2

Just curiosity, as follows: I was trying to explain/illustrate to a non-technical friend that physics is just "mathematical models", which may or may not represent/correspond_to some "underlying reality". And we can't infer it does just because the calculated numbers work (correspond to observed measurements).

And the example that crossed my mind was this: classical Greeks (mostly) thought planets revolve around the earth. But to explain retrograde motion, etc, they introduced epicycles. And when that didn't exactly work, they introduced epicycles on epicycles. Now, I suggested, if they'd known a little more math, they could've "expanded" the observed motion in epicycles (if epicycles are "complete" for describing such orbital curves). And then they could've argued along the lines, "Look, our calculated numbers are accurate to 16 significant decimal digits. So our epicycle model of planetary motion must be right. How could we obtain such incredible accuracy otherwise???"

So how good/bad an illustration is this? Are epicycles complete in this sense? And, of course, an underlying question I didn't mention to my friend: how can you "protect" QED, etc, from such objections? Or can't you?

Conifold
  • 5,293

2 Answers2

2

Yes they could, here is a fun YouTube video where Homer Simpson is sketched by a tower of epicycles. We can think of epicycles as sums of $a_ne^{i\lambda_n t}$ by identifying the plane of the ecliptic with the complex plane. The $a_n$ then code their radii and phases and $\lambda_n$ are the frequencies. This is a generalization of the Fourier expansion, in which $\lambda_n$ must all be integers. Any continuous almost periodic function admits such an expansion, the epicyclic exponents are complete in their space (some discontinuous functions can also be approximated). Such functions were introduced and studied by Harald Bohr, the famous physicist's brother. In particular any continuous periodic motion can be accomodated by epicycles. For more details see Mathematical Power of Epicyclical Astronomy by Hanson.

As for QED, it is at least protected from the Homer Simpson objection, unlike epicyclic astronomy it only has finitely many parameters to "adjust". And it made predictions of (even today) remarkable accuracy without any analog of mounting epicycles upon epicycles that Islamic astronomers engaged in at the end of middle ages, see Ancient Planetary Model Animations, especially the Arabic models for outer planets.

Conifold
  • 5,293
  • Has it been demonstrated that QFT cannot be modified ad infinitum to fit anything? I hear all the new particles that appear on modifications of the standard model. Can't you keep doing this forever? –  Jun 08 '17 at 02:24
  • @WillyBillyWilliams Anything can be modified ad infinitum to fit anything. However, QED produced its predictions without such retrofitting. There was no need, it already matched the measured values orders of magnitude better than epicyclic astronomy did its after centuries of adjustments. – Conifold Jun 08 '17 at 03:07
  • Thanks, Conifold . I guess I should have realized that there exists no goofy question I could possibly conjure up that hasn't been exhaustively studied:) It didn't even cross my mind to try googling it. Re comments with @WillyBillyWilliams , I'd been thinking more or less along those same lines, with AdS/CFT as a concrete example. Two very different pictures/interpretations of "underlying reality" that come up with the same answers for observable phenomena through very different mathematical approaches, each corresponding to its own different picture. Maybe nobody's noticed a different QED yet –  Jun 08 '17 at 03:26
  • 1
    @JohnForkosh Dawid discusses just this http://philsci-archive.pitt.edu/3584 and comes to an interesting conclusion. The paradox results from the naive classical inferring of the "underlying reality" from incidental mathematical machinery of abstract theories:"Duality does not just spell destruction for the notion of the ontological scientific object but in a sense offers a replacement as well. By identifying theories with different sets of elementary objects it reduces the number of independent possible theories eventually down to one." It's the invariant structure that's real. – Conifold Jun 08 '17 at 03:54
  • @JohnForkosh Any theory can ultimately be reduced to set theory. So any theory has at least a dual one based on sets. That is all we are, proper sets of the class of sets. –  Jun 08 '17 at 04:24
  • Thanks again, Conifold, that's indeed an interesting article, though (obviously) I've so far only had a chance to skim it. The "Demise of Ontology" (pg33) section seems to be more or less along the lines I was trying to think. But all these philosophy guys seem to go on and on and..., way beyond any reasonable inferences that can be drawn from available facts. Physicists, when talking about such stuff, seem to know about where to draw the line, before going too far out on any very shaky limbs. Although "shut up and calculate" is probably way too conservative. –  Jun 08 '17 at 05:24
  • @WillyBillyWilliams Well, the Chu Space people (not to mention the better-known category theory people) might argue with you about what any theory can "ultimately be reduced to". But, in any event, it's irrelevant here. You'd be "reducing" the mathematical theory, or more to the point, the mathematical model. But we're not talking about the theory/model, per se, rather its (ontological) interpretation. –  Jun 08 '17 at 05:31
  • 1
    @JohnForkosh I agree with you, set theory was just an example. I am not sure though, that we share the meaning for "underlying reality"/ontological interpretation. I might not even know the meaning. –  Jun 08 '17 at 06:13
  • @WillyBillyWilliams Yes, I agree "underlying reality" may have no well-defined meaning. Or, like Supreme Court Justice Stewart (ain't google great:) said about pornography, "I can't define it, but I know it when I see it." Of course, even that may be overstating the case about underlying reality (i.e., not sure we'd know it when we see it, either). More likely, there's a large (maybe countably or even uncountably infinite?) equivalence class of interpretations that all give rise to mathematical models from which the same set/collection of observable measurements can be calculated. –  Jun 08 '17 at 07:37
1

A mathematical model that could explain anything, doesn't actually explain anything.

The trouble with epicycles is that they can be used to fit anything. See the Wikipedia article on epicycles. Newton's theory of gravity is much better because the orbits that it predicts must be ellipses (in the first approximation) and the rest of Kepler's Laws also follow.

As with a lot of questions of this sort (the philosophy behind physical reasoning) I suggest you check out Richard Feynman's lecture on physics.

David Elm
  • 1,911
  • 1
    Yeah, once you've seen Newton's laws, and can choose "this or that" (epicycles or Newton), then Newton's the clearly preferable winner. But in their time, those classical Greeks making my proposed argument wouldn't have had any such choice. They'd have one possible explanation they could accept or reject, and no explanation at all if rejected. And given its accuracy, any "reasonable person" (using the legal phrase) would preferably accept that one-and-only epicycle explanation. So, could, e.g., QED, etc, just be today's analogous situation, just waiting for its own AdS/CFT-like correspondence? –  Jun 08 '17 at 03:37