I need help understanding Frege's definition of number I have really been trying to understand Frege's definition of a number or at least gain a strong intuition of it. However, my attempts have not been fruitful. If someone could help me it would be much appreciated. I will write his definition down explicitly, then give write down a little bit of what he writes around this definition. 

Number - The content of a statement of number is an assertion about a concept. 

[source : The Foundations of Arithmetic, §55, Engl.transl. page 67]

Here is what he writes before and immediately after stating his definition:

Immediately Before: While looking at one and the same external phenomenon I can say with equal truth both "It is a copse" and "It is five trees," or both "Here are four companies" and "Here are 500 men." Now what changes here from one judgement to the other is neither any individual object, nor the whole, the agglomeration of them, but rather my terminology. But that is itself only a sign that one concept has been substituted for another. 

[source : The Foundations of Arithmetic, §46, Engl.transl. page 59]

Immediately After: This is perhaps clearest with the number $0$. If I say "Venus has $0$ moons", there simply does not exist any moon oor agglomeration of moons for anything to be asseted of; but what happens is that a property is assigned to the concept "moon of Venus", namely that of including nothing under it. If I say "the King's carriage is drawn by four hourses", then I assign the number four to the concept "horse that draws the King's carriage"

[source : The Foundations of Arithmetic, §46, Engl.transl. page 59]

Edit: For those who are more inclined to reading logic. On part 2.5 of the following link it explains what Frege was doing. Unfortunately, I'm having difficulties understanding this as well. I have been exposed to some first order logic, but no second order logic whatsoever. I am not sure if that is necessary though. 

http://plato.stanford.edu/entries/frege/#NatNum

Thank you in advance for anyone who can help!
Some hints

The context of Frege's assertion is his analysis of the "number assertions"; see :


Die Grundlagen der Arithmetik: eine logisch-mathematische Untersuchung über den Begriff der Zahl (1884), translated as The Foundations of Arithmetic: A logico-mathematical enquiry into the concept of number by J.L. Austin (1950).


Roughly speaking, Frege denies that when we use number-words in natural language assertions we express a property of objects; i.e. when I say "my hands are white" I'm asserting that each of my hands is white, while when I'm saying  "I have two hands", I'm not asserting that each of my hands "is two".

Thus, Frege's conclusion is that the "number-words" are like adjectives for concepts, i.e. they express a property of a concept.

Example : there are two satellites of Mars : Phobos and Deimos. When we say "Mars has two satellites", we express the fact that the concept "satellite of Mars" is instantiated by two objects.

In modern expression, we have that :


  $\exists x \exists y [Sat_M(x) \land Sat_M(y) \land x \ne y \land \forall z(Sat_M(z) \to (z=x \lor z=y))]$.


This is also clear from the analysis of the number $0$ : 


  "if I say "Venus has $0$ moons", there simply does not exist any moon or agglomeration of moons for anything to be asseted of; but what happens is that a property is assigned to the concept "moon of Venus", namely that of including nothing under it."


i.e.


  $\lnot \exists x Moon_V(x)$.


This point has fundamental consequences in Frege's view : numbers are akin to quantifiers.

In Frege's terminology, both number and quantifiers are second-level concepts, i.e. they are concepts that express a property of first-level concepts (like the concept : "moon of Venus", i.e. concepts "applying" to objects).

This fact can be highlighted by the "formal" analysis; we have that Phobos satisfy the concept "Satellite of Mars", i.e. $Sat_M(Phobos)$ holds. 

In Frege's terminology, the object Phobos "falls under" the concept "Satellite of Mars".

The concept "Satellite of Mars" is instantiated, i.e. there are objects that "fall under" it. Thus the second-level concept "existence" applies to the first-level concept "Satellite of Mars", i.e. $\exists x Sat_M(x)$ holds.



We can "formalize" Frege's view (in a "philological" inaccurate way) as follows :

1) assume the universal quantifier $\forall$ as primitive;

2) define the existential quantifier $\exists$ as usual, as : $\lnot \forall \lnot$;

3) introduce the "numerical" quantifiers :

i) $\exists^0$ defined through : $\lnot \exists Fx$ : "there are no $F$'s"

ii) $\exists^1$ defined as the "usual" $\exists !$, i.e. as $\exists x Fx \land \forall y (Fy \to x=y)$ : "there is (exactly) one $F$"

iii) $\exists^2$ as $\exists x \exists y (Fx \land Fy \land x \ne y \land \forall z(Fz \to z=x \lor z=y))$ : "there are two $F$'s"

and so on.

These "numerical" quantifiers are the formal counterpart of "number-words".



You can see :


Kevin Klement, Frege’s Changing Conception of Number, Theoria (2012).
Frege's description of number (which is a philosophical and not mathematical question, yet essential to mathematics) is problematic because it appeals to the relationship "falls under". Yet there is no relationship, at least not a logical one, that is given by "falls under". There is merely a grammatical tradition. We say "objects fall under concepts", but this only expresses the way we casually talk about objects and concepts. It leaves us in the dark regarding both concepts and objects and their relationship. Yet it is their supposed relationship that Frege uses in his definition of number. 

But the question of what Frege meant by number can be dismissed as secondary to a higher problem. The question we should be asking is not what sort of things numbers are, or in what domain they exist, but what are the conditions for the creation of numbers. Frege takes numbers to be independent of any constructional conditions. He takes what is called a trancendentally real line, a posture common to all scientists and mathematicians, and most philosophers. Instead, he should be taking (like Wittgenstein) a transcendentally ideal line, which shows that there are no independent numbers, nor any domain that supports their existence. Logic and mathematics are unable to distinguish between transcendentally real and transcendentally ideal objects. The syntax of their expression looks the same in both models, a fact that astonishingly has gone unnoticed throughout the history of science and logic.