154 THE PEINCIPLES OF SCIENCE. [CHAP. 



throughout the range of thought, and mathematical reason- 

 ing is cogent only when it conforms to these conditions, of 

 which logic is the first development. And if it be 

 erroneous to suppose that all certainty is mathematical, it 

 is equally an error to imagine that all which is mathe- 

 matical is certain. Many processes of mathematical 

 reasoning are of most doubtful validity. There are points 

 of mathematical doctrine which must long remain matter 

 of opinion ; for instance, the best form of the definition and 

 axiom concerning parallel lines, or the true nature of a 

 limit. In the use of symbolic reasoning questions occur on 

 which the best mathematicians may differ, as Bernoulli 

 and Leibnitz differed irreconcileably concerning the exis- 

 tence of the logarithms of negative quantities. 1 In fact we 

 no sooner leave the simple logical conditions of number, 

 than we find ourselves involved in a mazy and mysterious 

 science of symbols. 



Mathematical science enjoys no monopoly, and not even 

 a supremacy, in certainty of results. It is the boundless 

 extent and variety of quantitative questions that delights 

 the mathematical student. When simple logic can give 

 but a bare answer Yes or No, the algebraist raises a score 

 of subtle questions, and brings out a crowd of curious 

 results. The flower and the fruit, all that is attractive 

 and delightful, fall to the share of the mathematician, who 

 too often despises the plain but necessary stem from which 

 all has arisen. In no region of thought can a reasoner 

 cast himself free from the prior conditions of logical cor- 

 rectness. The mathematician is only strong and true as 

 long as he is logical, and if number rules the world, it is 

 logic which rules number. 



Nearly all writers have hitherto been strangely content 

 to look upon numerical reasoning as something apart from 

 logical inference. A long divorce has existed between 

 quality and quantity, and it has not been uncommon to 

 treat them as contrasted in nature and restricted to 

 independent branches of thought. For my own part, I 

 believe that all the sciences meet somewhere. No part of 

 knowledge can stand wholly disconnected from other parts 

 of the universe of thought ; it is incredible, above all, that 



1 Montucla. Histoire des MatMmatiques, vol. iii. p. 373. 



