space. When we say natural law, we mean generalization of this type. The laws of science are generalizations of the relations between phenomena. According to Poincaré 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 verifiable nor unverifiable, such as the hypothesis of atoms or of a continuous 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 observed 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 mathematical generalization. It might in broad terms be characterized as the purely scientific type. The second type, which might be broadly characterized 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 N 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 consider that the new objects have (in mathematics) been created or discovered, is merely a matter of psychologic point of view. For example, in one of Poincaré's last papers[1] he explains the apparently irreconcil-
- ↑ Scientia, 12 (1912), pp. 1-11.
