THEORIES AND MODELS 215 



as non-mathematicians we conceive Euclidean geometry as some- 

 thing more than a formal system. Very much as Euclid himself pre- 

 sumably did, we conceive his system as highly endowed with ex- 

 periential relevance— as quite literally a "geometry" usefully appli- 

 cable to the measurement of land. And Euclid's formulation of his 

 geometry does, indeed, embody one element so far unmentioned. 

 He furnishes a set of explicit definitions of his posits. Formally these 

 definitions are wholly superfluous: the Euclidean theorems stem not 

 at all from them but only from the axioms. Once stated, these defini- 

 tions make no subsequent appearance in the system, and they may 

 well seem quite meaningless. For example, Euclid defines a point 

 as "that which has no parts"; and a straight line as "that in which all 

 points lie evenly on one another." But how are we to understand 

 "point" if we are not told what is meant by "parts"? And an attempt 

 to define parts involves us in either a potentially infinite regress or 

 a circularity of greater or less extent. Not knowing the meaning of 

 "lie evenly on," we are in no better position to recognize a straight 

 line than a point. Purely verbal reformulation is of no assistance. 

 Were we, for example, to define a straight line as the shortest dis- 

 tance between two points, then— waiving the difficulty of defining 

 "points"— w^e would have still to grasp the meaning of "shortest dis- 

 tance." The attempt to define this in terms of the straight line is ob- 

 viously an immediate and hopeless circularity. 



Apparently nonoperative and entirely inoperable, Euclid's defini- 

 tions are yet of major importance. For some two millennia, until 

 mathematics had reached a very high level of sophistication, the in- 

 adequacy of Euclid's statement of syntactic rules was automatically 

 made good by the intuition of geometers who never failed to assume 

 what Euclid had himself taken for granted. And in much the same 

 way, for much the same period, men have affiliated with his posits 

 experiential denotations his definitions perhaps imply or evoke but 

 certainly do not state. We may then, for example, take a "point" to be 

 the point of a needle, or the fine hole left thereby in a thin sheet of 

 metal, a "straight line" to be the path of a ray of sunlight selected by 

 passage through that hole, and so on. Proceeding in this fashion, we 

 identify certain of the hitherto wholly abstract terms of the formalism 

 with concepts that have for us reasonably clear denotations, estab- 

 lished perhaps around such tools as the ruler and compass with 

 which we now proceed to "construct" the other entities of Euclidean 



