INDUCTION IN ITS VARIOUS SENSES 25 



The processes we have just described as abstraction, intuition^ 

 and generalization, of simple, self-evident, necessary axioms, are 

 usually described by modern logicians as &quot;geometrical&quot; or &quot; mathe 

 matical&quot; induction}&quot; And the reason assigned for the superior 

 cogency of the evidence and certitude we have in regard to these 

 truths, as compared with those we reach by &quot;physical&quot; induc 

 tion, is stated to be this : that in the former the objects compared, 

 being abstract, have their essential qualities fixed for certain by 

 ourselves, in the definitions we impose upon them ; while in the 

 latter the essential qualities of the objects which are now con 

 crete things and agencies existing and acting in physical nature 

 are not fixed by definitions which we impose on them, but &quot;have 

 to be discovered and proved &quot;. Thus Dr. Mellone writes : 2 



&quot;The universality of the result [that &quot;the angles at the base of an isosceles 

 triangle are equal &quot;] depends upon our being absolutely certain of what are 

 the essentials of the kind of triangle in question ; and we can be certain of 

 these because in geometry definitions have not to be discovered. The geome 

 trician can frame his own definitions, and change them, if necessary . . . the 

 mathematician makes his own definitions of what is essential and argues from 

 them. But in Nature the essential conditions have to be discovered and 

 proved. This is the great difference between mathematical and physical in 

 duction, and all the difficulties of physical induction result from it.&quot; 



This explanation of the difference between metaphysically or 

 absolutely necessary and universal truths on the one hand, and 

 physically or contingently necessary and universal truths on the 

 other, differs from the scholastic account of them only in one par 

 ticular, but one which is all-important. Our definitions of the 

 abstract objects of thought with which the mathematical sciences 



(quoad se), yet we may not have grasped the intension of the notions sufficiently to 

 see the necessity of the connexion between them : they may not be clear to us 

 (quoad nos) (C/.MAHER, Psychology, p. 289, n. 33 ; JOYCE, Logic, p. 239). The truth 

 of these we learn by Demonstration, i.e. by gradually tracing their rational, logical 

 connexion with the former ones. 



The demonstration of a remote geometrical conclusion is simply the process of 

 showing how and why it is true, by revealing the rational connexions it has with 

 simpler antecedent truths, and ultimately with first principles (195). This process 

 is essentially deductive : a diagram may be necessary in order to help the imagina 

 tion, and to serve as a concrete illustration or instance : but it is not from the 

 diagram, from the instance, but, through it, from wider and simpler necessary prin 

 ciples that the conclusion is derived. It is only by an improper use of language 

 that this process can be described as &quot;Geometrical Induction&quot;. Cf. Palaestra 

 Logica, pp. 103-104. 



1 It is in the mathematical sciences we find the simplest and most obvious 

 examples of such axioms. Cf. MELLONE, op. cit., pp. 265-70. On the nature of 

 mathematical reasoning, cf. JOSEPH, op. cit., chap. xxv. ; infra, 258. 



2 ibid. pp. 267, 269. 



