Posts Tagged ‘Bentley’s paradox’


Newton grew up in England, and he went Cambridge University, and right as he was beginning his education, the plague struck, the college closed, and he had to go home.

It is a fact that his mother had an orchard, and it is more than likely that he sat in that orchard, reading some of the books he would have read at school, including a brand new book, just published: Galileo’s last book, translated into English at long last. He read that Galileo believed that the orbit of the Moon was following the very same law of gravity as the fall of a spoon or anything else, such as the apples around him. He worked out the math and – with a little push from his friends – he published it thirty years later

A little push: his first thought was that this knowledge was too sacred for the common man. Robert Boyle and others were aghast. “You must publish,” they said.

“Anyway, it’s too expensive,” he answered.

“I’ll fund it!” exclaimed Boyle. And that is why we have Newton’s thoughts.

Since that publication, we see that it isn’t just a mathematical convenience to locate the Sun in the center of the solar system; rather this location is due to its more substantial mass. Gravity, the most familiar and fundamental of all physical powers, is now understood to govern the orbital motions of the entire solar system. That the Sun was vastly larger than the Earth was already known, and this relationship with gravity was the final disposition of the Ptolemaic system of cycles and epicycles and all the embroideries on it. Geocentrism after Newton amounts to a denial of gravity.

Deepening space

Time moved on. Telescopes improved with the passing of years, and with their improvement, more and more stars appeared. Still no parallax, but with so many stars invisible to the naked eye but visible to telescopes, might it not be that they really were just farther away? The impression of deep space became stronger and stronger.

How deep?

Newton had thought the universe was finite, but where did it end? Bruno’s riddle about his arrow was still unanswered, and the impression of utterly endless spaces was growing.

Yet there were very good reasons for thinking the universe finite.

One became known as Olbers’ Paradox – named for a scientist who lived in the 19th century, but the idea was already around here in the 18th century, 100 years before it got his name. It went like this:

Suppose the universe is infinite, and suppose it is everywhere sprinkled with stars, just as we see – thus with an infinite number of stars. In that case, if you should go into a field at night, the light of an infinite number of stars must fall about you. Only a very small amount of light from each one, but… infinite is infinite. There would be no dark sky in an infinite universe with stars throughout.

There was a similar line of reasoning about gravity: an infinite number of stars would have an infinite gravitational pull and the universe must collapse into its own weight. This was called Bentley’s Paradox. It should be called Bentley’s proof of a finite universe, just as Olbers’ Paradox should be called the light proof of a finite universe, but some people were so convinced that the universe was infinite that they called it a paradox, as if they were dealing with a trick of thought, not with an impossibility.

Read Full Post »

Richard Bentley was a younger contemporary of Isaac Newton. After Newton had formulated his law of gravitation, he observed, in a letter to Richard Bentley, that if all the stars are drawn to each other by gravitation, they should collapse into a single point. One will be drawn to another; that star will grow and pull in still more and more. In time, everything must be drawn in.

How is the universe constructed so as to prevent this from happening?

It did not occur to Newton, or perhaps to anyone before the 20th century, that the universe is a changing space-scape. It has a history, at the start of which (in Big Bang theory) its matter was ejected apart; so far, it does not have the energy to re-gather everything in a universe-crushing event. Or maybe I should say that its momentum is still too great to be overcome by its gravity. Either way, it could face gravity collapse, but not yet.

Bentley’s Paradox (which maybe should be called Newton’s Paradox, since it is odd to name an insight after its first recipient, rather than its author) is similar to Olbers’ Paradox in that it does suggest a finite universe: an infinite universe would have infinite gravity and would certainly collapse.

Well, again, perhaps you’d have the Kelvin/Boltzmann suggestion of a universe with pieces far enough apart not to respond. But this suggestion only serves to indicate the distant (really distant!) possibility of a kind of multiverse. Unless it can be tested, it is not a scientific hypothesis, though it is an interesting thought experiment.

Read Full Post »