Comment by timschmidt
2 months ago
Maybe I'm just a filthy computationalist, but the way I see it, the most accurate model of the universe is the one which makes the most accurate predictions with the fewest parameters.
The Newtonian model makes provably less accurate predictions than Einsteinian (yes, I'm using a different example), so while still useful in many contexts where accuracy is less important, the number of parameters it requires doesn't much matter when looking for the one true GUT.
My understanding, again as a filthy computationalist, is that an accurate model of the real bonafide underlying architecture of the universe will be the simplest possible way to accurately predict anything. With the word "accurately" doing all the lifting.
As always: https://www.sas.upenn.edu/~dbalmer/eportfolio/Nature%20of%20...
I'm sure there are decreasingly accurate, but still useful, models all the way up the computational complexity hierarchy. Lossy compression is, precisely, using one of them.
The thing is, Lagrangian mechanics makes exactly the same predictions as Newtownian, and it starts from a foundation of just one principle (least action) instead of three laws, so it's arguably a sparser theory. It just makes calculations easier, especially for more complex systems, that's its raison d'être. So in a world where we don't know about relativity yet, both make the best predictions we know (and they always agree), but Newton's laws were discovered earlier. Do they suddenly stop being natural laws once Lagrangian mechanics is discovered? Standard physics curricula would not agree with you btw, they practically always teach Newtownian mechanics first and Lagrangian later, also because the latter is mathematically more involved.
I will argue that 'has least action as foundation' does not in itself imply that Lagrangian mechanics is a sparser theory:
Here is something that Newtonian mechanics and Lagrangian mechanics have in common: it is necessary to specify whether the context is Minkowski spacetime, or Galilean spacetime.
Before the introduction of relativistic physics the assumption that space is euclidean was granted by everybody. The transition from Newtonian mechanics to relativistic mechanics was a shift from one metric of spacetime to another.
In retrospect we can recognize Newton's first law as asserting a metric: an object in inertial motion will in equal intervals of time traverse equal distances of space.
We can choose to make the assertion of a metric of spacetime a very wide assertion: such as: position vectors, velocity vectors and acceleration vectors add according to the metric of the spacetime.
Then to formulate Newtonian mechanics these two principles are sufficient: The metric of the spacetime, and Newton's second law.
Hamilton's stationary action is the counterpart of Newton's second law. Just as in the case of Newtonian mechanics: in order to express a theory of motion you have to specify a metric; Galilean metric or Minkowski metric.
To formulate Lagrangian mechanics: choosing stationary action as foundation is in itself not sufficent; you have to specify a metric.
So: Lagrangian mechanics is not sparser; it is on par with Newtonian mechanics.
More generally: transformation between Newtonian mechanics and Lagrangian mechanics is bi-directional.
Shifting between Newtonian formulation and Lagrangian formulation is similar to shifting from cartesian coordinates to polar coordinates. Depending on the nature of the problem one formulation or the other may be more efficient, but it's the same physics.
You seem to know more about this than me, but it seems to me that the first law does more than just induce a metric, I've always thought of it as positing inertia as an axiom.
There's also more than one way to think about complexity. Newtownian mechanics in practice requires introducing forces everywhere, especially for more complex systems, to the point that it can feel a bit ad hoc. Lagrangian mechanics very often requires fewer such introductions and often results in descriptions with fewer equations and fewer terms. If you can explain the same phenomenon with fewer 'entities', then it feels very much like Occam's razor would favor that explanation to me.
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Laws (in science, not government) are just a relationship that is consistently observed, so Newton's laws remain laws until contradictions were observed, regardless of the existence of or more alternative models which would predict them to hold.
The kind of Occam’s Razor-ish rule you seem to be trying to query about is basically a rule of thumb for selecting among formulations of equal observed predictive power that are not strictly equivalent (that is, if they predict exactly the same actually observed phenomenon instead of different subsets of subjectively equal importance, they still differ in predictions which have not been testable), whereas Newtonian and Lagrangian mechanics are different formulations that are strictly equivalent, which means you may choose between them for pedagogy or practical computation, but you can't choose between them for truth because the truth of one implies the truth of the other, in either direction; they are the exactly the same in sibstance, differing only in presentation.
(And even where it applies, its just a rule of thumb to reject complications until they are observed to be necessary.)
Newtownian and Lagrangian mechanics are equivalent only in their predictions, not in their complexity - one requires three assumptions, the other just one. Now you say the fact that they have the same predictions makes them equivalent, and I agree. But it's clearly not compatible with what the other poster said about looking for the simplest possible way to explain a phenomenon. If you believe that that's how science should work, you'd need to discard theories as soon as simpler ones that make the same predictions are found (as in the case of Newtownian mechanics). It's a valid philosophical standpoint imho, but it's in opposition to how scientists generally approach Occam's razor, as evidenced eg by common physics curricula. That's what I was pointing out. Having to exclude Newtownian mechanics from what can be considered science is just one prominent consequence of the other poster's philosophical stance, one that could warrant reconsidering whether that's how you want to define it.
> Do they suddenly stop being natural laws once Lagrangian mechanics is discovered?
Not my question to answer, I think that lies in philosophical questions about what is a "law".
I see useful abstractions all the way down. The linked Asimov essay covers this nicely.