12 UNIVERSITY OF MISSOURI BULLETIN 



and logic, if, indeed, mathematics and logic are not the same 

 thing. As to their subject-matter, their peculiarity is that 

 they deal merely with the relations of implication between 

 certain ideas or types of ideas or propositions, irrespective of 

 the special nature of the concrete objects for which the ideas 

 stand or to which the propositions refer. This 'abstractness' 

 is usually outwardly manifested by the use of symbols; the 

 truths of mathematics or logic can be as well exemplified by 

 talking about x and 3; as by considering real things. "Pure 

 Mathematics", in the words of Mr. Bertrand Russell," con- 

 sists entirely of asseverations that if such and such a proposi- 

 tion is true of anything, then such and such another proposi- 

 tion is true of that thing." But it makes no difference 

 whether anything really exists or actually could be exper- 

 ienced which the first proposition could be about, or whether 

 that proposition be itself true (unless, indeed, the proposi- 

 tion be itself one of the fundamental principles of logic or 

 mathematics with respect to the laws of implication). Thus, 

 to quote Mr. Russell again, "Mathematics may be defined as 

 the subject in which we never know what we are talking 

 about, or whether what we are saying is true". This epigram 

 does not mean quite what it at first may seem to say; but if 

 you get its real meaning, you will have a pretty precise state- 

 ment of the character of the abstract sciences with respect 

 to their subject-matter and the nature of their results. Yet 

 in these respects they do not, after all, differ from the con- 

 crete sciences by so perfectly clear-cut and radical a differ- 

 ence as is often supposed and as even Mr. Russell seems to 

 suppose. There is a sense in which, in every science deserv- 

 ing of the name, you do not ever talk of any particular thing 

 or event or know whether the truth you affirm will hold good 

 of any particular thing or event. 



But with respect to their methods of acquiring knowl- 

 edge, the distinction between the abstract and concrete sciences 

 is clear and sharp. The abstract sciences do not get their 

 conclusions through the observation of facts that are to be 



