The interesting feature of this ‘derivation’ is the cavalier treatment of differentials. It was customary in classical physics to treat infinitesimals as vanishingly small quantities. A differential can be added to a quantity to get an infinitesimally larger quantity, x + dx. One can multiply or divide both sides of an equation by a differential, and even interpret the ratio of two differentials as a derivative, as in d /d . Most mathematicians would flunk any calculus student who offered such definitions. It was and remains a common practice in classical physics. The issue of justifying such practices will be treated briefly later. The basic reason why physicists concerned with incorporating and accommodating experimental data, rely on sloppy mathematics is that the mathematical forms build on and expressed the physical significance of the quantities treated. As Heisenberg expressed it:
When you try too much for rigorous mathematical methods, you fix your attention on those points which are not important form the physics point and thereby you get away from the experimental situation. If you try to solve a problem by rather dirty mathematics, as I have mostly done, then you are forced always to think of the experimental situation; and whatever formulae you write down, you try to compare the formula with reality and thereby, somehow, you get closer to reality than by looking for rigorous methods. (Heisenberg, 1990, p. 106)
The nineteenth century arithmetization of analysis initiated by Cauchy led to a separation of mathematics from such physicalistic foundations. If mathematical formulations are to supply a foundation for physical theories, then they cannot rely on physical considerations to justify mathematics. Reconstructions of theories, and philosophical analyses based on actual or possible reconstructions, see syntax as foundational and impose semantics on independently developed syntactical structures. We will consider theories briefly later. The pertinent point here is that this method of analyzing physics leaves no role for informal inference, or inferences based on the meanings of concepts and their role in complex changing conceptual systems.