The most significant thing must be the laws of quantum mechanics, but in order to get to that, there is more fundamental stuff.
This is the lesson of Newtonian mechanics. If you have a good computer, and you are given the initial conditions, the initial positions and velocities, and you know the force law, you can figure out how the objects move. You don't even need to work hard today, you can just let the computer do the work. This is the fundamental insight of the 17th and 18th centuries. The computer came later, of course, but Leibnitz already was thinking about similar things during the early days.
This is the insight of Mach, and it evolved into logical positivism. This is how you know when you are asking a sensible question--- when you can discrimate between the answers through some sort of observation.
Sometimes, in order to answer the meaningful questions, you need to introduce some extra concepts, for example, you might need to introduce a wavefunction, or space-time coordinates, or a gauge-choice to define the vector potential. All this extra framework is fine, so long as you understand that any two choices are ultimately equivalent, so you can translate between these choices freely.
It is logical positivism that allows you to understand that two coordinate systems for a physical problem are not different, they are really the same, when the predictions for all the sense-impressions predicted by the two systems are equivalent. So if someone insists that the world is really in polar coordinates (perhaps because rotations are important), and someone else insists that the world is really in rectangular coordinates (because translations are important), they are both being silly. Each coordinate system is as good as any other.
It might sound like this isn't a big deal, it's obvious. But there are so many questions people who are not trained in positivism think are meaningful, and these constitute the majority of questions a physicist gets from the public! Questions like "where did the universe come from?" "Is the world really made of particles or fields?" "Is the universe infinite beyond the cosmological horizon?" even "How many dimensions are there in string theory?" (because there are descriptions in different dimensions which reproduce the same sense-impressions, the question is not as meaningful as it sounds). These questions are annoying, because they don't have an answer and do not need an answer, because they are not questions! They are just the brain fooling.
One perspective on positivism is that the irreducible things are the sense-impressions, and the physical law is predicting the relation of these. But the sense-impressions are related to brain states, so you could also say that the physical law is predicting relations of the brain states, and then you need a map to identify brain-states with sense-impressions, to understand what the physics means really.
The positivism allowed 19th century physicists to convince themselves that fields are actual objects, as real as a chair, because there is a procedure to check if a field is present, just like there is a procedure to check if a chair is present. So it makes the abstract things concrete, whenver you can figure out a solid experimental test to see how the abstract thing is configured. But it also made other questions meaningless, like what is the true rest-frame of the ether? Or where is the electron exactly in the ground state of a Hydrogen atom?
The positivism is so essential for physics, that physicists automatically reduce the debates in other fields into the logically positivistically inequivalent positions, and tune out when the debate is meaningless. This is most debates, unfortunately, at least in the way that they are framed (usually there is a way to frame them so that the debates are meaningful too, it's just that if you aren't used to positivism, you don't tend to do that)
The idea of time is something that appears in our psychology, and we think we understand that time is something that "goes forward" somehow. This psychological idea is difficult to relate to the coordinate notion of time, which appeared after Einstein formulated the space-time picture. The space-time picture is very counterintuitive, because people are weirded out by the idea of time being like space, precisely because space is psychologically "all at once", while time is "a little bit at a time".
But it is exactly through positivism that one resolves these problems. The positivism tells you that the questions you can meaningfully ask are those which relate the measurements on clocks to the behaviors of people carrying brains. These questions can't resolve the question of whether time is "all at once" or "a little bit at a time", so slowly, you come to understand that really there is no sense in asking whether time is "all at once" or "a little bit at a time", only in asking whether a person's memories and experience will produce answers to questions that are consistent with time feeling like it is "a little bit at a time" to the person. The psychological feeling is not something that physical law cares about at all.
This insight is enormous, because probability is a type of thing that one thinks is necessarily restricted by logic to be exactly how it is normally, you can't fiddle with probability. But in quantum mechanics, the laws of probability are modified so that probabilities do not add for alternative ways, amplitudes add. It's not the probabilities that multiply for consecutive events, it's the amplitudes.
