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INTRODUCTION 9 

way to become acquainted with new ideas is to study the 
way in which knowledge about them grew up. This, then, is 
what we will do in the first place, and it is here that I must 
bring my own views forward. Briefly stated, they are these. 
Every great advance in mathematics with which we shall be 
concerned here has arisen out of the needs shown in natural 
science or out of the need felt to connect together, in one 
methodically arranged whole, analogous mathematical processes 
used to describe different natural phenomena. The application 
of logic to our system of descriptions, which we may make 
either from the motive of satisfying an intellectual need (often 
as strong, in its way, as hunger) or with the practical end in 
view of satisfying ourselves that there are no hidden sources of 
error that may ultimately lead us astray in calculating future 
or past natural events, leads at once to those modern refine- 
ments of method that are regarded with disfavour by the 
old-fashioned mathematicians. 

In modern times appeared clearly what had only been 
vaguely suspected before the true nature of Mathematics. 
Of this I will try to give some account, and show that, since 
mathematics is logical and not psychological in its nature, all 
those petty questions sometimes amusing and often tedious 
of history, persons, and nations are irrelevant to Mathematics 
in itself. Mathematics has required centuries of excavation, 
and the process of excavation is not, of course, and never will 
be, complete. But we see enough now of what has been 
excavated clearly to distinguish between it and the tools which 
have been or are used for excavation. This confusion, it should 
be noticed, was never made by the excavators themselves, but 
only by some of the philosophical onlookers who reflected on 
what was being done. I hope and expect that our reflections 
will not lead to this confusion. 

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