894 



SCIENCE 



[N. S. Vol. XXXVII. No. 963 



for the moment, that Lucretius, Pascal and 

 the normally educated mind are right: let 

 us suppose that space is infinite, as they 

 assert and believe. In that case the bounds 

 of the universe are indeed remote, and yet 

 we may ask: are there not ways to pass in 

 thought the walls of even so vast a world? 

 There are such ways. But where and how ? 

 For are we not supposing that the walls to 

 be passed are distant by an amount that is 

 infinite? And how may a boundary that 

 is infinitely removed be reached and over- 

 passed? The answer is that there are 

 many infinites of many orders; that infi- 

 nites are surpassed by other infinites; that 

 infinites, like the stars, differ in glory. 

 This is not rhetoric, it is naked fact. One 

 of the grand achievements of mathematics 

 in the nineteenth century is to have defined 

 infinitude (as above defined) and to have 

 discovered that infinites rise above infi- 

 nites, in a genuine hierarchy without a 

 summit. In order to show how we can in 

 thought pass the Lucretian and Pascal 

 walls of our universe, I must ask you to 

 assume as a lemma a mathematical proposi- 

 tion which has indeed been rigorously es- 

 tablished and is familiar, but the proof of 

 which we can not tarry here to reproduce. 

 Consider all the real numbers from zero to 

 one inclusive, or, what is tantamount, con- 

 sider all the points in a unit segment of a 

 continuous straight line. The familiar 

 proposition that I am asking you to assume 

 is that it is not possible to set up a one-to- 

 one correspondence between the points of 

 that segment and the positive integers (in 

 the sequence above given), but that, if you 

 take away from the segment an infinitude 

 (Aleph) of points matching all the posi- 

 tive integers, there will remain in the seg- 

 ment more points, infinitely more, than you 

 have taken away. That means that the 

 infinitude of points in the segment infi- 



nitely surpasses the infinitude of positive 

 integers; surpasses, that is, the infinitude 

 of mile posts on the radius of our infinite 

 (Pascal) sphere. Now conceive a straight 

 line containing as many miles as there are 

 points in the segment. You see at once 

 that in that conception you have over- 

 leaped the infinitely distant walls of the 

 Lucretian universe. Overleaped, did I 

 say? Nay, you have passed beyond those 

 borders by a distance infinitely greater 

 than the length of any line contained 

 within them. And thus it appears that, 

 not our imagination, indeed, but our reason 

 may gaze into spatial abysses beside which 

 the infinite space of Lucretius and Pascal 

 is but a meager thing, infinitesimally smaU. 

 There remain yet other ways by which we 

 are able to escape the infinite confines of 

 this latter space. One of these ways is 

 provided in the conception of hyperspaces 

 enclosing our own as this encloses a plane. 

 But that is another story, and the hour is 

 spent. 



The course we have here pursued has not, 

 indeed, enabled us to answer with final 

 assurance the two questions with which we 

 set out. I hope we have seen along the 

 way something of the possibilities involved. 

 I hope we have gained some insight into 

 the meaning of the questions and have seen 

 that it is possible to discuss them profitably. 

 And especially I hope that we have seen 

 afresh, what we have always to be learning 

 again, that it is not in the world of sense, 

 however precious it is and ineffably won- 

 derful and beautiful, nor yet in the still 

 finer and ampler world of imagination, but 

 it is in the world of conception and thought 

 that the human intellect attains its appro- 

 priate freedom — a freedom without any 

 limitation save the necessity of being con- 

 sistent. Consistency, however, is only a 

 prosaic name for a limitation which, in 



