I think this post must be moved to chatbox. I dont't know how this should be done though.
Surface, before you go, I wonder if we might have your opinion, if you have one, on Maxwell's equations (in their common form due to Heaviside). I've often thought them an example of what physics should be - in their empirical soundness, their elegant simplicity which anyone can grasp with a bit of study, and the vast practical uses to which they've been put in improving civilization
(as opposed to the ongoing junk-science revolution - in and out of academia - which has the opposite intent). These four equations are the theoretical basis of the characteristic inventions that made the modern world of, say, 1930 possible: motor transport, household electrification, refrigeration, wireless communication, cinema, phonographs, etc.
However, I of course remain willing to listen to criticism of Maxwell from the rare legitimately thoughtful person (as opposed to mere destructive agitator) with an opinion on such matters. So, if you've discovered something untoward here, say on please.
Ah, very relevant point and should I say a classic. Actually, speaking off the top of my head, Maxwell while known for his classic equations, has also some other not so simple contributions to science. One is , in my current state of mind, the rather weird introduction of transcendental functions into probability, and if you read between the lines, mixing of real numbers, essentially a "continuous number system" with discrete mathematics! This equation used in probabilistic approaches in chemistry is hailed as the genius of the man. Hmmmm I am feeling uncomfortable already. But I am talking off the top of my head and from what I remember from my own direct studies at the time when there was no Wikipedia. So much for the character of the man and his contributions, one of which I now see as a deliberate attempt to force "transcendental functions" into probability and statistics, an essentially discrete field of mathematics concerned only with "integers". Now that I am thinking about this point, it looks to me like something done as a matter of principle rather than based on necessity!
I cannot remember when but some years ago I read an article claiming that Maxwell equations are the ultimate explanation we need in modern physics. At that time I dismissed this idea as too simplistic, and still hoped, based on the apparent "rate" at which our "science" was progressing, that soon some dramatic result will be published that will effectively explain all the obvious anomalies we meet in our explanations of the "infinitesimal world" as opposed to our "human scale" world, and also be able to merge all these with "cosmic scale" too. So obviously I didn't look into that claim at that time, now I am curious about it.
But before I go and have a look, I should point out that really, ironically, things are not so simple so that a set of equations can explain everything, and somehow everyone is missing that! There are quite a lot of "alternative" explanations in existence that attempt to complement the newtonian mechanics. Just to name one, Lagrange - Euler mechanics. I think it is called Lagrangian mechanics. The point is that these alternative mechanics, actually relativity is one among them, these are playing with equations. Really it is not hard when you get the hang of it. You essentially knock together equations to make them compatible with the "observed" phenomenon. Fair enough. The problem is that the observed phenomena, is reported by the likes of NASA! Then we have a problem. How can we test theories then?
The other problem is that digging further into these "advanced" topics, you will find a lot of "assumptions". As soon as something weird is observed the method of choice is "modeling", creating empirical models of the phenomenon, creating "systems of differential equations", which is the standard approach, then getting a computer to apply finite elements method "FEM" to these and get awesome 3d graphs out of it! Here the idea is that some stuff is just probabilistic and simply "too complex" to explain simply! Not only that, but in fact they are fully understood as probabilistic or chaotic, or "unexplainable". An example off the top of my head. Fluid dynamics, neat systems of differential equations. But how can you explain the chaotic dynamics? Don't worry there is a guy who understands it and wrote a 2000 page absolutely incomprehensible book. He is Chinese and unlike me has two PhD so I am intimidated and won't even expect to understand it!
Another trick, seen in the heavily "formula-based" approaches. They pick something out of pure mathematics, bring it into science, watch it grow! Why is this tricky? Because it is impossible to be both a scientist and a mathematician, specifically master of that particular mathematical branch! You simply have to rely on the introduced mathematical entity as perfectly valid. Why perfectly valid? Because we assume mathematics has a life of its own. We are not aware of the completely artificial nature of a large body of pure mathematics. Have you ever heard of those games where someone asks you to choose a number in your head and then add it with itself etc and then he can tell you the original number by knowing the end result? Yeah it works like that. They use a circular idea that relies on itself. This is the source of perfection of pure mathematics! They start with an idea, an assumption. Move backwards, making it incredibly complicated. Move forward and add it to itself, again move back make it complex. After a while they have created a web of impressively perfect yet absolutely useless knowledge! But what tricked me was the thought that such a degree of "complexity" must surely be signs of ultimate importance and relevance of our pure mathematics! That was a big mistake! These things are completely artificial, by no means have a life of their own. There is a philosophical debate about whether mathematics is actually a life of its own, is it real, is it part of the universe? Ha! What mathematics? The artificial one or the real one? It is easy to drop a stone in a well, but it takes much effort to take it out again.
So with this in mind, I suggest that we just don't assume that our universe must necessarily be expressed in systems of differential equations, or using vector calculus, or any known and discovered mathematics. One example. College physics explain the fact that two current carrying conductors exert force on each other. Why? College students will answer, because both are creating "fields" and these fields interact, attracting or repelling the conductors! Perfect. As soon as you pick up "advanced modern physics" you will see this: "sorry kids we have been kind of lying to you. Forget about the field, the real reason the conductors are moving is that errr the electrons in one conductor know that the other electrons are moving and so they try to move also perpendicular to the normal direction, so relativity remain valid!" The book admits that this giving "consciousness" to electrons is a bit weird, but that is how it is! In one experiment, observing one phenomenon immediately changes the other! Observing the other one changes the first! See the pattern? Giving ESP to the physical world, electrons, and physical phenomena. These are examples from official textbook physics, not Internet articles!
Well, to recap. Our problems almost certainly will not be solved with just one set of equations. Sorry. Neither with a whole bunch of them. There is simply too many problems to explain, both real and maybe hoaxes. Our mathematics is not to be blindly trusted, sometimes it is based on circular assumptions! Systems of differential equations, while used extensively in structural engineering, while being "simple", and while beautifully explaining some phenomena, like newtonian mechanics, are ultimately just that, systems of differential equations, and the real world, the infinitesimal world, does not seem to be explainable with them at all!
I will check recent views regarding Maxwell 's equations, but really I have zero hope for these even without looking into them again. Engineering works with pure trial and error. Electric motors work regardless. The explanations we read in a standard "Electromagnetic theory" book, which is a very tough subject I should admit, are simplifications. They are just models, like what I mentioned about attractions and repelling of current carrying conductors. That is how electric motors work and the relativistic explanation about this is nonsense. Engineering doesn't wait for theoretical physics. It is trial and error based.
What really bothers me is that no one is seriously looking into these issues. Is it that because for some reason we can never hope to understand or even explain the world fully? Do we live in a purely empirical world at our human macroscopic scale, and on infinitesimal scale the picture changes so much, that it becomes incomprehensible? That explains why engineering systems can be explained so simply, but physics is still stuck. Hmmmm that also explains why theoretical physicists are happily playing with equations.
Anyway I think I typed a lot. Will post a comment if I found anything recently interesting about those Maxwell equations.
I will post in chatterbox. I hope you will see that.