Parts of Kurt Gödel’s biography resemble the celebrity stories you find in a modern day tabloid. The Austrian mathematician rocketed to fame at a young age (completing his incompleteness theorem aged 25), hung out with other celebrities at cool places (one of his closest friends was Albert Einstein who he worked with at Princeton), rebelliously married someone of disputable reputation (a former nightclub dancer who was divorced and 6 years his major), and died under extraordinary and somewhat mysterious consequences.
Gödel's work shook the foundations of mathematics for all time. This and his tragic biography not only make him a legend but also the topic of our second post in our series “Tragic deaths in science”.
Setting the stage — The world and science in turmoil
The old political order of Europe and the consistency of mathematical frameworks showed severe cracks
Gödel entered the scientific stage at a time of great unrest and uncertainty. In the world of science, Georg Cantor had stretched the limits of math and logic by trying to define infinity because “God had told him so”, while Boltzmann had shown that the world was not determined from a deity above, but steered by random bumps of atoms within. Scientific certainty had been replaced by mere probability, which may or may not depend on God-given structures anymore. Russell had shown inconsistencies in the set theory, laid out in Russell’s Paradox, thereby destroying the hopes of mathematicians that this theory could provide a new strong foundation for arithmetics.
All of these events shook the foundations of mathematics, and its belief in a finite and consistent framework, at times which saw the old order of Europe disappear and the Austro-Hungarian empire fall apart. The upheaval of fundamental scientific beliefs therefore tragically reflected the growing concerns that even God-given powers can shift.
Searching for order
Gödel, who was brought up in the midst of World War I, believed that all is not lost and that order can be reinstated again in this mathematical chaos. So he set out to prove that arithmetic was consistent and complete. However, it was not meant to be. His innate curiosity and stubbornness in finding the answers gave him the nickname Herr Warum (Mr Why) in his childhood.
He set out to prove that arithmetic was consistent and complete. It was not meant to be.
As part of the Vienna Circle led by Moritz Schlick in the years between the wars, Gödel strongly believed in logical positivism and the idea that every mathematical problem must stand to be true or false. His philosophical principle was: “First, the world is rational”. Therefore, everything should be provable, including mind-boggling ideas such as Cantor’s paradox. This paradox is derived from Cantor’s theorem that there is no greatest cardinal number, and that even infinity is not the end, as there is always a larger infinity that lies beyond, formed by power sets of the first one.
Facing the limits of logic
His “failure” to identify a unifying axiom led to the groundbreaking discovery that completeness at arithmetic was not possible
Gödel’s attempt to find the ultimate logic underlying any mathematical paradox backfired in the most impressive way, as his “failure” to identify a unifying axiom led to the groundbreaking discovery that completeness at arithmetic was not possible, and that Cantor’s paradoxes can never be proved. The core idea of this incompleteness theorem is best described by the simple sentence “I am not provable”. Here, two options are possible: a) the sentence is right - and therefore it is not provable; or b) the sentence is false, and it is provable - in which case the sentence itself is false.
What can be derived from this core idea is that no consistent theory can prove its own consistency, and that there will always be statements that are true, but that are unprovable within the system. However, it is not predictable which statements this applies to until it is investigated.
While at first glance to the non-mathematical eye this theorem appears to not only shake the foundations of mathematics, but to pretty much close down the shop — what’s the point if nothing can be proved anyway? — the fall is slightly softened by viewing it as a philosophical-mathematical problem, while business as usual goes on in everyday life. While Gödel’s theorem is widely viewed as fundamental in mathematics, its impact on the field is, to say the least, limited. Maybe it is just that little bit too hard to grasp, even for fellow mathematicians?
An Austrian biography
After the Anschluss in 1938, the political situation in Austria changed and Gödel lost his lectureship at the University of Vienna. Because of his connections to Jewish mathematicians, he struggled to find a position in the new national socialist education system. Gödel left for Princeton where a mathematician of such exceptional talent was welcomed with open arms.
Today, long after his death, Kurt Gödel is welcome in Vienna again. He is honored and remembered by a street in the 10th Vienna district “Gödelgasse”, the “Kurt Gödel Research Center for Mathematical Logic”, and numerous plaques. None of the plaques mentions that the “greatest mathematician and logician of his time” — as Gödel is generally referred to in these inscriptions — did not find a job in Vienna at his time.
Kurt Gödel shares this part of his biography with numerous other Austrian scientists of the time such as Sigmund Freud or Erwin Schrödinger.
A mind of a kind
The brilliance of Gödel is reflected by his impact on the field of mathematics and science in general — but was he just lucky and painstakingly persnickety, or a mental genius? While we will probably never really find out, two things indicate that his mind was rather extraordinary — and not one to live with easily.
First, his close friendship with Albert Einstein, who he worked with at Princeton after emigrating from Nazi Germany to the US. Einstein, not a simpleton himself, claimed in later years that “his work didn’t mean much any more, and that he mainly came to the institute to have the privilege of walking home with Gödel.”
In later years Einstein only came in to work “to have the privilege of walking home with Gödel”
Following from this close friendship and their discussions, Gödel provided a model of Einstein’s equation that showed the universe as a large rotating disc with closed time-like world lines following the trajectory of a particle through space-time. As these lines are closed (i.e. they come back to themselves), his model provided an argument for the logical possibility of time travel. According to his model, a rocket launched into the right direction should return to itself at an earlier time. His level of comprehension of a new field as complex as the space-time continuum gives a startling glimpse into his capacities and shows just how distinct he was from most other scientists.
After his death, a letter from 1956 surfaced in which Gödel formulates and points out its significance of what is now known as the famous P-NP-Problem.
The dark side
The second side of the coin however depicts a mind with certain psycho-pathological traits that should in the end get the better of him. He suffered several mental breakdowns during his life. He spent several months in a psychiatric clinic in 1935.
His death certificate reported “malnutrition and inanition caused by personality disorder”
Since the assassination of his close friend Moritz Schlick by one of his own students, Gödel lived in paranoid fear of being poisoned. This fear became obsessive in later years, and he only ate food prepared by his wife Adele. When she was hospitalized for a few months after a stroke, Gödel refused to eat and died of starvation in 1978. His death certificate reported “malnutrition and inanition caused by personality disorder”.
Gödel and God
Paradoxically, Gödel may be more widely known — among non-mathematicians — for an attempted proof of God that he himself, a theist, had kept quiet until the end of his life, when he shared it with some of his students, who later published it.
Based (in a very simplistic view) on the assumptions that (i) positive traits are necessarily positive, (ii) being God is inherently positive, (iii) existing is a positive trait, and (iv) God comprises all positive traits, Gödel deducted that God could possibly exist and deducted further that God necessarily must exist.
This formalization has recently been used to demonstrate the capabilities of automatic theorem solvers. The theorem was revisited with the help of a computer, which delivered a 300 line result confirming it.
However, what Gödel’s intentions were or to what extent he believed the proof himself will never come to light; whether this proof may actually change the thinking of any atheist is debatable.
About this series
This series is about the tragic lives and deaths of three Austrian scientists who shook the foundations of physics, mathematics and biology.
They opened their mind to a world beyond the boundaries of what was accepted at their time. Their controversial and dangerous ideas led them to the question: is this world still worth living if you’re alone in your grasp of something ungraspable?