A productive purpose of giving definitions to your interlocutor is to incite cognitive activity that’s relevant to your own thoughts, as levers that bring about thinking on the right topic. Apt definitions are great, when available, but demanding them when they are not is losing sight of the purpose of the whole activity. Formulating definitions (or researching existing ones) might be a good subproblem to focus on though.
For vague ideas it makes little sense to use definitions that sharply delineate their instances. A definition should instead give a degree of aptness (centrality) to its potential instances, and definitions should match based on the whole distribution of aptness they assign.
With free will, and similarly with possible worlds, I think a crucial point missing from discussion on LW is that these are semantic notions, and it’s possible to consider radically different semantics for the same syntactic thing. A great illustration is semantics of programming languages, where it’s possible to understand what a given program is doing in terms of very different semantic constructions. To give some examples, there’s the straightforward sets of possible values, related worlds in Kripke semantics, formal theories of observations (that are not even maximal) in Scott domains, plays and strategies in game semantics. Closer to normal mathematics, there’s internal languages of categories that let us interpret a term as something incarnated in very different situations, notably sheaf toposes where you can interpret a construction (as in discussion, argument, proof, term, type) as varying continuously on some space and saying different things for different places, all at the same time.
So when considering a thought experiment, assuming that what’s going on is that there is some discrete set of possible worlds is very limiting. There are other ways to think about things, and thus prematurely locking in a discussion under a technical definition can be a problem, manifesting as failure to notice or urge to ridicule the notions arising from other definitions and inapt for this one. (Not formulating technical definitions is another problem.)
A productive purpose of giving definitions to your interlocutor is to incite cognitive activity that’s relevant to your own thoughts, as levers that bring about thinking on the right topic. Apt definitions are great, when available, but demanding them when they are not is losing sight of the purpose of the whole activity. Formulating definitions (or researching existing ones) might be a good subproblem to focus on though.
For vague ideas it makes little sense to use definitions that sharply delineate their instances. A definition should instead give a degree of aptness (centrality) to its potential instances, and definitions should match based on the whole distribution of aptness they assign.
With free will, and similarly with possible worlds, I think a crucial point missing from discussion on LW is that these are semantic notions, and it’s possible to consider radically different semantics for the same syntactic thing. A great illustration is semantics of programming languages, where it’s possible to understand what a given program is doing in terms of very different semantic constructions. To give some examples, there’s the straightforward sets of possible values, related worlds in Kripke semantics, formal theories of observations (that are not even maximal) in Scott domains, plays and strategies in game semantics. Closer to normal mathematics, there’s internal languages of categories that let us interpret a term as something incarnated in very different situations, notably sheaf toposes where you can interpret a construction (as in discussion, argument, proof, term, type) as varying continuously on some space and saying different things for different places, all at the same time.
So when considering a thought experiment, assuming that what’s going on is that there is some discrete set of possible worlds is very limiting. There are other ways to think about things, and thus prematurely locking in a discussion under a technical definition can be a problem, manifesting as failure to notice or urge to ridicule the notions arising from other definitions and inapt for this one. (Not formulating technical definitions is another problem.)