio of any two integers, while there is no problem in representing