66 SCEPTICISM. 



ideally straight lines, perfect circles, etc., such as do 

 not exist in real Space, and which, for all we know, 

 may be incapable of so existing, because real Space 

 is " pseudo - spherical " or "four-dimensional." If, 

 therefore, mathematical demonstrations are supposed 

 to apply to figures in real Space, they are not true, 

 and if not, to what do they apply ? It seems easy 

 to reply, to the ideal space in our minds ; but what if 

 there be no relation between real and ideal Space ? 

 And if mathematical truths exist only in our heads, 

 what and where are they before they are discovered ? 

 Surely the truth that the angles of a triangle are 

 equal to two right angles did not come into being 

 when it was first discovered ? 



Such considerations may justify the Sceptic in his 

 doubt whether the ideal certainty of mathematics is 

 after all relevant to reality, and in his denial of the 

 self-evidence of the assumptions which underlie the 

 scientific treatment of Space. 



§ 8. Motion also is feigned for scientific purposes 

 to be something different from what it is : it can be 

 calculated only on the assumption that it is discrete 

 and proceeds from point to point, and yet the ancient 

 Zeno's famous fallacy of the Arrow warns us that 

 the Real moves continuously} 



Our conception, too, of Rest is illusory ; for all 

 things seem to be in more or less rapid motion. 

 And yet motion is calculated only by the assumption 

 of fixed points, i.e, of Rest. But these fixed points 



1 If the arrow really moved from point to point, it would 

 be at rest at each point, i.e, would never move at all. But of 

 course it never is at the points at all, but moves through them. 

 Only unfortunately our thought and our speech refuses to express 

 a fact which our eyes behold, and we must continue to say one 

 thing, while meaning another. 