This is a difficulty, because in the macroscopic scale, the laws of probability work. The transition between the microscopic realm, with the amplitudes, and the macroscopic realm, where it's probability, is the measurement thing in quantum mechanics. When you interact with a quantum system in such a way that you end up in a situation where you should end up having different amplitudes to be in different situations, you don't "feel" that you are amplitudinous over different alternatives, rather you see that exactly one possibility occurs with a probability proportional to the absolute square of the amplitudinosity of you ending up in this situation.
This is mysterious, because amplitudes are different from probabilities, but when they are happening to big things, they become probabilities according to this squaring rule. But they never quite become identical to probabilities in quantum mechanics, they only become indistinguishable in the infinite system limit, which we are never precisely at.
I think there are exactly two possibilities here. Maybe quantum mechanics is exactly correct, and this is another philosophical problem, like the psychological time. It is again just due to the fact that what things "feel like" is a notion of experience, not a notion of physics, or rather, it is in the map from the physical description to our experience, the map that tells you what this or that brain state is supposed to feel like psychologically. This would mean that the superposed brain-states just don't "feel" superposed, they feel like a definite outcome to an individual, and the outcome just is one or the other according to the laws of probability, or at least, that's what it "feels like", and if you ask why, you are asking a meaningless question, because what the "feels like" stuff is supposed to be is only answered through experimental probing, and this point of view is consistent with experimental probing of all sorts of entities that feel all sorts of things. This philosophical point of view makes quantum mechanics complete, and it is the Everett many-worlds idea.
Because Everett is self consistent (at least assuming people are big enough to be infinite systems, which is probably a safe assumption) the physical law can be this crazy amplitude thing, and it's just psychologically that one cannot feel superposition.
The other possibility is that the real laws of nature are the standard laws of probability, and quantum mechanics is only approximate and emergent from a large distributed system, where the transitions are nonlocal. In order to be reasonable, such a theory, which no one has convincingly formulated, should not be so enormous that it reproduces quantum mechanics exactly, then it would be a philosophical thing, like Bohm's theory. It would have to be of physical size, and reproduce quantum mechanics only approximately, for small things, and be exactly probability for the big things (because it's exactly probability for the small things too, but only approximately quantum mechanics). Nobody knows how to make something that is probabilitiy reproduce something that looks approximately like quantum mechanics, but it might be possible.
Both ideas are reasonable today, because the notion of locality is out the window now that we know about holography and string theory. The proofs that it is impossible to reproduce quantum mechanics, at least the correct and stringent one due to John Bell, assumes the classical probability theory underneath, if there is one, is local. Then Bell shows that this doesn't work.
This is important--- it is something that is engrained in physics culture. You don't produce truth by putting stuff and holding a vote on whether it is correct. Rather you do it by individually debating the merits of proposals, checking them for yourself, and coming together after independent analysis to share your conclusions, with scathing criticism whenever everyone else is wrong and only you are right, until you each get the same answer, every single one of you, regardless of who ends up looking stupid.
This is the only way to ensure that politics is kept at bay, otherwise the things that are socially mediated to feel correct beat out the things that are correct. Physics is very good training to make sure you do this, because if you don't do it all the time, you can't understand anything. There are lots of things that no one is going to ever explain to you, and you need to just work out for yourself, so that you build up your own intuition for it, until it is obviously true, because you thought of it, not because somebody said so.
This process is not followed in other fields, where people take stuff that they personally doubt for granted, because some big-shot said so. This makes physicists dismissive of other fields, and rightly so! The mechanisms in other fields are completely busted. In mathematics, people also check for themselves, so there is nothing wrong, but in mathematics, the criterion of deciding whether something is "important" or "not so important" is political (but nothing can be done about it). In philosophy, the whole thing, including what constitutes a solid argument, is decided entirely politically, based on who wins more converts in a popularity contest. This makes the whole field useless. In other fields, the politics and the honesty mix to a different degree.
