20 THE GROUNDWORK OF SCIENCE

It may be replied that Mathematics deals with abstractions

and considers numerical relations of things apart from the

things themselves. The assertion is most true, but from that

very fact it must be applicable to all things, and would be

mere nonsense apart from the implication that there really

are things, be it only thoughts, to which the idea of number

can be really and truly applicable. And if thoughts are to be

capable of enumeration they must have existence, distinction,

similarity, unity and truth, just as a pair of sheep (as above

pointed out) must possess those attributes. But this degree

of similarity between things so essentially dissimilar as

" thoughts " and " sheep," suggests the further question, " What

is likeness " ?

Now a moment's reflexion must make it evident to any

thinker that not everything can be defined or explained.

If there were not some things capable of being understood

without definition and explanation, then nothing whatever

could ever be understood at all ; for in that case the pro-

cesses of definition and explanation would have to be

carried on for ever. Now " likeness," like " number," can

be clearly seen to imply ideas of existence, distinction,

unity and truth ; but that, of course, is no explanation

of it. It is one of those primary, ultimate, fundamental

ideas which (like the idea of "existence" or "being") is

incapable of definition or explanation just because it is

so simple. For to say that two things are "alike" when

they are identical in some respect, or respects, does not

deserve to be called an explanation. For to recognize

that two objects are identical in certain respects we must

be aware that their other respects are alike in not being

identical. Any one who thinks he cannot understand what

he means when he says two things are " alike," or when he

declares, " there is a 'likeness' between them," may as well give

up the attempt to understand any branch of science and,

It may be replied that Mathematics deals with abstractions

and considers numerical relations of things apart from the

things themselves. The assertion is most true, but from that

very fact it must be applicable to all things, and would be

mere nonsense apart from the implication that there really

are things, be it only thoughts, to which the idea of number

can be really and truly applicable. And if thoughts are to be

capable of enumeration they must have existence, distinction,

similarity, unity and truth, just as a pair of sheep (as above

pointed out) must possess those attributes. But this degree

of similarity between things so essentially dissimilar as

" thoughts " and " sheep," suggests the further question, " What

is likeness " ?

Now a moment's reflexion must make it evident to any

thinker that not everything can be defined or explained.

If there were not some things capable of being understood

without definition and explanation, then nothing whatever

could ever be understood at all ; for in that case the pro-

cesses of definition and explanation would have to be

carried on for ever. Now " likeness," like " number," can

be clearly seen to imply ideas of existence, distinction,

unity and truth ; but that, of course, is no explanation

of it. It is one of those primary, ultimate, fundamental

ideas which (like the idea of "existence" or "being") is

incapable of definition or explanation just because it is

so simple. For to say that two things are "alike" when

they are identical in some respect, or respects, does not

deserve to be called an explanation. For to recognize

that two objects are identical in certain respects we must

be aware that their other respects are alike in not being

identical. Any one who thinks he cannot understand what

he means when he says two things are " alike," or when he

declares, " there is a 'likeness' between them," may as well give

up the attempt to understand any branch of science and,