What is a concept? There are many definitions, but I use the term here with the meaning a set of abstractions that all satisfy a common set of requirements. Though this is a technical definition (one which can be made considerably more rigorous), I hope to show that it corresponds well to many everyday uses of "concept."
The requirements on a concept are explicitly stated in textual form in what is called a concept description. Thus concept descriptions and concepts are distinct notions, the former being a piece of text (possibly with links to other pieces of text), and the latter being a set of abstractions. We say the concept--the set of abstractions--is the denotation of the concept description. Often though it is convenient, if not quite precise, to blur the distinction between a concept and its description by referring to the description itself as a concept.
It is certainly conceivable that concept descriptions could be stated in a completely formal mathematical language, so that discussions of concepts could be carried out with a very high degree of mathematical rigor. It is also conceivable that processing of this language could be automated to a great degree using algorithmic decision procedures or heuristic artificial intelligence approaches. For development of concept webs over the next few years, under present limitations of computer science theory and computer hardware, it seems best to state concept descriptions in informal natural language, though with a concerted effort to be precise.