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84 THE NATURE OF MATHEMATICS 

Grammatical usage makes us use a capital letter even for 
"mathematics" in the psychological sense when the word 
begins a sentence, but in this case I have guarded and will 
guard against ambiguity. 

That particular function of history which I wish here to 
emphasize will now, I think, appear. In mathematics we 
gradually learn, by getting to know some things about Mathe- 
matics, to know that there is such a thing as Mathematics. 

We have, then, glanced at the mathematics of primitive 
peoples, and have seen that at first isolated properties of 
abstract things like numbers or geometrical figures and of 
abstract relations between concrete things like the relations 
between the weights and the arms of a lever in equilibrium. 
These properties were, at first, discovered and applied, of 
course, with the sole object of the satisfaction of bodily needs. 
With the ancient Greeks comes a change in point of view which 
perhaps seems to us, with our defective knowledge, as too 
abrupt. So far as we know, Greek geometry was, from its 
very beginning, deductive, general, and studied for its own 
interest, and not for any applications to the concrete world it 
might have. In Egyptian geometry, if a result was stated as 
universally true, it was probably only held to be so as a result 
of induction the conclusion from a great number of particular 
instances to a general proposition. Thus, if somebody sees 
a very large number of officials of a certain railway companj r , 
and notices that all of them wear red ties, he might conclude 
that all the officials of that company wear red ties. This 
might be probably true : it would not be certain : for certainly 
it would be necessary to know that there was some rule accord- 
ing to which all the officials were compelled to wear red ties. 
Of course, even then the conclusion would not be certain, since 
these sort of laws may be broken. Laws of Logic, however, 
cannot be broken. These laws are not, as they are sometimes 
said to be, laws of thought : for logic has nothing to do with 
the way people think, any more than poetry has to do with 
the food poets must eat to enable them to compose. Some- 
body might think that 2 and 2 make 5 : we know, by a process 
which rests on the laws of Logic, that they make 4. 

This is a more satisfactory case of induction : Format stated 
that no integral values of x, y, and s can be found such that