400 



SCIENCE 



[N. S. Vol. XXXII. No. 821 



of any special geometrical scheme is known 

 when the group of transformations which 

 leave unaltered certain relations of figures 

 has been assigned. Two schemes in which 

 the space elements may be quite different 

 have this fundamental identity, provided 

 they have the same group ; every special 

 theorem is then capable of interpretation 

 as a property of figures either in the one or 

 in the other geometry. The mathematical 

 physicist is familiar with the fact that a 

 single mathematical theory is often capable 

 of interpretation in relation to a variety of 

 physical phenomena. In some instances a 

 mathematical formulation, as in some fash- 

 ion representing observed facts, has sur- 

 vived the physical theory it was originally 

 devised to represent. In the case of elec- 

 tromagnetic and optical theory, there ap- 

 pears to be reason for trusting the equa- 

 tions, even when the proper physical 

 interpretation of some of the vectors ap- 

 pearing in them is a matter of uncertainty 

 and gives rise to much difference of opin- 

 ion; another instance of the fundamental 

 nature of mathematical form. 



One of the most general mathematical 

 conceptions is that of functional relation- 

 ship, or "functionality." Starting orig- 

 inally from simple cases such as a function 

 represented by a power of a variable, this 

 conception has, under the pressure of the 

 needs of expanding mathematical theories, 

 gradually attained the completeness of 

 generality which it possesses at the present 

 time. The opinion appears to be gaining 

 ground that this very general conception 

 of functionality, born on mathematical 

 ground, is destined to supersede the nar- 

 rower notion of causation, traditional in 

 connection with the natural sciences. As 

 an abstract formulation of the idea of de- 

 termination in its most general sense, the 

 notion of functionality includes and trans- 

 cends the more special notion of causation 



as a one-sided determination of future phe- 

 nomena by means of present conditions ; it 

 can be used to express the fact of the sub- 

 sumption under a general law of past, 

 present and future alike, in a sequence of 

 phenomena. Prom this point of view the 

 remark of Huxley that mathematics 

 "knows nothing of causation" could only 

 be taken to express the whole truth, if by 

 the term "causation" is imderstood "effi- 

 cient causation." The latter notion has, 

 however, in recent times been to an in- 

 creasing extent regarded as just as irre- 

 levant in the natural sciences as it is in 

 mathematics; the idea of thoroughgoing 

 determinancy, in accordance with formal 

 law, being thought to be alone significant 

 in either domain. 



The observations I have made in the 

 present address have, in the main, had 

 reference to mathematics as a living and 

 growing science related to and permeating 

 other great departments of knowledge. 

 The small remaining space at my disposal 

 I propose to devote to a few words about 

 some matters connected with the teaching 

 of the more elementary parts of mathe- 

 matics. Of late years a new spirit has 

 come over the mathematical teaching in 

 many of our institutions, due in no small 

 measure to the reforming zeal of our gen- 

 eral treasurer, Professor John Perry. The 

 changes that have been made followed a 

 recognition of the fact that the abstract 

 mode of treatment of the subject that had 

 been traditional was not only wholly un- 

 suitable as a training for physicists and 

 engineers, but was also to a large extent a 

 failure in relation to general education, 

 because it neglected to bring out clearly 

 the bearing of the subject on the concrete 

 side of things. With the general principle 

 that a much less abstract mode of treat- 

 ment than was formerly customary is de- 

 sirable for a variety of reasons, I am in 



