398 



SCIENCE 



[N. S. Vol. XL VI. No. 1191 



writers, the theory of function (without 

 any hint of its manifold connections with 

 physics), the calculus of variations (de- 

 natured, without a hint of its vast impor- 

 tance in mechanics and elsewhere), projec- 

 tive geometry (with no mention of descrip- 

 tive geometry nor the representation of 

 space forms). 



In all these, tradition has been leading us 

 as far astray as it has in those more elemen- 

 tary courses of the secondary school, which 

 we are wont to criticize. Shall we not 

 search our own house? Shall we not ask 

 if our own collegiate and graduate courses 

 in mathematics demonstrate to students the 

 real significance of the theory they cover? 

 Have we denatured each subject until in- 

 sight is eliminated and only formalism and 

 logical tricks remain ? 



So long as this blight remains, we must 

 expect and we shall deserve public disdain 

 and sincere doubt of our value to humanity. 



It should be unnecessary for me to ex- 

 plain my own deep interest in the logical 

 and cultural side of mathematics. Cer- 

 tainly I would be the last to belittle its 

 great spiritual values. But this is for the 

 specialist rather than for the usual student. 

 Values to the world at large must be stated 

 in terms of more concrete realities. Shall 

 we hide the fact of the immense service of 

 mathematics to society? To emphasize 

 beauty and pleasure to the entire exclusion 

 of the more convincing argument of benefit 

 to mankind is as quixotic and short-sighted 

 as is the corresponding formalization of our 

 courses of instruction. To ignore the sig- 

 nificance of our great subject is to spurn 

 our birthright. 



Let me then, in retiring from office in the 

 association, leave with you the sincere hope 

 that a part of the work of this association 

 be to impress upon the public the great 

 value of mathematics in its direct effects 

 upon life and upon human society. To ac- 

 complish this end, a most effective means, 



and one ready to hand, is to bring out to 

 our own students, not halfheartedly, but 

 with vigor, not a few but all available facts 

 that shed light on the real meaning of what 

 we teach. Let this association be a focus 

 from which such doctrine may emanate, a 

 forum in which such views may be empha- 

 sized and detailed. Thus I to-day have 

 mentioned to you a few samples of our 

 neglect, in haste and by name only. Shall 

 we not discuss among ourselves these and 

 other means toward the end, other topics 

 whose significance is commonly lost or 

 neglected, other points of view that will in- 

 crease insight, even if it be at the expense 

 of a few formulas or theorems that we tra- 

 ditionally treasure. 



To the same end, may I now emphasize 

 what seems to me a great if not the great- 

 est function of this association 1 In Amer- 

 ica, up to recent years, the beauty and in- 

 terest centering in pure mathematics has so 

 absorbed all mathematical talent that we 

 have almost if not quite neglected that 

 other phase of mathematics in which the 

 significance of all we do is so self-evident: 

 applied mathematics. This association has, 

 through its journal and through its meet- 

 ings, already demonstrated its willingness 

 and its ability to foster mathematics of this 

 tj'pe. On this side of mathematics, not 

 only discussion of the mathematics taught 

 or to be taught, but even research papers of 

 high grade have had in the past no ade- 

 quate means of exposition. The wonderful 

 work of Gibbs was for this reason long 

 buried in the obscure Connecticut Academy 

 and mathematical advancement along the 

 important lines that he laid down was de- 

 layed or wholly prevented. The great work 

 of G. W. Hill, which included profound in- 

 vestigations on infinite determinants, was 

 for the same reason unknown and unap- 

 preciated by many mathematicians in this 

 country until near his death, and work by 

 others along the lines he mapped out was 



