HENRI POINCARE A8 AN INVESTIGATOR 217 



space. When we say natural law, we mean generalization of this type. 

 The laws of science are generalizations of the relations between phenom- 

 ena. According to Poincare there are three classes of hypotheses in 

 science: (1) Natural hypotheses, which are the foundations of the 

 mathematical treatments, such as action decreases with the distance, 

 small movements follow a linear law, effect is a continuous function of 

 the cause, physical phenomena are discontinuous functions; (2) Neutral 

 hypotheses, which enable us to formulate our ideas, and are neither veri- 

 fiable nor unverifiable, such as the hypothesis of atoms or of a continu- 

 ous medium; (3) Generalizations, invariantive relationships, which are 

 valuable, may be verified by experiment and lead to real progress. In 

 " Science and Hypothesis " his thesis is, that science consists of ob- 

 served facts organized according to these three classes of hypotheses. 

 In "Value of Science" the thesis is, that the objective value of science 

 consists in the laws, that is, in the generalizations, discovered. In 

 " Science and Method " the thesis is, that the discovery of laws is by 

 methods substantially the same as those of mathematical investigation, 

 deducing from a significant particular a wide-reaching generalization, 

 selecting our facts because of their significance. 



This type of generalization, however, is only a part of the mathe- 

 matical generalization. It might in broad terms be characterized as the 

 purely scientific type. The second type, which might be broadly char- 

 acterized as the purely mathematical type, is that in which there is a 

 distinct widening of the field of a conception, usually by the addition 

 of new mathematical entities. Examples are the irrational numbers, 

 negative numbers, imaginary numbers, quaternions and hypercomplex 

 numbers in general. The name imaginary indicates the fact that the 

 actual existence of these was once open to question in the minds of some. 

 Other examples are the non-euclidean geometries, the non-archimedean 

 continuity, transfinite numbers, space of four and of iV dimensions. 

 The ideal numbers of Kummer and the geometric numbers of Minkowski 

 are generalizations mainly of this type. It is not possible to separate 

 sharply this kind of generalization from the other, and it would often 

 be difficult to say whether a given mathematical investigation belongs 

 primarily to the one kind or the other. However, when an investigation 

 does not merely utilize material that is already known, but introduces 

 new objects for study whose properties are not known, we can classify it 

 under the second type. Usually the second type arises from inversion 

 processes. We have given certain properties to find the class of things 

 satisfying them. If they do not exist we create them. Whether we con- 

 sider that the new objects have (in mathematics) been created or dis- 

 covered, is merely a matter of psychologic point of view. For example, 

 in one of Poincare's last papers 8 he explains the apparently irreconcil- 



s Scientia, 12 (1912), pp. 1-11. 



