Confusion about what exactly is meant by "Field" Does the "field" in physics same as the "field" in mathematics? Which one those is more abstract in Nature?
 A: The terminology addressing the meaning of field, in physics, tends to change according to the area and the context one is dealing with. 
In general a field is a map $p\mapsto T(p)$ that assigns a quantity to each point in the domain of definition of the theory, in order to define the states of the theory itself and their evolution.


*

*In point particle classical mechanics a field is a map
$$
t\in \mathbb{R}\to (\mathbf{r}(t),\mathbf{v}(t))\in\mathbb{R^6}
$$
giving the position and the velocity of the particle at any time $t$. More generally, for a system of classical particles a field is a map
$$
s\in \mathbb{R}\to \gamma(s)\in T^*Q
$$
with $T^*Q$ being the phase space as cotangent bundle of the configuration space.

*In quantum mechanics a field is a map
$$
t\in \mathbb{R}\to |\psi(t)\rangle\in\mathcal{H}
$$
that assigns an element of a Hilbert space to any point $t$ in time.

*In classical electromagnetism a field is a map
$$
(x,t)\in \mathbb{R^4}\to A^{\mu}(x,t)\in\mathbb{R^4}
$$
that assigns a vector potential to every point in space and time $(x,t)$. More generally, for gauge theories, a field is a map
$$
m\in \mathcal{M}\to T(m)\in\chi(\mathcal{M})
$$
that assigns a tensor to any point $m$ of a smooth manifold $\mathcal{M}$. Likewise for quantum gauge theories, where the above classical gauge fields are integrated in the path integral to give rise to correlation functions.

*In thermodynamics a field is a map
$$
(X_1,\ldots,X_N)\in\mathbb{R^N}\to S(X_1,\ldots,X_N)\in\mathbb{R}
$$
that assigns the entropy given a set of extensive variables $(X_1,\ldots,X_N)$
and many more examples can be provided.
