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.

Solutions Collecting From Web of "Does intrinsic mean existing regardless of some bigger space?"

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

$${g_{ij}} = \left( {\begin{array}{*{20}{c}}
E&F \\
\end{array}} \right) = \left( {\begin{array}{*{20}{c}}
{\left\langle {\frac{{\partial f}}{{\partial u}},\frac{{\partial f}}{{\partial u}}} \right\rangle }&{\left\langle {\frac{{\partial f}}{{\partial u}},\frac{{\partial f}}{{\partial v}}} \right\rangle } \\
{\left\langle {\frac{{\partial f}}{{\partial v}},\frac{{\partial f}}{{\partial u}}} \right\rangle }&{\left\langle {\frac{{\partial f}}{{\partial v}},\frac{{\partial f}}{{\partial v}}} \right\rangle }
\end{array}} \right)$$

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
$${g_{ij}} = \left( {\begin{array}{*{20}{c}}
{{{\cos }^2}\theta }&0 \\
\end{array}} \right)$$

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:
$$L = \int\limits_a^b {\sqrt {g} } $$

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.