Read The Nature of Mathematics Online

88 THE NATURE OF MATHEMATICS 

classes, and the notation " 1 + 1 " makes it look as if there 
were two. Remembering that 1 is a class of certain classes, 
what the above proposition means is : If a; and y are members 
of 1, and x differs from y, then x and y together make up a 
member of 2. 

At last, then, we arrive at seeing that the nature of 
Mathematics is independent of us personally and of the world 
outside, and we can feel that our own discoveries and views do 
not affect the Truth itself, but only the extent to which we or 
others see it. Some of us discover things in science, but we do 
not really create anything in science any more than Columbus 
created America. Common sense certainly leads us astray 
when we try to use it for the purposes for which it is not 
particularly adapted, just as we may cut ourselves and not our 
beards if we try to shave with a carving knife ; but it has the 
merit of finding no difficulty in agreeing with those philosophers 
who have succeeded in satisfying themselves of the truth and 
position of Mathematics. Some philosophers have reached the 
startling conclusion that Truth is made by men, and that 
Mathematics is created by mathematicians, and that Columbus 
created America ; but common sense, it is refreshing to think, 
is at any rate above being flattered by philosophical persuasion 
that it really occupies a place sometimes reserved for an even 
more sacred Being. 



BIBLIOGRAPHY 

THE view that science is dominated by the principle of the economy 
of thought has been in part 1 very thoroughly worked out by Ernst 
Mach (see especially the translation of his Science of Mechanics, 
Chicago ; Open Court Publishing Co., 3rd ed., 1907. On the history 
of mathematics, we may mention W. W. Rouse Ball's books, A 
Primer of the History of Mathematics (3rd ed., 1906), and the fuller 
Short Account of the History of Mathematics (4th ed., 1908, both 
published in London by Macmillan), and Karl Fink's Brief History 
of Mathematics (Chicago, 3rd ed., 1910). 

As text- books of mathematics, De Morgan's books on Arithmetic, 
Algebra, and Trigonometry are still unsurpassed, and his Trigonometry j 
and Double Algebra contains one of the best discussions of complex 
numbers, for students, that there is. As De Morgan's books are 
not all easy to get, the reprints of his Elementary Illustrations of the 
Differential and Integral Calculus and his work On the Study and 
Difficulties of Mathematics (Chicago, 1899 and 1902) may be recom- 
mended. Where possible, it is best to read the works of the great 
mathematicians themselves. For elementary books, Lagrange's 
Lectures on Elementary Mathematics, of which a translation has been 
published at Chicago (2nd ed., 1901), is the most perfect specimen. 

The questions dealt with in the fourth chapter are more fully dis- 
cussed in Mach's Mechanics. An excellent collection of methods 
and problems in graphical arithmetic and algebra and so on, is 
contained in H. E. Cobb's book on Elements of Applied Mathematics 
(Boston and London : Ginn & Co., 1911). 

Finally, the best discussion of the nature of Mathematics is 
contained in B. Russell's Principles of Mathematics (Cambridge 
University Press, 1903). 

1 Cf. above, pp. 7, 12, 16, 20, 57.