There was a reason why I constantly failed physics at school and university, and that reason was, apart from the fact I was immensely lazy, that I mentally refused to "believe" more advanced stuff until I understand the fundamentals (which I, eventually, never did).
As such, one of the most fundamental things in physics that I still don't understand (a year after dropping out from the university) is the concept of field. No one cared to explain what a field actually is, they just used to throw in a bunch of formulas and everyone was content. The school textbook definition for a field (electromagnetic in this particular case, but they were similar), as I remember it, goes like:

An electromagnetic field is a special kind of substance by which charged moving particles or physical bodies with a magnetic moment interact.

A special kind of substance, are they for real? This sounds like the authors themselves didn't quite understand what a field is so they decided to throw in a bunch of buzzwords to make it sounds right. I'm fine with the second half but a special kind of substance really bugs me, so I'd like to focus on that.
Is a field material?
Apparently, it isn't. It doesn't consist of particles like my laptop or even the light.
If it isn't material, is it real or is it just a concept that helps to explain our observations? While this is prone to speculations, I think we can agree that in scope of this discussion particles actually do exist and laws of physics don't (the latter are nothing but human ideas so I suspect Universe doesn't "know" a thing about them, at least if we're talking raw matter and don't take it on metalevel where human knowledge, being a part of the Universe, makes the Universe contain laws of physics). Any laws are only a product of human thinking while the stars are likely to exist without us homo sapiens messing around. Or am I wrong here too? I hope you already see why I hate physics.
Is a field not material but still real?
Can something "not touchable" by definition be considered part of our Universe by physicists? I used to imagine that a "snapshot" of our Universe in time would contain information about each particle and its position, and this would've been enough to "deseralize" it but I guess my programmer metaphors are largely off the track. (Oh, and I know that the uncertainty principle makes such (de)serialization impossible — I only mean that I thought the Universe can be "defined" as the set of all material objects in it). Is such assumption false?
At this point, if fields indeed are not material but are part of the Universe, I don't really see how they are different from the whole Hindu pantheon except for perhaps a more geeky flavor.
When I talked about this with the teacher who helped me to prepare for the exams (which I did pass, by the way, it was before I dropped out), she said to me that, if I wanted hardcore definitions,

a field is a function that returns a value for a point in space.

Now this finally makes a hell lot of sense to me but I still don't understand how mathematical functions can be a part of the Universe and shape the reality.

I'm going to go with a programmer metaphor for you.


The mathematics (including "A field is a function that returns a value for a point in space") are the interface: they define for you exactly what you can expect from this object.
The "what is it, really, when you get right down to it" is the implementation. Formally you don't care how it is implemented.

In the case of fields they are not matter (and I consider "substance" an unfortunate word to use in a definition, even though I am hard pressed to offer a better one) but they are part of the universe and they are part of physics.

What they are is the aggregate effect of the exchange of virtual particles governed by a quantum field theory (in the case of E&M) or the effect of the curvature of space-time (in the case of gravity, and stay tuned to learn how this can be made to get along with quantum mechanics at the very small scale...).

Alas I can't define how these things work unless you simply accept that fields do what the interface says and then study hard for a few years.


Now, it is very easy to get hung up on this "Is it real or not" thing, and most people do for at least a while, but please just put it aside. When you peer really hard into the depth of the theory, it turns out that it is hard to say for sure that stuff is "stuff". It is tempting to suggest that having a non-zero value of mass defines "stuffness", but then how do you deal with the photo-electric effect (which makes a pretty good argument that light comes in packets that have enough "stuffness" to bounce electrons around)? All the properties that you associate with stuff are actually explainable in terms of electro-magnetic fields and mass (which in GR is described by a component of a tensor field!). And round and round we go.
