[oggy]

In my experience, finding the "correct" specification for a problem is usually very difficult for realistic systems. Generally it's unlikely that you'll be able to specify ALL the relevant properties formally. I think there's probably some facet of Kolmogorov complexity there; some properties probably cannot be significantly "compressed" in a way where the specification is significantly shorter and clearer than the solution.

But it's still usually possible to distill a few crucial properties that can be specified in an "obviously correct" manner. It takes A LOT of work (sometimes I'd be stuck for a couple of weeks trying to formalize a property). But in my experience the trade off can be worth it. One obvious benefit is that bugs can be pricey, depending on the system. But another benefit is that, even without formal verification, having a few clear properties can make it much easier to write a correct system, but crucially also make it easier to maintain the system as time goes by.

[awesomeMilou]

I'm curious since I'm not a mathematician: What do you mean by "stuck for a couple of weeks"? I am trying to practice more advanced math and have stumbled over lean and such but I can't imagine you just sit around for weeks to ponder over a problem, right? What do you do all this time?

[namibj]

Experience counter examples for why a specific definition is not going to work. Many times, at various levels of "not going to", usually hovering slightly above a syntactic level, but sometimes hovering on average above a plain definition semantic level, i.e. being mostly concerned with some indirect interaction aspects.