pervect said:
epicycles, or as i would call them, Fourier series, are only part of the issue. The paths of planet(s), in some particular coordinate system, can be modeled with this technique. But this is just part of the big picture, which is space-time geometry. The "big picture" allows you not only to plot the orbits of planets, but to evaluate any sort of measurement (for instance distances as measured by radar time delays) that you might choose to make.
What you say is conventionally true, and I agree in that sense, but...
I'm sorry, I misled you when I spoke of epicycles. I meant them in a metaphorical sense rather than literal. In other words, an "epicycle" in a helio/geo-centric system can be a metaphor for any superfluous mathematics required to describe a needlessly complicated system, and, with a little frantic searching, one can sometimes add metaphorical "epicycles" indefinately to give a preferred theory bootstrap logic-support that is hard to detect.
The later helio-centric model, against a supposed static background, was
supposedly simpler mathematically, but more importantly, simpler
to imagine, or picture from afar. Had not Copernicus and Galileo fortunately preceeded Newton, he, wrapped up in the latest actual epicycle, may never have formed his laws of motion and gravity and our relativity science would still be (a mere) 500 years behind. That far-flung heavenly bodies would be hurling around violating inertia would be of no consequence without Newton's Laws and would likely be explained away in otherwise fashion. Do not be so sure that someone else, rather than these three men, would have surely discovered these consecutive pre-relativity laws so soon. Even seasonal star parallax, a seeming proof of helio-centricity, may have gotten assigned it's own "epicycle" in more stagnant circumstances. I hope it makes more sense why I used epicycles as a
"bad example of science"* and how nature fools us to construct them.
pervect said:
But even with perfect, beautiful math proof, we can still apparently get an inside-out object that blueprints perfectly, yet does not meet the criteria of sensible geometry.
i would disagree in what you literally said. If the math works out right, it meets the criterion of geometry, at least in this area of application. "sensible" is an odd word here, which i will interpret as self-consistent. The math may describe an unfamiliar or non-euclidean geometry, for instance the lorentzian geometry of special relativity, but if the math works out, the geometry will be self-consistent. It might not match the reader's intuition of geometry, though. I'm afraid that i can't think of any simple cure for this - an intuition for an unfamiliar geometry can be developed, but it will take time and effort.
I use the word ,"sensible", as in meaning both h/g-centric "picture" systems cannot be preferable aesthetically. The word "Self-consistant" is not a bad synonym. The geometry of geo-centricity and helio-centricity are quite different by intuitive visual perspective, but the pre-relativistic math very similar, because they are in fact, the same mathematical solar system. A hunter could calculate a travel path, shot from orbiting earth, to another orbiting solar system body, to:
* either be a path to intercept one another, both in solar orbit, as a distant observer viewed the sun from a solar pole, without significant star rotation (helio-centric),
* or be a path observed from a supposed non-sun-orbiting earth, to pierce the other non-sun-orbiting planet mid-epicycle (geo-centric).
pervect said:
the real test, though, comes not in just getting a theory that is self-consistent, though that's obviously necessary. The real test requires getting a theory that is not only self-consistent, but matches experiment.
There may be a problem defining the exact difference in h/g-centrics. The physical laws are the same in either reference system, so what will be different? Yes, the discovery of seasonal parallax with distant stars seems proof enough after the fact, but the observation could almost certainly be labeled a new exciting epicycle if one did not know better beforehand. There may no conclusive mathematical proof today
between h/g-centric systems other than strong circumstantial evidence and, I believe, our
usual sincere belief that GR superceeds the Mach Principle.
pervect said:
i must admit that wn (in my opinion) sometimes goes a bit overboards on the math, assuming a great deal of specialized knowledge in the recipient. But it's a hard problem to try to match the right response to the right poster. The more sophisticated math responses can be vary valuable to the right poster, even as they fly over the head of others.
That is ok, and I understand. I worked for a railroad and they have their own jargon built over 100 years. There is a language barrier to laymen in modern science primarily because of advanced math. A picture is worth a thousand words, but only half a good equation. I am not being critical of WN, as it is his noble intent to say much with few words and there is no other way to do it considering the deep concepts discussed here. As ever the devils advocate in this thread, I am biased to envisioning geometry foremost because that is what my quirky mind excels at compared to my meager other skills.
That, and frankly, I believe we just may someday do the helio/geo-centric debacle again concerning who, or what, is at rest or some other semi-deceiving intuition starting us off on the wrong track however true it must seem at first. Not the sun vs Earth this time, but perhaps matter vs energy, the latest canon to run the gauntlet. As you say, pervect,
"an intuition for an unfamiliar geometry can be developed, but it will take time and effort". I can only say it has.
I want to thank everyone, especially the OP, DaleSpam, Nugatory, WannabeNewton and yourself, pervect. It has been a very enlightening and stimulating discussion for me and I hope others as well.
The most important thing I have learned here is that, in the essence, our temporary theories are all built upon axioms, which are all inevitably built upon intuition. Next, proper logic flawlessly proceeds accordingly. If a theory,
any theory such as geo-centricity or otherwise, ever should stumble or be incomplete, the flaw will be, if not improper logic, then because of prior faulty intuition, and be found lurking in the raw axiom.
Many students and faculty are on this forum, class is out and I wish all a happy holiday.
Thanks,
Wes
