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In a seventeenth century world the Newtonian model did mostly very well to describe how gravity works in the universe and did well with most empirical evidence of that time. Of course now we know that gravity is not that Newtonian but described more correctly by General Relativity. My question is does there exist an ad hoc hypothesis that we could add to the Newtonian model to arrive at the Einsteinian model and more generally can we always find an ad hoc hypothesis to result in a more sound generalization ? Because if we can I think this will violate the heuristic of parsimony that the simpler theory is more likely to be correct since this likelihood decreases as we have more chances of finding an ad hoc hypothesis that will result in a more general theory. Does this mean a particular over-fitted extrapolation (which would contain that particular ad hoc hypothesis) out of many such over-fitted model will stand a chance of predicting with higher accuracy thus disproving the bias-variance trade-off ?

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    People generally use the letter K to indicate something they don't quite understand. So, say you know something is roughly an inverse square law, but experiments differ ever so slightly from mathematical predictions, you can just bang a K in there. So we'd go F = K * G(m1m2/r^2). Of course nobody at the time could have discovered any discrepancy between prediction and results, they are so small. But to answer your question.. what would K be in this case? – Richard Mar 25 at 0:01
  • No, we can not in this case. Einstein's theory does not have a fixed background space required by Newton's theory, space evolves and can even change topology. It is possible in some other cases. For instance special relativity can be reproduced in classical mechanics by adding the so-called hypothesis of molecular forces of Lorentz-Poincare, it enforces length contraction and time dilation dynamically. But the resulting theory is not "simpler", so it does not violate the parsimony. Ad hoc hypotheses almost never make anything simpler in any useful way, epicycles are another classical example. – Conifold Mar 25 at 0:06
  • @Conifold it would make no sense to, in this case.. because einstein's field equations replace newtons classical equations completely. But the result of both is still a force in newtons. We could write a field equation fiddle factor into newtons law, but it.wouldn't be elegant (like say, adding higgs to the standard model was). – Richard Mar 25 at 1:00
  • @Richard You could for tame enough curvatures. When a black hole forms, or some topological alteration happens, fiddle factor would not be enough. The background assumptions that make Newtonian forces meaningful break down, there is no even local inertial frame. Standard model, and even most of string theory, still assume fixed spacetime background, which is one reason why it is so hard to merge them with general relativity. Logunov constructed a variation of GR in fixed spacetime, but it not equivalent to GR. – Conifold Mar 25 at 2:57
  • @Conifold wouldn't you say that since the LET is a better model than Newton's and as you stated can be derived by adding a so-called hypothesis that it would not violate the parsimony as it predicts what Newton does except better. There is of course no comparison with SR and of course the geometry of GR changes the meaning of space – katipra Mar 25 at 3:01
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My question is does there exist an ad hoc hypothesis that we could add to the Newtonian model to arrive at the Einsteinian model

No. General Relativity is not Newtonian mechanics plus a bit. Newtonian mechanics is a special case of General Relativity.

Now, if you question was:

does there exist an ad hoc hypothesis that we could add to the Newtonian model that produces the same results as the Einsteinian model

Then the answer is: possibly. Or more precisely, one may be able to find an adhoc hypothesis that, for most practical purposes, makes the same corrections as General Relativity. I don't believe this approach was ever taken with General Relativity (Einstein was attempting to reconcile Special Relativity and Newtonian Gravity, not explain anomalous experiments) but is fairly common in physics and is known as phenomenology. It is considered to be somewhat of a hack, though, and not a good substitute for a more general theory.

can we always find an ad hoc hypothesis to result in a more sound generalization ?

Historically, this has rarely been the case. It has been much more prevalent that a generalization has been found for which the previous theories were special cases.

this will violate the heuristic of parsimony that the simpler theory is more likely to be correct

But that's ok. The bit that's often left out with Occam's Razor is the "all other things being equal" bit. It's only useful as a tie breaker if the hypotheses you're comparing have equivalent explanatory power. As Einstein (may have) said, a theory should be as simple as possible but not simpler.

  • I completely missed the "all other things being equal" part. A good reminder – katipra Mar 25 at 19:29
  • Hmm, are you sure that your interpretation about Newtonian being a particularization of GE makes time-relative sense? What I mean is that in time Newtonian ideas maybe preceded GE ideas, thus from perspective of "thought formation", Newtonian could be seen to lead to the building of GE. And only later on it's understood that Newtonian is now a special case of the GE, when GE is "completed"? But this observation might not have been done prior to forming GE based on Newtonian ideas. This would then be an example of "scientific revolution", where "a new idea revolutionalizes the earlier ones". – mavavilj Apr 24 at 20:22
  • @mavavilj Yes I'm sure. For GE to be a scientific theory of any value, it was essential that it contained Newtonian mechanics. The reason is simply due to Newton's astonishing success as a scientific theory. If GE had substantially contradicted Newton it would have been binned. – Alex Apr 25 at 16:06

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