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SCIENCE 



[N. S. Vol. XXVII. No. 688 



here, and the vital question of legitimacy of 

 method looms up. 



M. Poincare's tendencies in this connection 

 are familiar already to readers of, say, his 

 " Electricite et Optique," or " La Theorie de 

 Lorenz et le Principe de Reaction" (Arch, 

 neerland., ser. 2, v. 5), where his criticism 

 of mechanical constructs ends in the affirma- 

 tion that, not these, but unity, do men really 

 seek. In a word, the empirical references of 

 mechanics must be expressed in mental terms 

 if we would estimate their value for a human 

 esperient. The way in which men regard 

 things, the way of thought, calls for consid- 

 eration just as close as the grasp they may 

 have obtained upon particular objects. In 

 fact, thus the value of alleged grasp must 

 needs be estimated. In -a more concrete sense 

 than Grassmann's, and with no necessary ref- 

 erence to prudence, " a doctrine of Porms 

 should precede a doctrine of Magnitude," as 

 H. Hankel pointed out forty years ago. In 

 short, two questions, long subordinated, thanks 

 to preoccupation in special discovery, thrust 

 themselves forward. What basis does scien- 

 tific thought possess in the sphere of logical 

 reasoning? What value can be assigned to 

 scientific thought in the complexus of human 

 experience? That is to say, M. Poincare con- 

 fronts the Sphinx, who asks. What validity, 

 if any, does the scientific view of the universe 

 hold in its own right. And, naturally, his 

 interest being what it is, his achievements 

 heing what they are, he presupposes the work 

 of such earlier masters as Eiemann and Weier- 

 Btrass, of such recent scholars as Kronecker, 

 Paul du Bois-Reymond and F. Klein, to say 

 nothing of the remarkable group of his own 

 fellow countrymen. I mean his approach is 

 from this side, not from that of the philos- 

 opher puA sang. But this matters little, for 

 he has been gifted with a double portion of 

 that Gallic wit which, in our time, stands 

 for Attic salt — the wit to refine ideas of all 

 ■dross, and to present them crisp from the 

 •crucible of thought. 



The contrasted, yet complementary, nature 

 of the labors of Eiemann and Weierstrass, as 

 noted by M. Poincare himself (Acta Math., 

 xxii.), serves to hint the general scope of the 



problem with which he wrestles in this book. 

 " By the instrument of Eiemann we see at a 

 glance the general aspect of things — ^like a 

 traveler who is examining from the peak of 

 a mountain the topography of the plain which 

 he is going to visit, and is finding his bear- 

 ings. By the instruments of Weierstrass 

 analysis will, in due course, throw light into 

 every corner, and make absolute clearness 

 shine forth." What is this but the age-old 

 IJuzzle of the universal and the particular? 

 What kind of author have we but one who, 

 being a marvelous analyst, is also an ornament 

 of the school of synthetic mathematics ? And 

 we must be prepared to learn, accordingly, 

 that rule of thumb may turn out no rule. A 

 theory may never render a more valuable 

 service to science than when it breaks down, 

 as M. Poincare has himself said (" La Science 

 et I'Hypothese," p. lYO). 



M. Poincare's conclusions are dominated by 

 considerations like the following : " A reality 

 completely independent of the mind which 

 conceives it, sees or feels it, is an impossibil- 

 ity " (p. 14). "We have not a direct intui- 

 tion of simultaneity, nor of the equality of 

 two durations. If we think we have this 

 intuition, this is an illusion. We replace it 

 by the aid of certain rules which we apply 

 almost always without taking count of them " 

 (pp. 35-6). 



" Space is a mathematical continuum, it is 

 infinite, and we can represent it to ourselves 

 only by physical continua and finite objects. 

 . . . Absolute space is nonsense" (p. 56). 

 " Experience does not prove to us that space 

 has three dimensions; it only proves to us 

 that it is convenient to attribute three to it, 

 because thus the number of fillips is reduced 

 to a minimum" (p. 69). ■ "I believe, there- 

 fore, that if by space is understood a mathe- 

 matical continuum of three dimensions, were 

 it otherwise amorphous, it is the mind which 

 constructs it, but does not construct it out of 

 nothing ; it needs materials and models " (p. 

 72). " The invariant laws are the relations 

 between the crude facts, while the relations 

 between the ' scientific facts ' remain always 

 dependent on certain conventions" (p. 128). 



" A philosopher really anti-intellectualistic 



