I have watched "Contact" several times and was watching it again the other day. Carl Sagan got a lot of things right in it, including the truth that even scientists have "faith" in matters disconnected with science. But one of the key parts of the film hasn't aged well for me.
For those who haven't seen it or read the book, Ellie Arroway, a brilliant astronomer played by Jodie Foster, is on a shortlist of people selected to be passengers on an interstellar machine constructed according to blueprints received by radio transmission from the Vega constellation. As earth's first ambassador to space, she is interviewed by a panel on her views on different topics. What would be the most important question she would ask the alien civilization?
An old flame who is on the panel - and who has a personal vested interest in not having her go since he still has romantic feelings for her - asks her squarely if she believes in God. The other members of the panel think that it would be unwise to pick as earth's first interstellar ambassador, someone who does not believe what 95% of the world's population believes. They think that one of the foremost questions Ellie should ask the aliens, should she meet them, is, "What God do you worship?". Elie being a scientist naturally says that she can't believe anything without demonstrated evidence. Candidate rejected.
It seems to me that Sagan really had an opportunity here, if not in the film then in the book, to showcase the theological and intellectual debates and problems concerning religion. The first question Ellie should have asked the panelists is: "When you ask whether I believe in God, I would ask you, *What* God? Those 95% of people you are referring to worship a zillion different Gods, from Jesus to Brahma. But there's even more, now-extinct Gods that their ancestors believed in, including Odin and Huitzilopochtli. Which God am I supposed to believe in? And do we think the aliens wouldn't ask me which one of these many Gods I believe in? What if I say the wrong name?". That would have driven home the central dilemma with believing in God right there.
But there might have been another, much more important question regarding religion that Arroway could have asked, and it would have been one that is independent of specific Gods. Religion clearly serves an important biological and evolutionary purpose, one explicated by numerous scientists. Instead of asking what God the aliens worship, the scientifically relevant question would be, "What are your deepest beliefs and how do you satisfy them?". This would have been a relevant question that is science, and yet one that would have provided an important answer about religion as, in Daniel Dennett's words, a "natural phenomenon".
As it turns out, the answer Arroway gives regarding the question is one I would have given myself: she says she would have asked the aliens how they did it; how they avoided blowing themselves up while developing such advanced technology. Especially in our present circumstances, asking a technologically advanced civilization that seems to have lasted much longer than us how they prevented self-extinction would be perhaps the most question we can ask.
But I can understand why Sagan had his character ask that question: it sets her up for the climax. After being transported to another world, Arroway sees and has a conversation with her loving father, one who had done everything he could to develop her interest and skills in science before tragically dying of a heart attack when Ellie was ten. When Ellie comes back after having that heartrending conversation, she comes to know that from the point of view of people here on earth, she was gone for only a short time, and her audiovisual equipment recorded nothing but noise. She is kept holding on to her vision of what is effectively an out-of-body experience and conversation with her father by the same slender thread which she had rejected before - faith. Sagan's point is that even scientists can have powerful experiences which they have to take on faith because there's no other way to explain them.
But upon watching that part again I still wasn't convinced of what Sagan was trying to say. If he was trying to propose reconciliation between science and religion, he was picking the wrong argument based on faith here. A scientist's "faith" that the sun will rise tomorrow is very different from faith that Jesus was born of a virgin. The former is predicated on well-understood laws of science that result in a probabilistic model which we can believe with high confidence; if the sun indeed failed to rise tomorrow, not just common sense but much of our understanding of physics, astronomy and planetary science would suddenly be called into question. That means that other phenomena that depend on this understanding would also be called into question. A scientist may take some things on "faith", but this is not really faith so much as it is informed judgement based on confidence limits and well-constructed models of reality.
Ultimately though, as much as I think Sagan could have done a much better job with these matters, I think the most important point he makes is still valid: that point simply is that, as monumentally useful and important science is, holding on to it is very hard and needs a lot of rock-solid conviction. That's a message we can all be on board with
We seem to live in a time when skepticism of science and its experts runs deep and where political mandarins of all persuasions are all too eager to make out science as a villain. It is at times like this that we must remind ourselves that science has not just been the greatest force for practical good that we have discovered but the most moral one as well.
It's easy to make the mistake of thinking of this statement as controversial, especially in a time when science is knocked for its perceived evils. But think about it in simple terms, and in fact in terms of a sphere where the practical and moral improvements are not just obvious but coincident. This sphere is the conquest of disease. I have been reading a fantastic book recently - Frank von Hippel's "The Chemical Age". The title betrays the content. The book is actually an amazing journey through various diseases that literally ended civilizations and destroyed the lives of millions, the lifesaving drugs and other public health measures science devised to end them and the heroic efforts of dogged individuals ranging from Louis Pasteur to Ronald Ross who defeated these implacable foes through blood, sweat and tears, sometimes quite literally so; Ross, during his efforts to prove that the malarial parasite was spread through the mosquito, worked so hard day and night at his microscope that the hinges rusted because of his sweat, the eyepiece cracked and he almost lost his eyesight. Another brave and almost otherworldly soul, a University of Pennsylvania doctoral student named Stubbins Ffirth, injected blood, urine and saliva from yellow fever patients into his body to rule out direct patient to patient transmission. These were the heroic deeds of heroic men.
But look at what they accomplished. Diseases like yellow fever, malaria and typhus, killers whose death toll easily exceeds the lives taken by all the wars of the world combined, which were endemic and a fact of life in ever city and village, which were literal destroyers of armies and even civilizations and scourges of families whose children they took away, were tamed, drastically reduced in intensity and fatal reach and finally contained. They haven't disappeared from our planet, but we, at least those of us who live in most developing and developed countries, hardly even think of any of these maladies any more, let alone know someone who has died of them.
This is not just a practical triumph of science but a profoundly moral one. Think of all the men, women and children numbering in the millions whose lives were saved, extended and enriched because of the innovations of chemistry and medicine, who could love and help and be there for each other and enjoy the blessings of precious life which in earlier ages was cruelly snatched away from them on a regular basis. In all these cases the "practical" and "moral" impact of science is indistinguishable.
