Why is introductory physics not taught in a more “axiomatic” way?

I am engineer who was taught the standard four semester, two-year physics courses from Halliday and Resnick's book. However, after reading the insightful answers here (e.g. Ron Maimon's awesome explanation of $v^2$ term in kinetic energy) and some good books (e.g. QED and Weyl's fantastic Space-Time-Matter), my eyes have opened to the possibilities of explaining deep physical knowledge from simple first principles.

So, my question is:

Why is undergraduate Physics not taught more in an axiomatic fashion, e.g. rigorously introduce the concepts of time and space (like Weyl does), then introduce the Galilean transformation and the concept of fields. Then proceed to show some powerful consequences of these.

One can argue the classic approach is more suited for engineers and related disciplines, which probably is true. But future physicists learn from the same books, too, which I find to be amazing.

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Aren't Newton's laws, the conservation of energy and momentum axioms? I'd say it's the complete opposite to what you claim where people are initially taught to accept the axioms as true and later on, they come to understand how they are so using other principles. –  John McVirgo Feb 6 '12 at 20:17
Physics cannot be reduced to a limited set of axioms, at least in its present state. And methodologically it is as harmful as axiomatizing Bohr orbits. It would prevent us from developments. –  Vladimir Kalitvianski Feb 6 '12 at 21:47
Leave the axiomatic systems to the mathematicians. Progress in physics must come from experimental evidence. –  wim Feb 7 '12 at 0:50
Just saw this one. Interesting. But beware: Who says time and space are the deepest axioms? I can construct a program in which space, at least, is an emergent property of rules that restrict how a globally accessible set of points interact with each other. That sounds pretty abstract, but if you look carefully at how quantum mechanics works, there are some uncomfortable similarities in it to just that sort of situation. Also, defining time when you are always in "now" requires you to model of the past, not touch it directly. That implies intelligence just to define t meaningfully. –  Terry Bollinger Feb 23 at 1:39
@TerryBollinger: Uncomfortable similarities? Stop worrying and learn to love the bomb. ;) –  NikolajK Feb 23 at 1:43

Building off dmckee's answer, even students who are interested in physics generally like to take a "reality-based" approach to it. For example, time and space: to students beginning their physics education, it's obvious what time and space are, and trying to axiomatically define them is just a waste of time that could be spent learning about things that they can do with time and space. Plus, the students will wonder why you're putting so much effort into this axiomatic definition, when there is a (to them) perfectly satisfactory intuitive definition. Only later on, when they get into more advanced physics where the intuitive notions of time and space don't have enough detail, do they see the need for a rigorous (or semi-rigorous), axiomatic definition. That's the time to introduce it.

Of course, there are some physics students who take nothing from intuition, and who want the rigorous, axiomatic approach right from the beginning. Those students usually wind up being mathematicians. (This is also related to why mathematicians love to make fun of physicists: we're perfectly willing to work in a framework based on what makes sense, rather than what can be rigorously proven.)

To take the example from the comment: why don't we discuss the equivalence principle in introductory mechanics classes? Well, beginning physics students have an intuitive idea of what mass is: they know that more massive things are harder to push around, and that they are harder to hold up. So their intuition tells them that both gravity and inertia are dependent on what they know to be mass. That intuition is confirmed when they see $m$ appearing in both formulas. If you tell them at this stage that gravity and inertia could in principle depend on two different quantities, $m_g$ and $m_i$, they might remember it as an interesting bit of trivia, but it's going to seem pretty useless as far as actual physics goes. After all, they intuitively know that $m_g$ and $m_i$ are the same thing, namely $m$, so why would you bother to use two different variables when you could use one?

In fact, this particular concept is a bad choice to demonstrate why intuition is not always reliable, because it's a case in which your intuition does work. Learning to rely on intuition is a useful skill in physics. As FrankH said, unlike mathematics in which the foundation of any theory is an arbitrary set of axioms, the foundation of physics is the behavior of the physical world. We're all equipped with an innate understanding of that behavior, a.k.a. physical intuition, and it makes sense to use it when it is applicable. The process of learning physics involves not only learning how to use physical intuition, but coming to understand its limits, which usually entails being confronted with a "critical number" of phenomena in which physical intuition flat-out fails. Once students have reached that point, they are going to be in a better position to appreciate something like the equivalence principle.

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The question is: Do they have to be that advanced to see the failings of the simple, intuitive approach? For example, most students studying gravitation in the 2nd semester have no idea that the $m$ here may be different from that in $F=ma$, because most textbooks don't point this out specifically. Wouldn't it be great to show things like that (again, to students who're interested) to show how intuition fails? –  recipriversexclusion Feb 6 '12 at 20:48
Good example - I'll actually edit an explanation into my answer. –  David Z Feb 6 '12 at 20:58
I don't find this to be the best example. In my introductory physics course in the first semester, which covered classical mechanics and a bit of thermodynamics, it was specifically pointed out that one should a priori introduce two masses but they turn out to have the same value. Moreover, we were shown an experiment to confirm this. –  NikolajK Feb 7 '12 at 8:02

Even the way we teach it now, the leading complaint that I hear from student is that physics is too divorced from the real world for them to be interested. Making it even more abstract will make the situation worse.

