10/04/2018

Kurt Gödel: the mathematician compared to Aristotle

Reproduction of the G1 news report.

They were a duo that attracted attention for several reasons.

One of them, with unruly white hair, wore a wrinkled shirt, baggy trousers, and suspenders as he walked the streets of Princeton, in the United States, occasionally stopping for ice cream. This was Albert Einstein, who, by the 1930s, was already the most famous scientist in the world.

The other, younger man, dressed traditionally and wore dark-rimmed glasses, which gave him an austere expression. He was the mathematician Kurt Gödel, described as the most important philosopher in the world after Aristotle.

Although less famous than Einstein, he was well known in academic circles for "having shaken the foundations of our understanding of the human mind," Princeton University declared when awarding him an honorary doctorate.

Escape from Nazism

Both went to Princeton, in the United States, because of the rise of Nazism in Germany. One because he was of Jewish origin (Einstein), and the other to escape his fate as a soldier in Hitler's army (Gödel).

Both rejected quantum theory, going against the mainstream, and shared a fact that made them exceptional: they changed our perception of the world when they were only 25 years old.

Einstein, with his brilliant formula E= ^mc² . And Gödel, with the discovery that we can never be certain that 1 is not equal to 0.

'Sir, why?'

Gödel was born in Austria in 1906, a year after Einstein proved that time, as it was understood, is an illusion.

Gödel's family nicknamed him "Mr. Why" because he was extremely curious. From a very young age, he enjoyed studying different languages, religions, mathematics, and history.

When he entered the University of Vienna at the age of 18, he already knew so much about mathematics that the regular courses had nothing to add. Eventually, he became interested in mathematical logic, which he described as "a science prior to all others, containing the ideas and principles that underlie all sciences."

Gödel's 'revolution'

Until the last century, mathematics was known as the science capable of offering "certainties." It was a world where everything was either true or false, right or wrong. If applied correctly, it would always be possible to discover the right, exact answer.

But in 1900, when the International Congress of Mathematicians met in Paris, the atmosphere was one of both hope and questioning. The consistency of mathematics was being challenged. During the congress, a young man named David Hilbert launched a plan to reconstruct the foundations of mathematics, to make them consistent and free of paradoxes.

Hilbert was one of the world's greatest mathematicians, but his plan failed due to Kurt Gödel. With his doctoral thesis, Gödel put an end to this ambition, demonstrating that there were problems in mathematics that were impossible to solve. The dazzling clarity and precision of mathematics was, in reality, a labyrinth full of paradoxes.

Gödel proved that: 1) In any consistent axiomatic formal system that can express facts about basic arithmetic, there are true statements that cannot be proven; 2) That the consistency of the system cannot be proven within the same system.

These are the incompleteness theorems. If you're confused, don't worry, you're not alone.

There are more truths than we can prove.

Fortunately, there have been several attempts to explain the incompleteness theorems in a didactic way, so that everyone could understand the great achievement of "Mr. Why".

In short, what Gödel did was use mathematics to prove that mathematics cannot always be proven through calculations. In any system there are statements that are true, but that cannot be proven.

Paradigm shift

The incompleteness theorems revolutionized mathematics and inspired people like John von Newman, one of the creators of Game Theory, and Alan Turing, creator of the mathematical system that made the computers we use today possible.

They have also proven valuable for Information Technology. The recognition that there are things that cannot be proven has set a limit to what computers are capable of solving, preventing the wasted time of trying to achieve the impossible.

Many believe that Gödel's theorems will impact other fields. The physicist, mathematician, and philosopher Roger Penrose, for example, believes they could help uncover new physics that explains the mysteries of consciousness.

Fears and anxieties

Towards the end of his career, as he was about to retire, Einstein remarked that he continued to go to his office only to have the privilege of walking with Gödel, something he did until shortly before his death in 1955.

The two scientists chatted during the drive to the Institute for Advanced Study in Princeton. Some of Gödel's thoughts, however, were obscure. He had always lived tormented by fears and anxieties. He was, for example, terrified of being poisoned.

Therefore, he refused to eat unless his wife, Adele, tasted the food first. When she became ill and had to be hospitalized for an extended period, Gödel practically stopped eating. Fearing for his life, he eventually died of starvation in 1978.