This same overlap between the practical and the moral exists in other spheres. The discovery of cheap distillation methods for hydrocarbons not only enabled electricity and transportation but kept people warm in cold climates and cool in hot ones. Better chemical treatment of textiles led to similar, insulating material that protected the vulnerable and the young. And of course, far and away, the methods of artificial selection and genetic engineering have literally led to the feeding and saving of millions in parts of the world like India and China. If this existential improvement to humanity's basic predicaments by science isn't moral, I don't know what is.
The same book, von Hippel's, raises a counterargument when it talks about chemical weapons which disfigured and maimed millions. And yet the numbers don't compare. As hideous as thalidomide, sarin, phosgene and DDT are, the lives they claimed pale in significance and numbers compared to the lives saved by antibiotics, pesticides, disinfectants and the Haber-Bosch process; antibiotics for instance brought down the death toll due to infection on the battlefield from 200% (Civil War) to less than 10% in World War 2. Simpler measures like hand-washing and better sanitation were also the fruits of scientific discovery, and heretics like Ignaz Semmelweis who contributed to these measures were often hounded and ostracized; Semmelweis met a terribly tragic end when he died from beatings and possibly a self-inflicted wound in a mental asylum.
For me the conclusion is obvious. Science can indeed be used for good and evil, but the good outweighs the evil by an infinite amount. This is a timely reminder that the greatest force for practical improvement discovered by humanity is also the most moral one.
Werner Heisenberg was on a boat with Niels Bohr and a few friends, shortly after he discovered his famous uncertainty principle in 1927. A bedrock of quantum theory, the principle states that one cannot determine both the velocity and the position of particles like electrons with arbitrary accuracy. Heisenberg’s discovery foretold of an intrinsic opposition between these quantities; better knowledge of one necessarily meant worse knowledge of the other. Talk turned to physics, and after Bohr had described Heisenberg’s seminal insight, one of his friends quipped, “But Niels, this is not really new, you said exactly the same thing ten years ago.”
In fact, Bohr had already convinced Heisenberg that his uncertainty principle was a special case of a more general idea that Bohr had been expounding for some time – a thread of Ariadne that would guide travelers lost through the quantum world; a principle of great and general import named the principle of complementarity.
Complementarity arose naturally for Bohr after the strange discoveries of subatomic particles revealed a world that was fundamentally probabilistic. The positions of subatomic particles could not be assigned with definite certainty but only with statistical odds. This was a complete break with Newtonian classical physics where particles had a definite trajectory, a place in the world order that could be predicted with complete certainty if one had the right measurements and mathematics at hand. In 1925, working at Bohr’s theoretical physics institute in Copenhagen, Heisenberg was Bohr’s most important protégé had invented quantum theory when he was only twenty-four. Two years later came uncertainty; Heisenberg grasped that foundational truth about the physical world when Bohr was away on a skiing trip in Norway and Heisenberg was taking a walk at night in the park behind the institute.
When Bohr came back he was unhappy with the paper Heisenberg had written, partly because he thought the younger man seemed to echo his own ideas, but more understandably because Bohr – a man who was exasperatingly famous for going through a dozen drafts of a scientific paper and several drafts of even private letters – thought Heisenberg had not expressed himself clearly enough. The 42-year-old kept working on the 26-year-old until the latter admitted that “the uncertainty relations were just a special case of the more general complementarity principle.”
So what was this complementarity principle? Simply put, it was the observation that there are many truths about the world and many ways of seeing it. These truths might appear divergent or contradictory, but they are all equally essential in representing the true nature of reality; they are complementary. As Bohr famously put it, “The opposite of a big truth is also a big truth”. Complementarity provided a way to reconcile the paradoxes that seemed to bedevil quantum theory’s interpretation of reality.
The central scientific paradox was what is called wave-particle duality. In 1803, the British polymath Thomas Young had proposed that light, contrary to Isaac Newton’s view of it, consists of waves; an experiment like diffraction makes this wave nature clear. A hundred years later, in 1905, Einstein proposed that light in fact consists of particles, an idea he invoked in order to explain the photoelectric effect and which won him a Nobel Prize; these particles were later called photons. Soon it was found through other experiments that all subatomic particles and not just photons could display wave and particle behavior. In 1924, the French physicist and aristocrat Louis de Broglie saw a way through the impasse when he came up with a simple equation that related the momentum of a particle – a particle property – inversely to its wavelength – a wave property.
In spite of de Broglie’s insight, particles clearly don’t look like waves and waves don’t look like particles in real life. In fact the very names seem to put them at odds with one another. It was Bohr who saw both the problem and the solution. Particles and waves both exist and are equally valid and essential ways of interpreting the quantum world. Depending on what experiment you do you might see one or the other and never both, but they are not contradictory, they are complementary. Most crucially, you simply cannot make sense of reality without having both in hand. It was a powerful insight that cut through the complexities of intuition and language; it was not too different in principle from other counterintuitive truths that science has uncovered, for instance the truth that both lighter and heavier bodies fall at the same rate. Complementarity rationalized opposing tendencies of the physical world and indicated that they were one. It was what had made Bohr subsume the opposing quantities in Heisenberg’s uncertainty principle under the same rubric.
Complementarity was also pregnant with far more general interpretation. The most effective application of it to human affairs in Bohr’s hands was the problem posed by nuclear weapons. Even before the bomb had been used on Hiroshima, Bohr saw deeper and further than anyone else that the very fact that nuclear weapons are so enormously destructive might make them the most potent force for peace that the world has ever seen, simply because statesmen will realize that nobody can truly “win” a nuclear war if everyone has them. “We are in a completely new situation that cannot be resolved by war”, Bohr said. The complementarity of the bomb continues to keep the peace through deterrence.
