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THE NATURE OF MATHEMATICS 83 
CHAPTER VII 

THE NATURE OF MATHEMATICS 

IN the preceding chapters we have followed the development 
of certain branches of knowledge which are usually classed 
together under the name of "mathematical knowledge." These 
branches of knowledge were never clearly marked off from all 
other branches of knowledge : thus geometry was sometimes 
considered as a logical study and sometimes as a natural 
science the study of the properties of the space we live in. 
Still less was there an absolutely clear idea of what it was 
that this knowledge was about. It had a name Mathematics 
and few except " practical " men and some philosophers 
doubted that there was something about which things were 
known in that kind of knowledge called "mathematical." 
But what it was did not interest very many people, and there 
was and is a great tendency to think that the question as to 
what Mathematics is could be answered if we only knew all 
the facts of the development of our mathematical knowledge. 
It seems to me that this opinion is, to a great extent, due to 
an ambiguity of language : one word " mathematics " is used 
both for our knowledge of a certain kind and the thing, if 
such a thing there be, about which this knowledge is. I have 
distinguished, and will now explicitly distinguish, between 
" Mathematics," a collection of truths of which we know 
something, and " mathematics," our knowledge of Mathematics. 
Thus, we may speak of " Euclid's mathematics " or " Newton's 
mathematics," and say truly that mathematics has developed 
and therefore had history ; but Mathematics is eternal and 
unchanging, and therefore has no history it does not belong, 
even in part, to Euclid or Newton or anybody else, but is 
something which is discovered, in the course of time, by human 
minds. An analogous distinction can be drawn between 
" Logic " and " logic." The small initial indicates that we are 
writing of a psychological process which may lead to Truth ; 
the big initial indicates that we are writing of the entity the 
part of Truth to which this process leads us. The reason 
why mathematics is important is that Mathematics is not 
incomprehensible, though it is eternal and unchanging.