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SCIENCE 



[N. S. Vol. XLIV. No. 1134 



ods which you always employ ; and there is 

 the special answer, that logical forms and 

 logical implications are not so very simple, 

 and that the whole of mathematics is evi- 

 dence to this effect. 



One great use of the study of logical 

 method is not in the region of elaborate de- 

 duction, but to guide us in the study of the 

 formation of the main concepts of science. 

 Consider geometry, for example. What 

 are the points which compose space? 

 Euclid tells us that they are without parts 

 and without magnitude. But how is the 

 notion of a point derived from the sense- 

 perceptions from which science starts? 

 Certainly points are not direct deliver- 

 ances of the senses. Here and there we 

 may see or unpleasantly feel something- 

 suggestive of a point. But this is a rare 

 phenomenon, and certainly does not war- 

 rant the conception of space as composed 

 of points. Our knowledge of space proper- 

 ties is not based on any observations of re- 

 lations between points. It arises from ex- 

 perience of relations between bodies. Now 

 a fundamental space relation between bod- 

 ies is that one body may be part of another. 

 We are tempted to define the "whole and 

 part" relation by saying that the points 

 occupied by the part are some of the points 

 occupied by the whole. But "whole and 

 part" being more fundamental than the 

 notion of "point," this definition is really 

 circular and vicious. 



We accordingly ask whether any other 

 definition of "spatial whole and part" can 

 be given. I think that it can be done in 

 this way, though, if I be mistaken, it is 

 unessential to my general argument. We 

 have come to the conclusion that an ex- 

 tended body is nothing else than the class 

 of perceptions of it by all its percipients, 

 actual or ideal. Of course, it is not any 

 class of perceptions, but a certain definite 

 sort of class which I have not defined here, 

 except by the vicious method of saying 



that they are perceptions of a body. Now, 

 the perceptions of a part of a body are 

 among the perceptions which compose the 

 whole body. Thus two bodies a and b are 

 both classes of perceptions; and b is part 

 of a when the class which is b is contained 

 in the class which is a. It immediately fol- 

 lows from the logical form of this definition 

 that if & is part of a, and c is part of b, 

 then c is part of a. Thus the relation 

 "whole to part" is transitive. Again, it 

 will be convenient to allow that a body is 

 part of itself. This is a mere question of 

 how you draw the definition. With this 

 understanding, the relation is reflexive. 

 Finally, if a is part of b, and 6 is part of o, 

 then a and b must be identical. These 

 properties of "whole and part" are not 

 fresh assumptions, they follow from the 

 logical form of our definition. 



One assumption has to be made if we as- 

 sume the ideal infinite divisibility of space. 

 Namely, we assume that every class of per- 

 ceptions which is an extended body con- 

 tains other classes of perceptions which are 

 extended bodies diverse from itself. This 

 assumption makes rather a large draft on 

 the theory of ideal perceptions. Geometry 

 vanishes unless in some form you make it. 

 The assumption is not peculiar to my ex- 

 position. 



It is then possible to define what we mean 

 by a point. A point is the class of extended 

 objects which, in ordinary language, con- 

 tain that point. The definition, without 

 presupposing the idea of a point, is rather 

 elaborate, and I have not now time for its 

 statement. 



The advantage of introducing points into 

 geometry is the simplicity of the logical 

 expression of their mutual relations. For 

 science, simplicity of definition is of slight 

 importance, but simplicity of mutual rela- 

 tions is essential. Another example of 

 this law is the way physicists and chemists 



