I've thought of an analogy that demonstrates the difference between the two approaches: Consider an electrical engineer designing an analog circuit and a mechanical engineer designing a contraption made of gears, pulleys and springs. They can use the “standard” mathematical approach, plugging in the equations describing what the components do and get an equation for the system which they can then analyze, or they can invent an algebra of electronic components and an algebra of mechanical components and reason directly in a “language” of capcitors and resistors (one capacitor X plus one resistor Y is equal to capacitor S plus resistor T etc.) and in a language of gears and springs. Lamport’s approach is analogous to the former, while the PL approach is analogous to the latter.