At the age of twenty-four Kurt Gödel published two academic papers which overturned two thousand years of assumptions in the world of mathematical logic, ended the aspirations of a generation of mathematicians to prove the consistency of mathematics, and lead to his recognition as the greatest logician since Aristotle. Yet Gödel ’s theorems did not show the world to be illogical, instead – and importantly perhaps for Gödel and his belief in a rational God – they showed the world to be overflowing with reason: overflowing, in fact, with more consistent statements than could be proven. Mathematics turned out to be more full of reason – to be more full of true things – than mathematics itself could prove. For Gödel reason was the central component of the universe, and the central characteristic of God. Still, Gödel ’s God was personal, in the manner of Western theologies, rather than the impersonal God of pantheism or deism.
Gödel believed that numbers were real objects: like Plato he thought they existed in an objective world, and he studied the mindfulness techniques of the phenomenologist Edmund Husserl in an attempt to learn to actually see this world. This drew some contempt from his colleagues, who suggested that numbers were no more real than ghosts; something we could talk about, but which did not actually exist outside our minds. But Gödel, as it turned out, believed that ghosts were real as well, and part of his investigation of religion involved the study of the paranormal at the University of Vienna. He also believed in the afterlife, was an avid reader of the Bible, and spoke highly of other monotheistic religions. Gödel’s religious ideas began from deductions he made from the rational nature of the world and the rational God it implied. He also published what he believed to be a mathematical proof of the existence of God.
Gödel’s mental health has received much focus, though it may have been exaggerated by biographers and journalists keen to portray him as a “crazy genius”. During one suspected hospitalisation he was actually on an extended holiday at an alpine resort, and a cancelled trip to America turned out to be because he was secretly getting married to a Viennese cabaret dancer. He was certainly a shy man with some peculiar tendencies, but during his time at Vienna in which he produced his most famous work and investigated the paranormal with Carnap and Hahn, his mental and physical health seem to have been outstanding and certainly at least on a par with the average person.
Life in Vienna
Gödel was born in lands controlled by the dual monarchy of Austria and Hungary which fall within the modern-day Czech Republic. He was noted as a child for his curiosity and continual questioning. At school he would fill up pages of his books repeating the same sums over and over. He excelled in all subjects, and was particularly interested in mathematics, languages, and religion. By the time he enrolled at university in Vienna he had already masted university level mathematics, and turned his attention to mathematical logic, a branch of mathematics that investigates the logical relations that underpin the subject. He believed that mathematical logic was the master subject that determined the whole of conventional mathematics and the sciences.
As an undergraduate in Vienna Gödel was quickly recognized as an exceptional talent, and was soon invited to the meetings of the Vienna Circle academic group, where he contributed as an equal with colleagues with international reputations such as Carnap, Hahn, Neurath, Popper, Tarski, and Wittgenstein. Some comments on the impression Gödel made in Vienna:*
“He was a slim exceptionally quiet young man… his expression was always of the greatest precision and at the same time of exceeding brevity”,
“It became slowly obvious that he was incredibly talented… his help was much in demand and he offered it wherever it was needed… But he was very silent… he did not like to contribute to non-mathematical conversations”,
“his was clearly the mind of a genius of the very first order.”
Gödel’s doctoral thesis provided the basis for his incompleteness theorems. The theorems definitively ended attempts to find a consistent footing for mathematics, confounding centuries of attempts to do so. Gödel had achieved one of history’s most stunning intellectual achievements by the age of twenty-four. Von Neuman summarised:
“Kurt Gödel's achievement in modern logic is singular and monumental—indeed it is more than a monument, it is a landmark which will remain visible far in space and time. ... The subject of logic has certainly completely changed its nature and possibilities with Gödel's achievement.”
What was revealed was not that the universe was irrational, but rather the opposite, that the universe was overflowing with rational mathematical statements which could not be arrived at through mathematical reasoning but which still existed. The universe was not only rational, but contained far more rational things than mathematics could prove.
As well as mathematics, Gödel was fascinated with the paranormal, sharing the interest of his colleagues in Vienna Rudolph Carnap and Otto Hahn who attended and investigated seances. Seances were generally looked down on in academic circles, and subject to a great deal of fraud, fakery and financial exploitation. Gödel attended seances himself and believed that the great number of fairly obvious fake mediums obscured genuine occurrences of mediumship. He also had an interest in telepathy and precognition and his library receipts and notebooks reveal he had made extensive study of the early parapsychology literature. He believed in what he called “elementary psychic factors” which were the mental equivalent to the elementary physical particles that natural science had discovered.