Another noteworthy example was a speech delivered by Bohr in 1938 to the International Congress of Anthropological and Ethnological Sciences at Kronberg Castle in Denmark. Apologizing at the outset for presuming to speak about a topic on which he was not an expert, Bohr proceeded to provide a succinct summary of complementarity in the context of atomic physics. Turning to biology, he then made the perspicacious observation – still the subject of considerable debate – that reason and instinct which might appear to be opposed to each other are complementary, both providing a complete picture of a sentient being. Bohr then came to the crux of the matter, pointing out that complementarity is of the essence when people are judging other cultures which might seem divergent from their own but which turn out to be different and equally productive ways of looking at the world. As Bohr eloquently put it: “Each culture represents a harmonious balance of traditional conventions by means of which latent potentialities of human life can unfold themselves in a way which reveals to us new aspects of its unlimited richness and variety.” Bohr believed that contact between cultures can go a long way in not just dispelling biases but in mutually enriching both parties: “A more or less intimate contact between different human societies can lead to a gradual fusion of traditions, giving birth to a quite new culture”. This is as clear an appeal for internationalism and mutual understanding that one can think of; if everyone had understood complementarity, maybe we might have had less fascism, imperialism and genocide. The final goal of complementary views of societies, as Bohr pointed out powerfully in the same lecture, isn’t different from the goal of science as a whole – it is “the gradual removal of prejudices”.
As we approach what seem to be novel problems in the 21st century, Bohr’s complementarity is a message in a bottle from one fraught world to another, telling us that seeing these new problems through the lens of an old principle can be most rewarding. We seem to live in a time when many see social and political problems through a binary, black-or-white, zero sum lens. Either my viewpoint is right or yours, but not both. Complementarity bridges that division. For instance consider the problem of individualism vs communalism, a divide that also hints at the cultural divide Bohr spoke about, in this case largely an Eastern vs Western divide. Western society is fiercely individualistic; self-interests guide people’s lives and most people don’t want others to tell them that they should live their lives for others. Meanwhile, Eastern and some European societies are much more communal; community interests often override self-interest and individuals are told that their self-development should take a backseat to the development of their community and society. Bohr’s complementarity tells us that this divergent view should not exist. Communal and individualistic views are both essential for looking at the world and building a more productive society; in fact one can gain self-knowledge and wisdom by working for a community, and likewise a community can be improved when people engage in individualistic self-improvement that helps everyone.
There are other problems for which complementarity provides a potential solution. I will speak mostly for the United States since that’s where I live, but these problems are in fact global. Consider the problem of immigration, one to which Bohr’s 1938 address is directly applicable. People criticized as “globalists” think that unfettered immigration is a net good. The opposing camp thinks that preserving a nation’s culture is important, and too much or too rapid immigration will weaken this culture. But complementarity tells us that nationalism is in fact strengthened when immigrants work together for the common good of the country. At the same time, immigrants should put their country first and prioritize work that will strengthen their nation’s economy, military and social institutions. We are global citizens, but we are also shaped by evolution and culture to take care of our immediate own. The opposite of a big truth is also a big truth.
Even scientific debates like the nature (genes) vs nurture (environment) conundrum can benefit from complementary views. People criticized as biological “essentialists” believe that genes dictate a lot of an individual’s physical and psychological makeup while the opposing “nurture” camp believes that much of the effect of genes can be changed by the environment. But complementarity says that just like the joint wave-particle view of reality, individuals are whole and complete, and this wholeness arises from a combination of genes and environment. In that sense, how much of a person’s mental and physical constitution we can control by either manipulating their genes or their environment is almost irrelevant. What’s relevant is the basic understanding in the first place that both matter; even if both camps agree with this baseline, they would already be talking a lot more with each other.
A third application of complementarity to international affairs, one which stems directly from Bohr’s view of the complementarity of the bomb, is to the relationship between United States and China, a relationship which will likely be the single-most important geopolitical determinant of the 21st century. China clearly has an autocratic regime that is not likely to yield to demands for more democratic behavior, both internally and externally, anytime soon. This has led to many in the United States to regard China as an implacable foe, almost a second Soviet Union. An important consequence of this view has been to see almost every technological development in the two countries, from gene editing to artificial intelligence to new weapons, as a contest.
But irrespective of the moral wisdom of engaging in this contest, complementarity tells us that such contests are likely to lead to the mutual ruin of both China and the United States, and by extension the rest of the world; for the same reason that any arms race would hollow out countries’ coffers and ramp up the specter of mutual annihilation. The reason is simple: both computer code and the physics of nuclear weapons are products of the fundamental laws of science and technology discovered or invented by human minds. Both can be divined and implemented by any country with smart scientists and engineers, which basically means any developing or developed country. An arms race in AI between China and the United States, for instance, would be as futile and dangerous as was the nuclear arms race between the United States and the former Soviet Union. Both countries would be fooling themselves if they think they can write better computer code and keep it secret for a long time. In that sense the fact that computer code, just like nuclear fission, is essentially a discovery of the human mind poses inconvenient truths for both countries. Whether we like it or not, we need to realize the complementarity of artificial intelligence akin to how we grudgingly realized the complementarity of the bomb: we need to realize that AI is powerful, that it is dangerous, that its secrets cannot stay secret for very long, that there are no real defenses against it, that the very dangers of AI cry for peaceful solutions to AI, and therefore that mutual cooperation between China and the United States under an umbrella of an international organization like the United Nations would be the only solution to avoid mutual cyber-destruction. China might be autocratic, but it has self-interests and wouldn’t want to see its own ruin. In the end, harmony between the United States and China might not be forced by bridging the moral divide between the two countries’ social and political systems: it would be forced by the very laws of science and technology.
Without oversimplifying the issue, it’s clear to me that Bohr’s complementarity provides a mediating middle ground for almost any other social or political issue I can think of; it’s not so much that it offers a solution but that it will compel each side to see the importance of the other side’s argument in providing a complete view of reality that cannot be provided by either viewpoint by itself. Pro-life or pro-choice? One can respect both the life of an unborn child and the life of the mother; the two are complementary. Socialism or capitalism? One can certainly have a mixed market economy – of the kind found in Niels Bohr’s home country for instance – that would give us the benefits of both. Climate legislation or rapid economic growth? One can create jobs related to new climate technologies that will result in economic growth. Science or religion? They address complementary aspects of the world, Stephen Jay Gould’s “non-overlapping magisteria”.
