“Journey to the Edge of Reason: The Life of Kurt Godel” by Stephen Budiansky

Godel finished his career working as a mathematician and logician at the Institute for Advanced Study in Princeton, New Jersey. While there, he was praised by Albert Einstein as the greatest logician since Aristotle. He was also good friends with John von Neumann and Oskar Morgenstern, the inventors of game theory. He studied under Hans Hahn at the University of Vienna and became part of the famed Vienna Circle led by Moritz Schlick, although he was never a fellow traveller in their positivist epistemology. He also became friendly and sparred intellectually with Karl Menger, Karl Popper, Olga Taussky, and Rudolf Carnap, among many other towering academic luminaries during the inter-war years in Vienna.

From early on in his student days, Godel was pegged as a genius. “In his address [at the quadrennial Congress of Mathematics] entitled “Probleme der Grundlegung der Mathematik”—“Problems of Laying Foundations for Mathematics”—[David] Hilbert now posed four problems whose solution he believed would at last place all of mathematics on an unshakable, rigorous footing. The first two involved proving the consistency of mathematical systems: that they contain no contradictions. The third was to prove the parallel property of completeness: that every valid statement within a system can be derived from its basic axioms. The fourth and last was to prove the completeness of the fundamental system of logic known as first-order, or predicate, logic…. The twenty-two-year-old Godel wasted no time answering the call. Within six months he would, in his PhD thesis, solve Hilbert’s fourth problem. The following year, even more astonishingly, he was to prove the impossibility of anyone’s ever solving the first three.”

Despite his years attending the weekly Vienna Circle gatherings, Godel was far from a positivist. “Godel scholars have debated at length how far back the undeniable Platonism of his later mathematical ideas went…. Godel told Carnap in 1928, in one of their many coffeehouse conversations, that he did not see why abstract mathematical concepts like infinity had to be justified on grounds of their empirical application to physical reality: he believed they possessed a reality of their own.” Years later, Godel would write, “It is true that my interest in the foundations of mathematics was aroused by the “Vienna Circle,” but the philosophical consequences of my results, as well as the heuristic principles leading to them, are anything but positivistic or empiristic.”

Perhaps Godel’s greatest achievement was his proof of the Incompleteness Theorem. He wrote, “There are mathematical problems that can be expressed in Principia Mathematica, which cannot be solved by the logical means of Principia Mathematica.” Budiansky explains the ramifications, “As Godel pointed out, the existence of undecidable propositions was not just a matter of incompleteness but a threat to the integrity of the whole works. If a statement F were true but not provable, then the statement not-F would not cause any contradictions to arise if it were added as an axiom to the system, as there was nothing in the system to disprove it. In such an unsettling circumstance, “One obtains a consistent system in which a contentually false proposition is provable.”” Budiansky continues, “While a pair of statements of the form A and not-A cannot both be false, they can both be unprovable…. Whether true or false, a statement that says, “This statement is unprovable,” is trouble. If true, it is itself an instance of a true but unprovable proposition. But if it is false, that means it can be derived from the axioms of the system, and is thus an instance of a false but provable statement, which is arguably even worse. Godel slammed shut the final hope of salvation by showing that any formal system which contains arithmetic, not just Principia Mathematica, will suffer from the identical flaw.”

Godel also contributed in other ways to the progress of math. In a draft to a talk titled, “The Present Situation in the Foundation of Mathematics,” he wrote, “The problem of giving a foundation for mathematics (and by mathematics I mean here the totality of the methods of proof actually used by mathematicians) can be considered as falling into two different parts. At first these methods of proof have to be reduced to a minimum number of axioms and primitive rules of inference, which have to be stated as precisely as possible, and then secondly a justification in some sense or other has to be sought for these axioms, i.e., a theoretical foundation of the fact that they lead to results agreeing with each other and with empirical facts.”

