I've read rather too much of the fascinating comments thread you've indirectly linked to, but this nugget by Mike Shulman in that thread seems especially releva

I've read rather too much of the fascinating comments thread you've indirectly linked to, but this nugget by Mike Shulman in that thread seems especially relevant to the present one: > If we distill it down from the categorical heights, this brings us back to John Baez’s pithy quote about equality: “Every interesting equation is a lie.” The only obviously true equation is x=x, but it carries no (or little) information. Any other equation, like x=y, is “a lie” because x and y are not the same thing; yet what is interesting about the equality is that they are nevertheless the same in some way. But knowing the equation doesn’t mean that we should forget all about y and use only x; the point of having the equation is that we can pass back and forth between x and y, according to which is most appropriate in any given context.