Morgan Arnold on Nostr: npub1ky223…rghvc sure, but i'm not sure that containment, in this geometric sense, ...
npub1ky223zcc4q69d8t0me4vg5uw8mw0yxeukjgvz6h92laqnenr0ajs5rghvc (npub1ky2…ghvc) sure, but i'm not sure that containment, in this geometric sense, is really sufficient to make the argument that points are somehow more fundamental than lines
even though lines contain points in this context, lines are not somehow easily eliminable in favour of points here, at least i don't think so. lines are not mere agglomerations of points, defined by the rules which govern points, at least no more than points are pair of non-parallel lines, defined by the rules which govern lines. presumably one could rewrite hilbert's axioms into an equivalent set of axioms which eliminates lines in favour of points, and vice versa, by the addition of additional axioms governing points and lines, respectively
i don't see a compelling reason to claim that points are somehow the primary concept, and that lines (or strings!) ought to be viewed as agglomerations of points which are governed by axioms, or forces, or what have you. points certainly may be the physically correct descriptor, but i would tend to disagree with the argument that this follows from any mathematical or geometric reasoning
unrelated, but i am somewhat curious if hilbert's axioms allow one to prove that a line contains uncountably many distinct points. certainly a line must contain countably many distinct points, but i can't seem to tell, at least easily, if a line in this context must contain uncountably many points
even though lines contain points in this context, lines are not somehow easily eliminable in favour of points here, at least i don't think so. lines are not mere agglomerations of points, defined by the rules which govern points, at least no more than points are pair of non-parallel lines, defined by the rules which govern lines. presumably one could rewrite hilbert's axioms into an equivalent set of axioms which eliminates lines in favour of points, and vice versa, by the addition of additional axioms governing points and lines, respectively
i don't see a compelling reason to claim that points are somehow the primary concept, and that lines (or strings!) ought to be viewed as agglomerations of points which are governed by axioms, or forces, or what have you. points certainly may be the physically correct descriptor, but i would tend to disagree with the argument that this follows from any mathematical or geometric reasoning
unrelated, but i am somewhat curious if hilbert's axioms allow one to prove that a line contains uncountably many distinct points. certainly a line must contain countably many distinct points, but i can't seem to tell, at least easily, if a line in this context must contain uncountably many points