I understand why definite descriptions like 'The King of England' are denoting phrases, but I am confused by differing between concepts and what they 'denote'.
In Principles of Mathematics:
A concept denotes when, if it occurs in a proposition, the proposition is not about the concept, but about a term connected in a certain peculiar way with the concept. If I say “I met a man,” the proposition is not about a man: this is a concept which does not walk the streets, but lives in the shadowy limbo of the logic-books. What I met was a thing, not a concept, an actual man with a tailor and a bank-account or a public-house and a drunken wife. Again, the proposition “any finite number is odd or even” is plainly true; yet the concept “any finite number” is neither odd nor even.
For example 'a man' seems equivalent to 'one man', why is this treated as denoting anything but the idea of there being a single entity and that entity being of a class 'man'?
For example, I could easily say 'one man lives next door', It is a simply a statement about how many men are next door I do not need to be using 'one man' to refer to any individual, in this case it denotes a concept.
This is different to 'The King of England' where I could argue that 'the king of england' is a person and the idea of what it is to be the king of england is a different entity which needs it's own reference.
Do these phrases have two meanings and what is the difference between the concepts and what they denote? Take 'two' does it have a concept and denotion because it's denotation is a concept?
With 'The king of England', I could suggest that the only 'concept' it gives me is that of the man, the denotation in this case which to me is clearly a particular person.
Clearly I am misunderstanding, but I'd appreciate if anyone could suggest where, and if there is other sources for which I can clear up my confusion.