John W. Dawson jr
Dilemmi Logici: La vita e l'opera di Kurt Gödel

Summarizing the intellectual and psychological journey of the life of mathematician Kurt Gödel (Brünn 1906 - Princeton 1978) in a book must not have been easy. Despite the immense difficulties, Dawson, professor of mathematics at Pennsylvania State University, logician, and historian of science, a world authority on Gödel's work, has brilliantly surpassed the challenge. The work is dense with mathematical formulas of immense complexity. It also includes bibliographic notes, quotes, appendices, biographical profiles of famous mathematicians, photographs, and a family genealogy chart. Gödel is not only known for his incompleteness theorems, which significantly contributed to the field of logic, but also because he dealt with cosmology and the philosophy of mathematics. His rooted beliefs were these: the universe is rationally organized and comprehensible to the human mind. It is deterministic from a causal point of view; there is a mental and conceptual domain external to the physical world, and through introspection one can find conceptual understanding. He was a pioneer who tackled big problems by making fundamental discoveries and leaving others to develop his ideas in detail. To 'reconstruct' his personality, Dawson used Gödel's most important collection of documents, the "Nachlass," kept at the Firestone Library of Princeton University. "Nachlass" is an immense collection of information containing almost every piece of paper that passed through Kurt’s hands. The other main sources are: the memories of Menger, his brother Rudolf, Gödel's letters to his mother, and for the last thirty years, the essential diaries of his friend Morgenstern. The volume meticulously divides and analyzes Kurt's life into three periods. His childhood in Brno, the years in Vienna as a university student and then a lecturer, where he obtained his most important mathematical results. Finally, the years after emigrating to America, during which he was especially interested in physics and philosophy. Since he was a child, Kurt never stopped asking questions to the point where he was nicknamed 'Mr. Why.' During his studies, even from gymnasium, he amazed teachers and students with an extraordinary mathematical knowledge. He read works by philosophers such as Bacon, Schopenhauer, Descartes, Leibniz, Spinoza, Hobbes, Locke, Rousseau, Kant, and Hegel. He studied the classics of mathematics, including Euclid's "Elements," Euler's "Introductio in Analysin Infinitorum," and Lagrange's "Mécanique Analytique." He was even curious about parapsychology, telepathy, and ghosts. He was a silent and calm person, speaking softly with extreme precision, and led a life dedicated to study. In '28 he was significantly influenced by four events. The publication of Hilbert and Ackermann’s volume "Grundzüge der theoretischen Logik," Hilbert's conference "Problems of the Foundations of Mathematics" held in Bologna, and two lectures given by the iconoclastic mathematician Brouwer in Vienna. His reasoning showed that: "arithmetic is 'incomplete,' that is, there exist formally undecidable arithmetical statements and even demonstrating that its consistency is not only expressible within the theory itself, but is a particular example of an undecidable statement." "Where formalized arithmetic was consistent and nevertheless unable to formally prove this fact, then, under the assumption that the theory is valid (proves no false statements), its consistency must be formally undecidable, that is, neither provable nor refutable." However, the opposite is not clear. One might think that the consistency of a theory should be provable within it, even if other statements within the theory were undecidable. This is the distinction between the first and the second incompleteness theorems. Gödel observed that whatever philosophical stance is taken towards mathematics, in some form 'the phenomenon of its inexhaustibility' will necessarily emerge. It is likely Gödel's fundamental discovery of the first theorem was achieved by him just before revealing the result to Carnap at Café Reichstrat on August 26, 1930. Today, the term 'incomplete' is widespread, but in the original German text, Gödel referred to it as 'entscheidungsdefinit,' translated as 'capable of decision,' suggesting that the theory can decide (prove or refute) each of its statements. The incompleteness theorems do not set limitations on the possibilities of human reasoning, but reveal that 'the necessary reasoning in mathematics cannot be completely mechanized.' Over the years, thanks to their protean nature, his theorems have lent themselves to continuous reinterpretations. For example, in computer science, no program designed to detect computer viruses can be simultaneously safe and 100% efficient (diagonalization). At the Conference on the Epistemology of the Sciences, Gödel waited until late to expound his observations. Few present managed to grasp the importance of Gödel's revelations, apart from Neumann, due to his legendary quickness of mind. After the congress, Neumann, reflecting on Gödel's considerations, wrote to him that he had reached a remarkable result: in a consistent system, every effective proof of the non-provability of the statement 0=1 could be turned into a contradiction. The incompleteness theorems encountered opposition in several circles. Gödel's seemingly ambiguous remarks led him to say: "Should we think that 2+2 is not 4, but 4.001?" Russell, in a letter from '63, admitted he had realized that Gödel's work was fundamental, but at the same time was astonished because he had found a contradiction in arithmetic. Gödel also actively engaged as a reviewer in the mathematical field, with as many as 33 writings. In the early thirties, the Institute for Advanced Study (IAS) was founded, initially structured only for a School of Mathematics. In '32 Einstein and Veblen were the first professors appointed. Among the 24 scholars invited as guests in the first year of the IAS, there was Gödel. In '34, his physical and mental health began to 'crack,' and he was admitted to the Purkersdorf clinic. Wagner-Jauregg, a world-renowned psychiatrist (Nobel prize in '27 for the treatment of general paresis), diagnosed him with a nervous breakdown caused by overwork. Gödel soon recovered and in '35 obtained another fundamental result: a proof on the relative consistency of the axiom of choice with the other axioms of set theory. On June 19, he presented a paper titled: "On the Length of Proofs," in which it would be 'possible to shorten, in a remarkable measure, an infinite quantity of already available proofs.' With the development of computational complexity, Gödel's result