How would someone who really knows how to take advantage of dynamic programming languages approach programming differently than someone working in a static language?
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My biggest gains are in mapping between databases and objects (ORM).
If there is no concept of a type, it becomes very easy to say assign every column in a row to a value in an object. Of course the trade off is that there can be a mismatch between the type of value you think is there and what type the computer does.
I can't cite this right now (my memory is failing me), but I've heard something along the lines of:
The closest the programming industry has come to a silver bullet is managed languages – freeing the programmer from having to worry about the details of memory management and letting them focus more energy on solving the problem at hand.
So, I might venture a guess and say it's not so much that you program differently, it's that you can devote more of your brain to "solving the problem" rather than the solution's implementation details.
Dynamic languages are capable of executing code which was created at run-time. This is very dangerous if malicious code is injected. But very powerful if you can sanitize the environment.
I think Javascript people do this by executing JSON files.