Frank - > There is a rigorous definition of curvature that doesn't depend on the > manifold's being embedded in Euclidean space. Right, Jon? I'll give you "curved" but not "bent" as something other than metaphor. > By the way, I was a private pilot during the 70s. Hywel was a more > experienced and more cautious pilot. I think there are others in Friam.
So plotting a cross-country course would require at least a mechanical accommodation for the curvature of the earth (not to mention the distortion of the magnetic field), and/or with enough practice a "feel" for navigating on the surface of a spheroid (if not ellipsoid)? I never got past the mechanical in spite of staring at globes and trying to "feel" the difference. My flight paths were never long enough to matter really, but I *did* sometimes have an intuitive feel for "shape of space" implied by the winds aloft. But surely not as much as a hang-glider or sailplane pilot. Long before I flew in an airplane I dreamed of soaring like a raven, especially those "surfing" on the uplift currents flowing over the ridge behind my house. I would expect that (high flying) birds and ocean dwellers live in an ever-changing (based on currents) manifold onto which our euclidean is nearly a fiction? Any specifics about that I might feel are surely wrong. The tennis court did not remain "rectangular" for me for very long after I began to play as a youth... it quickly took on a "shape" in phase space, moderately asymmetric due to my right-handed reach and changing with the style of play of my opponent. My own strategy with a new player in competition was to try to quickly gain control of the "shape" of that space, and a match could "turn" on one of us putting an unexpected "kink" in the other's playable space. This was well before I had a word for phase space or even a conception of manifolds or non-euclidean geometries. I'm belaboring this because I think those experiences (internalizing/direct-apprehension of the non-euclidean) ARE grounding out in the direct-experience/sensorium, and do not require (allow for?) a stacking of linguistic mappings (previously "metaphor"), but the way such things are normally taught in school ARE stacked on top of the conventional idiom we have for "the shape of space" (i.e. euclidean) so we DO use terms (and conceptions) like "bent" space. A whale or highly intelligent bird might *develop differential/integral calculus" as a method for managing the abstractions in *their world* that they don't experience directly (euclidean like straight lines and flat surfaces). I don't know if this addresses (well) Dave's insistence on "other ways of knowing", but that is where *I* go when he speaks of that. Learning to play tennis well was not a science for me, it was an art and involved practicing my body and reflexes and strategery into a direct apprehension of the phase space (post-hoc name for it) I described above. I ONLY talk about it in terms of "phase space" because it is a common mathematical abstraction that we both share, not because I think or feel IN phase space. It is just the "dynamic spacetime of tennis playing"? I've never talked to other tennis players much, and certainly not in these terms. To the rest of you, maybe a tennis court is a rectangular region within which you must keep the ball to remain in play and within which there are ballistic trajectories modified by (mostly) the varying lift/drag on the ball based on it's rate and direction of spin. A naive tennis player (including extremely good ones) surely don't have strong conceptions of the abstractions of physics, but instead a strong intuitive command of the behaviour of the coupled system of their body, the racquet, the ball, the air, the surface of the court, etc. In all this rambling I'm arguing against myself on the "metaphors all the way down"... and for "metaphors all the way down until you can A) use more formal analogy and mathematical mappings if that is your training, and/or B) until you have internalized those mappings and feel them intuitively. - Steve - Steve -- --- .-. . .-.. --- -.-. -.- ... -..-. .- .-. . -..-. - .... . -..-. . ... ... . -. - .. .- .-.. -..-. .-- --- .-. -.- . .-. ... FRIAM Applied Complexity Group listserv Zoom Fridays 9:30a-12p Mtn GMT-6 bit.ly/virtualfriam un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com archives: http://friam.471366.n2.nabble.com/ FRIAM-COMIC http://friam-comic.blogspot.com/