Rogers — The Logical Basis of Mathematics. 183 



The belief that mathematical reasoning is independent of Logic, and 

 proceeds altogether by experiments with images or formal intuitions, is due to 

 the following causes (and probably others) : — 



1. To confounding developed methods of modern mathematics with primi- 

 tive methods of suggestive discovery (like mensuration). 



2. To confounding the psychological conditions required for the suggestion 

 and retention of premisses with the process of inference from those premisses. 

 This is a fallacy of the same kind as identifying words with the things they 

 represent, and is thus a type of Nominalism. This fact escapes notice because 

 the language of geometrical figures, though inexact, is far better representative 

 of its concepts and more suggestive than is the case with ordinary language. 



3. The suspicion against Logic is increased by the erroneous belief that all 

 syllogistic reasoning, regarded as proof, is a jjetAtio lyrindpii. Mill adopts 

 this view in one chapter of his Logic, and rejects it in the next.* The fallacy 

 is due partly to ignoring the hypothetical nature of syllogistic reasoning, 

 and ultimately means that it is unnecessary to hang a convicted murderer, 

 because he has been hanged already by the law of the State ! 



4. It is supposed by some mathematicians that the conclusions of logic 

 are self-evident in the premisses ; whereas this is not the case in mathematics. 

 This, however, is true only of smgle elementary syllogisms. By the use of 

 thirteen axioms, Desargues' theorem of perspective triangles may be proved ;t 

 it is not to be discovered in any lesser number of these, but only through 

 their careful combination, in which the conclusion follows gradually, but not 

 immediately. It is mainly in the selection of the appropriate or interesting 

 propositions, and of the premisses required to prove or disprove an asserted 

 proposition, that mathematics differs from general logic, not in the nature 

 of the reasoning. Logic, in fact, is creative in the most literal sense ; it 

 actually discovers new truths latent in the premisses, and extends our 

 experience. If the premisses are given by generalisation of experience, 

 the conclusions are commonly materialised truths ; but in every case they 

 are hypothetical truths. The productive power of logic in science is analogous 

 to the productive power of methodical habits in practical life. 



5. Elementary geometry often conveniently picks up its premisses as it 



* Bk. II., Chaps, iii. and iv. 



t See The Axioms of Projective Geometry, by A. N. Whitehead, Chap. ii. 