marked a first example of what are now known as 'speed-up' theorems, a central subject in theoretical computer science. Shortly after, he relapsed into depression, and the subsequent recovery required time. On September 20, '38, he married Adele Nimbursky. For Morgenstern, the existing attraction between the two was an enigma. He described Adele as a talkative, uneducated, willful, ordinary, and coarse woman. She became Kurt's food taster because he was obsessed with the fear of being poisoned. He was even terrified of gas leaks from the refrigerator. Out of these fears, he became interested in toxicology, pharmacology, normal and pathological physiology, psychiatry, and neurology. These obsessions deepened over the years, eventually turning into severe mental distress, along with hypochondria, paranoia, and anorexia. For various reasons, the couple did not have children. With the outbreak of the Second World War, Kurt began facing difficulties with immigration; moreover, he was summoned for a medical examination to check his suitability for Nazi military service. If the German authorities had known about his episodes of mental instability, they could have sent him to a concentration camp or clinic, but this didn't happen. Kurt realized how critical the situation had become for him when one day, while walking with Adele, he was attacked by a gang of Nazi thugs. They mistook him for a Jew or an intellectual, and only Adele managed to fend them off with her umbrella. They decided to leave for a long and exhausting journey to America... A fundamental relationship was with Einstein, whom he first met in '33, but only got to know well in '42. Becoming friends raised many questions, as their characters were diametrically opposed, even though they shared superior intelligence. Einstein was sociable, cheerful, with a rich sense of humor, while Gödel was serious, reflective, and solitary, but they esteemed and understood each other. They were often together during walks to and from the Institute to discuss philosophy, physics, and politics. Both rejected the idea of indeterminacy or chaos in physics, and in later years, they became passionate about questions that addressed the essential center of things. Kurt's research was interrupted by a hearing on December 5, '48, to obtain approval for American citizenship. For Gödel, the event was serious because, confiding in Morgenstern, he said he had found a contradiction in the Constitution. Morgenstern, though amused, understood that his citizenship might be at risk, so he consulted Einstein. Einstein tried to distract him before the car ride, saying to Kurt, "Well, are you ready for the penultimate test?" "What do you mean by penultimate test?". Einstein replied, "Very simple. The final one will be when you enter the grave." Despite Einstein's 'macabre' humor, Gödel remained worried. When the judge asked Kurt if he thought a dictatorship like the one in Germany could arise in the United States, Gödel replied with a yes. The judge, immediately understanding what could happen, quickly changed the subject. Citizenship was granted to him anyway. In '49 he demonstrated, through the general theory of relativity, the existence of universes where 'time travel' into the past is possible. In '50 he gave a lecture on his results in cosmology, but at such a level that it surpassed the comprehension of almost all the listeners. Like the incompleteness theorems, his cosmological results also upset deeply-rooted preconceptions, irrelevant to most of the scientific community's interests. In '51, for several days, his life was in danger due to a duodenal ulcer, but he recovered. To lift Kurt's spirits, Morgenstern thought that an award, even with money, would comfort him. After some difficulty, he was finally awarded jointly with Schwinger the Einstein Prize. When Einstein presented the medal, he jokingly said to Kurt, "And this, dear friend, for you. - And you don't need it!". Despite the highest honors Gödel received over the years, Princeton continued to ignore his genius presence. Within five years, Einstein, Neumann, and Veblen died. Einstein's death devastated him. Kurt described Einstein as 'the personification of friendship.' In the last ten years, he withdrew more and more from the outside world, refusing invitations to scientific conferences and continuing to work on revising his previous articles, but also on new results, particularly on determining the 'true' power of the continuum and on a formulation of Anselm's 'ontological proof' for the existence of God. When his office was relocated to the back of the IAS structure housing the library, his contacts with colleagues became increasingly infrequent. The growing conflict in Vietnam and internal controversies at the IAS increased his depression, while the space program intrigued him. Over time, his psycho-physical state, although in cycles, worsened more and more. Decisive factors were the absence of his wife due to health issues and the death of his friend Morgenstern. Unfortunately, his continuous and unstoppable curiosity to ask questions, while enabling him to achieve fundamental discoveries in logic and beyond, also led him to progressive social isolation. Shortly before dying, he said, "Every disorder is a mistaken appearance." Among his papers was found a note listing 14 principles he regarded as fundamental. The first of these was: "Die Welt ist vernünftig," 'the world is rational.' Throughout his life, Gödel refused to accept the idea of random events. But to explain our life rationally, there are insurmountable problems to face. Persistently seeking rationality in our world and everything around it is, in essence, irrational behavior. He would die from anorexia resulting from personality disorders on January 14, '78. Gödel unequivocally believed in life after death. He argued that we would begin a new existence with 'latent memories' of past experiences. For him, it would not make sense to create a being like man with all his development potentials if only some of them out of thousands were allowed to expand! He believed that the powers of the human mind were potentially unlimited, and consequently, so was our understanding. To discover the truth about 'everything,' it is necessary to study one's self, as the mind is not static but constantly evolving. However, his introspection was so intense that it turned into paranoia. The obsessive concern for himself made him a hypochondriac. Kurt was a genius; his prodigious intellect could operate with a superior perception regarding certain topics, but at the same time, he remained a child. Behind the boundless intellectual passion, profound turmoil, and inscrutable gaze, a great humanity was concealed.