A PARADOX is a self contradictory proposition that is or may be true. For example, "I always tell lies" is a paradoxical statement. The history of science is peppered with paradoxes arising from prevailing scientific theories.
Nature abhors a paradox, and when one arises from a theory it either means that the theory is wrong or that we have an inadequate understanding of the axioms upon which the theory is built. Resolving paradoxes is a very useful way to advance our understanding of nature.
The first systematic use of the paradox in philosophy is attributed to the Greek, Zeno of Elea (5th century BC). Zeno outlined many paradoxes, the resolution of which can still fruitfully occupy mathematicians and philosophers to this day. Zeno's best known paradox goes as follows: In order to reach his goal a runner must first cover half the distance, then half the remaining distance, and so on. Because an infinite number of bisections exist in any length, it is impossible to cover the distance and arrive at the goal.
Zeno was philosopher Parmenides's disciple. Permenides taught that natural phenomena have no real existence; their apparent existence is due to human error as registered by our senses. He held that reality is to be found only in reason and is not known to the senses. Motion, and the steady progress of a runner from start to destination is readily apparent to our senses. Zeno's argument is aimed at demonstrating the logical impossibility of motion and therefore the illusory nature of the senses. My resolution of Zeno's paradox is as follows: Although there is an infinite number of bisections in any distance, each successive bisection is half the previous one and will be traversed in half the time. As the runner nears his goal he is soon traversing successive bisections infinitely quickly, which cancels out the effect of their infinite number. He therefore truly reaches his goal in the finite time registered by the senses. There is no paradox.
Have you ever wondered why the night sky is dark and not brilliantly bright? Most schoolchildren will know the answer - when the sky darkens it simply means that the Earth has rotated so that the observer's position is out of sight of the sun's light and we must wait until we rotate back into this light before the sky again brightens at dawn.
But are things that simple - what about the light from the stars? Our Earth rotates around a star (the sun), in the middle of an infinitely large universe containing an infinite number of stars, each shining as brightly as our sun. Why does the light from all these innumerable stars not add up so that our sky is blindingly bright both day and night? This in essence is the dark night sky paradox, one of the best known paradoxes to have arisen in the physical sciences. Its resolution eluded the powers of the best astronomers for 3 1/2 centuries.
The dark night sky paradox was first raised in 1610 by Johannes Kepler, imperial mathematician to the Emperor of the Holy Roman Empire. Kepler's solution was to conclude that the universe is finite, ends at an edge and contains a finite number of stars. According to this interpretation, when you look at the starry night sky what you see are faint stars shining against a background of black void that lies beyond the edge of the universe.
The problem became particularly acute with the rise of the infinite Newtonian universe. The astronomer Edmond Halley considered the matter in 1720. He concluded that the night sky was dark because individual distant stars were too faint to be seen. However, this would not explain why their collective light cannot be seen.
In 1823, Heinrich Oblers, a German astronomer, argued that starlight from far distant stars is absorbed by interstellar gas and never reaches us. But in 1848 John Herschel shot down this theory, explaining that any absorbing gas would heat up and would soon emit as much light as it absorbed.
In order to resolve the paradox, let me first of all state it in a more formal manner. Then I will outline the axioms that under pinned the paradox, one of which is wrong.
Consider a series of imaginary spheres of increasing radius, each with its centre on the Earth. The radius of each successive sphere increases by a fixed amount. The spaces between the spherical surfaces are therefore shells of equal thickness. The volume of each shell increases as the square of the radius and, because stars are uniformly distributed in space, their number in each shell also increases as the square of distance.
The light received on Earth from each star is inversely proportional to the square of its distance. Therefore, when we multiply the number of stars in "each shell by the amount of light received from each star, we see that the amount of light received from each shell is the same and is independent of the distance of the shell from Earth. The total amount of light reaching Earth is the fixed quantity per shell multiplied by the number of shells. In a universe of stars stretching away endlessly there is an infinite number of shells, and therefore an infinite amount of light should fall upon Earth.
When the dark night sky paradox was formulated, it was underpinned by four axioms - (a) that the laws of physics are everywhere and always the same; (b) that the universe is flat, i.e. it goes on indefinitely in all three directions - forward/backwards, up/down, left/right; (c) that we live in a typical part of the universe, and (d) that the universe is static, i.e. its large scale density does not change with time.
The first axiom, that the laws "of physics are the same everywhere, we must retain as otherwise we can make no headway. The second axiom, that the universe is flat, may not be true. If the universe is curved in one way so that it forms a closed curve, then its size becomes finite and the total stellar energy is finite. However, it can be shown mathematically that the dark night sky paradox remains. And there is no evidence, either theoretical or observational, that space is curved in the opposite direction.
The third axiom, that we live in a typical part of the universe must be retained since it is borne out by observation.
The fourth axiom is the one that will not hold up. We now know that the universe is not static, but is expanding. An observer on Earth notes that distant bodies in the universe are receding from us, and that the further away they are, the faster they are receding. In an expanding universe, in which all the matter is spreading out in every, direction, and getting less dense with time, the energy reaching us from the successive star shells is, progressively weaker. Therefore, the total energy we receive remains not only finite, but small enough to allow for the observed difference between night and day.