310 INTRODUCTION TO PHILOSOPHY OF SCIENCE 



the now-then dimension refuses obstinately to become 

 identified with any of the spatial dimensions. In a strictly 

 Euclidean space the dimensions would be interchangeable. 

 Furthermore, in a four-dimensional Euclidean system the 

 additional dimension would be interchangeable with the 

 other three. For example, the hypotenuse of a right triangle 

 is represented by the formula, 



s 2 = (x 2 - Xi) 2 + (y 2 - yi) 2 . 



Analogously, in a four-dimensional world the equation would 

 be 



s 2 = (x 2 - x,) 2 + (y 2 - y0 2 + (z 2 - z,) 2 + (t 2 - ti) 2 . 



But it turns out that space-time is not strictly Euclidean, 

 though only a slight modification is required. If the / is 

 changed from positive to negative, the required alteration 

 is introduced. Hence the formula for distance in the theory 

 of relativity becomes 



s 2 = (x 2 - x,) 2 + (y 2 - y0 2 + (z 2 - z,) 2 - (t 2 - t,) 2 . 



This expresses the lack of homogeneity in the space-time of 

 relativity; the before-after dimension is always distinguish- 

 able from the spatial dimension. 



From the directional point of view, space- time is isotropic 

 as to its spatial dimensions but not as to its temporal dimen- 

 sion. "The common impression that relativity turns past 

 and future altogether topsy-turvy is quite false." l Rela- 

 tivity, in fact, makes a distinction between past and future 

 which is quite as fixed as it was for the classical physics. The 

 basic notion is again that of causal influences, which is 

 associated with the idea of messages. Since the maximum 

 speed of messages is that of light, which is finite, causal 

 relations are possible only between events so located that a 

 message may be sent from one to the other. If an event B 

 occurs at such a time that it may receive a message from 

 another event A, then B may be the effect of A, and B must 

 succeed A. Two such events lie absolutely in the past and 



1 A- S. Eddington, Nature of the Physical World, p. 48. 



