Genady

I didn't say anything about proof being required or offered. The above sentence has nothing to do with my claims.

No, the G.M.m.ln(x) is NOT a solution for the Newton's law. The -G.M.m/x is. My question is, what is wrong with it? In 2D, I mean. This is what I mean by "the idea that the Newton's law holds as is."

Our ability or inability to prove something or to construct something in math doesn't affect its existence. The same with ethics. I am not religious. Just logical. The existence of a "chart" mentioned by @Peterkin above, as a list, set of rules, result of a procedure, ..., is "Genady's conjecture." I don't think it is OT. It is my answer to the OP question.

I don't care about Plato. Mathematical sense is not the same as philosophical sense, whatever that could mean (as I said, I don't care.) It is very specific. For example, the number pi has a one billionth digit. That digit exists. It is unique. It doesn't matter if we have it calculated already or not. Being or not being a human creation also doesn't matter for the existence. An equation, for example, might be a human creation. Nevertheless, it has or it doesn't have a solution, regardless of humans. Same with ethics. A Platonic ideal is not required. Isolation is not required.

And, answering my question would be nice, too. Here it is again: what is 'wrong' when you try to integrate the Newton's gravitational force? it seems to obey all your conditions.

None of the above. It exists in mathematical sense. Like a solution of an equation exists. Regardless we know it or not.

No, I don't have such a chart. I claim that such a chart exists.

The concept is not movable. A set of torture cases is large. Some of them fall into the constant concept of Wrong, the others fall into the constant concept of Right.

I consider myself, in this respect, a theoretical ethicist. And torture in my fixed, non-negotiable notion of Right and Wrong, is not always wrong.

This looks like an opinion survey rather than a discussion. Then, I have an opinion, too: Yes, it is not always wrong / unethical / not-right / immoral.

What is wrong when we try to integrate?

in this case, for your 4 statements to hold, you have proved that the idea of field "dilution" can't hold. But what is wrong with the idea that the Newton's law holds as is?

But, as you and @joigus said, Newton did it from the Kepler's law, not from the idea of field "dilution" that you've described in the OP. If you want to do it like Newton did, you need to start with a 2D version of the Kepler's law.

How many solutions like this do you want? Here is one, for example, -G.M.m/(ex-1). Or, -G.M.m/x4

This is incorrect. Force is a gradient of potential energy and its direction is the direction of the gradient. Does not matter what a sign or even the value of the potential energy is, only its gradient. Again, this is incorrect. Since the force is a gradient it would change direction only if the gradient changes direction. Changing the sign of the potential energy doesn't change the direction of force.

In my response above I refer to the OP "fantasized" gravitational force that is inversely proportional to distance rather than to distance squared. Not to the Newtonian gravity.

It will take an infinite amount of energy to remove a body from any given position to infinity.

I thought you got the answer, even more than one. Maybe you have missed this one, for example:

Not necessarily. The 2D world doesn't have to be flat.

Not necessarily. Imagine a 3D sphere with the gravitating mass in the center and the gravitational field lines going from there. The force goes inversely proportional to the distance squared. Now cut the sphere in the middle and look at the cross section. This gives you the 2D image of the field. The force still goes inversely proportional to the distance squared.

Because physical 2D world is a pure fantasy, I guess one can make up any 2D physics they like. However, they still cannot make up mathematics.

When calculating gravitational potential energy near the Earth surface, where the gravitational acceleration g is almost constant, we choose it to be 0 on the floor and the energy is mgh where h is height above the floor. It changes sign, too. And it works just fine.

In fact, you can get a nice formula in your 2D case as well: since adding a constant to energy doesn't change anything, you can simply remove the ln(L) part and get the general equation for absolute value for any distance, GMm*ln(d). Is there a magnetic field in 2D?

This list of physical and technical difficulties and potential points of failure can go on and on. It seems that self-replication algorithm is an easy part - it is all the rest that is difficult or impossible. Self-replication doesn't help in solving other issues.

The value calculated from infinity is as absolute as values calculated from any other choice of distance. You can choose the energy being 0 at some finite distance L. Then in 3D, the general equation for absolute value for any distance is GMm(1/L-1/d). In 2D, it would be GMm*ln(d/L). They all are absolute values. You can choose L to be 1 km, 1 parsec, diameter of the Milky Way galaxy, radius of the observable universe, whatever. It doesn't matter because in all physical calculations the constant L will cancel out. The difference between the formulas for 3D and for 2D is that the former allows you to choose L being infinity and to get a nicer formula without L to start with, since it will cancel anyway, -GMm/d, while in 2D you can't do this, although it will cancel there as well.