When Newton contemplated what the universe is like, he considered three possibilities:
- Aristotle's idea: the universe is finite
- Stoicism's idea: the universe is a finite island of stuff floating in an infinite vacuum.
- Epicurus's idea: the universe is infinite and uniformly filled with stuffs.
He settled on the infinite uniform universe.
Now he faced a question: what is the gravitational force felt by a test mass in this infinite uniform universe? He thought that, by spherical symmetry, it has to be zero, because if the force isn't zero, it would break the spherical symmetry.
Now, a quick thought would show that there is a serious problem. It's known that a uniform, infinite slab attracts a mass with the same force, no matter how far.
So cut the universe into a infinite series of equally thick slabs, on the left and right of the test mass. We see that to calculate the total force on the test mass, we must do this infinite sum:
$$F + F - F + F - F ...$$
which is indeterminate.
In short, these three conditions are incompatible:
- The universe is infinite and uniform
- The universe is Newtonian (Newton's three laws of motion, plus Newtonian gravity)
- The Newtonian laws of physics should fix a unique solution.
This was what Seeliger argued in 1895. Since then, many solutions have been proffered, by giving up some of the conditions. One interesting solution is by Milne and McCrea in 1934, see Newtonian Cosmology (1955). It gives up the uniqueness, but shows that there are some very interesting solutions: an expanding and contracting universe with finite lifespan and a "center point" (despite being infinite and uniform), something that Newton could have discovered!
