Weighing in here because this is a suboptimality I’ve often encountered when speaking with math oriented interlocutors (including my past self):
The issue here is an engineering problem, not a proof problem. Human minds tend to require lots of cognitive resources to take provisional definitions for things that have either no definition or drastically different definitions in their minds outside this specific context.
Structuring your argument as a series of definitions is fine when making a proof in a mathematical language, since comprehensibility is not a terminal goal, and (since each inferential step can be trusted and easily verified as such) not a high-priority instrumental goal either.
But when you’re trying to accurately convey a concept and it’s associated grounding into someone else’s mind, it’s best to minimize both the per-moment attempted deviation from their existing mentality (to maximize the chance that they both can and will maintain focus on your communications) and the total attempted deviation (to minimize the chance that the accumulated cognitive costs will lead them to (rightly!) prioritize more efficient sources of data).
This gives us a balance between the above two elements and the third element of making the the listener’s mind be as close as possible to the conveyed concept. The efforts of all involved to maintain this balance is key to any successful educational effort or productive argumentative communication.
PS: If you’re familiar with math education, you may recognize some of it’s flaws/inefficiencies as being grounded in the lack of the above balance, by the way. I’m not an expert on the subject, so I won’t speak to that.
Weighing in here because this is a suboptimality I’ve often encountered when speaking with math oriented interlocutors (including my past self):
The issue here is an engineering problem, not a proof problem. Human minds tend to require lots of cognitive resources to take provisional definitions for things that have either no definition or drastically different definitions in their minds outside this specific context.
Structuring your argument as a series of definitions is fine when making a proof in a mathematical language, since comprehensibility is not a terminal goal, and (since each inferential step can be trusted and easily verified as such) not a high-priority instrumental goal either.
But when you’re trying to accurately convey a concept and it’s associated grounding into someone else’s mind, it’s best to minimize both the per-moment attempted deviation from their existing mentality (to maximize the chance that they both can and will maintain focus on your communications) and the total attempted deviation (to minimize the chance that the accumulated cognitive costs will lead them to (rightly!) prioritize more efficient sources of data).
This gives us a balance between the above two elements and the third element of making the the listener’s mind be as close as possible to the conveyed concept. The efforts of all involved to maintain this balance is key to any successful educational effort or productive argumentative communication.
PS: If you’re familiar with math education, you may recognize some of it’s flaws/inefficiencies as being grounded in the lack of the above balance, by the way. I’m not an expert on the subject, so I won’t speak to that.