November 12, 1915] 



SCIENCE 



671 



I turn now for a moment to the pros- 

 pects of one who might choose to devote the 

 hour to an exposition or an indication of 

 modern developments in what it is cus- 

 tomary to call the foundations of mathe- 

 matics — to a characterization, that is, and 

 estimate of that far-reaching and still ad- 

 vancing critical movement which has to do 

 with the relations of the science, philo- 

 sophically considered, to the sciences of 

 logic and methodology. What can he say 

 on this great theme that will be intelligible 

 and edifying to the multitudes of men and 

 women who, though mathematically inex- 

 pert, yet have a genuine humane curiosity 

 respecting even the profounder and subtler 

 life and achievements of science? He can 

 point out that mathematics, like all the 

 other sciences, like the arts too, for that 

 matter, and like philosophy, originates in 

 the refining process of reflection upon the 

 crude data of common sense; he can point 

 out that this process has gradually yielded 

 from out the raw material and still con- 

 tinues to yield more and more ideas of ap- 

 proximate perfection in the respects of pre- 

 cision and form; he can point out that 

 such ideas, thus disentangled and trimmed 

 of their native vagueness and indetermi- 

 nation, disclose their mutual relationships 

 and so become amenable to the concatena- 

 tive processes of logic; and he can point 

 out that these polished ideas with their 

 mutual relationships become the bases or 

 the content of various branches of mathe- 

 matics, which thus tower above common 

 sense and appear to grow out of it and to 

 stand upon it like trees or forests upon the 

 earth. He will point out, however, that 

 this appearance, like most other obvious 

 appearances, is deceiving; he will, that is, 

 point out that these upward-growing sci- 

 ences or branches of science are found, in 

 the light of further reflection, to be down- 

 ward-growing as well, pushing their roots 



deeper and deeper into a dark soil far be- 

 neath the ground of evident common sense ; 

 indeed, he will show that common sense is 

 thus, in its relation to mathematics, but as 

 a sense-litten mist enveloping only the mid- 

 portion of the stately structure, which, like 

 a towering mountain, at once ascends into 

 the limpid ether far above the shining 

 cloud and rests upon a base of subter- 

 ranean rock far below; he will point out 

 that, accordingly, mathematicians, in re- 

 spect of temperamental interest, fall into 

 two classes — the class of those who culti- 

 vate the upward-growing of the science, 

 working thus in the upper regions of 

 clearer light, and the class of those who de- 

 vote themselves to exploring the deep- 

 plunging roots of the science; and it is, he 

 will say, to the critical activity of the lat- 

 ter class — the logicians and philosophers of 

 mathematics — that we owe the discovery of 

 what we are wont to call the foundations 

 of mathematics — the great discovery, that 

 is, of an immense mathematical SM6-struc- 

 ture, which penetrates far beneath the 

 stratum of common sense and of which 

 many of even the greatest mathematicians 

 of former times were not aware. But 

 whilst such foundational research is in the 

 main a modern phenomenon, it is by no 

 means exclusively such; and to protect his 

 auditors against a false perspective in this 

 regard and the peril of an overweening 

 pride in the achievements of their own 

 time, our speaker may recommend to them 

 the perusal of Thomas L. Heath's superb 

 edition of Euclid's "Elements" where, 

 especially in the first volume, they will be 

 much edified to find, in the rich abundance 

 of critical citation and commentary which 

 the translator has there brought together, 

 that the refined and elaborate logico- 

 mathematical researches of our own time 

 have been only a deepening and widening 

 of the keen mathematical criticism of a few 



