D&D 5E Could we do away with numbers for the six stats?

I think you'd then want to open up the abilities into a wider, longer list. We'd care much less about how Strong a character is if it didn't impact how Athletic he was (unless you doubled up the proficiency there). So then, it seems to me that we'd be likely to treat the abilities as just more skills. And then tack weapons onto the list of tools.
 

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Stat bonuses and proficiency bonuses are in the same ballpark, so could we just replace the former with the latter?

For instance, being Proficient in Str would let you add Str to your attack and damage rolls, while being a Fighter might bump that to Expertise. Being Proficient in Con would give you bonus HP (= HD * Prof. Bonus). Hit points would be as now. Similarly, being Proficient in Int would let you add your proficiency mod to the DC of arcane spells whereas someone not proficient (a dabbler like an Eldritch Knight, perhaps) would not. And so on.

Characters could gain stat proficiencies as they advance.
Yes you could take them out, the actual numbers play no role.

Which is why in Low Fantasy Gaming RPG, I made the uncertain action roll - "roll equal or under stat with mods", instead of an scaling target number. Makes attribute numbers relevant again (which is what earlier versions of D&D did).

It works surprisingly well, partly because players already know their rough chance of success without needing to ask the GM for the TN.
 

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