Proof is of critical importance - It's how you find those junctures, for example, how you optimize, how you design and how you reason about runtime, etc. But programming as practiced in the field is engineering first, mathematics second.
> Beware of bugs in the above code; I have only proved it correct, not tried it.
Deep thinking and mathematical proofs are tremendously useful when thinking about a well-defined and scoped problem at a more or less fixed level of abstraction.
The problem with software engineering is that in practice all abstractions are leaky, and so you inevitably find yourself dealing with issues that are too far-flung and random to be mathematically tractable. The best you can do is be rigorous about the core problem you are solving, but there is always some amount of hammering and duct tape to build it into a non-trivial real-world system. It's possible to attack this problem asymptotically with a rigorous engineering process such as NASA employs, but we don't because it's simply not cost effective for the majority of software.
That's a great point. We utilize best practices, like automated tests, to narrow the gap between pure experimentation and mathematical proof. Proofs in most code would be difficult to impossible because most of use libraries that use other libraries, etc., so we just do the best we can given the time and monetary constraints that we have.