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82 THE NATURE OF MATHEMATICS 

theory not touched upon here of ordinal numbers (denoted by 
"first," "second," . . .) which apply to sets arranged in some 
order, known at present. 

To illustrate (1), think of this. According to the above 
definition 2 is the general idea we call " couple." We say : 
" Mr. and Mrs. A are a couple " ; our definition would ask us 
to say in agreement with this : " The class consisting of Mr. 
and Mrs. A is a member of the class 2." We define " 2 " as 
" the class of classes u such that, if x is a u, u lacking a; is a 
1 " ; the definition of " 3 " follows that of " 2 " ; and so on. 
In the same way, we see that the class of fingers on your right 
hand and the class of fingers on your left hand are each of 
them members of the class 5. It follows that the classes of 
the fingers are similar in the above sense. 

Out of the striving of human minds to reproduce con- 
veniently and anticipate the results of experience of geometrical 
and natural events, mathematics has developed. Its develop- 
ment gave priceless hints to the development of logic, and then 
it appeared that there is no gap between the science of 
number and the science of the most general relations of objects 
of thought. As for geometry and mathematical physics, it 
becomes possible clearly to separate the logical parts from those 
parts which formulate the data of our experience. 

We have seen that mathematics has often made great strides 
by sacrificing accuracy to analogy. Let us remember that, 
though mathematics and logic are the highest forms of certainty 
within the reach of us, the process of mathematical discovery, 
which is so often confused with what is discovered, has led 
through many doubtful analogies and errors arriving from the 
great help of symbolism in making the difficult easy. Fortu- 
nately symbolism can also be used for precise and subtle 
analysis, so that we can say that it can be made to show up 
the difficulties in what appears easy and even negligible 
like 1 + 1 = 2. This is what much modern fundamental work 
does.