Gödel was a Platonist, and as such believed that numbers, and mathematical relations such as sets, actually existed as objects in a mental realm. Like ghosts or spirits, for Gödel numbers had a real existence, and were not just imagined in the mind. His colleague Wang wrote that Gödel studied Edmund Husserl’s methods of mindfulness and introspection in an attempt to actually see the world in which numbers were real.
He was also interested in cars – according to his brother he was a fast driver; and in women – and perhaps surprisingly he seems to have been quite successful in this area. His letters reveal several attractions and relationships, one at the age of eighteen with a Viennese women ten years older than him described as an “eccentric beauty.” Unhappy with the age difference, his parents stepped in and forcefully prevented it progressing towards marriage.
Despite his quietness Gödel was not socially reclusive and was a regular in the Viennese coffee houses, at the theatre, and also in the nightclubs. Perhaps in one of these clubs at the age of twenty-one he first spotted Adèle, married at the time, who became friends with him, and eventually his wife. Once again this was not a match his mother approved of – Adèle was not only of low social class, six years older, divorced, Catholic, with an unusual birth mark on her face – but was also employed as a nightclub dancer and, it is likely, an escort. The pair kept their relationship secret for many years, before finally marrying when he was thirty-two. Despite loneliness on Adèle’s part when Gödel’s career took him to America, they appear to have been happily married. Adèle was an animal lover and talented clothing designer, and was given an award for her efforts in making children’s clothing to donate to the war time relief effort in Europe. Gödel considered her to have strong telepathic abilities – a trait traditionally associated with her facial birth mark. He remained dedicated to Adèle for the duration of his life, to the extent that on her entering hospital severely unwell he effectively ended his life through suicide by refusing to eat.
Gödel’s views were monotheistic and fitted reasonably well with Judaism, Christianity, and Islam. Gödel did not go to church, but according to his wife read the Bible in bed every Sunday morning. He also spoke highly of Islam, and admired its “consistency.” A questionnaire on religion found in his office read “baptized Lutheran (but not member of any religious congregation). My belief is theistic, not pantheistic, following Liebniz rather than Spinoza.”
He was keen to align science and religion. He wrote, “science confirms the apocalypse prophesied in the last book of the Bible and allows for what follows: ‘And God created a new heaven and a new earth’”. This appears to refer to the eventual ending of the universe predicted in modern cosmology. Much as the universe would experience an “afterlife” in Christian theology, through the new heaven and earth, so Gödel believed the individual would experience an afterlife. This enveloping structure of smaller within larger was in common with Leibniz’s notion of monads, which held a lasting appeal for Gödel – worlds within worlds, souls within souls.
Much of his discussion of the afterlife comes from a series of informal letters to his mother. Gödel’s parents were not religious and raised Gödel and his brother as atheists. In the letters he appears to be attempting to persuade her of an afterlife. He begins with these general remarks, which really should not be profound, but because the simple point they make has been so overlooked, they do seem rather profound despite their simplicity:
“Since we understand neither why this world exists, nor why it is constituted exactly as it is, nor why we are in it, nor why we were born into exactly these and no other external relations: why then should we presume to know this to be all there is, that there is no other world, that we shall never be in another world?”
Gödel believed in an afterlife for reasons independent of theology, and pointed out that even some atheists believed in an afterlife. His belief in an afterlife was related to his belief in the rationality of the universe. He continues:
“Now one may of course ask: Why didn’t God create man so that he would do everything correctly from the very start? But the only reason that this question appears justified to us could very well be the incredible state of ignorance about ourselves in which we find ourselves today. Indeed, not only do we not know where we’re from and why we’re here, we don’t even know what we are (in essence and as seen from the inside).”
An afterlife will make sense of what happens in the current life:
“Above all we must envision the greater part of “learning” as first occurring only in the next world, namely in the following way: that we shall recall our experiences in this world and only then really understand them; so that our experiences are, so to speak, only the raw material for this real learning.”
He then gives the example of a cancer patient undergoing suffering as a seemingly meaningless learning experience in this life, but one that might very well make enormous sense in the next. In this way a whole manner of seemingly meaningless experiences of suffering might very well become meaningful and fruitful.
He adds that these points are as relevant to academics as they are to laymen. In fact, academics more than anyone need to open their minds to this kind of possibility: “ninety percent of today’s philosophers see their main task as getting religion out of people’s heads.” It was one of Gödel’s great strengths to be able to look at the same mathematical ideas that everyone else looked at and think about them in a way that was very different (but with hindsight was really fairly obvious) and he seems to have the same ability with more general philosophical ideas.
