228 POPULAR SCIENCE MONTHLY 



order and classification are the ones to yield results. If we designate 

 any limited combination of things of this nature a group, we may 

 arrange a group in different ways, that is, we may determine for 

 each thing the relation in which it is to stand toward neighboring 

 things. Such an arrangement produces not merely the relations pre- 

 scribed above, but in addition a large number of new ones; and it be- 

 comes plain that, given the first relations, the others may be observed 

 at once. This gives us the type of a law of nature : the possibility of in- 

 ferring from the presence of a definite classification-relation the pres- 

 ence of others which we have not yet tested. 



To illustrate by an example : Let us imagine the things arranged 

 in a simple series formed by choosing one thing for the first member; 

 placing another next to this one; then a new one next to the latter; 

 another next to the last, etc. The position of each thing in the series 

 is determined in relation to the immediately preceding one. Never- 

 theless the position of every member of the entire series is determined ; 

 and thus its relation to every other member. This fact appears in a 

 number of special laws. If we distinguish between preceding and suc- 

 ceeding members one of the laws we may observe is : If B is a succeed- 

 ing member in relation to A, and if C is a succeeding member in rela- 

 tion to B, then C is also a succeeding member in relation to A. 



The correctness and universal validity of this proposition seems to 

 us beyond any possible doubt. It depends, however, merely upon the 

 fact that we are able to test it with the greatest ease in innumerable in- 

 dividual instances and have so tested it. We know none other than 

 instances agreeing with this proposition and none that contradict. 

 'Therefore, to designate such a statement as a necessity of thought 

 seems to me misleading. Now the expression necessity of thought can 

 only be based upon the fact that each time one thinks this proposition, 

 that is, remembers having tested it, one always has in mind its con- 

 firmation. Any wrong proposition is, however, conceivable as the fact 

 that so much that is wrong is actually thought indisputably shows. 

 To base the proof of the truth of a proposition upon the inconceiv- 

 ability of its converse is an undertaking that can not be carried out, 

 because it is possible to think any sort of nonsense. Whenever this 

 proof was believed to have been demonstrated, thinking was always 

 confused with considering, demonstrating or proving. 



Of course the theory of groups is not exhausted with this single 

 statement. We do not, however, care to develop this theory here, but 

 rather to give an example of the nature of the problems of science. Of 

 the other questions only the method of coordination will be briefly 

 treated. 



Given two quantities A and B, one may assign to every member of 

 A a member of B, that is, one determines that certain operations which 