Throughout his career, Godel would fight against the ideas of Brouwer and Wittgenstein that mathematics was a human invention. In his Gibbs Lecture of 1951, he comments that the “creation” of math “shows very little of the freedom a creator should enjoy.” He continues, “Even if, for example, the axioms about integers were a free invention, still it must be admitted that the mathematician, after he has imagined the first few properties of his objects, is at an end with his creative ability, and he is not in a position also to create the validity of the theorems at his will…. If mathematical objects are our creations, then evidently integers and sets of integers will have to be two different creations, the first of which does not necessitate the second. However, in order to prove certain propositions about integers, the concept of [a] set of integers is necessary. So here, in order to find out what properties we have given to certain objects of our imagination, [we] must first create certain other objects—a very strange situation indeed!” To Godel’s mind, his Incompleteness Theorems also lent credence to the fact that humans could not have created math ex novo. “Both of his Incompleteness Theorems proved that no finite process of inference from axioms within a well-defined system can capture all of mathematics. But that, Godel pointed out, leads to an interesting either-or choice: either the human mind can perceive evident axioms of mathematics that can never be reduced to a finite rule—which means the human mind “infinitely surpasses the powers of any finite machine”—or there exist problems that are not merely undecidable within a specific formal system, but that are “absolutely” undecidable. Both choices point to a conclusion “decidedly opposed to materialistic philosophy.”” Godel expands, “So this alternative seems to imply that mathematical objects and facts (or at least something in them) exist objectively and independently of our mental acts and decisions, that is to say some form or other of Platonism or ‘realism’ as to the mathematical objects.”

Godel believed that philosophy was susceptible to being axiomatized in the same way as math, even if its current epistemic state was primitive. He quipped to Morgenstern, “Philosophy today is—at best!—where the Babylonians were with mathematics.” However, Godel was confident, “One could establish an exact system of postulates employing concepts that are usually considered metaphysical: ‘God’, ‘soul’,’idea’.” He expanded to his friend Hao Wang, “The beginning of physics was Newton’s work of 1687, which needs only very simple primitives: force, mass, law. I look for a similar theory for philosophy or metaphysics. Metaphysicians believe it possible to find out what the objective reality is; there are only a few primitive entities causing the existence of other entities.” Later, he compared his philosophy to Leibniz, “My theory is rationalistic, idealistic, optimistic, and theological.”

Some of his colleagues at the Institute thought Godel was wasting his time studying Leibniz’s more esoteric works and neglecting furthering pure math. However, Godel did find time to ruminate on his first love, particularly when buoying a friend in need. Instead of pleasantries, he would often cut to the bone writing on a math problem he was currently engaged with. Gerald Sacks noted, “I noticed over the years that Godel’s way of cheering up a dying person was to send him a logical or mathematical puzzle.” These weren’t simple brain teasers. When Von Neumann sat paralyzed, dying of cancer, Godel wrote to him, posing what was to become one of the most fundamental problems of computer science to this day. “Godel’s letter to his dying colleague was apparently the very first formulation of the so-called “P vs. NP” problem, which offered a striking analogy of his Incompleteness Theorem to the field of computing. “P” is the set of problems easy to solve, for example multiplication and addition. “NP” is the set of problems for which an efficient algorithm exists for checking a given solution, but finding the solution may or may not be easy…. Godel pointed out that one could readily build a machine that works through every possible series of proof steps to discover whether a proof of n steps exists for a given formula. The crucial question is how rapidly the time required for the calculation increases as n gets larger. If it grows slowly—linearly, or even as a square of n—then in principle every problem that is easily checked (“NP”) is also easily solved (“P”). If it grows exponentially, however, that means there will be a set of verifiable but, as a practical matter, forever uncomputable problems—just as his Incompleteness Theorem showed there are true but undecidable propositions within formal mathematical systems.”

In 1963, Godel drafted a lecture to the American Philosophical Society that he would never end up giving. It was a critique of positivism and reductionism. Budiansky relates, “On the “left” stand skepticism, materialism, empiricism, and positivism—the values of Mach, the Vienna Circle, and most of modern science and philosophy. On the “right” are spiritualism, ideology, apriorism, and theology. Godel made no bones about belonging to the “right,” even as it placed him “in contradiction to the spirit of the time.”” Godel revealed to his friend Wang that he thought of the brain as “a computing machine connected with a spirit.” Through idealism and theology, he saw “sense, purpose and reason in everything.” Budiansky expands, “Rather than regarding his Incompleteness Theorem as a pessimistic limitation on mathematical knowledge, the idealist viewpoint upholds the belief “that for clear questions posed by reason, reason can also find clear answers.”” Finally, Godel compared the reality of mathematical perception to that of sense perception. “Despite their remoteness from sense experience, we do have something like a perception also of objects of set theory, as is seen from the fact that the axioms force themselves upon us as being true. I don’t see any reason why we should have less confidence in this kind of perception, i.e., in mathematical intuition, than in sense perception, which induced us to build up physical theories and to expect that future sense perceptions will agree with them, and, moreover, to believe that a question not decidable now has meaning and may be decided in the future. The set-theoretical paradoxes are hardly any more troublesome for mathematics than deceptions of the senses are for physics.”

