1873] HEGEL AND THE CALCULUS 35 



&quot; half-hour &quot; which he had devoted to the calculus had 

 not sufficed to give him any just idea of that great method. 

 It is certainly much to be regretted that so able a man 

 did not study mathematics thoroughly, for such a course 

 might have proved useful to the theory of mathematics, 

 and could not have failed to be profitable to himself. As 

 it is, he has only given us criticisms such as we have seen, 

 and an attempt to which we now proceed to establish the 

 calculus on a new and very inadequate basis. 



The point which we have always found Hegel urging 

 is, that mathematical functions, when they become 

 quantitatively indefinite or &quot; infinite,&quot; may still have 

 a real qualitative value. Passing over the fact that this 

 is not the technical sense of infinite in mathematics, we 

 may grant that there is a kind of meaning, however vague, 

 that may be attached to the view. Thus an incom 

 mensurable is infinite in the Hegelian sense, not because 

 it can be expressed arithmetically only by an infinite 

 series, but because it is essentially not a sum of units, but, 

 as Hegel vaguely says, a &quot; relation.&quot; For relation we 

 should say function, and then we should be able to read in 

 Hegel s words some meaning like this. Algebraic and 

 geometrical functions are qualitatively different from 

 mere arithmetical functions. They imply an entirely 

 different way of looking at quantity, expressing, in fact, 

 steps in time or space (or in kinematics, both in time and 

 space). So, again, the differential coefficient which takes 



the form - ceases to be intelligible on the mere arithmetical 



view, but gives us a real result of a different quality, when 

 we understand it as equivalent to a proposition about the 

 rates of the vanishing quantities. But then Hegel does 



not seem to have seen that - has a real quantitative value, 



expressing accurately a definite quantity of a different 

 quality. And further, there was in Hegel a rigid deter 

 mination not to see the real qualitative difference between 



