I agree with almost everything in this post, except that (ironically) I think it draws too narrow a boundary around the concept of “mathematics.” I do very little formal mathematics, but use mathematical styles of reasoning very often, to good effect. To my understanding, math is the study of patterns, and to point other people at useful patterns, definitions can be a valuable starting point. This is especially true if you explicitly point out that the definition is approximate or fuzzy. If you’re trying to inform or educate or advise people, then you need to do it (in part) with words, and you’ll need to give enough definition (with examples, yes, but not only with examples) to get the process started.
What you shouldn’t do is use definitions to debate someone else when they have a good underlying point.
That said, there have also been a few times in my career where the most valuable thing I’ve been able to observe is, “this word shouldn’t exist because it doesn’t refer to a natural category,” or “people stop using this word to describe a thing when the thing starts working properly, so they always think things in the category don’t work.” My main personal examples of these are smart materials, metamaterials, and nanotech. There is a useful underlying concept in each case, but real-world usage can be so inconsistent that it needs definition at the start of any conversation for the conversation to be useful.
I agree with almost everything in this post, except that (ironically) I think it draws too narrow a boundary around the concept of “mathematics.” I do very little formal mathematics, but use mathematical styles of reasoning very often, to good effect. To my understanding, math is the study of patterns, and to point other people at useful patterns, definitions can be a valuable starting point. This is especially true if you explicitly point out that the definition is approximate or fuzzy. If you’re trying to inform or educate or advise people, then you need to do it (in part) with words, and you’ll need to give enough definition (with examples, yes, but not only with examples) to get the process started.
What you shouldn’t do is use definitions to debate someone else when they have a good underlying point.
That said, there have also been a few times in my career where the most valuable thing I’ve been able to observe is, “this word shouldn’t exist because it doesn’t refer to a natural category,” or “people stop using this word to describe a thing when the thing starts working properly, so they always think things in the category don’t work.” My main personal examples of these are smart materials, metamaterials, and nanotech. There is a useful underlying concept in each case, but real-world usage can be so inconsistent that it needs definition at the start of any conversation for the conversation to be useful.