“Critique and Crisis: Enlightenment and the Pathogenesis of Modern Society” by Reinhart Koselleck

Koselleck was a German historian who wrote this book as his dissertation in 1959. His book makes the case that the Absolute State, propounded by Hobbes and optimized by pre-revolutionary France, successfully cleaved morality from the political sphere, following the religious civil wars that engulfed Europe after the Reformation. The Peace of Westphalia created spheres of authority between sovereign States, but also gave their subjects freedom of conscience within the borders of each Absolutist State. Morality was effectively divorced from public policy. The Enlightenment, with its primacy of reason, beginning with Locke, began to chip away at this wall between politics and conscience. It was this very private space, carved out by the State, which allowed the Enlightenment to first flourish. “The bourgeois intelligentsia set out from the private inner space within which the State had been confining its subjects. Each outward step was a step towards the light, an act of enlightenment…. The public, without surrendering its private nature, became the forum of society that permeated the entire State…. With every step the Enlightenment was shifting the jurisdictional borderline which the Absolutist State had carefully sought to draw between politics and the moral interior.” Koselleck continues, “Locke virtually requires citizens to proclaim their private opinions to be generally binding laws, for it is only in their independent judgment that the power of society is constituted, and only in the constant exercise of moral censure will this censure prove to be a law…. Reason needs the perpetual process of critique to establish itself as supreme authority, so it is only in constant performance of censure that the moral views of Locke’s citizens rise to the State of generally binding laws.” Locke introduces public morality into the Absolutist State. “Locke gave a political charge to the interior of the human conscience which Hobbes had subordinated to State policy. Public actions were now not merely subject to the authority of the State but at the same time to the moral authority of the citizens. What Locke had thus put into words was the decisive breach in the Absolutist order, the order expressed in the relationship of protection and obedience. Morality was no longer a formal matter of obedience, was not subordinated to the politics of Absolutism, but confronted the laws of the State.”

Koselleck goes on to describe the Republic of Letters and its role in spreading the idea of criticism across Europe. He focuses on Pierre Bayle, “By including all areas of human knowledge and history in the critical method and involving them in an infinite process of relativisation, Bayle turned criticism into the essential function of reason…. If criticism is the ostensible resting place of human thought, then thought becomes a restless exercise in movement. Criticism is the activity that marks reason as a factor of judgment, a constant goad to the process of pro and con…. Criticism was not confined to philological, aesthetic or historical concerns but became, more generally, the art of arriving at proper insights and conclusions via rational thought.” Bayle posited that criticism was still separate from politics, but only because it was supra-politics. “The critic stands above the parties; his task is not to ‘destroy’ but to ‘establish’ the truth. He enters into competition with a rational State that sets itself above the religious groupings. Not that he creates a new order here and now; the reign of criticism is non-partisan only in an infinite process of renewal…. The self-assurance of criticism lay in the connection of the critic to the yet-to-be-discovered truth.” For Bayle, truth and freedom were intimately connected. “The critical process for seeking out the truth can be set in motion only under conditions of absolute freedom. In the Republic of Letters everyone is therefore the master of all and subject to the judgment of all. The civil war that was unexpectedly did away with by the State reappears; and, moreover, in that private inner realm which the State had to grant man as man. There, absolute freedom, the bellum omnium contra omnes, reigns; truth is the common goal of all, and criticism—which all engage in and to which all are subject—is the true sovereign in the spiritual conflict. Sovereignty rules relentlessly, and all share it. Bayle’s Republic of Letters, extended to the State, is the total democracy which Rousseau conceived of half a century later. It provides the model of a form of government in which civil war, even if only one of minds, has legitimacy and is the basis of legitimacy…. The King as ruler by divine right appears almost modest alongside the judge of mankind who replaced him, the critic who believed that, like God on Judgement Day, he had the right to subject the universe to his verdict.”

Commingling with the advancement of morality into the political sphere was the idea of the public moving as one in an ever-forward march of progress. “Although the divine plan of salvation was secularised into a rational plan of history, it also became the philosophy of history that assured the course of the planned future. The philosophy of progress offered the certainty (neither religious nor rational but historico-philosophical) that the indirect political plan would be realised and, conversely, that rational and moral planning determines the course of history.”

