TITLE: How is "point" in geometry undefined? And What is a "mathematical definition"?
QUESTION [11 upvotes]: How is "point" in geometry undefined? I mean, when we say "A point in geometry is a location. It has no size, i.e., no width, no length, and no depth," is it not a definition?  If it is not a definition, then how can we know whether some statement is definition or not? What are the characteristics of a definition in math?

REPLY [5 votes]: Concerning your definition of "point" in geometry, what littleO said in a comment is enough:

You have only replaced one undefined term with other undefined terms.

To be more explicit, if you define "point" in terms of "location", I would simply ask you to define "location".
Now this does not address your other questions.

If it is not a definition, then how can we know whether some statement is definition or not?

Informally, we say that a valid definition is a way of describing something that is precise and only involves previously defined concepts. This is good enough for informal mathematics, but there is in fact a completely precise definition of "valid definitions" in first-order logic, which is variously called definitorial expansion or full abbreviation power. This rule basically allows one to name and later use any constant-symbol or predicate-symbol or function-symbol that can be represented uniquely by some first-order formula. For example, if you work within first-order Peano Arithmetic plus full abbreviation power, you can define:

$even(n) \overset{def}\equiv \exists k\ ( k+k = n )$.

And from then on you can reason about objects that satisfy the now defined predicate $even$, and can prove theorems involving it such as:

$\forall n\ ( even(n \times n) \to even(n) )$.

Suffice to say that such a technical device is necessary in practice so that we do not have pointless duplication of content. Of course the above theorem could have been written in a plain arithmetic sentence without using $even$, but clearly it would be much longer and less informative.

What are the characteristics of a definition in math?

The above concerns the technical details of how to formally define "valid definition" in first-order logic, and hence most of mathematics (which is based on a first-order set theory called ZFC). The concept of abbreviation power extends easily to other logics anyway. But there is a second issue of the different kinds of definitions in mathematics.
The first kind involves defining concepts within an existing framework. The example of $even$ is one instance of this. The second kind involves defining an entire framework (as a single concept)! For instance, we can define a structure to be a model for Peano Arithmetic iff it obeys all the axioms of PA. Note carefully that such a definition does not define what a single natural number is, but what is the collection of natural numbers together with the arithmetic operations as a whole.
Similarly, in any usual axiomatization of Euclidean geometry one does not define what points are, but rather defines that a structure is a Euclidean geometry iff it consists of lines and points (and usually numbers) that together satisfy certain axioms.