Was there a formula-first hypothesis for gravity?

Question

Being a mathematician I wonder, has there been an attempt to blindly approximate a formula for gravitational attraction that would match the data, based on "normal" matter we can see plus that which we can't, but can reasonably expect, e.g. black holes, dust etc? I'm talking about an unrestrained approximation, one that can take a negative value, regardless of how preposterous that might seem, just to see if we can find one that fits and try to make sense of it.

It seems to my uneducated brain that Dark Matter hypothesis is essentially identical to the planet Vulcan hypothesis, i.e. we assume we understand gravity so we explain weird behavior with mass we didn't yet detect (only this time we cheat by hypothesizing the mass is undetectable). Also, we know of forces that don't abide by inverse square laws and can either effectively pull or push depending on the distance (nuclear forces). So we have a precedent of us being wrong about gravity (despite prior experimental success e.g. finding of Neptune) and examples of forces that behave differently to gravity... so maybe let's just assume gravity grossly defies our intuition, try to match a formula to observations and see if we can figure it out from there?

Sorry if it's not the space for such amateur questions - please let me know where I could go with that. Thanks!

EDIT

Just to clarify, I am specifically asking about approaches that do not derive equations out of logical interpretations of the data, but out of the data directly. As such, for example, MOND does not qualify because it derives from Newtonian interpretations and just adds an extra piece on top, inheriting all of the limitations of the Newtonian formulae (like the idea of gravity being always attracting, never repelling).

I am literally asking if somebody (recently, as new measurements are made) tried to feed bulk data like below to an array of approximation algorithms to see if any of them produces something that fits the data:

Object position	Object mass	Apparent G vector
$p_0$	$M_0$	$\vec v_0$
$p_0$	$M_1$	$\vec v_1$

... where position and acceleration vector are in any coordinate system that makes calculations easier and objects are aggregated as needed (e.g. instead of 400 billion stars per galaxy, put clusters of stars as singular objects instead).

Answer 1

Yes, this is pretty much exactly how Modified Newtonian Dynamics (MOND) works. It is observed that Newtonian mechanics works very well for large accelerations, but an ad hoc correction function to Newtonian gravity is proposed at small accelerations. This function can be tuned such that the rotation curves of galaxies can be explained without the need for dark matter.

Edit in response to edited question:

Note that explaining rotation curves is of course only one empirical constraint. And MOND does poorly at matching others.

If you are looking for some universal gravitational law that explains all possible empirical observations of stellar and galaxy motions (which is only one of the constraints any theory must satisfy), without dark matter, then I'm afraid there isn't one (afaik).

Any approach to a universal theory of how gravity works must of course be identical (or at least similar to a very great extent) to inverse-square Newtonian gravity outside the regime of very strong or very weak accelerations (as indeed General Relativity is), because there is a wealth of observational evidence that shows simple Newtonian gravity works very well in those cases.

Answer 2

In the emergent gravity theories, dark matter is thought to be inherent to normal matter and spacetime. It's a consequence of the interplay between conformal fields in 5d anti-de Sitter spacetime and the 4d-spacetime (ours) it encloses. This theory (contrary to MOND) is there even if (conjectured) dark matter-induced motion wasn't observed. In this respect, it is preferable to dark matter theories, which indeed cause Vulcan-like situations. See for example this article. Even normal gravity itself is explained in this way.

Answer 3

I'm not sure if it's appropriate to add another answer but as the question has been edit to make its intention more clear, I suppose it is.

Let me explain why this is not the way it's done. To simplify the situation, consider the motion of a particle that falls down on Earth. One can take a number of these particles, with various masses and do measurements on these.

Somewhat like Galileo once did in his tower experiment. Galileo, who said that the only way to read the universe was to learn the language of mathematics, is thought to have thrown two massive spheres from Pisa's tower and thereby showed that all masses fall in the same way. The spheres reached the ground in the same time, hence the conclusion. The experiment falsified Aristotles' conjecture that the falling of objects depend on their mass. If two objects are dropped the heavier would reach the floor earlier as the light one, according to him. Now this is indeed the case for many different objects when you let them fall on the ground. A feather will end up with less velocity than a small lead ball with the same mass. Heavy objects, compared to light ones of comparable size, take generally less time to reach the Earth than light ones (because of air resistance). Aristotle didn't know anything about air resistance (like we don't know a dark matter) and because of this he thought gravity acting differently for different masses.

The (alleged) measurements of falling times done by Galileo were done though after the workings of gravity were already anticipated. The measurements were an experiment to confirm (or contradict) these.

Now he could have done a lot of experiments on a lot of falling objects. He could have used different masses and different shapes, different initial velocities and positions, and performed the experiment at different locations and different circumstances. This would have resulted in a huge amount of data: rows of initial positions, initial velocities, rows of masses and shapes, rows of times, velocities, and positions, rows of circumstances, and rows of locations. If the data were ordered in the right way (for which no theoretical assumption about gravity has to be made), and if he had quantum super computer like capabilities, he could search for a connection between the data. How to find the algorithm? Well, he could have made a visualization of the data. If he had the mathematical knowledge of today (differentiable functions only saw the daylight after the birth of Newton) he could use his knowledge of the graphic form of functions to see if any of them (or a function of a function, say a sum) matched the data. If he had found any functional correspondence he could have started theorizing about gravity (or the circumstances of the experiment).

I doubt that this is done in practice. Now this is exactly your question, but I merely answer to state that this way of proceeding isn't followed in practice. Pure naked observation cannot be detached from theory. That is, empiricism and theory cannot be separated. Even in the case of measuring times, distances, positions, velocities, or masses, you already make a theoretical assumption (for example, that all velocities refer to displacement in space, while it could be that space itself expands).

Now, as seen, it could be done in this way. That is, measuring masses, velocities, and positions (and distances) of different celestial objects, after which you look for a functional relationship between them, but mostly it's done the other way round. On top of that, it would be very time-consuming, although a computer could do the job. So in the case of dark matter, the computer will spit out a functional relation between the data (the one that fits these data best). This relationship will be undisputable (though this is disputable!). To find out how these functional relationships between the data comes about a theory is needed. This can change the very Nature of the quantities that you measured (like mass, velocity, and space), thereby changing the Nature of your assumptions made in the measuring (like space time being Euclid).

So again, it could be done, but to separate empiricism completely from theory is impossible.