He says that we are a long way from a scientific proof of the afterlife but it is instead “possible to perceive through pure reasoning” that it is “entirely consistent with all known facts.” He then raised the example of Democritus, who arrived at the theory of atoms 2500 years ago based purely on reasoning, which turned out to be essentially correct. He finished with a reminder that the idea that everything in the universe happens for a reason is analogous to the accepted scientific idea that everything in the universe has a cause.
Gödel was prone to what we might call unusual behaviour which led to speculations about his mental state – though these might have been over blown. At the age of eight he was ill with a severe bout of rheumatic fever, and since then was prone to caution about his health. Despite this he performed extraordinarily well in school, in university, and in his early career – huge powers of concentration where evident during this time and no noted mental or physical health problems. He still had these powers of concentration into his sixties. There was an acute episode of what sounds like general anxiety which developed from physical ill health issues following a jaw bone infection in 1934, and his mental health tailed off in the 1970s in the last years of his life. But generally his mental health issues may have been exaggerated, perhaps due to the appeal to journalists of the unstable genius characterisation, and often due to wrong assumptions by colleagues when he failed to attend or excused himself from planned events. He seems to have had sustained periods of years or even decades of excellent mental and physical health, and maintained a demanding schedule of teaching, lecturing internationally, and research, fairly consistently.
Gödel certainly had some unusual tendencies. At Princeton he was noted to be suspicious of everything from poisoning from carbon monoxide emitted by central heating systems to people tampering with his food. He also became increasingly suspicious of far-right elements taking hold in the United States, a subject he even raised at his immigration interview. He believed a “secret power” had been involved in the death of the Democratic president Roosevelt. In university holidays he would often take himself off to guest houses along the Coast of Maine remaining in his room to work during the day and only venture outside in the night to walk the coastal paths through the pines, hands clasped behind his back, deep in thought. With his Eastern European accent, and with the United States now at war with Germany, this led to rumours among the equally suspicious local community of retirees that he was a German spy trying to signal to submarines in the bay!
Incompleteness theorems and ontological argument
Gödel’s most famous work in mathematics was the incompleteness theorems. The two incompleteness theorems worked together, and created a similar structure to the “liar paradox”, but set the paradox within a complex mathematical framework, which allowed every possible mathematical statement to be expressed as a natural number. Gödel found a way of assigning a number to every mathematical statement (of which there are infinite) and then substituting certain mathematical statements into their own terms so that they referred to themselves, at which point some of the statements became paradoxical.
Consider the notion of unprovability. Can we prove that X is unproveable? The only way to prove it unproveable would be to prove it, but by proving it unprovable we would have proved it and so it wouldn’t be unproveable! So X must be unproveable, as the contradiction involved in proving it means that it cannot be proved. With a similar structure at the core of his argument Gödel showed that there are many mathematical statements which are true but which cannot be proved. The quest to set mathematics on a consistent footing in which all true statements could be proved from within the system, which had existed since the time of Aristotle, was over.
Gödel spent much time thinking about and trying to prove the Leibnizian view that all souls in the universe inhered within a central soul known as a monad. Part of his thinking involved him creating a proof of the existence of God, called an ontological argument. Ontological arguments had been a part of Western philosophy since the Middle Ages, but Gödel’s ontological argument improved on the logic of previous ontological arguments by introducing modern logical components, and is accepted by a greater number of philosophers.
Key to the argument is the idea that existence is a “positive property”. If we are to imagine the best possible painting, then of all the positive qualities that painting possesses, one of them must be existence – because a painting that exists is better than a painting that does not exist. The assumption that existence is a positive property tends to be a polarising one in philosophy, with philosophers either completely embracing it or rejecting it outright. Gödel then defines God as the being that has the maximum possible positive properties, and from this it follows that one of those properties will be existence. He then argues, by an extension of the same principle, that the being that exists in all possible worlds is greater than the one that only exists in some, and deducts that God must therefore exist in all possible worlds, which by definition includes our own.
In a conversation with his friend and Princeton scholar Hao Wang, Gödel said that the God his ontological argument pointed to was not necessarily the same God he believed in, and that the ontological argument was really an exercise to see what God arose from logic alone.
*These comments are taken from Dawson, J. Logical Dilemmas: The life and work of Kurt Gödel.