One issue here is that (as I understand the history) epicycles were not originally proposed in the context of "just add them until you've got something that looks right"; it was proposed that, occasionally, one needed to add one epicycle. After a while it was found that even this didn't work, and we needed sometimes to add another. I forget whether the theory collapsed before instances were found where we'd need a third.
This, I think, is 'wrong' in the sense of physics: if your model is constantly scrambling to catch up with newly acquired data, then it has no predictive, and hence little scientific, value. Further, the very ability of epicycles constantly to scramble to catch up—the ability of trigonometric polynomials to approximate continuous functions arbitrarily well over arbitrary bounded intervals, as you mention—is also a mark against them: a scientific theory that is so general that it could be tuned to predict anything can, in fact, for that reason actually predict nothing.
To be sure, there is nothing wrong with changing your model in response to data—what would be wrong would be the opposite. What I was calling 'wrong' was two things: (1) a model that can only be tuned post hoc to observations (that is, it's always being changed to fit them, and never predicting new ones); and (2) a model that is so flexible that it can be tuned to fit any observations (again, because it will never have the ability to predict future observations).
I don't really get the thing about "too flexible". According to Wikipedia there is apparently some myths floating around regarding the science history of epicycles, but I assure there is nothing magical about them - at least they are not any more magical than Newtonian mechanics.
I think we may be talking at cross purposes here, but I wanted to give it another shot. I do not think that there is anything magical about epicycles, and my objection to them is scientific; it has nothing to do with their history. My claim is that a scientific tool is valid science only to the extent that it has predictive power. If your model has so many parameters that, by setting them to appropriate values, you can fit any outcome, then (absent some other data telling you which values of the parameters to use) you'll never actually predict anything, only scramble setting them to 'post-dict' existing observations—which is, effectively, useless.
(For example, any finite collection of observations can be 'post-dicted' by a high-degree polynomial, and we can accommodate more observations simply by increasing the degree of the interpolating polynomial. This is not magical, and mathematically totally sound; but its very flexibility means that it is scientifically useless, because there's no reason to believe that our perfect fit to the `n` values we've seen so far says anything about the (`n + 1`)st value we'll see next.)
Just to be clear, I'm not in any way insulting the intelligence of those who proposed them. It was a clever mechanism for dealing with what seemed at the time to be a blemish on an otherwise beautiful theory, and the fact that it turned out to be (factually) wrong does not mean that it was wrong to propose and test it.
With that said, here's my stab at what I see as the difference. The Keplerian model predicts elliptical orbits. To be sure, one must do some tuning of parameters to find the semimajor and semiminor axes of these ellipses; there is no way for someone sitting in a windowless room, with access only to Kepler's laws, to predict exactly, say, the length of a Jovian year, or even the existence of Jupiter.
However, what is crucial is that that person sitting in a windowless room can say, definitively, that, if Jupiter has been observed in these five positions at these five times, then it will be in this position at this sixth time. If it's not there, then the Keplerian model is wrong; it has been falsified.
Now suppose instead that another person in another windowless room knows about, not just Kepler's laws, but also epicycles. Given the same five measurements, he or she can make a prediction (which won't require epicycles). This prediction might or might not agree with the sixth measurement, but, if it doesn't, then that's no problem; just add on an epicycle to account for that measurement. If the result of that correction doesn't agree with the seventh measurement, then just add another epicycle.
To me, these two systems are meaningfully different. It is not the reliance on parameters to which I object, but the extreme flexibility; a model that can literally never be falsified, because it can always be adjusted to match new observations, is worthless as a scientific model.
Is that really how you think epicycles were used?
Yes. So, if that's not true, then you've been right all along, and my objection was mostly founded on a misunderstanding of the history. Do you have a reference I could read?
(I should say that, even if it's not actually how they were used, it's still how they could have been used, and I think that the spirit of my objection applies; but the actual practicalities of it would certainly be undermined by my historical misunderstanding.)
The thing is, this model is not wrong and makes reasonable correct predictions about the motion of the planets as observed from the earth. The Copernican model is just simpler because by describing the planets as orbiting the sun rather than the earth, you don't need as many epicycles in the model. The Tychonian model, by the way, solves the same issue by having the planets orbit the sun, but the sun oribiting the earth, thereby additionally solving the parallax problem which the Copernican model could not explain given the knowledge at the time. All three models needed additional epicycles, which Kepler simplified by describing the orbit as a elliptical.
You will note these models are actually equivalent in modern understanding, where neither the sun or the earth is at the center of the universe.
The current model of the solar system still have epicycles (eg. for moons), so if the presence of epicycles somehow turns a model into pseudoscience, then what?
There is nothing magic about the the word "dark matter" either. Yes indeed it describe a phenomen we don't understand, this does not mean it is magic.