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SCIENCE 



[N. S. Vol. XLII. No. 1089 



but what is the point of it all? It is ob- 

 viously true enough but what, pray, can be 

 its bearing upon the matter in hand? 

 "What light does it throw upon the human 

 significance of mathematics? The ques- 

 tion is timely and just. The answer, which 

 will grow in fullness and clarity as we pro- 

 ceed, may be at once begun. 



How long our human ancestors, in re- 

 mote ages, may have groped, as some of 

 their descendants even now grope, among 

 the things of sense, in the hope of finding 

 there the desiderated good, we do not 

 know — past time is long and the evolution 

 of wisdom has been slow. "We do know that, 

 long before the beginnings of recorded his- 

 tory, superior men — advanced representa- 

 tives of their kind — must have learned that 

 the deliverance sought was not to be 

 found among the objects of the mobile 

 world, and so the spirit's quest passed 

 from thence ; passed from the realm of per- 

 ception and sense to the realm of concept 

 and reason: thought ceased, that is, to be 

 merely the unconscious means of pursuit 

 and became itself the quarry — mind had 

 discovered mind; and there, in the realm 

 of ideas, in the realm of spirit proper, in 

 the world of reason or thought, the great 

 search — far outrunning historic time — has 

 been endlessly carried on, with varying 

 fortunes, indeed, but without despair or 

 breach of continuity, meanwhile multiply- 

 ing its resources and assuming gradually, 

 as the years and centuries have passed, the 

 characters and forms of what we know to- 

 day as philosophy and science and art. I 

 have mentioned the passing of the quest 

 from the reabn of sense to the realm of con- 

 ception: a most notable transition in the 

 career of mind and especially significant 

 for the view I am aiming to sketch. For 

 thought, in thus becoming a conscious sub- 

 ject or object of thought, then began its 

 destined course in reason: in ceasing to be 



merely an unconscious means of pursuit 

 and becoming itself the quarry, it definitely 

 entered upon the arduous way that leads to 

 the goal of rigor. And so it is evident that 

 the way in question is not a private way; 

 it does not belong exclusively to mathe- 

 matics ; it is public property ; it is the high- 

 way of conceptual research. For it is a 

 mistake to imagine that mathematics, in 

 virtue of its reputed exactitude, is an insu- 

 lated science, dwelling apart in isolation 

 from other forms and modes of conceptual 

 activity. It would be such, were its rigor 

 absolute ; for between a perfection and any 

 approximation thereto, however close, there 

 always remains an infinitude of steps. But 

 the rigor of mathematics is not absolute — 

 absolute rigor is an ideal, to be, like other 

 ideals, aspired unto, forever approached, 

 but never quite attained, for such attain- 

 ment would mean that every possibility of 

 error or indetermination, however slight, 

 had been eliminated from idea, from sym- 

 bol, and from argumentation. We know, 

 however, that such elimination can never be 

 complete, unless indeed the human mind 

 shall one day lose its insatiable faculty for 

 doubting. "What, then, is the distinction of 

 mathematics on the score of exactitude? 

 Its distinction lies, not in the attainment 

 of rigor absolute, but partly in its excep- 

 tional devotion thereto and especially in 

 the advancement it has made along the end- 

 less path that leads towards that perfec- 

 tion. But, as I have already said, it must 

 not be thought that mathematics is the sole 

 traveler upon the way. It is important to 

 see clearly that it is far from being thus a 

 solitary enterprise. First, however, let us 

 adjust our imagery to a better correspond- 

 ence with the facts. I have spoken of the 

 path. We know, however, that the paths 

 are many, as many as the varieties of con- 

 ceptual subject-matter, all of them eon- 

 verging towards the same high goal. "We 



