The axiomatic method is one of the greatest inventions of mankind, without which the development of mathematics, physics and technology over the centuries wouldn't have been possible.
The method is very powerful. It works like a snowball and allows constructing complex theories from easy to understand basic axioms.
The "Little Bird Theory" is nonsense, since it disagrees with our daily experience. But apart from some technical details, which are unimportant here (eg. we haven't defined how exactly we derive new theorems using logical steps), there is nothing to complain about this theory.
The success of the mathematics, physics and computer sciences based on the axiomatic method lies in the clever choice of axioms at the very beginning of a new theory. The better the choice of axioms, the more likely the axiomatic method will produce a theory which has better applications in the real world or makes better predictions about the real world. For instance, the axiom "The speed of light is constant whether the ray be emitted by a stationary or by a moving body" allowed Albert Einstein at the beginning of the 20th century to develop his Theory of Relativity. Now, we harvest this theory by applications like the GPS system, which wouldn't work, if the axiom was wrong.
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