Is there a mathematical proof about The general law of attraction between two point masses
$m_1$ and $m_2$ which represented as: $$F=G\frac{m_1m_2}{r²} ,$$where $r$ is the distance between the centers of the masses, and $G$ the gravitational constant?
NOTE: I'm not familiar in physics but just I'm confused why $r²$ not $r^3$ or $r^n$, where $n$ is a natural number and what are the mathematical setps let us up to the above law?
There is not any mathematical but beside that Newton combined his laws of motion with Kepler's laws and deduced the law of gravitation. Path of planets around the sun are elliptical so for simplicity we can assume the orbit to be circular. Let us consider $a$ planet of mass $m$ moving with constant speed $v$ in a circular orbit.
$$T=\frac{2\pi r}{v}$$ and $v>0$
Where $r$ is the radius of circular path
And from Kepler;s 3rd law, where $k$ is a constant of proportionality. $$T^2=kr^3$$
From above two equations we can rewrite
$$\frac {4\pi ^2r^2 }{v^2}=kr^3$$ We know that an object in a circular path is accelerated and its acceleration towards the centre is $$\frac {v^2}{r}$$,and $r>0$ because the distance between two point masses should be no nul
$$F=\frac {mv^2}{r}$$ $$F = \frac{m.4\pi ^2}{kr^2}$$
$$F \propto \frac {1}{r^2}$$
$$F \propto m$$
Now force on the planet due to the sun = force on the sun due to the planet. If the force is proportional to the mas of the planet, it should be proportional to the mass of the sun.
$$F \propto \frac{Mm}{r^2}$$ $$F = G \frac{Mm}{r^2}$$
We can't expect to be able to take the principles of mathematics and come to conclusions about physics problems. At some point observation of phenomena is required. We then take these observations and use them to conclude whatever we can to find is true about the world. Historically, two different people played the two roles of data collection and then conclusion making. Tycho Brahe made good careful observations about the solar system. He made no general principles or ways to predict what the coming positions of the planets and the moons would be, but he recorded their locations well. Johannes Kepler then came and used Brahe's data to make his three Kepler's laws of planetary motion. He made these with the use of mathematics. He took a wide array of data and fit them to general mathematical principles. With the use of the first and second law an inverse square gravity law can be arrived at.
For the specifics of this look at the sections on the first law, second law and the inverse square on this wikipedia page https://en.wikipedia.org/wiki/Kepler's_laws_of_planetary_motion#First_law
It is easy to see how we could only conclude inverse-square dependence after making observations about the world.
