470 DR. A. N. WHITEHEAD ON 



deduction of those properties of the defined entities which do not depend upon the 

 axioms. The third stage is the selection of the group of axioms which determines 

 that concept of the material world. The fourth stage is the deduction of propositions 

 which involve among their hypotheses some or all of the axioms of the third stage. 

 Psychologically the order of study is apt to be inverted, by first choosing propositions 

 of the second and fourth stages because of their parallelism with the propositions of 

 sense-perception and then by considering the first and third stages. The essential 

 part of our task in passing concepts in review is the exhibition of the first and third 

 stages. The second and fourth stages will only be so far touched upon as seems 

 desirable for the purposes of elucidation. 



Thus in respect to each concept considered the investigation will proceed as 

 follows: A certain relation R (the essential relation of the concept in question), 

 which holds between a certain definite number of entities, is considered. The class 

 of entities, between sets of which this relation holds, is called the "field" of R. 

 Definitions of entities allied to R and to entities of the field of R are then given. 

 These definitions involve no hypotheses as to the properties of R, but are of no 

 importance unless R has as a matter of fact certain properties. For example, it may 

 happen that the classes, thus defined, are all the null class (i.e., the class with no 

 members) unless R has the requisite properties. Again deductions (in the second 

 stage), made without any hypothesis as to the properties of R, may be entirely trivial 

 unless R has certain properties. If R has not the requisite properties the deductions 

 often sink into the assertion that a certain proposition which is false implies some 

 other proposition. This is true* but trifling. The " axioms " respecting R are then 

 given. These are the hypotheses as to the properties of R which are required in the 

 concept under consideration. Finally such deductions are given as are necessary to 

 elucidate the concept. 



None of the reasoning of this memoir depends on any special logical doctrine which 

 may appear to be assumed in the form in which it is set out. Furthermore certain 

 contradictions recently discovered have thrown grave doubt upon the current doctrine 

 of classes as entities. Any recasting of our logical ideas upon the subject of classes 

 must of course simply issue in a change of our ideas as to the true logical analysis of 

 propositions in which classes appear. The propositions themselves, except a few 

 extreme instances which lead to contradictions, must be left intact. Accordingly the 

 present memoir in no way depends upon any theory of classes. 



The above considerations as to method must essentially hold for any investigation 

 respecting axioms of geometry or of physics, viewed purely as deductive sciences, and 

 apart from the question of experimental verification. 



In Concepts I., II., and III. the members of the " field" of R are to be considered 

 as points, except those members of the field which are instants of time. In these 

 concepts the lines and planes are classes of points. In Concepts IV. and V. the 



* Of. KUSSELL, ' The Principles of Mathematics,' 16. 



