>I've heard and observed that the Ph.D teaches you one crucial skill - how to take a vaguely defined research problem that nobody has done before, where you're

>I've heard and observed that the Ph.D teaches you one crucial skill - how to take a vaguely defined research problem that nobody has done before, where you're not even sure that a solution exists, and make tangible, rigorous progress on it. I’d like to elaborate on this a bit further. In my experience as an academic researcher in a technical field (math), one of the most important skills is the ability formulate precise statements/hypotheses that satisfy several criteria. First, they should relate to the existing body of research in your field in a meaningful way. Second, they should have non-trivial answers (in the context of math, i mean that the proof of your hypothesized theorem should be legitimately nontrivial, not merely obfuscated to appear that way, as good mathematicians will be able to tell) or reveal something that was previously unexpected (such as a connection between two previously different topics). Finally, they must be testable/verifiable/provable/etc using your available methodologies and within a predictable timeframe.