The Mathematician Who Discovered Math's Greatest Mystery

Kurt Gödel once confessed that he believed he was  a “failure”. . .

despite revolutionizing mathematics. He was so extraordinary that Albert Einstein  once said he would go to his office every day “just to have the privilege of being  permitted to walk home with Kurt Gödel. ” Gödel’s most famous contribution,  his Incompleteness Theorem, shook the foundation of mathematics, revealing  that there are truths within any mathematical system that cannot be proven by the system itself. 

This statement cannot be proven within the system. Yet despite his groundbreaking work, he  couldn’t shake his own feelings of inadequacy. In his final days, he refused to eat, and  when he died, he weighed only 65 pounds.

To understand Gödel’s profound discovery and deep-seated insecurities, we  must go back to his beginnings. Gödel entered the world on April 28, 1906 in  Brünn, now known as Brno in the Czech Republic, which was then part of the safe  and stable Austro-Hungarian Empire. His family lived a privileged  life thanks to his father’s success in running a large textile company.

As a young boy, Gödel was known as “Mr Why”, always questioning everything around him. He excelled in school, earning a “very good” in all subjects with the exception  of a “good” in mathematics, ironically. Gödel enrolled at the University of Vienna  where he had planned on studying physics.

But, he changed his mind after encountering the  renowned math professor Philipp Furtwängler who was paralyzed from the neck down  and lectured from a wheelchair. Inspired by the charismatic professor, Gödel  switched his focus to pure mathematics. He soon became a member of the Vienna  Circle, a group of intellectuals who met every other Thursday to discuss  philosophy, science, and mathematics.

Most members were strong proponents of  logical positivism, the belief that only truths that can be empirically verified  or logically proven are meaningful. But Godel challenged this view. His famous theorem revealed that some meaningful mathematical truths cannot  be proven within any formal system.

When he was just 24 years old, he announced  his groundbreaking Incompleteness Theorem, declaring at a conference in September 1930: “. . .

there are mathematical problems that  can be expressed in Principia Mathematica, which cannot be solved by the logical  means of Principia Mathematica. ” Although his theorem is complex  and abstract, it can be understood through a simple analogy: the Liar Paradox. Consider the statement: This statement is false.

If the statement is true, then  it must be false, as it claims. But if it’s false, then it must be true. This creates a paradox where the statement  cannot be consistently labeled true or false.

Similarly, Gödel’s theorem showed that  within any formal mathematical system, there will always be true statements that  cannot be proven using the system’s rules. His theorem applies to any system  capable of expressing arithmetic. Even with the help of a computer, there will always be truths that  the system itself cannot prove.

Gödel’s revelation, published in 1931,  shocked the mathematical community. The brilliant mathematician John von Neumann  immediately grasped its significance, later declaring: “. .

. it is a landmark which  will remain visible far in space and time. ” Author Stephen Budiansky who  wrote a remarkable biography of Godel captured his theorem beautifully: “The proof itself was a feat of mathematical wizardry so extraordinary that its conception  and execution were as awe-striking as the result.

Gödel’s proof has that air of unconstructed  inspiration, of having seen the whole in a glance. Like the countersubjects of a Bach fugue  that work perfectly together at the moment when they meet halfway through, or the turn of a  sonnet where thought, rhythm, and rhyme combine in one perfect word at the end of the stanza…” However, not everyone welcomed Gödel’s discovery. It directly challenged the philosophy of  David Hilbert, a leading mathematician of the time, who famously proclaimed, “In  mathematics, there is nothing unknowable!

” Gödel proved this was not true. Despite Gödel’s monumental discovery, he struggled to secure a stable academic position. The economic devastation of the Great Depression, coupled with the political turmoil  in Austria following the collapse of the Austro-Hungarian Empire, made it nearly  impossible for him to find a permanent post.

His position as a Privatdozent at the University  of Vienna allowed him to lecture but without a salary - he only received a small fee  for each student enrolled in his courses. As Austria descended into chaos under the  influence of the Nazis, Gödel's mental state began to unravel, exacerbated by the sudden  death of his mentor, Hans Hahn, following surgery. Godel’s depression was so severe that  his mother feared he’d become violent and locked her bedroom door at night, as author  Stephen Budiansky described in his book.

Gödel checked himself into sanatoriums where he refused to eat, convinced that the  doctors were trying to poison him. His girlfriend Adele, a former  nightclub dancer, became his lifeline. Despite his parents’ disapproval of  their relationship, Adele stood by Gödel, even taking the first bite of his food  to reassure him that it was safe to eat.

Decades later, he confided to his  psychiatrist that he had lived in constant fear of being arrested and  charged after Adele had an abortion. They were not married; Adele was previously  married, and under Austrian law at the time, civil divorce was not permitted for Catholics. The final nail in the coffin that caused  Gödel to spiral out of control came in 1936, when his former professor, Moritz Schlick  was murdered by a deranged former student.