If we accept the idea of complementarity, we are in essence accepting the validity of all ways of looking at the world, and not just one. This does not mean that all ways are equally right – we can’t accept the germ theory of disease and the “theory” of diseases as a punishment from God on equal terms – but it is precisely through placing them on a level playing field and letting them play out their logical flow that we can even know how much of which view is right. In addition, Bohr realized that the world is indeed gray, that even flawed visions may contain snatches of truth that should be acknowledged as potential building blocks in our view of reality. But ultimately, Bohr’s plea for complementarity was a plea for what he called an “open world”, an ideal that for him was the highest that the peoples of the world could aspire to, an ideal that arose naturally from the democratic republic of science. If we accept complementarity, we automatically become open to examining every single approach to a problem, every way of parsing reality. Most importantly, we become open to true, unfettered communication with our fellow human beings, a tentative but lasting step toward Bohr’s – and science’s – “gradual removal of prejudices”. That seems like an important message for today.
I was quite saddened to hear about the passing of Steven Weinberg, perhaps the dominant living figure from the golden age of particle physics. I was especially saddened since he seemed to be doing fine, as indicated by a lecture he gave at the Texas Science Festival this March. I think many of us thought that he was going to be around for at least a few more years.
Weinberg was one of a select few individuals who transformed our understanding of elementary particles and cemented the creation of the Standard Model (he coined the name), in his case by unifying the electromagnetic and weak forces; for this he shared the 1979 Nobel Prize in physics with Abdus Salam and Sheldon Lee Glashow. His 1967 paper which heralded the unification, "A Model of Leptons", was only 3 pages long and remains one of the most highly cited articles in physics history.
But what made Weinberg special was that he was not only one of the most brilliant theoretical physicists of the 20th century but also a pedagogical master with few peers. His many technical textbooks, especially his 3-volume "Quantum Theory of Fields", have educated a generation of physicists; meanwhile, his essays in the New York Times Book Review and other avenues and collections of articles published as popular books have educated the lay public about the mysteries of physics. But in his popular books Weinberg also revealed himself to be a real Renaissance Man, writing not just about physics but about religion, politics, philosophy, history including the history of science, opera and literature. He was also known for his political advocacy of science. Among scientists of his generation, only Freeman Dyson had that kind of range.
There have been some great tributes to him, and I would point especially to the ones by Scott Aaronson and Robert McNees, both of whom interacted with Weinberg as colleagues. The tribute by Scott especially shows the kind of independent streak that Weinberg had, never content to go with the mainstream and always seeking orthogonal viewpoints and original thoughts. In that he very much reminded me of Dyson; the two were in fact friends and served together on the government advisory group JASON, and my conversation with Weinberg which I describe below ended with him asking me to give my regards to Freeman, who I was meeting in a few weeks.
I had the good fortune of interacting with Steve on two occasions, both rewarding. The first time I had the opportunity to be with him on a Canadian television panel on the challenges of Big Science. You can see the discussion here:
The next time was a few years later when I contacted him about a project and asked whether he had some thoughts to share about it. Steve didn't know me personally (although he did remember the Big Science panel) and was even then very busy with writing and other projects. In addition, the project wasn't something close to his immediate interests, so I was surprised when not only did he respond right away but asked me to call him at 10 AM on a Sunday and spoke generously for more than an hour. I still have the recording.
Steve was a great physicist, a gentleman and a Renaissance Man, a true original. We are unlikely to see the likes of him for a long time.
One of the reasons I feel particularly wistful with his passing is because he was among the last of the creators of modern particle physics. He worked in an enormously fruitful time in which theory went hand in hand with experiment. This is different from the last twenty years in which fundamental physics and especially string theory have been struggling to make experimental connections. In cosmology however, there have been very exciting developments, and Weinberg who devoted his last few decades to the topic was certainly very interested in these. Hopefully fundamental physics can become as involved with the productive interplay of theory and experiment as cosmology and condensed matter physics are, and hopefully we can again resurrect the golden era of science in which Steven Weinberg played such a commanding role.
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.
The overarching goal of mathematics was to prove completeness and consistency starting from a fundamental, minimal set of axioms, much like Euclid had built up the grand structure of plane geometry starting with a handful of axioms in his marvelous ‘Elements’.
Mathematicians had good reasons to be optimistic. The 19th century had perhaps been the most important for the development of the discipline, solidifying results in analysis, geometry and other key mathematical domains. The mathematical giants of that time, textbooks names like Gauss, Dedekind, Cantor and Riemann, had put mathematics on a solid foundation. It was against this background that Bertrand Russell and Alfred North Whitehead wrote their magnum opus, the dense ‘Principia Mathematica’ that sought to put mathematics on a solid foundation of logic. Unnecessary axioms of mathematics would be discarded, the superstructure trimmed, and mathematics would be put on a sound basis of symbolic logic. One of the major goals of their work was to resolve any paradoxes in mathematics that would lead to statements akin to the famous Liar’s Paradox – “I am lying” that are false when they are true and true when they are false. Russell and Whitehead thought that paradoxes were merely a consequence of not clarifying the axioms and the deductions from them well enough.
David Hilbert, perhaps the leading mathematician of the early 20th century
The intellectual godfather of the mathematicians was David Hilbert, perhaps the leading mathematician of the first few decades of the twentieth century. In a famous 1900 address at the International Congress of Mathematics in Paris, Hilbert set out 23 open problems in mathematics that he hoped would engage the brightest minds of the next few decades; it is a measure of Hilbert’s perspicacity in picking these problems that some of them are still unsolved and pursued. The second among these problems was to prove the consistency of arithmetic using the kind of axiomatic approach developed by Russell and Whitehead. Hilbert was confident that within a few decades at best, every question in mathematics would have a definite answer that could be built up from the axioms. He famously proclaimed that there would be no ‘ignorabimus’ (a statement whose truth or falsity could never be known) in mathematics. Mathematicians soon began to make themselves busy in carrying out Hilbert’s program.
When Hilbert gave his talk, Kurt Gödel was still six years away from being born. Thirty years later he would drive a wrecking ball into Hilbert’s dream, showing that even this most exact, pristine of all human intellectual endeavors contained truths that are fundamentally undecidable. And he did it in such a final manner that there could be no debate about it. That is what left brilliant men like Einstein and von Neumann with their mouths agape.
