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In a book Quantum Mechanics with Applications (1970) by D.B. Beard and G.B. Beard, the authors wrote on page 34:
"By methods beyond the scope of this text, one could state the uncertainty principle as a fundamental postula and derive quantum mechanics therefrom."
Were they trolling or was that a common knowledge back in 1960's or even earlier?
They are referring to something serious, but stating it too briefly to be useful.
The starting point here is that no wide-ranging theory of physical phenomena can be deduced from any single principle or observation. However, if you start out with a lot of ideas of what kind of theory you wish to construct (e.g. continuous quantities handled by partial differential equations, maybe a Lagrangian or something like that, maybe some notion of a state space and an equation of motion, etc. etc.) then some axioms can serve as a useful way to constrain your theory and the Heisenberg uncertainty principle is one of those. For example, you could show, by fairly well-constrained analysis but possibly without the kind of rigor which mathematicians seek, that it leads to the commutation relation between position and momentum (once you have already decided to represent those quantities by operators or something similar). But you cannot deduce from the uncertainty principle alone the notion of Hilbert space or a state vector and you certainly cannot deduce Schrödinger's equation. Nor can you make any significant progress in figuring out any quantum field theory.
I think what those authors probably had in mind was something like "starting out from classical mechanics and Poisson brackets and going towards operators or matrix mechanics, the uncertainty principle will show you what is the minimum adjustment or ingredient you need to make a matrix mechanics that might correspond to empirical observations."
