TITLE: What is a field, really?
QUESTION [158 upvotes]: 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.

REPLY [0 votes]: Mathematical functions aren't real, so your intution is correct there. However, they model somethimg that is real. Your struggle with the field concept is quite natural, after all, physicists attempted to construct a mechanical description of the field - they called this the aether - in early modern physics after Newton but continually ran into road-blocks. Its only beginning with Faraday that we begin to see the field concept shorn of mechanics.
This raises the question of the reality of the field concept.
The EM field is taken to be real, for example, because it is able to carry momentum and energy, just like particles of matter.
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