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.
Answer
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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