Now we have a biography of Gödel and his times written by veteran science and history writer Stephen Budiansky that is the most evocative and comprehensive biography of the logician written so far for a general audience. The book is really about Gödel and his times rather than his work. There have been some fine books on Gödel until now, including the detailed “Incompleteness” by Rebecca Goldstein, the impressionistic “Gödel: A Life in Logic” by John Casti and most notably, John Dawson’s “Logical Dilemmas” which is perhaps the most complete exploration of the man. But Budiansky’s book is the best one so far that situates Gödel in the magical time that was turn of the century Austria-Hungary, a time that was tragically shattered with a totality approaching anything in mathematics by the onslaught of totalitarianism. Budiansky also sensitively investigates Gödel’s dark side; the same mind that could not tolerate anything that was not precisely defined fell prey to its own exacting standards and unleashed demons that would lead to a life punctuated by paranoid delusions and extreme starvation. When Gödel died in 1978, he weighed 65 pounds.
Gödel’s end was a far cry from his beginning in the glorious years of the Austro-Hungarian empire. The emperor Franz-Joseph, an arch Habsburg, prized order above everything else. The town of Brünn that Gödel was born in was one of the most industrialized towns in the empire, and his father Rudolf was a well-to-do managing director of a textile firm. But it was from his mother Marianne who he was closer to and who was well-versed in music and the arts that Kurt got much of his intellect; throughout his life, Marianne would be a crucial link and lifeline through letters. A brother who was a doctor, Rudi, extended the family. When Gödel was born, the empire was perhaps the foremost fountain of intellect in Europe and possibly the world. In art and philosophy, music and architecture, science and mathematics, Vienna and Budapest led the way. Names like Freud, Wittgenstein, Klimt and Zweig trickled out of the fin de siècle city in a steady stream. They exemplified Vienna and Eastern Europe’s cafe culture, with places like the Café Reichsrat and Café Josephinum becoming battlegrounds of fervent intellectual debate on the deepest questions of epistemology, fueled with marathon shouting matches lasting into the night, strong black coffee topped with whipped cream and scribbles on the marble tabletops.
It was in this heady intellectually milieu that Gödel grew up. He was an outstanding student at the Realgymnasium and showed a meticulous attention to detail that was to be both his biggest strength and his ruin. He was also often the poster child for the head-in-the-clouds intellectual, and throughout his life, as brilliant as his mathematical acumen was, he often remained oblivious to the state of politics around him. A fondness with what many would consider childish preoccupations like children’s toys and kitschy household objects would punctuate his otherwise fanatical commitment to the most abstract reaches of human thought.
Under the facade of Vienna’s intellectual beehive lay a rotten foundation of class and religious inequality, bitterly growing nationalism and, most fatally, anti-Semitism. Viennese Jews had been liberated by Franz Joseph in 1867, and centuries of bottled up ambition and talent in the face of their persecution found a release that led to unprecedented success not just in the practical arts like medicine and law but also in the most abstract realms of mathematics and philosophy. This success bred resentment among Vienna’s growing middle class gentiles. The collective philosophical talent of both Jews and non-Jews culminated in the creation of the famous Vienna Circle and their philosophy of logical positivism. Logical positivism asked to reject anything that could not be rigorously scientifically verified and the philosophers sought to outdo their fellow scientists and place metaphysics on a solid scientific foundation. The philosophers Hans Hahn who was Gödel’s PhD advisor at the University of Vienna and Moritz Schlick were the leaders of the movement; their patron saints were the mysterious, penetrating Ludwig Wittgenstein and Bertrand Russell. Wittgenstein deigned to speak to the circle only once and remained a distant figure, while the philosopher of science Karl Popper tried to become an official member but was spurned.
Into this milieu entered Kurt Gödel, only 24 years old. He became a regular member of the Vienna Circle but spoke up rarely, preferring to instead listen and occasionally interject with a penetrating comment. But even then Gödel’s predilections ran counter to the circle’s. While the circle emphasized the existence only of propositions that could be verified by grounding in the real world, Kurt became a staunch Platonist whose belief that mathematical objects existed in a world of their own without any human intervention only became deeper during his life. For Gödel, numbers, sets and mathematical axioms were as real as planets, bacteria and rocks, simply waiting to be discovered and existing independent of human effort. A large part of this existence stemmed from the sheer beauty of mathematical structures that Gödel and his colleagues were uncovering: how could such beautiful objects exist only under the pre-condition of discovery by ordinary human minds?
By 1930 the Platonist Gödel was ready to drop his bombshell in the world of mathematics and logic. In September 1930, a big conference was going to be organized in Königsberg. German mathematics had been harmed because of Germany’s instigation of the Great War, and Hilbert’s decency and reputation played a big role in resurrecting it. Just before the conference Gödel met with his friend Rudolf Carnap, a founding member of the Vienna Circle in the Cafe Reichsrat. There, perhaps scribbling a bit on the marble table, he told Carnap that he had just showed that Hilbert and Russell’s program to prove the completeness and consistency of mathematics was was fatally flawed. A few days later Gödel delivered his talk at the conference. As often happens with great scientific discoveries, few people understood the significance of what had just happened. The one exception was John von Neumann, a child prodigy and polymath who was known for jumping ten steps ahead of people’s arguments and extending them in ways that their creators could not imagine. Von Neumann buttonholed Gödel, fully understood his result, and then a week later extended it to a startling new domain, only to find through a polite note from Gödel that the former had already done it.
So what had Gödel done? Budiansky’s treatment of Gödel’s proof is light, and I would recommend the 1950s classic “Gödel’s Proof” by Ernest Nagel and James Newman for a semi-popular treatment. Even today Gödel’s seminal paper is comprehensible in its entirety only to specialists in the field. But in a nutshell, what Gödel had found using an ingenious bit of self-referential mapping between numbers and mathematical statements was that any consistent mathematical system that could support the basic axioms of arithmetic as described in Russell and Whitehead’s work would always contain statements that were unprovable. This ingenious scheme included a way of encoding mathematical statements as numbers, allowing numbers to “talk about themselves”. What was worse and even more fascinating was that the axiomatic system of arithmetic would contain statements that were true, but whose truth could not be proven using the axioms of the system – Gödel thus showed that there would always be a statement G in this system which would, like the old Liar’s Paradox, say, “G is unprovable”. If G is true it then becomes unprovable by definition, but if G is false, then it would be provable, thus contradicting itself. Thus, the system would always contain ‘truths’ that are undecidable within the framework of the system. And lest one thought that you could then just expand the system and prove those truths within that new system, Gödel infuriatingly showed that the new system would contain its own unprovable truths, and ad infinitum. This is called the First Incompleteness Theorem.
