The analysis and synthesis approaches to understanding systems have respectively been the driving forces for two major breakthroughs in 20th century physics: reductionist and emergent phenomena. Reductionism aims to understand a system by reducing it to component parts and assuming that the composition is simple. This attitude drives particle physics. On the other hand, one of the most important themes from condensed matter physics has been how "more is different" and collective phenomena can induce emergent behaviour which is very different from the behaviour of the constituent parts. In this perspective, the precise constituent parts don't really matter too much -- many substrates which like completely different to analysis can end up looking very similar in synthesis. This is the principle behind universality classes in critical phenomena. This patter of thought should also be familiar to folks who advocate for a "systems perspective".

In the language of differential vs integral calculus, you can have perfectly well behaved and physical functions whose derivatives can completely miss the global behaviour of the function i.e. smooth but not analytic eg exp(-1/x) at x=0. Funnily enough, this is exactly the form of the action taken by quantum mechanics and an argument for how the classical limit irrecoverably ignores the physics of quantum systems.