definition - What is the degree of freedom?

In here, https://en.wikipedia.org/wiki/Degrees_of_freedom_(mechanics),
the degree of freedom is defined as "the number of independent parameters that define its configuration." So, if $N$ particles are in the system, the degree of freedom is $3N$.

But here, https://en.wikipedia.org/wiki/Degrees_of_freedom_(physics_and_chemistry),
defined as "The set of all dimensions of a system is known as a phase space, and degrees of freedom are sometimes referred to as its dimensions." In this sense, the degree of freedom is $6N$.

What is the definition of the degree of freedom?

Answer

A degree of freedom is basically a system variable that's unbound (free).

We say "degrees of freedom" rather than just "variables" to clarify that we're referring to that freeness of the system rather than a specific count of variables.

For example, consider a 2-D grid with a particle at $\left(x,y\right)$. We can also refer to that particle's location in terms of polar coordinates, $\left(r,\theta\right)$. So that's 4 variables: $\left\{x,y,r,\theta\right\}$; however, at most we can only fill in 2 of them. This is what we mean by the system having "2 degrees of freedom": sure there're more than 2 variables, but only 2 of them are free.

Example: $3n$ vs. $6n$ from the question

If you have a system of $n$ particles, then their positions have $3n$ degrees-of-freedom:

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  1 for each $x$ coordinate;

  
  



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  1 for each $y$ coordinate; and

  
  


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  1 for each $z$ coordinate.

  
  

But what if you want to include their velocities? Then you need $3n$ more for the components of velocity: $v_x$, $v_y$, and $v_z$. That brings it to $6n$.

However, neither $3n$ nor $6n$ is particularly fundamental or worth memorizing. You'll generally want to think out the number of degrees of freedom every time you consider a physical situation.