An example of Gödel’s ingenious technique to transform mathematical symbols – and therefore statements – into numbers. (Source: Math stack exchange)
The Second Incompleteness showed that such a system cannot prove its own consistency, leading to another paradox and in effect saying that any formal system that is interesting enough to prove its own consistency can do so only if it’s inconsistent. This was an even more damning conclusion. Far from getting rid of the paradoxes that Russell and Whitehead believed would be clarified if only one understood the axioms and the deductions from them well enough, Gödel showed that such paradoxes are as foundational a feature of mathematical systems as anything else. As far as Hilbert was concerned, he had uncovered a rotten foundation underlying mathematics that doomed Hilbert’s program forever.
Ironically, just a day after Gödel’s talk, Hilbert gave a speech reinforcing his belief that there would be no ‘ignorabimus’ in mathematics and ending with a famous refrain: “Wir müssen wissen – wir werden wissen.” (“We must know – we will know.”). As sometimes happens when a great mind declares a truth in a scientific discipline with such finality, reactions can range from disbelief and denial to acceptance. Hilbert himself recognized the significance of Gödel’s results but held out hope that they wouldn’t be as far-reaching as they were thought to be. Von Neumann on the other hand is on record saying that after he heard of the incompleteness theorems, he decided to abandon his own productive work in set theory and the foundations of logic and move on to other topics. Gödel’s work had a seismic impact on that of many other thinkers. His proof that a system made up of purely mechanical, axiomatic procedures would contain undecidable propositions inspired Alan Turing’s own answer in the negative to the question of whether a mechanical computer could decide the truth value of an arbitrary proposition in a finite number of steps. Most notably, Gödel’s ingenious scheme of having numbers represent both themselves as well as instructions to specify operations on themselves is, without him ever knowing it, the basis of digital computing.
Thus by the time he was 24 years old, Gödel had established himself as a logician of the first rank and immortalized his name in history. In the next few years his friends and colleagues spread his gospel around the world, most notably in the United States. The noted mathematician Karl Menger was a close friend and was spending a semester in Iowa, sending Gödel periodic letters describing life in America (“Americans as a rule do not go for walks, they think that dashing around in their cars on Sundays is sufficient recreation.”). Not only did Menger give talks about Gödel’s results in the United States, but he performed a crucial service. Helped by a largesse from the brother-sister pair of Louis and Caroline Bamberger who sold their clothing business to Macy’s, the educator Abraham Flexner had established an institute in Princeton dedicated to pure thought, with no administrative and teaching duties. To populate this heavenly tank Flexner had bagged the biggest fish of them all – Albert Einstein. Along with Oswald Veblen, von Neumann and a few others, Einstein became one of the first faculty members at what came to be called the “Institute for Advanced Study”, although given the exorbitant money that Flexner dangled in front of his faculty in addition to the unique work environment, it quickly came to be christened the “Institute for Advanced Salaries”. Menger recommended that Flexner hire Gödel on a temporary basis. He would visit a few more times before permanently relocating in 1940.
Kurt and Adele at their wedding (Source: Institute for Advanced Study)
By this time, both the personal currents of life and the larger currents of history would steer Gödel’s destiny. In 1927 he had married Adele Porkert, an older woman who lived across from his street. Adele had worked as a nightclub dancer and was a masseuse, qualifications which neither Gödel’s colleagues nor his family considered worthy of his stature. But Adele was to be a true mother to Kurt until her own death. Her role became clear when Gödel started suffering from a kind of psychotic paranoia that would mark him as indelibly as his genius. Starting in the early 1930s, he spent time in sanatoriums, convinced that an apparently weak heart from a bout of rheumatic fever which afflicted him as a child would kill him. More ominously, he started suspecting the sanatorium staff of conspiring to poison him or inject him with lethal substances. He drastically lost weight, and Adele had to feed him food that she had prepared herself to convince him to eat it. In retrospect it is clear that the ultra-logical Gödel also suffered from what we now call obsessive compulsive disorder. He obsessed over his bodily functions, interpreting ordinary signs as signs of trouble – his letters to his mother from America are generously interspersed with accounts of the health of his bowels. Unsurprisingly, this obsession led to a detailed keeping of diaries recording his thoughts and real and perceived symptoms, along with miscellaneous hospital, travel and grocery receipts. It is to Budiansky’s credit that he has combed through these sources to reveal to us the vivid portrait of a methodical, detail-oriented stickler whose very commitment to logic and details would prove to be his undoing.
Political events were also clearly not evolving favorably by the time Gödel first made his way to America. Austrian anti-Semitism had already had a long history, and German-speaking Austrians were fanatically enthusiastic about embracing their former compatriot and army corporal Adolf Hitler. Hitler triumphantly marched into Austria to ecstatic, waving crowds in March 1938 during the Anschluss. But even while Gödel had been proving his famous theorems, the writing had been on the wall. The University of Vienna had been a venue for anti-Semitic demonstrations for a long time, and the Vienna Circle with its Jewish members and commitment to abstract thought and “Jewish science” like relativity was a brightly painted target. In 1936, Johann Nelböck, a mentally troubled former student of Moritz Schlick shot and killed Schlick on the steps of the university, seething under the illusion that Schlick was having an affair with a female student he was obsessed with. Supported by the Nazis and seen as a martyr to the cause of eradicating the foreign element from the body of the Teutonic intellect, Nelböck was sentenced to ten years in prison, only to be promptly released by right-wing authorities in 1938 after the Anschluss. After Schlick’s murder the Vienna Circle effectively dissolved, and with it a glorious intellectual age whose quick demise remains a reminder of how quickly totalitarianism can destroy what takes decades or even centuries to build. After Jewish professors were all dismissed throughout Germany and Austria, Hilbert was asked by the new Nazi minister of education what mathematics was like at the University of Göttingen where he taught. “There is no mathematics anymore at Göttingen”, Hilbert retorted.
