178 On the Fundamental Principles of Mathematics. 



sulphuric acid at present, but shall defer the consideration of the 

 subject until the publication of my report on the mineral springs 

 of Canada, which will be accompanied with the analyses of this 

 water as collected in different years. Hoping that my observa- 

 tions may resolve a hitherto unexplained problem in the geology 

 of this region, I beg leave to submit them to the notice of the 1 

 Association. ■ 



( 



Art. XV. — On the Fundamental Principles of Mathematics ; 

 by Stephen Alexander, Professor of Mathematics and Astron- 

 omy in the College of New Jersey. 



The object of this dissertation is to present, to some extent at 

 least, those ultimate principles and reasons, on which are founded 

 the conclusions of mathematics ; principles and reasons which 

 lie beneath the artificial symbols which the science employs ; i. e. 

 -to borrow a most expressive and beautiful figure — an attempt 

 will be made to seize upon and exhibit, in so far as may be, that 

 " central thread of common sense, on which the pearls of analyti- ' 

 cal research are invariably strung. 7 '* 



In pursuance of the object thus indicated, a definite arrange- 

 ment will be made of the several topics to be specially consid- 

 ered, each under its own descriptive title ; commencing with the 

 following : 



The Characteristics of Truth, especially Mathematical Truth. 



(1.) Truth, which may be employed to designate the great ob- 

 ject of all scientific research, is a term much too valuable to be 

 misunderstood, but withal so general as not to admit of a ready 

 definition. Yet, under its various aspects, truth will be found 

 to present the characteristic feature of consistency with some 

 great standard. Thus a careful consideration of its subjects of 

 research will show that 



That is true in Mathematics, which, under the existing system 

 of things, is supposable. It is in no case requisite that the sup- 

 position should have been realized. 



[No one, for instance, has ever seen a perfect circle ; yet every 

 one, who has carefully considered the matter, can clearly under- 

 stand what a perfect circle ought to be, and will concede that 

 its existence is entirely possible, perfectly supposable.f] 



That is true in Physics, or (with some restrictions of applica- 

 tion) in Metaphysics, which has been permitted to exist. 



* Sir J. F. W, Herschel- 1 - Cabinet Cyclopaedia, Treatise on Astronomy, (10). 

 \ Among the truths supposable, must even be classified those expressed by the 

 aid of what are termed imaginary quantities. 



