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Not having a useful type system introduces its own complexity.I agree, I am not promoting dynamically typed languages. My intuition about this is more that there is a sweet-spot between automation and expressivity that gives you the best software engineering experience in most practical programming tasks. Milner's let-polymorphism is closer to that sweet-spot than full-on dependent types a la Calculus-of-Constructions
> extract programs from the proofs.
In practise you don't have specifications in > 95% of programming tasks. What, for example, is the full specification of a climate simulation, or of TikTok? One could argue that the shipped product is the (first and only) specification. Every program I've ever been part of constructing started from an informal, intuitive, vague idea what the software should do. This includes safety-critical software. To quote from a famous paper [1]: "We have observed that the errors we find are divided roughly evenly between errors in the test data generators, errors in the specification, and errors in the program."
> If you can't think of the right theorems or specifications I doubt you will write a correct program.
I strongly disagree. I have written many programs that are, as far as I can see, correct, but I am not sure I fully know why. Here is an example: the Euclidean algorithm for computing the GCD. Why does it terminate? I worked out my informal understand why at some point, but that was a lot later than my implementation.
More importantly: in practise, you do not have a specification to start off with. The specification emerges during programming!
> how does a sound type theory end up this
I'm not sure what you are referring to. Type inference and type checking can be undecidable. For example System F, a la Curry.
> extract a full program from a specification
Let me quip: Proof are programs! In other words: if you want to be able to extract a full program from proofs, the specification / proof must already contain all the information the eventual program should have. So any bug that you can make in programming, you also must be able to make in proving. Different syntax and abstraction levels clearly corresponds to to different programmer error probabilities. But fundamentally you cannot get away from the fact that specifications can and often do contain bugs.
[1] K. Claessen, J. Hughes, QuickCheck: A Lightweight Tool for Random Testing of Haskell Programs.