| Jeremy – I’m a mega fan of your work. But think going deeper into this is quite fun Your post goes to the point at the heart of philsophical number theory. What does equality mean ? Yup – you got functinal equivalence, isomorphism, and temporary assignment of values. But I think you could prove – that all these types of equality – are “instances” of “different implementations” of “equivalence. They are no more equivalent than 1 = 1 is equivalent. I.e. 1 = 1 means I think we can define a bijective “counting function” that proves there’s the “same number” of “elemetns” in the “sets” I think (not sure) – if you define – counting fucntion / same number / elements / sets differently – you get the differing definitions of equivalence you enumerate. The interesting thing for me is that 1 = 1 is defined clear in 4 of peano’s axioms https://en.wikipedia.org/wiki/Peano_axioms#Formulation And you could mentally – try to develop different (and potentially) – more powerful notions of “equivalence” – with differing axioms A final point… the prevalence of several “similar” concepts of equivalence in computer science – may point to an underlying “platonic idea” of equivalence – that either exists dormant in the world awaiting for us to discover it; or is a useful “technologocial” construct – that has accelerated “progress” |