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8 THE NATURE OF MATHEMATICS 

I hope that I shall succeed in showing that the process of 
mathematical discovery is a living and a growing thing. Some 
mathematicians have lived long lives full of calm and unwavering 
faith for faith in mathematics, as I will show, has always been 
needed some have lived short lives full of burning zeal, and 
so on ; and in most of the faith of mathematicians there has 
been much error. 

Now we come to the second object of this book. In the 
historical part we shall see that the actual reasonings made by 
mathematicians in building up their methods have often not 
been in accordance with logical rules. How, then, can we say 
that the reasonings of mathematics are logical in their nature ? 
The answer is that the one word " mathematics " is habitually 
used in two senses, and so, as explained in the last chapter, I 
have distinguished "mathematics," the methods used to dis- 
cover certain truths, and " Mathematics " the truths discovered. 
When we have passed through the stage of finding out, by 
external evidence or conjecture, how mathematics grew up with 
problems suggested by natural events, like the falling of a 
jtone, and then how something very abstract and intangible 
but very real separated out of these problems, we can turn our 
attention to the problem of the nature of Mathematics without 
troubling ourselves any more as to how, historically, it gradu- 
ally appeared to us quite clearly that there is such a thing at 
all as Mathematics something which exists apart from its 
application to natural science. History has an immense value 
in being suggestive to the investigator, but it is, logically 
speaking, irrelevant. Suppose that you are a mathematician ; 
what you eat will have an important influence on your dis- 
coveries, but you would at once see how absurd it would be to 
make, say, the momentous discovery that 2 added to 3 makes 
5 depend on an orgy of mutton cutlets or bread and jam. 
The methods of work and daily life of mathematicians, the 
connecting threads of suggestion that run through their work, 
and the influence on their work of the allied work of others, 
all interest the investigator because these things give him 
examples of research and suggest new ideas to him ; but these 
reasons are psychological and not logical. 

But it is as true as it is natural that we should find that the