This article is written by Prof. S. D. Agashe. I greatly enjoyed it when I was a graduate student in department. I hope, some of you will enjoy reading this too.
Euclidean geometry and the axiomatic method
Euclid’s Elements constitutes the earliest extant substantial presentation of a body of material in the axiomatico-deductive form . Through it the subject of geometry got permanently associated with axiomatico-deductive formulation which was then viewed as a method, so much so that the expression ‘more geometrico’ (the geometric way) became synonymous with axiomatico-deductive formulation. Thus arose the general belief, especially in methodological quarters, that Euclid’s Elements and, in particular, Euclid’s geometry were merely instances of the application of a previously thought out/discovered/known method, and, thus, that the axiomatico-deductive method existed prior to the axiomatico-deductive formulation of geometry. Using Euclid’s Elements as my principal evidence, I want to suggest that the true state of affairs is the other way round. The axiomatico-deductive formulation of geometry emerged out of a successful attempt- most probably by some of Euclid’s predecessors – to solve some geometrical problems. Once this was done, it was seen by these geometers and also, of course, by Euclid as an instrument of open-ended discovery. Only, then, could the germs of a method be seen in it. My view of the genesis of the axiomatic method emboldens me to suggest further that in general a method, which is something consciously conceived, arises as the result of reflection on an activity that is already being pursued ‘intuitively’. Again, once the method is consciously conceived, it can engender new activity being pursued consciously in accordance with the method, i.e. methodically.