Sorry if the title sounds meta-sciency, allow me to clarify.
In physics, our goal is to understand how the universe works. To this end, we construct a theory, which hopefully makes falsifiable predictions, and then carry out experiments to test the theory.
In classical mechanics, for example, our theory allows us to determine the position and momentum of the particles in a system given some set of initial conditions. That is to say, in this theory we care about position and momentum, which we can measure.
In quantum mechanics, the situation is slightly more complicated, in that the whole theory revolves around the wave function, something which we can't measure directly. However, knowing the state of a system (its wave function), we can extract measurable information, like the probability of a particle having some position.
However, I'm starting to learn some classical field theory (to go on to QFT) and I'm completely lost from the very beginning. Here, the focus seems to be on some field $\phi(x)$. But, what does this field represent? I'm familiar with the mathematical notion of field, and could for example understand a scalar field like the temperature as a function of position. But what in the world does $\phi(x)$ represent? And how do we turn it into something we can measure?
Thanks in advance.