This cultural idea is very important, it is the one thing that physicists know that id just a universal social truth. The only way to get true propositions to win out is through rude bickering based on your own personal opinion, gotten to by half-baked personal thinking and rethinking, and independent of your authority position. It won't happen though polite analysis through socially mediated conversations that lead to gradual consensus. Those processes are guaranteed to produce bullshit. When consensus happens in physics, you can be sure that thousands of people checked it themselves, and all the diehards were either converted or died.
In addition to these big ideas (which are not big), physicists know all sorts of detailed things about the world that are interesting and important.
I don't just mean that this is in principle possible, I mean that physicists can actually see more or less how every single every-day phenomenon occurs, from the motion of quantum mechanical spread-out electrons bound to classical nuclei, where the nuclei move according to a potential energy which is the electronic energy of the configuration of electrons. This is the Born-Oppenheimer approximation, which is usually perfect. You walk around seeing how every material response is related to these microscopic constituents, more or less completely.
This is a nice thing to know! It makes life more interesting. You see a shiny metal, and you can feel how the electrons are distributed, they are spread out in long waves just like a cold Fermi gas. The light hits the electrons, and you know how they shake, and reflect the light, becaue they are delocalized. In an insulator, you know that there is a band-gap, so that the electrons can't shake at low energies. But at high enough photon-energies, enough to excite a molecule or atom, an individual electron can shake and retransmit the light, making a pretty color. And you can see how to break the metal, by disorder like Anderson described, or by making the electrons Mott-freeze into a lattice, or by a reconfiguration of the lattice into a periodic modulation, like Pieirls described.
You understand why metals are never brittle (because the electrons are delocalized and hold the atoms together), and ductile (because the spread out electrons only care about the total volume to first approximation, not the detailed association of neightboring atoms), why they conduct heat so fast (the electrons go far), why they are so fast at transmitting sound (because the delocalized electrons make them stiff).
You can see why heat makes water evaporate, and why there is a phase transition at a sharp temperature to a boiling state. You can understand how engineering mechanics works, through the flows of the momentum, and so on. You get a picture of the detailed functioning of the entire everyday world, and that's really fantastic.
This is the most important skill in physics, turning mathematical abstractions into concrete things, where you know what they will do without calculating anything, just from calculating a few simple things, and using your intuition, refined by first carefully checking for inconsistency, and later not having to check anything, because you just "know" what's going to happen.
This method is similar to what babies do when they first see water. It's weird stuff, it goes through your hands, but after a few experiences, you sort of know what it's going to do in any given situation, splash, pour, deform, get flat at the top, ripple, wave, etc. And you didn't need to calculate anything, even though the equations that describe water are hopelessly complicated, and babies don't know them.
Physicists do the same thing for everything in the physics literature. Really and truly. Even when it seems that they couldn't possibly know, they do. They don't calculate all the situations, of course, they calculate a few simple ones, and graually build up intuition, like a baby, for what will happen in the general case, by doing spot-checks, more complicated simulations, and getting a feel for when the range of phenomena has been more or less exhausted by the examples, when they are sufficiently general. This happens more quickly than you think--- the baby gets a sense for water almost immediately.
So if you are a decent General Relativist, and you are reading about a Schwartzschild black hole, you immediately see a one-way horizon in your mind's eye, you see the slow-time region nearby, the deforming of light paths, the range of orbits, and you also see a gooey lossy resistive membrane, and you immediately understand how it bends and deforms when you bring a massive charged object close and shake it. That's not because you did a calculation of all of these, but because you just KNOW what it does, because it's a physical object you are familiar with.
A good relativist is as familiar with a black hole as a baby in a bath is with water. That doesn't mean there ar eno surprises, if you show a baby capillary action, it will come as a surprise, but it means the surprises are rare, and reveal something new.
