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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).

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    $\begingroup$ Have you ever read into the history of the theoretical development of gravity? Literally so many people developed empirical relationships relating forces and distances (a similar history for Coulomb's law). Hooke was such a contemporary to Newton, but Newton had the mathematical know-how to formalize it with symbolic rigor. Your questions might be addressed better in the history of S&M stack exchange, for examples: hsm.stackexchange.com/questions/8094/… $\endgroup$ Jun 3 at 15:00
  • $\begingroup$ and hsm.stackexchange.com/questions/12915/… $\endgroup$ Jun 3 at 15:01
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    $\begingroup$ Also, people have done what you suggest for the dark matter hypothesis. Many lines of research explore this, in much more sophisticated ways than you've proposed, e.g. see this old review cambridge.org/core/journals/… I'll lastly add that the model known as MOND is another way of attempting this by modifying Newton's law. Dark matter is more of a paradigm than a theory, per say. $\endgroup$ Jun 3 at 15:04
  • $\begingroup$ Already Einstein added the cosmological constant... which does a similar thing at large distances (but does not solve all gravity-related problems either)... yet much more rigor and approaches have been applied to this issue in quite open-minded ways than you seem to assume. The problem is that a theory has to explain quite a lot, sometimes even so far contradictory observations. It's not just finding the right parameters for a known equation type. en.wikipedia.org/wiki/Cosmological_constant $\endgroup$ Jun 3 at 17:17
  • $\begingroup$ In order to solve this mystery or problem it might even require to come up with a theory of quantum gravitation - a subject where everyone so far came up with a blank. $\endgroup$ Jun 3 at 17:18
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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.

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  • $\begingroup$ I edited my question to clarify why MOND is not what I was asking for - thanks :) $\endgroup$ Jun 4 at 15:15
  • $\begingroup$ @JacekKołodziejek but it is what you are asking about. Repulsive gravity is not required $\endgroup$
    – ProfRob
    Jun 4 at 20:17
  • $\begingroup$ Well, MOND does not explain several things Dark Matter does from what I heard. Obviously I'm too dumb (and don't have the time) to read proper scientific papers but I do listen to scientific videos/podcasts (while I work) and e.g. based on this video youtube.com/watch?v=dtfEzDAlL5k (9:23-12:20) it seems that MOND simply replaces one issue with another that is better solved by Dark Matter. So it doesn't get it all right. Also, it is another theory based on something, instead of a clean slate. $\endgroup$ Jun 4 at 20:28
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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.

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  • $\begingroup$ Emergent Gravity seems very appealing to me and I very much like the section 8.1 of the paper you linked to: "the evidence in favour of dark matter is just as much evidence for the possible breakdown of the currently known laws of gravity". Spot on. However, this paper conjures a massive underlying theory to come up with a set of equations, whereas I asked if there was an attempt at just getting equations out of the data and seeing if it's at all possible. EG also seems to be strongly disputed, though I'm unqualified to opine on the arguments (see EG page on wiki). $\endgroup$ Jun 5 at 22:27
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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.

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    $\begingroup$ You can use a blank line to separate paragraphs, This would make your answer easier to read. $\endgroup$
    – James K
    Jun 5 at 0:03
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    $\begingroup$ @JamesK You edit at the same time I did! Thanks. $\endgroup$
    – Methadont
    Jun 5 at 0:08
  • $\begingroup$ So in short, the reason this is not done is because our measurements are affected by our understanding of the subject matter, i.e. how we calculate the apparent gravitational pull to feed into the approximation algorithm will be affected by e.g. General Relativity. $\endgroup$ Jun 5 at 7:17
  • $\begingroup$ continued cause I missed the edit window :P: That's a fair point, but not a reason not to try it. Applying a function on the measurements does not remove the relation. For example, if you pick a number of pairs (X,Y) where X=Y, relationship will approximate to linear. If you square all Ys and approximate again, the relationship will be the square function. If the underlying data is in a strict relationship and affected uniformly by some continuous function, the modified data is still in a strict relationship, just modified. $\endgroup$ Jun 5 at 7:26
  • $\begingroup$ @JacekKołodziejek Yes, exactly. you can make a list of data (as you have done in your question), but the interpretation of these data involves theoretical assumptions about the data. The data cannot exist on their own. You can let a computer do the job of finding data (by using a telescope connected to a computer) and finding the appropriate algorithm (function) relating the data. But the interpretation involves a theory of ours (maybe the computer can find a theory too, but then it has to be very advanced!). : D $\endgroup$
    – Methadont
    Jun 5 at 7:26

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