Johann Nelböck waited for Professor Schlick on a staircase at the University of Vienna and  shot him four times at point-blank range. Nelböck became a martyr of the anti-Semitic  right despite the fact that Professor Schlick wasn’t Jewish but was associated  with the Jewish intelligentsia. In March 1938, when the Nazis took  over Austria, Nelböck was paroled.

Many of Gödel’s colleagues fled to America, but Gödel naively believed his  future still remained in Austria. Despite invitations from prestigious American  institutions, he was reluctant to leave, still clinging to the hope of securing a  permanent position at the University of Vienna. However, the Nazis viewed Gödel with suspicion.

Although he wasn’t Jewish, many of  his friends and colleagues in his profession were, and he was associated with them. As financial support dwindled after his father’s  death, Gödel realized he had no future in Austria. John von Neumann recognized Godel’s immense  value and urged Abraham Flexner, the head of the Institute for Advanced Study (IAS), to hire him,  insisting, “Gödel is absolutely irreplaceable.

” Scholars at the IAS were free to pursue their research without the burden  of teaching regular courses. When the Institute extended an invitation,  von Neumann urgently sent a cable to Godel that read: “INVITATION STILL  VALID COME AS SOON AS POSSIBLE. ” To persuade Nazi officials to allow him to leave,  the new IAS head, Frank Aydelotte argued that Gödel’s presence in America would significantly  enhance the prestige of German science.

Godel and his now-wife Adele  arrived in New York in March 1940. Although the move to America brought safety  and stability, Gödel never shook his feelings of inadequacy, believing that his later work,  including his proof of the generalized consistency of the Continuum Hypothesis, fell short of his  early achievement on the Incompleteness Theorem. His psychiatrist, Dr Philip  Erlich, noted in their sessions, “Belief that he hasn’t achieved goals that he set out for himself - hence a  “failure” - therefore other people, particularly the Institute, will also regard  him as a failure & try to get rid of him.

” Despite Gödel’s self-doubt, one of the greatest  scientific minds held him in the highest regard. Albert Einstein, whose office was  on the same floor as Godel’s at the Institute for Advanced Study, cherished  their daily walks together, saying: “My own work by then did not amount  to much, but I came into the office just to have the privilege of being  permitted to walk home with Kurt Gödel. ” In 1947, Gödel prepared to  become an American citizen.

He thoroughly studied the U. S.  Constitution to prepare for his citizenship exam and even discovered what he  believed to be a logical loophole that could, in theory, allow a dictatorship  to arise in the United States.

During his interview, Gödel  began to explain this concern, but his friends Albert Einstein  and economist Oskar Morgenstern, who both accompanied him, quickly intervened  to steer the conversation back on track. Gödel passed the exam and  became a U. S.

citizen in 1948. The IAS also made Gödel a permanent professor a few years later despite concerns about his  mental state, which deteriorated further. When Gödel was treated for a  stomach ulcer in the early fifties, he insisted that his doctors  didn’t know what they were doing.

Gödel was extremely anxious about his health, as described in Budiansky’s book, taking  his temperature three times a day. He craved certainty and control over his  health—an inherently unpredictable area, much like the limitations and uncertainties  revealed by his Incompleteness Theorem. This deep-seated need for certainty  extended to every aspect of Gödel’s life, including his views on the meaning of life itself.

He was skeptical of evolution,  believing the complexity of life couldn’t be fully explained by  natural selection and random mutation. He once told a colleague that the human mind was  “. .

. a computing machine connected with a spirit. ” In a letter to his mother, he wrote: “You pose in your last letter  the momentous question, whether I believe we shall meet in the hereafter. 

About that I can only say the following: If the world is constructed rationally  and has a meaning, then that must be so. ” He even completed a logical proof of the existence of God - though it wasn’t empirical  proof but rather a logical argument. Even as his mind grappled with such  profound and abstract concepts, Gödel was increasingly besieged by paranoia.

He refused much-needed prostate surgery out  of fear that doctors were trying to harm him. He insisted to his psychiatrist  that the IRS was after him, the IAS was trying to take away his pension,  and his brother Rudi, who remained in Austria, had been sent to a concentration camp because  Godel had failed to return to Austria as promised. When Adele was hospitalized for  several months following an illness, Gödel slowly starved to death at home.

Adele finally persuaded him to enter a hospital. During his final two weeks in  hospital, he refused to eat. On January 14, 1978, Kurt Gödel, one of the  greatest logicians there ever was, passed away.

He was 71 years old. Gödel revealed a profound truth  about math and about life: even the most sophisticated systems have limits. There are truths that will  forever remain beyond our reach.

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