96 ROYCE— PRINCIPLES OF THEORETICAL SCIENCE. 



[April 19, 



own separate light, can the logical theory of science be founded. In 

 general what we call first principles are such merely in some cer- 

 tain respect, or from some special point of view. Otherwise viewed, 

 these same principles may appear as derived. And to discuss the 

 various ways in which such derivation may be brought to light, is 

 one of the principle problems of modern logical theory. 



V. 



The relative accomplishment of such a task in the case of any par- 

 ticular branch of logical theory involves a sort of study which the 

 recent discussion of the logic of geometry, as well as of the logic 

 of number theory, often exemplifies. Instead of setting forth cer- 

 tain self-evident axioms of geometry, or of arithmetic, the modern 

 logical investigator undertakes to do what Russell and Couturat call 

 defining a certain type of space, or a certain type of numbers. This 

 process of definition, also often called the process of definition by 

 postulates, consists substantially in saying : "I am going to describe 

 to you the properties of a certain class of ideal entities. I do not 

 say that these entities exist in the physical world, just as I do not 

 deny that they exist there. But I am going to treat them simply 

 as the entities which conform to the following definition. The 

 definition I will state in the form of a set of principles given in 

 order as first, second, third, and so on. I state the principles, and 

 I define the entities in question as a set of entities such that they 

 conform to these principles. If the principles involve no mutual 

 contradictions, such entities are possible." Thus Dr. Veblen, in his 

 recent essay on the so-called axioms of geometry, states twelve 

 different principles to which certain abstract entities named points 

 are to conform. He does not assert that these principles are self- 

 evident. Since he is talking about purely abstract entities, which 

 are the creatures of his definition, the principles could not be self- 

 evident. They are true only in the sense that the entities defined 

 are said by definition to conform to them. Dr. Veblen then shows 

 that the laws of our ordinary geometry can be deduced from 

 these principles as laws which hold for the defined entities. These 

 principles, then, are sufficient as a basis for geometrical science. 



