On the Continuity of Chance 5 



distribution. (Compare G. Cantor, Acta Mathematica II., 



308.) 



Let the distribution be perfected; and, now, note what is true 

 of any distance upon the line: 



It divides all distances into two sets such, that 



1. Every distance in the one set is larg'er than every dis- 

 tance in the other. 



2. But now, either there is a smallest distance in the set of 

 larger ones. 



3. Or else there is a largest distance in the set of smaller 

 ones, 



The very distance we have taken must be held to belong to 

 one or the other of our two sets. The row can, the line cannot, 

 be divided by a point that is not of it. Such is Herr Dede- 

 kind's definition of a continuum. 



But is there such a thing as a continuum? Can there be 

 aught save discrete points? The very thought of discrete 

 points in a row brings to mind something between them, else 

 they could not be discrete. What is this something but line ? 

 What is it but continuous? Continuity is an hypothesis the 

 mind is forced to make to explain discontinuity, for discontin- 

 uity of anything cannot exist save in a continuity of something. 

 If lines, space, time, motion, thought, feeling, are not contin- 

 uous, name the gap. The very attempt to do it brings to 

 mind, and inevitably, something that bridges the gap. The 

 discontinuous row of points is on a continuous line; the line, 

 in a continuous something we call space; discontinuous mo- 

 ments are in a continuous time; to cease thinking, to cease 

 feeling, is simply to think or feel otherwise than we did; there 

 is throughout a persistence of something, self, soul, mind, — call 

 it what you will. 



Throughout all this discussion, we have preserved a one-to- 

 one correspondence between numbers and points, to each point 

 its own number and to each number its own point. Had the 



135 



