Abstract

I elaborate in some detail on the First Book of Euclid's ``Elements'' and show that Euclid's theory of geometry is \underline{not} axiomatic in the modern sense but is construed differently. Then I show that the usual commonly accepted notion of axiomatic theory equally fails to account for today's mathematical theories. I provide some polemical arguments against the popular view according to which a good mathematical theory must be axiomatic and point to an alternative method of theory-building. Since my critique of the core axiomatic method is constructive in its character I briefly observe known constructive approaches in the foundations of mathematics and describe the place of my proposal in this context. The main difference of my and earlier constructive proposals for foundations of mathematics appears to be the following: while earlier proposals deal with the issue of admissibility of some particular mathematical principles and like choice and some putative mathematical objects like infinite sets my proposal concerns the very method of theory-building. As a consequence, my proposal unlike earlier constructive proposals puts no restriction on the existing mathematical practice but rather suggests an alternative method of organizing this practice into a systematic theoretical form. In the concluding section of the paper I argue that the constructive mathematics better serves needs of mathematically-laden empirical sciences than the formalized mathematics.