Gödel, as involved as he was with the search for mathematical truth, was not finely attuned to what was happening to politics in the country. Two days before Hitler’s takeover of Austria, Menger received a letter about mundane matters of conferences and mathematics from his friend which, as he put it, “may well represent a record for unconcern on the threshold of world-shaking events.” But even Gödel could not ignore what was happening to his colleagues at the university, and after some unpleasant episodes including one in which he was bullied on the streets by Nazi thugs and Adele fended off their taunts with her umbrella, the couple decided to emigrate to America for good. Bureaucratic snafus regarding Gödel’s visa and his new status as a German citizen led to intervention from the director of the Institute for Advanced Study at von Neumann’s goading: that is when von Neumann wrote the remarkable letter urging him to do everything he could to enable Gödel’s emigration, saying that Gödel was absolutely irreplaceable. Because German passengers crossing the Atlantic had to face the dual hazards of Nazi U-boats and potential arrest as enemy aliens by British authorities, Kurt and Adele took the long, scenic route, going through Eastern Europe through Moscow and then taking the Trans-Siberian railroad to Vladivostok, before finally boarding a steamer for San Francisco. Gödel would never leave the Eastern Seaboard of the United States again during his lifetime.
Kurt and Adele arrived in Princeton, a place puckishly described by recent resident Albert Einstein as “a quaint, ceremonial village, full of demigods on stilts”. Gödel had never known Einstein before coming to America, and yet it was Einstein who, along with the Austrian economist Oskar Morgenstern, provided him with the friendship of his Viennese colleagues which he so missed. Einstein and Gödel made for an unlikely pair: the former gregarious, generous, earthy and shabbily dressed, always eyeing the world through a sense of humor; the latter often withdrawn, hyper-logical, critical and unable to lighten up. And yet these exterior differences hid a deep and genuine friendship that went beyond their common background in German culture. Their families often visited each other, and Adele once knitted a woolen vest for Einstein. Animatedly conversing in German during their walks home together, Einstein communed with few others at the institute. It was Einstein who accompanied Gödel and Morgenstern to Gödel’s citizenship ceremony. At the ceremony the overtly pedantic and meticulous Gödel who had studied exhaustively for the citizenship test above and beyond the standard requirements, told the judge that he had found a flaw in the Constitution that would allow the United States to turn into a dictatorship. Einstein and Morgenstern hastily shut him up from saying anything further and the ceremony progressed smoothly.
But the real reason Einstein so admired Gödel was likely because he shared Gödel’s unshakeable belief in the purity of the mathematical constructs governing the universe. Einstein who was not formally religious nevertheless always harbored a deep belief that the laws of physics exist independently of human beings’ abilities to identify and tamper with them – that was one reason he was so uneasy with the then standard interpretation of quantum mechanics which seemed to say that there was no reality independent of observers. Gödel outdid him and went one step further, believing that even numbers and mathematical theorems exist independently of the human mind. It was this almost spiritual and religious belief in the objective nature of mathematical reality that perhaps formed the most intimate bond between the era’s greatest theoretical physicist and its greatest logician. It also helped that Gödel got interested in Einstein’s general theory of relativity, once playing with the equations and startling Einstein by concluding that the theory allowed for the existence of closed timelike curves – in other words, a universe without past and future, without time. For Gödel’s Platonic mind, this kind of result based purely on mathematics and without any physical basis was exactly the kind of absolute mathematical truth he believed in.
Oskar Morgenstern’s friendship with Gödel was even deeper, in part because he outlasted Einstein until Gödel’s own death. Morgenstern who combined worldly wisdom with brilliance in economics had made a name for himself by writing “Theory of Games and Economic Behavior” with von Neumann which established the field of game theory. Morgenstern worried about Gödel’s work, about Gödel’s health and Gödel’s marriage. One of the main sources of Gödel’s life is Morgenstern’s copious, often heartbreaking notes on Gödel’s worries and mental deterioration in his last years. He saw that Adele, while devoted to Kurt, was not a good fit in snobbish Princeton. A young Freeman Dyson vividly described an uncomfortable scene at a party where a very drunk Adele grabbed him and forced him to dance for twenty minutes while Kurt miserably stood by; Dyson could only imagine the horror of their lives. But Adele stayed utterly loyal to Kurt, feeding him, entertaining his paranoid health issues and generally taking good care of him.
After coming to the institute Gödel contributed one significant piece of work that added to the already hallowed place in mathematical history he enjoyed. In his famous 1900 address, the problem Hilbert had put at the top of his list was the so-called Continuum Hypothesis. The hypothesis deals with one of the most startling and deepest aspects of mathematics – a comparison of different kinds of infinity. The fact that there are in fact different kinds of infinity was discovered by Georg Cantor and came as a bombshell. Cantor showed that the “first” kind of infinity, called a countable infinity, was represented by the set of natural numbers. But there was another kind of much larger infinity, an uncountable infinity, represented by the real numbers. It may seem absurd to say that one infinity is larger or smaller than another, but using ingenious arguments Cantor showed that the real numbers cannot have a one-to-one mapping with the natural numbers and are much bigger. The Continuum Hypothesis asked if there is a third kind of infinity between that of the natural numbers and the real numbers.
The problem is still unsolved, but Gödel made a significant dent by showing that the contradiction of the hypothesis could not be proved by standard set theory. This is not the same as showing that the hypothesis is true, but it does result in one strike in favor of it. A bigger advance came in 1963 when mathematician Paul Cohen showed that the hypothesis is independent of standard set theory; that is, either the hypothesis or its negation can be added to standard set theory without destroying its consistency and axioms. For all of Gödel’s scathing remarks and frequent silence about other mathematicians’ work, he was profusely generous toward Cohen when Cohen sent him his proof of the independence of the Continuum Hypothesis, a problem that Gödel himself had tried and failed to solve for more than twenty years.
