@kennethamy,
Alongside the quotation from Wittgenstein, it might be worth setting an extract from section 63 of Frege's
The Foundations of Arithmetic, not to resolve the present perplexity, but to add to it:
Quote:It is not only among numbers that the relationship of identity is to be found. From which it seems to follow that we ought not to define it specially for the case of numbers. We should expect the concept of identity to have been fixed first, and that then, from it together with the concept of Number, it must be possible to deduce when Numbers are identical with one another, without there being need for this purpose of a special definition of numerical identity as well.
As against this, it must be noted that for us the concept of Number has not yet been fixed, but is only due to be determined in the light of our definition of numerical identity. Our aim is to construct the content of a judgement which can be taken as an identity such that each side of it is a number. We are therefore proposing not to define identity specially for this case, but to use the concept of identity, taken as already known, as a means for arriving at that which is to be regarded as identical. Admittedly, this seems to be a very odd kind of definition, to which logicians have not yet paid enough attention; but that it is not altogether unheard of, may be shown by a few examples.
As his main other example of the same kind of definition, he gives the definition that two lines have the same direction if and only if they are parallel. In section 68, he attempts to use this idea to define the direction of a line
a as the extension of the concept "line parallel to line
a", and similarly to define the number belonging to a concept
F as the extension of the concept "equal to [i.e. in one-to-one correspondence with] the concept
F".
However, is it notorious that Frege's programme foundered on Russell's paradox; and although it's not clear (at least, not clear to me right at this moment) that Russell's paradox depends in any essential way on the belief that every concept has an extension (I think it can be expressed in terms of concepts themselves), Frege himself seemed to think that this was the worm in his apple, and he should know! But there seems to me to be a more immediate difficulty in the notion of the extension of a concept. I wrote in the margin of my copy of Frege (on 29 Oct 1986):
Quote:There may be a difficulty here. If A and B are concepts [I used the Greek letters alpha and beta, but never mind] (say, A is the concept "parallel to line a" and B is "parallel to line b"), then I know what is meant by the sentence, "the extension of A is the same as the extension of B". But does it follow that I know what is meant by the phrase, "the extension of A"? Obviously I can't get out of the difficulty by defining the extension of A to mean the extension of the concept "having the same extension as A"!