The uniform infinite Newtonian universe (the standard solution to the Seeliger paradox)

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!

Survey of moral realism and irrealism

Note: philosophers use "ethics" and "morality" as the same word.

The moral realism vs irrealism contrast is the most important contrast in metaethics, the study about what ethics is, such as whether morality is objective, subjective, emotion-based, rationality-based, changeable over time, etc.

Realism says morality is real in some sense, while irrealism says it isn't.

Infinity

For an example of such a contrast, we can consider the positions on infinity in mathematics. Realism would simply say that infinity exists, is an actual mathematical object, and there's a symbol of it, ω. Irrealism would claim that ω is just a convenient fiction, and merely stands for the endlessness of natural numbers.

There are very fine-grained notions of realism vs irrealism in the study of infinity, too, used by expert set theorists. Most set theorists consider the first few infinities unproblematic, but as they climb up the ladder of infinities, they encounter more and more abstract infinities. They often draw a line somewhere and proclaim infinities beneath some line real and above unreal.