An in-depth introduction to a new subject should be an investigation- a guided tour of the questions which originally inspired the concepts in question. Proofs are an important thing to include as appendices, for the more independent and argumentative folks to throw themselves against. But they are unimportant for both the understanding and practice of science, and ought to be seen as a form of recreational debate.
As we said earlier, there are two parts to any successful solution: the investigation and the argument. Commonly, the investigation is obscured by the polished formal solution argument. But almost always, the investigation is the heart of the solution. Investigations are often tortuous, full of wrong turns and silly misconceptions. Once the problem is solved, it is easy to look over your prolonged investigation and wonder why it took you so long to see the light. But that is the nature of problem solving for almost everyone: you don’t get rewarded with the flash of insight until you have paid your dues by prolonged, sometimes fruitless toil.
The Art of Problem Solving, 2nd ed., p. 25
There’s a perfect example in Schaum’s Outline of Modern Physics, where it explains why Einstein redefined the concept of mass for special relativity. There is a thought experiment where two observers, one standing and one moving along the x-axis, both watch the same bullet fired along the y-axis into a block of wood. Carrying out this thought experiment forces the observers to assign contradictory values for the momentum of the bullet. Thus, one of the assumptions of the thought experiment must be incorrect.
Because classical momentum is mass*velocity, and Einstein had already given velocity the special relativistic treatment, he decided to redefine mass such that there would be no contradiction in the thought experiment. When you plug this new definition into the work-energy relationship, it spits out the familiar (but still remarkable) mass-energy equivalence that we’ve all come to know and love: E = mc^2.
Teaching concepts via proofs instead of a guided investigation obfuscates the intuition, and fails to produce reasonable belief. It is NOT the case that science has produced an axiomatic system that can predict most physical phenomena. Quite the contrary. And until it has done so, it will be necessary to train the physical intuition of scientists more than their ability to form mathematical proofs. I conclude that the historical presentation of scientific ideas is the correct one for producing understanding, with a special emphasis on informal investigation rather than formal argument. The universe is not a court of law- it is a crime scene.