Kurt Gödel’s Open World

by Ashutosh Jogalekar

Gödel and Einstein in Princeton (Source: Emilio Segre Visual Archives)

Two men walking in Princeton, New Jersey on a stuffy day. One shaggy-looking with unkempt hair, avuncular, wearing a hat and suspenders, looking like an old farmer. The other an elfin man, trim, owl-like, also wearing a fedora and a slim white suit, looking like a banker. The elfin man and the shaggy man used to make their way home from work every day. Passersby and motorists would strain their heads to look. Everyone knew who the shaggy man was; almost nobody knew who his elfin companion was. And yet when asked, the shaggy man would say that his own work no longer meant much to him, and the only reason he came to work was to have the privilege of walking home with the elfin man. The shaggy man was Albert Einstein. His walking companion was Kurt Gödel.

What made Gödel, a figure unknown to the public, so revered among his colleagues? The superlatives kept coming. Einstein called him the greatest logician since Aristotle. The legendary mathematician John von Neumann who was his colleague argued for his extraction from fascism-riddled Europe, writing a letter to the director of his institute saying that “Gödel is absolutely irreplaceable; he is the only mathematician about whom I dare make this assertion.” And when I made a pilgrimage to Gödel’s house during a trip to his native Vienna a few years ago, the plaque in front of the house made his claim to posterity clear: “In this house lived from 1930-1937, the great mathematician and logician Kurt Gödel. Here he discovered his famous incompleteness theorem, the most significant mathematical discovery of the twentieth century.”

The author in front of the house in Vienna where Gödel was living with his mother and brother when he proved his Incompleteness Theorems

The reason Gödel drew gasps of awe from colleagues as brilliant as Einstein and von Neumann was because he revealed a seismic fissure in the foundations of that most perfect, rational and crystal-clear of all creations – mathematics. Of all the fields of human inquiry, mathematics is considered the most exact. Unlike politics or economics, or even the more quantifiable disciplines of chemistry and physics, every question in mathematics has a definite yes or no answer. The answer to a question such as whether there is an infinitude of prime numbers leaves absolutely no room for ambiguity or error – it’s a simple yes or no (yes in this case). Not surprisingly, mathematicians around the beginning of the 20th century started thinking that every mathematical question that can be posed should have a definite yes or no answer. In addition, no mathematical question should have both answers. The first requirement was called completeness, the second one was called consistency. Read more »

Black Holes and the Curse of Beauty: When Revolutionary Physicists Turn Conservative

by Ashutosh Jogalekar

Main-qimg-da0bd0564345ac4af20890fb6dc10820-cOn September 1, 1939, the leading journal of physics in the United States, Physical Review, carried two remarkable papers. One was by a young professor of physics at Princeton University named John Wheeler and his mentor Niels Bohr. The other was by a young postdoctoral fellow at the University of California, Berkeley, Hartland Snyder, and his mentor, a slightly older professor of physics named J. Robert Oppenheimer.

The first paper described the mechanism of nuclear fission. Fission had been discovered nine months earlier by a team of physicists and chemists working in Berlin and Stockholm who found that bombarding uranium with neutrons could lead to a chain reaction with a startling release of energy. The basic reasons for the large release of energy in the process came from Einstein's famous equation, E = mc2, and were understood well. But a lot of questions remained: What was the general theory behind the process? Why did uranium split into two and not more fragments? Under what conditions would a uranium atom split? Would other elements also undergo fission?

Bohr and Wheeler answered many of these questions in their paper. Bohr had already come up with an enduring analogy for understanding the nucleus: that of a liquid drop that wobbles in all directions and is held together by surface tension until an external force that is violent enough tears it apart. But this is a classical view of the uranium nucleus. Niels Bohr had been a pioneer of quantum mechanics. From a quantum mechanical standpoint the uranium nucleus is both a particle and a wave represented as a wavefunction, a mathematical object whose manipulation allows us to calculate properties of the element. In their paper Wheeler and Bohr found that the uranium nucleus is almost perfectly poised on the cusp of classical and quantum mechanics, being described partly as a liquid drop and partly by a wavefunction. At twenty five pages the paper is a tour de force, and it paved the way for understanding many other features of fission that were critical to both peaceful and military uses of atomic energy.

The second paper, by Oppenheimer and Snyder, was not as long; only four pages. But these four pages were monumental in their importance because they described, for the first time in history, what we call black holes. The road to black holes had begun about ten years earlier when a young Indian physicist pondered the fate of white dwarfs on a long voyage by sea to England. At the ripe old age of nineteen, Subrahmanyan Chandrasekhar worked out that white dwarfs wouldn't be able to support themselves against gravity if their mass increased beyond a certain limit. A few years later in 1935, Chandrasekhar had a showdown with Arthur Eddington, one of the most famous astronomers in the world, who could not believe that nature could be so pathological as to permit gravitational collapse. Eddington was a previous revolutionary who had famously tested Einstein's theory of relativity and its prediction of starlight bending in 1919. By 1935 he had turned conservative.

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James Ensor curated by Luc Tuymans at the Royal Academy of Art, London

by Sue Hubbard

ScreenHunter_2386 Nov. 21 12.27In 1933 the Belgium artist, James Ensor, met up with Einstein, when the latter was on his way to the States, for lunch on the coast near Ostend. Walking along the beach Einstein tried to explain the theory of relativity to the bemused artist. “What do you paint?” Einstein asked. To which the painter of masks replied “Nothing”. Whether this response was existential, bombastic or simply bloody minded it's hard to say but it does illustrate something of the enigmatic complexity of one of Belgium's most celebrated artists who, despite a British father, is barely known in the UK.

That father was a bit of a wastrel and a drunkard who married beneath him and, with his Belgium wife, ran a souvenir and curiosity shop in Ostend filled with an array of parrots, exotic masks, and even a monkey. These curios were to have a profound influence on his son's later imagery, imagery that has continued to intrigue as well as baffle. Opposed to ideas of classical beauty, James Ensor was equally infuriated by any notion that an artwork might need to have a social function. An outspoken exponent of ‘the prestige of the new', he considered the greatest artistic sin to be banality. Although he'd go on to have a profound effect on Expressionism and Surrealism, the orthodoxies of Modernism held little interest for him and, when he spoke of them, it was with limited understanding. Yet he produced many stunningly original works. Now the Belgium artist, Luc Tuymans, has curated a show at the Royal Academy that brings this enigmatic artist to a wider international public.

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