| > Well, we can say 'an equivalence' (or adjoint) which would be accurate and still using natural language, no? No, their maps could be VERY different but be used in roughly the same way and therefore useful in the same way. There is an operator involved -- reading/navigating/interpreting -- which is why I chose "isomorphism" instead of "equivalent". Also, "isomorphism" IS natural language that is used in many fields outside of mathematics and also has a vernacular sense to it. The word happens to also be used to describe certain formal constructions by mathematicians from time to time, but it is natural language. Again, this is all very pedantic and silly. I insist on bloogity-blop. If we're going to have silly arguments, let's use silly words :) > Onto my second point, you don't need an isomorphism, an equivalence works fine if you find that the axioms hold in both cases, no? So now you're not just working with a 'formalism' that has no basis in reality. Reality isn't constrained by maps. Useful maps are constrained by reality. > Otherwise, if reality wasn't constrained by axioms (or even meta axioms) we'd use it to do things we couldn't with our formalisms. Huh? I can't use reality the way I use maps because I'm not always able to fly into the air and look around before navigating to the supermarket or a trail head. Reality is not constrained by my inReach. I promise you it's the other way around. And I promise you I can't fly, which means I need maps, even if flying high into the air would make it way easier to find a trail head than following a not-great trail map. Maps are useful. Very useful. But they DO NOT constrain reality. A belief to the contrary in the case of mathematics and physics is quite spiritual. Which was kind of my original point :) |