Morgan Arnold on Nostr: npub1ky223…rghvc this feels, at least partially, like more of an artefact of a ...
npub1ky223zcc4q69d8t0me4vg5uw8mw0yxeukjgvz6h92laqnenr0ajs5rghvc (npub1ky2…ghvc) this feels, at least partially, like more of an artefact of a particular (set theoretic) axiomatization of math, and thereby geometry, than anything that might be physically "fundamental", so to speak
i would tend to say that a set of points satisfying some properties is a way of realizing the mathematical concept of a line (or a curve, or what have you), but that a line is defined by its properties, and that decomposability into a set of points is not an inherent property of lines, but rather something that emerges from a particular (in our case, set-theoretic) model of geometry. in the euclidean tradition of geometry, for instance, lines are primitive objects, in some sense, and don't admit this kind of decomposition that they do in a set-theoretic model of geometry
i think that there is a more proper way of phrasing this, in terms of formal theories and models, but this is more or less how i think of it
i would tend to say that a set of points satisfying some properties is a way of realizing the mathematical concept of a line (or a curve, or what have you), but that a line is defined by its properties, and that decomposability into a set of points is not an inherent property of lines, but rather something that emerges from a particular (in our case, set-theoretic) model of geometry. in the euclidean tradition of geometry, for instance, lines are primitive objects, in some sense, and don't admit this kind of decomposition that they do in a set-theoretic model of geometry
i think that there is a more proper way of phrasing this, in terms of formal theories and models, but this is more or less how i think of it