Surely there are some student who would appreciate that approach, but I suspect that they are few and far between...I've seen grad students drop a thermodynamics course---using Callen---because the approach was too abstract (at least that's what he said his motivation was).

By the time I was in grad school I really liked the formalistic style, but in high school and as an undergrad not so much.

Aside: Perhaps I'll put this question to some of the PER folks I know. If I do I'll report back.

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Agreed. This sort of thing, obviously, will not be for everyone, it would be geared only to students who are interested in physics rather than having it shoved down their throats. –  recipriversexclusion Feb 6 '12 at 19:51

To quote Richard Feynman

In physics we need the Babylonian method and not the Euclidean/Greek method. (...) The problem in the Euclidean method is to make something about the axioms a little bit more interesting or important; but the question that we have is, in the case of the gravitation is is it more important, is it more basic, is it more fundamental, is it a better axiom, to say that the force is directed towards the sun, or to say that equal areas are swept in equal time? Well, from one point of view...

...the explanation goes on a lot longer.

Sure axioms are very useful if you want a consistent formal model, but physics is not a closed consistent model. One should be always aware of the multitude of possible points to start which may be equivalent today (allowing one to make the one an axiom that gives rise to the mathematically most elegant theory), but may become different when new experimental results have changed physics. If you've only ever learned one "definite", certainly mathematically appealing but not necessarily extendable enough model, you'll have difficulties adapting your view. If you've learned piece-by-piece, it's much easier, even though the theory seems less profound; but after all it isn't so it would be dangerous to train yourself to believe there's anything irreplacable about it.

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I would have to say that an axiomatic approach is not used because physics is not mathematics.

In mathematics, you start with axioms and it is possible to proceed from axioms to prove theorems even if there is no "intuitive" meaning to the terms used in the axioms. The terms used in the axioms are just abstract symbols that are manipulated by the rules of logic that are used to prove theorems. I am not saying that intuition is not needed in mathematics, just that it is theoretically possible to do mathematics as a completely abstract symbolic exercise.

Physics, on the other hand is rooted in the physical world. Your theories have to have some contact with the physical world or else you are not doing physics. The "proof" that a theory is correct does not depend on the mathematical proof that the theorem follows from the axioms - rather the proof is that the theory makes predictions that are confirmed by experiment.

Also theoretical physicists sometimes even use mathematics in sloppy or non rigorous ways that are properly criticized by mathematicians. Sometimes mathematicians then go back and develop more mathematically correct formulations of the arguments that the theoretical physicists use.

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True, I probably pushed the axiomatic analogy too far. What I meant was not a truly axiomatic system, but a rigorous definition of basic terms, i.e. what is energy, time, space, then build fundamental notions, e.g. Galilean transformation, homogeneity of space and time, concepts of symmetry, Newton's Laws. –  recipriversexclusion Feb 6 '12 at 22:01
Another issue is that axioms can be switched around. For example, inear and angular momentum. One can prove one from the other.. So, there are no fundamental axioms of physics, just a large set of axiom multiplets of which you can choose any multiplet. –  Manishearth Feb 7 '12 at 2:02
Also, if physics is taught that way in its introductory phase (middle and high school, nobody will like it and thus they won't understand it at all. Intuition is essential to develop a new topic.. –  Manishearth Feb 7 '12 at 2:05

Even in mathematics its understood that there is no ULTIMATE set of axioms, axioms are just useful starting points which can be more or less changed arbitrarily without disrupting the edifice of knowledge which the axioms purport to rigorously prove. For example, in Geometry one can start with the old Euclidean axioms, or with Hilbert's axioms and arrive, essentially, at the same subject matter, planar geometry. Often what is an "axiom" in one system is a "theorem" in the other or vice-versa. Physics should be seen as taking advantage of mathematical principles and not limited by mathematics, it doesn't really make sense to try and start from axiomatic principles because the results of physics are true only insofar as they agree with predictions to within certain statistical limits, this is almost the exact opposite method as what is done in axiomatic mathematics.

Having said that, nearly all methods used in mathematics are just ways to tease out implications from results which are, at least during the derivation, assumed to be absolutely true. When physicist use mathematics they implicitly assume that whatever equation they are working with is absolutely true and when they tease out implications from their equations they are producing 100% mathematical truth. But, these implications that are teased out of the equations must again be tested against reality and if they come up short we no longer take the assumption for granted and the mathematical truth which was produced under the tentative assumption is actually a physical falsity when compared to the empirical data, and its only false insofar as it fails to meet some kind of statistical threshold. So, a mathematical truth can be a physical falsity.

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