I actually agree with Sussman and Wisdom (and the author of this article) that physics is taught in a very sloppy manner, and that the computational notation ha

I actually agree with Sussman and Wisdom (and the author of this article) that physics is taught in a very sloppy manner, and that the computational notation has huge advantages. But that's not the point I (or they) were making. Lagrangian and Hamiltonian mechanics were perfectly rigorous before Sussman and Wisdom came along, even if they weren't taught very well. If mechanics had really been deep-down sloppy, then Sussman and Wisdom would now be heralded alongside Newton and Einstein for their incredible contribution to science! Rather than just creating a new notation, they would have advanced science incalculably, turning vague and untestable theories into hard empirical science for the first time! But that's obviously not what happened. Physicists were always capable of making exact calculations and predictions from Lagrangian mechanics. SICM contains no new theories, theorems or proofs, just a more explicit way of representing old ones. Nothing new was discovered and no old notions were clarified. Instead, they just found a better way of teaching mechanics, one that didn't rely on the implicit knowledge that masters of the subject already possessed. They were only capable of doing this because Lagrangian and Hamiltonian mechanics were well-defined in the first place. If they hadn't been then they would have had to advance a new theory of mechanics to replace them, rather than just re-presenting an old one.