Mathematician John von Neumann was one of Gödel’s biggest supporters (Source: Totally History)
Gödel’s peculiar obsessions and pedantry made him a difficult colleague, and his promotion of to full professor was held up until 1953 because the faculty feared he would be challenging to deal with when it came to the obligatory administrative matters that full professors had to busy themselves with. Once again von Neumann came to his friend’s rescue, asking, “If Gödel cannot call himself Professor, how can the rest of us?” But even after Gödel got promoted his insecurities did not leave him, and he kept on feeling a mixture of self-pity and suspicions of conspiracy on the part of the institute to demote or fire him. He could nonetheless be a very loyal friend and colleague, testifying against having Oppenheimer removed as director for instance after Oppenheimer’s enemy Lewis Strauss tried to oust him after his infamous security hearing. Especially in his later years, young mathematicians like Martin Davis and Hao Wang observed a Gödel who was friendly, curious and funny.
After Einstein’s death in 1955 and von Neumann’s excruciatingly painful death in 1957, Gödel began to increasingly rely on Adele and Morgenstern (as a measure of how startlingly original his mind remained, in March, 1956, as von Neumann was dying, Gödel sent him a letter that is supposed to contain the first statement of a famous problem in computer science, the P=NP hypothesis). His exalted mind often delighted in the simplest of objects, including trinkets and cheap children’s toys bought from convenience stores. The fear that his colleagues had about his obsession evolved, if anything, in the opposite direction: he would meticulously labor over member applications, exhaustively analyzing them and offering suggestions on points others had missed.
But the spark of genius that had lit the mathematical world on fire seemed to have gone missing. In his last few years, Gödel became obsessed with not just believing that there was a conspiracy against him but also one against a hero of his, the 18th century mathematician and polymath Gottfried Wilhelm Leibniz. He became convinced that there was a plot to keep Leibniz’s work hidden from the world. Beginning in the 1970s, he began to see a psychiatrist whose detailed notes Budiansky opens the book with: “Believes he has been declared incompetent and that one day they will realize he is free and take him away…fear of destitution, loss of position at institute because he hasn’t done anything for past year…brought out delusional ideas, including that brother is the evil person behind plot to destroy him…believes he wants to take his wife, house and position at the institute.” Clearly, having his mother and brother Rudi visit him in America, while welcomed initially, had also turned into a plot to take over his world. It didn’t help that by this time Gödel’s work had been popularized enough that he received the Einstein Prize from Einstein himself , the National Medal of Science from President Gerald Ford and that crowning sign of fame – letters from all over the world from fans and crackpots.
There was little that anyone could do to help. In 1977 Morgenstern himself received a diagnosis of terminal cancer and became paralyzed. His tragic last notes and letters indicate the struggle he was facing as Gödel increasingly came to rely on him, phoning him two or three times every day to communicate his latest worries, even as he himself was facing his own mortality. The last straw was when Adele fell sick and had to spend several months in a hospital. After Morgenstern, she had been his last link to the sane world, and in spite of neighbors and colleagues trying to help out, he stopped eating, convinced that he was being poisoned through his food and, unlike in Vienna in the 1930s, not having Adele around to feed him with tender, loving care. When Adele came back the end was already there, and Gödel entered the hospital for the last time. The cause of death was “malnutrition”, although most people believed that slow suicide was the more likely explanation.
How do we deal with the legacy of someone like Gödel? Philosophically, Gödel’s theorems had such a shattering impact on our thinking because, along with two other groundbreaking ideas of 20th century science – Heisenberg’s Uncertainty Principle and quantum indeterminacy – they revealed that human beings’ ability to divine knowledge of the universe had fundamental limitations. But while Heisenberg and the quantum pioneers found limits to understanding rooted in the physical world, Gödel found these limits even in the rarefied world of pure ideas. Nonetheless, mathematics continued to thrive within the boundaries of his theorems, gathering Fields Medals and revolutionizing new fields like algebraic topology and category theory. The deeper significance of Gödel’s work therefore, as he explained in a lecture, is that it’s hard to avoid a connection between them and a Platonic world of numbers and ideas existing independent of our efforts. This is because if human beings are fundamentally incapable of finding out all the results of axiomatic systems, it means there will always be some results outside the grasp of even our most exalted intellects. In our limitations lies mathematics’s freedom.
But that also says something about human minds and points to a debate still raging – whether the mind itself is some kind of Turing machine. The implication of Gödel’s proof is that if the mind is indeed a machine, it will be subject to the incompleteness theorems and there will always be truths beyond our grasp. If on the other hand, the mind is not a machine, it frees it up from being described through purely mechanistic means. Both choices point to a human mind and a world it inhabits that are “decidedly opposed to materialistic philosophy”. Beyond this possible truth is another one that is purely psychological. We can either feel morose in the face of the fundamental limits to knowledge that Gödel revealed, or we can revel, as the historian George Dyson put it, to “celebrate his proof that even the most rigid numerical bureaucracy contains the tools by which higher truth will always be able to effect an escape.”Gödel offers us an invitation to an open world, a world without end.
But what about the paradoxes of the man himself, someone devoted to the highest reaches of rational thought in the most logical of all fields of inquiry, and still one who seemed to have had an almost mystical belief in the spiritual certainty of mathematics and often gave in to the worst impulses of irrationality? I think a clue comes from Gödel’s obsession with Leibniz in his last few years. Leibniz was convinced that this is the best of all possible worlds, because that is the only thing a just God could have created. Like his fellow philosophers and mathematicians, Leibniz was religious and saw no contradictions between science and faith, between teasing out the truths of the world rationally and believing in a hereafter. A few years before his mother Marianne’s death in 1961, Kurt wrote to her in a letter his belief that a God probably exists: “For what kind of sense would there be in bringing forth a creature (man), who has such a broad range of possibilities of his own development and of relationships, and then not allow him to achieve 1/1000 of it?” Like his fellow philosopher Leibniz, Kurt Gödel could perfectly reconcile the rational and the transcendental. In doing this, he proved himself to be much more at home in the 18th century than the 20th. Perhaps that vision of a reconciliation between rational thought and seemingly irrational human frailty and belief will be, even more than his seminal mathematical discoveries, his enduring legacy.
