Does intrinsic mean existing regardless of some bigger space? How is the arc-length of a regular parametrized curve in a surface $S\subset\mathbb{R}^3$ intrinsic? Let $\bf{x}\rm(u,v)$ be a parametrization of $S$. Letting $E,F,G$ denote the coefficients of the first fundamental form, the arc length of a curve $\alpha:U\to S$ is said to be intrinsic because it can be computed with knowledge of only these coefficients as $$\int_0^t\sqrt{E(u^\prime)^2+2Fu^\prime v^\prime +G(v^\prime)^2}.$$But an ant living on the surface could not compute this value since $E,F,G$ are computed using points described by $3$-coordinates. The $3^{\text{rd}}$ coordinate does not exist to the ant.
If one lived in $\mathbb{R}^2$ (or a surface which they think is $\mathbb{R}^2$), and did not know of the existence of $\mathbb{R}^3$, they would have no way to compute this value. Would they?
I understand intrinsic meaning invariant under isometries, but I don't see how an intrinsic property can be computed or can exist without reference to some bigger space. To compute the Gaussian curvature at a point, you explicitly use the fact that the point is described by $3$ coordinates.
 A: An intrinsic property is one that can be defined only in terms of the first fundamental form. 
Taking our surface element to be $$f:U \to {\mathbb{R}^3}$$
We can parameterize f in terms of just two arbitrary coordinates (Even though the surface element is embedded in 3-space you don't need 3 coordinates to describe it)
$$f(u,v) = \left( {x(u,v),y(u,v),z(u,v)} \right)$$
We can than use this parametrization to define the first fundamental form
$${g_{ij}} = \left( {\begin{array}{*{20}{c}}
  E&F \\ 
  F&G 
\end{array}} \right) = \left( {\begin{array}{*{20}{c}}
  {\left\langle {\frac{{\partial f}}{{\partial u}},\frac{{\partial f}}{{\partial u}}} \right\rangle }&{\left\langle {\frac{{\partial f}}{{\partial u}},\frac{{\partial f}}{{\partial v}}} \right\rangle } \\ 
  {\left\langle {\frac{{\partial f}}{{\partial v}},\frac{{\partial f}}{{\partial u}}} \right\rangle }&{\left\langle {\frac{{\partial f}}{{\partial v}},\frac{{\partial f}}{{\partial v}}} \right\rangle } 
\end{array}} \right)$$
The arc length is than defined by just the first fundamental form and is therefore an intrinsic property
$$d{s^2} = Ed{u^2} + 2Fdudv + Gd{v^2}$$
$$L = \int\limits_a^b {ds} = \int\limits_a^b {\sqrt {Ed{u^2} + 2Fdudv + Gd{v^2}} } $$
Conceptual Example:
If you want to think about it terms of some ant that exists only in 2-space: This ant could travel the distance of some arc on a surface element and measure this distance without any knowledge of the ambient space, only knowledge of a 2-coordinate system it defines on the surface element.
Mathematical Example: 
The unit 2-sphere can be parametrized in terms of only 2 coordinates, the angles phi and theta.
$$f(\varphi ,\theta ) = \left( {\cos \varphi \cos \theta ,\sin \varphi \cos \theta ,\sin \theta } \right)$$
We can than define the first fundamental form as 
$${g_{ij}} = \left( {\begin{array}{*{20}{c}}
  {{{\cos }^2}\theta }&0 \\ 
  0&1 
\end{array}} \right)$$
And therefore the arclength as 
$$L = \int\limits_a^b {ds}  = \int\limits_a^b {\sqrt {{{\cos }^2}\theta d{\varphi ^2} + d{\theta ^2}} } $$
Definition of arc length independent of any specific ambient space:
To define the length of some arc on a hyper-surface element independent of the ambient space in which the element is embedded you can use Riemannian Geometry.
Take the metric tensor: $${g_{ij}} = {e_i} \bullet {e_j}$$
And the Line Element/Riemannian Metric:
$$d{s^2} = {g_{ij}}d{x^i} \otimes d{x^j} = g$$
We can than define Arc Length in a way that is the same for all spaces in which the hyper-surface element could be embedded:
$$L = \int\limits_a^b {\sqrt {g} } $$
A: The components $E$, $F$, and $G$ of the first fundamental form on a smooth surface $S$ are indeed defined using an embedding of $S$ into a Euclidean space, but you can imagine a situation (such as a "physical theory" in the surface) in which three functions $E$, $F$, and $G$ (or $g_{11}$, $g_{12}$, and $g_{22}$) are associated to each coordinate patch on an abstract surface. For instance, on $\mathbf{R}^{2}$ you might have
$$
E = G = \frac{4}{(1 + x^{2} + y^{2})^{2}},\quad F = 0.
$$
In this setting, arc length makes sense, even though no embedding is specified.
