MATHEMATICS AND THE SCIENCES — LASLEY 191 



This invariant incorporates the unity, if any, jji-esent in the differ- 

 ences of action in the situation in question. This unity answers the 

 question as to how we may see the permanent in the transitory. The 

 scope of it tells us how we may see the general in what is particular. 

 In civil law we have to make the statute. Whether we like it or 

 not, that statute may be broken. Still in the changing pattern of 

 civil law the statute formulates what unity is possible in the diversity 

 of action which it seeks to control. In natural law these differences 

 in action take the form of the great dissimilarities in observed phe- 

 nomena. The unity is the common part, if such there be. The 

 natural law expresses this unity amid the action differences. Its 

 form is never final until it partakes of the form of the mathematical 

 invariant. 



POSTULATION 



Keflections on the nature of law bring forcibly to our minds the 

 postulational character of our thinking. We do well to examine the 

 meaning of our most fundamental concepts as well as the lines of 

 argument leading to our most important conclusions. Every science 

 has its undefined terms. Aught else is an infinite regression. When 

 analysis fails, we rely on the properties of our concept to define it 

 for us. In setting up the discipline for a science, some of the prop- 

 ositions must be accepted without proof for similar reasons. The 

 criterion for choice is simplicity. This is not as simple as the name 

 indicates. By simplicity, as used here, is meant logical simplicity. 

 As Einstein so aptly words it, "By 'simplest' we mean that system 

 which contains fewest possible mutually independent postulates, or 

 axioms." This attitude of modern science is far removed from New- 

 ton's hypotheses non fingo. It is an attitude undoubtedly provided 

 by the mathematician. Einstein continues, "Nature is the realiza- 

 tion of the simplest conceivable mathematical ideas. I am convinced 

 that we can discover by means of purely mathematical constructions, 

 the concepts and the laws connecting them with each other, which 

 furnish the key to the understanding of natural phenomena. Exper- 

 ience may suggest the appropriate mathematical concepts, but they 

 most certainly cannot be deducted from it. Experience remains, of 

 course, the sole criterion of the physical utility of a mathematical 

 construction. But the creative principle resides in mathematics." 



Such a postulational approach to mathematical thinking was seen 

 by Euclid insofar as our inability to define satisfactorily all our 

 terms. The fact that even a mathematician cannot prove every- 

 thing was not formulated until Pasch, almost in our own time. Now 

 the necessity of a postulational approach to both definitions and 

 theorems is a universally accepted tenet of the mathematician. 



