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24 THE NATURE OF MATHEMATICS 

The generality of character has been attained by the effort to 
put mathematics more into touch with natural sciences in par- 
ticular the science of motion. The main difficulty was that, 
to reach this end, the way in which mathematicians expressed 
themselves was illegitimate. Hence philosophers, who lacked 
the real sympathy that must inspire all criticism that hopes 
to be relevant, never could discover any reason for thinking 
that what the mathematicians said was true, and the world 
had to wait until the mathematicians began logically to 
analyse their own conceptions. No body of men ever needed 
this sympathy more than the mathematicians from the revival 
of letters down to the middle of the nineteenth century, for 
110 science was less logical than mathematics. 

The ancient Greeks never used the conception of motion 
in their systematic works. The idea of a locus seems to imply 
that some curves could be thought of as generated by moving 
points ; the Greeks discovered some things by helping their 
imaginations with imaginary moving points, but they never 
introduced the use of motion into their final proofs. This was 
because the Eleatic school, of which the principal representative 
was Zeno (495-435 B.C.), invented some exceedingly subtle 
puzzles to emphasize the difficulty there is in the conception of 
motion. We shall return in some detail to these puzzles, 
which have not been appreciated in all the ages from the time 
of the Greeks till quite modern times. Owing to this lack of 
subtlety, the conception of variability was freely introduced 
into mathematics. It was the conceptions of constant, variable, 
and function, of which we shall, from now on, often have 
occasion to speak, which were generated by ideas of motion, 
and which, when they were logically purified, have made both 
modern mathematics and modern logic, to which they were 
transferred by mathematical logicians Leibniz, Lambert, 
Boole, De Morgan, and the numerous successors of Boole and 
De Morgan from about 1850 onwards into a science much 
more general than, but bearing some close analogies with, the 
ideal of Greek mathematical science. Later on will be found a 
discussion of what can be meant by a " moving point." 

Let us now consider more closely the history of modern 
mathematics. Modern mathematics, like modern philosophy