In mathematics, a successful definition is one that has the following properties:

1. It is unambiguous. It leaves no room for argument or even discussion about what it means.

2. It is useful for what we are trying to achieve. For instance we could define a zooblad to be any number that, when written out using English words (no numerals), takes a number of characters that is divisible by three. But it is not likely to be useful for any interesting work.

3. Uniqueness. It doesn't use a name that is already used for some other concept that is often used in the same subject area. Mathematics sometimes fails this standard itself. For instance the number of different meanings of 'normal' is staggering.

I suggest the same principles can be applied to definitions in philosophy.

But what are the criteria for a succesful definition? When is a definition of X correct or adequete? — PossibleAaran

Since defining a concept is essentially a consensus development process, one can never consider it correct as though there was some absolute agreement. At best, there is an agreed upon symbolic representation that can be utilized by a group to communicate with each other for some practical purpose. However, even within this group, if inquired for more details and precision, there will arise vast differences in understanding or description.

However, even within this group, if inquired for more details and precision, there will arise vast differences in understanding or description. — Rich

Hence, the continued effort of a philosopher to refine his definition.
Mill and Wittgenstein must be one of the most prolific when it comes to defining concepts.

Hence, the continued effort of a philosopher to refine his definition.
Mill and Wittgenstein must be one of the most prolific when it comes to defining concepts. — Caldwell

It's hopeless, but it is a way to pass life.

Definitions without agreement on some concrete ontology are just floating phrases waiting for an argument. I have come to understand that most of academic philosophy centers around arguments. I am in a different space. My primary interest is understanding the nature of nature and the nature of Life. Among all the philosophical arguments, I do pick up a nice new idea now and then but never within the such discussions as definitions. All there is are bullets flying in space.

Russell wrote somewhere that the more general a term is, the harder to define. Very specific things are very easy to define - screwdrivers, oranges, running. Very general things are of extremely difficult to define - consciousness, love, knowledge, for example. So in the latter cases, it’s how the terms are used that is more important than a definition as such. Such general terms get their meanings within specific domains of discourse, and in the way in which they are applied in various contexts.

This is much more the case in philosophy than in mathematics or almost any other discipline, because philosophy must deal with extremely general or wide-ranging ideas. That is why in some respects seminal philosophical works, like those of Aristotle, are might be largely devoted to saying what key terms mean and what the entailments are carried by it.

In the classical sense, philosophy is the pursuit of wisdom, namely, living the good life as informed by the truth. In the professional sense, as Anthony Quinton says, philosophy is thinking about thinking. It is the analysis of second order concepts, like truth, goodness, beauty, causality, etc. Indeed, it conceptually analyzes conceptual analysis itself, so we might say it analyzes third order concepts as well. In other words, philosophy analyses not only other things but itself and not only analyzes itself but analyzes the analyzing of itself. This naturally leads to an infinite regress, which is why philosophy never seems to progress, unlike other disciplines. One must simply accept a set of axioms in order to do philosophy but also to disavow it. So there is no real escape from it.

But what are the criteria for a succesful definition? When is a definition of X correct or adequete? — PossibleAaran

To me the correct but slippery answer is utility. How best to define utility? In a useful way, surely. With words like 'utility' and 'good,' I think we approach something like a bark or a meow. One feels success or goodness.