The ability to picture an abstract-seeming things in completely physical terms allows you to make leaps that are very difficult for rigorous folks, and this is really the only way that humans can understand things like elementary particles, or astrophysical structures. This is because we have only a limited amount of time to compute, and only the simplest calculations are tractable, but with familiarity, you really get to know everything, except for a few surprises, which then feed back to modify your intuition, and then you eventually run out of surprises.
But you don't get any intuition until you do enough simple calculations, and reproduce the things that are known for youself from scratch (because you need to get a feeling for all the signs, all the inequalities, and why they occur, what types of perturbations lead to qualitative differences, and so on and so on, it's tedious). This procedure makes you very very fast at getting the answer to certain mathematical questions in ways that look like magic to people who think about these things by proving theorems.
The thing is, this intuition business is very easy to fake! Write down a bunch of complicated equations, and say you understand that this and so is going to happen, and nobody can prove you wrong. This is the major reason that honesty is so important in physics, because you really need to be honest with yourself, so you know when you really have understood something from a rock-solid picture, where you have solid intuition for each of the parts, and when you just fooled yourself by a crappy analogy into thinking you know more than you do. And then you have to communicate this picture to other people, and this is hard too, but it is always possible, from experience, because every physicist today has no problem with Einstein's intuitions, or Feynman's.
The interesting thing about the intuition thing is that an abstract equation eventually turns into a physical picture in the head, an experience nearly as direct as water splashing on your hands. This makes it that each and every mathematical equation or physical argument in a good physics paper is as full of imagery as the greatest of novels, because you SEE the equation, and it makes complicated pictures in your head, that are true, because they work to tell you what happens in the equation. These pictures are always slightly different in each physicists's mind, but they are more or less the same (as you can tell by asking questions), and eventually they converge, and everyone has the exact same picture independently. The mathematical expressions don't do justice to the images, but they are pretty much the best one can do. But a person who hasn't carefully trained doesn't see the picture meaning, and cannot follow the arguments in the literature. This can be imporved perhaps by drawing more of the pictures using computer visualization, but sometimes the pictures are very abstract, and even a detailed computer picture doesn't help.
Good stuff on the whole. Regarding your #5, I think your observation that math requires politics to decide what's "important" is very apt. But I'm not sure the same thing doesn't also apply to physics. It seems funding is so limited in physics (compared to the number of physics PhDs) that there has to be a degree of politics in where that funding ultimately goes. I haven't seen it first- or second-hand like I have in math, but it's gotta be there.
I guess I have fundamental physics research in mind, things like the LHC, when I compare them to math. If you get into applied physics or biophysics, then those areas merge into various types of engineering or biology/chemistry, so it isn't clear to me why they're so much better than related fields. Every field needs socially nice methods for coming to truth; it's an inescapable problem.
Yes, it's there, but there's a nice thing that there is an independent probe, by experimental phenomena. In pure theory, where there is little experimental guide, there are also a few more or less objective tests: can you predict a (likely correct) new phenomenon? For example, Callan and Rubikov predicted monopole catalyzed proton decay. Gribov and Pomeranchuk predicted p-pbar cross sections asymptotically equal p-p cross sections, modern string theory predicts the precise microstates and emissions of black holes, and so on.
I realize now that mathematics has such an exactly analogous probe too--- you guys pick a random subset of elementary problems which are considered extremely difficult but not insurmountable, which you have absolutely no idea how to do, and turn them into famous conjectures, and when someone solves them, that's an objective test. These serve as benchmarks,
These serve as an antidote to politics, and they work. Some other fields have such things also.
I never did physics in my life, so i'm trying to understand this intuition thing. Does it mean roughly a probabilistic algorithm in your mind that allows to arrive to an approximately correct answer in short time?
It's things you know about equations without calculation or proof, but because your mind can "simulate" the equation qualitatively in your head.
For example, I have a top, and it is spinning fast, and I suddenly kick it sideways, How does it wobble?
Ok, i can see how it works, but how does this allow me to come up with new physics?
For example, you have Bekenstein's argument that black holes are thermodynamic objects.