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34 THE NATURE OF MATHEMATICS 

has developed. For centuries mathematicians used " negative " 
and "positive" numbers, and identified "positive" numbers 
with signless numbers like 1, 2, and 3, without any scruple, 
just as they used fractionary and irrational " numbers." And 
when logically-minded men objected to these wrong statements, 
mathematicians simply ignored them or said : " Go on ; faith 
will come to you." And the mathematicians were right, and 
merely could not give correct reasons or at least always gave 
wrong ones for what they did. We have, over again, the 
fact that criticism of the mathematicians' procedure, if it 
wishes to be relevant, must be based on thorough sympathy 
and understanding. It must try to account for the Tightness 
of mathematical views, and bring them into conformity with 
logic. Mathematicians themselves never found a competent 
philosophical interpreter, and so nearly all the interesting part 
of mathematics was left in obscurity until, in the latter half of 
the nineteenth century, mathematicians themselves began to 
cultivate philosophy or rather logic. 

Thus we must go out of the historical order to explain what 
"negative numbers" means. First, we must premise that 
when an algebraical expression is enclosed in brackets, it sig- 
nifies that the whole result of that expression stands in the 
same relation to surrounding symbols as if it were one letter 
only. Thus, "a (b c) " means that from a we are to take 
6 c, or what is left after taking c from 6. It is not, there- 
fore, the same as a 6 c. In fact we easily find that a (b 
c) is the same as 06 + c. Note also that " (a + 6) (c+d) " 
means (a + b) multiplied by (c + d). 

Now, suppose a and 6 are numbers, and a is greater than 6. 
Let a 6 be c. To get c from a, we carry out the operation 
of taking away 6. This operation, which is the fulfilment of 
the order: "Subtract 6," is a "negative number." Mathe- 
maticians call it a "number" and denote it by " 6" simply 
because of analogy : the same rules for calculation hold for 
" negative numbers " and " positive numbers " like " -f 6," 
whose meaning is now clear too, as do for our signless numbers ; 
when "addition," "subtraction," &c., are redefined for these 
operations. The way in which this redefinition must take 
place is evident when we represent integers, fractions, and 
positive and negative numbers by points on a straight line.