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SCIENCE 



[N. S. Vol. XXXII. No. 821 



the ordinary traditions of the science. Mr. 

 Eussell writes r "Pure mathematics is the 

 class of all propositions of the form 'p im- 

 plies q,' where p and q are propositions 

 containing one or more variables, the same 

 in the two propositions, and neither p nor 

 q contains any constants except logical con- 

 stants. And logical constants are all no- 

 tions definable in terms of the following: 

 Implication, the relation of a term to a 

 class of which it is a member, the notion of 

 such thai, the notion of relation, and such 

 further notions as may be involved in the 

 general notion of propositions of the above 

 form. In addition to these, mathematics 

 uses a notion which is not a constituent of 

 the propositions which it considers — 

 namely, the notion of truth. ' ' 



The belief is very general amongst in- 

 structed persons that the truths of mathe- 

 matics have absolute certainty, or at least 

 that there appertains to them the highest 

 degree of certainty of which the human 

 mind is capable. It is thought that a valid 

 mathematical theorem is necessarily of 

 such a character as to compel belief in any 

 mind capable of following the steps of the 

 demonstration. Any considerations tend- 

 ing to weaken this belief would be discon- 

 certing and would cause some degree of 

 astonishment. At the risk of this, I must 

 here mention two facts which are of con- 

 siderable importance as regards an estima- 

 tion of the precise character of mathe- 

 matical knowledge. In the first place, it is 

 a fact that frequently, and at various 

 times, differences of opinion have existed 

 among mathematicians, giving rise to con- 

 troversies as to the validity of whole lines 

 of reasoning, and afi:ecting the results of 

 such reasoning; a considerable amount of 

 difference of opinion of this character ex- 

 ists among mathematicians at the present 

 time. In the second place, the accepted 

 ' " Principles of Mathematics," p. 1. 



standard of rigor, that is, the standard of 

 what is deemed necessary to constitute a 

 valid demonstration, has undergone change 

 in the course of time. Much of the rea- 

 soning which was formerly regarded as 

 satisfactory and irrefutable is now re- 

 garded as insufficient to establish the re- 

 sults which it was employed to demon- 

 strate. It has even been shown that results 

 which were once supposed to have been 

 fully established by demonstrations are, in 

 point of fact, affected with error. I pro- 

 pose here to explain in general terms how 

 these phenomena are possible. 



In every subject of study, if one probes 

 deep enough, there are found to be points 

 in which that subject comes in contact with 

 general philosophy, and where differences 

 of philosophical view will have a greater or 

 less influence on the attitude of the mind 

 towards the principles of the particular 

 subject. This is not surprising when we 

 reflect that there is but one universe of 

 thought, that no department of knowledge 

 can be absolutely isolated, and that meta- 

 physical and psychological implications 

 are a necessary element in all the activities 

 of the mind. A particular department, 

 such as mathematics, is compelled to set up 

 a more or less artificial frontier, which 

 marks it off from general philosophy. 

 This frontier consists of a set of regulative 

 ideas in the form of indefinables and 

 axioms, partly ontological assumptions, 

 and partly postulations of a logical char- 

 acter. To go behind these, to attempt to 

 analyze their nature and origin, and to 

 justify their validity, is to go outside the 

 special department and to touch on the 

 domains of the metaphysician and the 

 psychologist. Whether they are regarded 

 as possessing apodictic certainty or as 

 purely hypothetical in character, these 

 ideas represent the data or premises of the 

 science, and the whole of its edifice is de- 



