THE NATURE OF NECESSARY TRUTH 347 



careful consideration reveals that it is an inference ; but the given 

 premises, the reasoning, and the inference are always there. An 

 axiom may seem self-evident to us who have unconsciously drawn 

 the inference a thousand times a day for years ; but it is not self- 

 evident to an infant who must not only acquire a knowledge of 

 the data on which it is founded, but also skill in thinking sufficient 

 to enable it to perceive their relations and how they affect one 

 another. 1 



585. To take other examples ; we do not regard a line as 

 necessarily straight unless we have reasons for doing so, unless we 

 draw an inference, unless we think it traverses the shortest distance 

 between two points and apprehend what that means. Uncon- 

 sciously we reason out that a line that does not traverse the 

 shortest distance cannot be straight. A straight line is not 

 necessarily perpendicular to another unless it makes equal angles 

 with it. Straight lines are not necessarily parallel unless they make 

 equal angles with a third straight line on the same plane. Straight 

 lines on the same plane will necessarily meet if produced far 

 enough unless they are parallel. Evolution through Natural 

 Selection may be true ; but it is not necessarily true unless off- 

 spring resemble their parents on the whole but differ from them 

 in details, unless they outnumber them, unless only the fittest 

 survive on the average, unless the fittest belong to types, and so 

 on. In brief, for us a necessary truth is always an effect or cause 

 which we have been able to trace so perfectly to its cause or 

 effect that no doubt of the nature and reality of the connexion 

 remains in our mind. It is a truth which we think we not only 



1 Dr Whewell and others have held that "it is not experience which proves 

 the axiom ; but that its truth is perceived a priori by the constitution of the 

 mind itself, from the first moment when the meaning of the proposition is 

 apprehended, and without any necessity of verifying it by repeated trials, as is 

 requisite in the case of truths really ascertained by observation " (Mill, Logic, v. 4). 

 This view is opposed by Mill who thinks that we learn the truth of axioms wholly 

 through experience. The two opinions do not appear to be absolutely contra- 

 dictory. Whewell admits that we must have experience that the meaning of 

 propositions and all that that implies in the way of experience, must be appre- 

 hended. Mill admits as necessary truths those " necessarily following from 

 hypothesis." I think I learned originally through experience that two halves 

 make a whole ; but now that I have learned " the meaning of the proposition " 

 I hold that belief on other grounds as a necessary truth. In this way, at first 

 mere experience led me to believe in the mathematical axioms and most of the 

 other necessary truths that I now hold as such ; but they did not become necessary 

 truths to me till I had reached them in another way. The view I suggest differs 

 from those of Whewell and Mill in that axioms, as such, seem to me products of 

 reasoning, not of intuition or of mere repeated experience. 



