This is (my understanding of) Wittgenstein in a nutshell. Definitions in the real world don't have distinct borders. Words' meanings depend on how they're used,

This is (my understanding of) Wittgenstein in a nutshell. Definitions in the real world don't have distinct borders. Words' meanings depend on how they're used, how we teach them to each other, how we experiment with them. Most of their utility comes "far from the border". Ideally the border is little-trafficked and needs little policing, and if it drifts a little one way or another between speakers few important concepts are left on different sides. Tying technical definitions to existing words is fine, obviously. Words can have several meanings, and where the contours of a technical concept approximate some lay word it can make sense to reuse the term. If mathematicians want to call something a "hole" (or a "set" or a "knot"), good for them. It might even in turn influence the lay definition, but it doesn't necessarily supersede it just by virtue of its well-defined borders (and indeed may not be suitable for general use because of its hard borders and low entropy.)