Rousseau was the man who finally explicitly developed the idea of the General Will. “As a member of Bayle’s Republic of Letters, Rousseau could imagine the neutralisation of the bruising contradiction of subject and man in no other way than through the subjugation of all to one and one to all…. In the ‘miraculous state’ in which no one rules yet everyone obeys and is at the same time free, the revolution is sovereign…. Rousseau’s crucial step was to apply the concept of sovereign will, which the Enlightenment had excluded from its purview, specifically to the moral autonomy of society. He claimed the sole, unconditional will, the accepted basis of the sovereign decision of the absolute ruler, for society. The result was the volonte generale, the absolute general will as a law unto itself.” At this point, society had been completely subsumed by the State and the State had been completely subsumed by society. There was no longer a private sphere for individual morality and conscience. The State had become a moral beast. “More and more, the will to link political rule—via the parlements or in the form of a constitution—to the eternally valid laws discovered and proclaimed by society, joins the theoretical link of the supreme State power to the interests of society.”

“The Concept of Irony” by Soren Kierkegaard (translated by Howard V. Hong and Edna H. Hong)

This 1841 work by Kierkegaard was the dissertation for his college degree. It deals with the irony espoused by Socrates, as he was portrayed by Xenophon, Plato, and Aristophanes. Kierkegaard begins by defining the ironic questioning style ubiquitous with Socrates. “One can ask with the intention of receiving an answer containing the desired fullness, and hence the more one asks, the deeper and more significant becomes the answer; or one can ask without any interest in the answer except to suck out the apparent content by means of the question and thereby to leave an emptiness behind. The first method presupposes, of course, that there is a plenitude; the second that there is an emptiness. The first is the speculative method; the second the ironic. Socrates in particular practiced the latter method…. Socrates’ questioning was essentially aimed at the knowing subject for the purpose of showing that when all was said and done they knew nothing whatever.” Kierkegaard is careful to separate, as best he can, the views of Socrates from those of the author who puts words into his mouth. “Just as Socrates’ philosophy began with the presupposition that he knew nothing, so it ended with the presupposition that human beings know nothing at all; Platonic philosophy began in the immediate unity of thought and being and stayed there. The direction that manifested itself in idealism as reflection upon reflection manifested itself in Socrates’ questioning. To ask questions—that is, the abstract relation between the subjective and the objective—ultimately became the primary issue for him.”

Kierkegaard furthers dissects Plato’s dialogues involving Socrates by analyzing the interplay of irony and the dialectic. “That irony and dialectic are the two great forces in Plato everyone will surely admit, but that there is a double kind of irony and a double kind of dialectic cannot be denied, either. There is an irony that is only a stimulus for thought, that quickens it when it becomes drowsy, disciplines it when it becomes dissolute. There is an irony that is itself the activator and in turn is itself the terminus striven for. There is a dialectic that in perpetual movement continually sees to it that the question does not become entrapped in an incidental understanding, that is never weary and is always prepared to set the issue afloat if it runs aground—in short, that always knows how to keep the issue in suspension and precisely therein and thereby wants to resolve it. There is a dialectic that, in proceeding from the most abstract ideas, wants to let these display themselves in more concrete qualifications, a dialectic that wants to construct actuality with the idea.” Kierkegaard continues on with the purpose of irony. “Because irony cuts the bonds that restrain speculation, helps to shove off from the purely empirical sandbanks and to venture out upon the ocean, this is a negatively liberating activity. Irony is in no sense a partner in the expedition. But insofar as the speculating individual feels liberated and an abundance spreads out before his observing eyes, he may readily believe that he also owes all this to irony and his gratitude may wish to owe it everything…. Thus a personality is a necessary point of departure in the positions of both irony and subjective thinking; the activity of this personality becomes liberating in both of these positions, but the position of irony is negative, whereas that of reasoning is positive.” 

Kierkegaard returns to the role of Socrates in the Platonic dialogues. “If Plato’s view of Socrates were to be expressed in a few words, it could be said that he provides him with the idea. Where the empirical ends, Socrates begins; his function is to lead speculation out of finite qualifications, to lose sight of finitude and steer out upon Oceanus where ideal striving and ideal infinity recognize no alien considerations but are themselves their infinite goal. Thus, just as lower sense perception turns pale before this higher knowledge—indeed, becomes a delusion, a deception by comparison—just so every consideration of a finite goal becomes a disparagement, a profanation of the holy. In short, Socrates has gained ideality, has conquered those vast regions that hitherto were a terra incognita [unknown land]. For this reason, he disdains the useful, is indifferent to the established [Bestaaaende], is an out-and-out enemy of the mediocrity that in empiricism is the highest, an object of pious worship, but for speculation a troll changeling.” 

Kierkegaard makes the convincing case that Socrates’ views were indeed antithetical to the polis. “The negative relation of the daimonian to Socrates… had the express effect of making him relate negatively to actuality or, in the Greek sense, to the state…. It was a polemical relation to the state religion to substitute a silence in which a warning voice was audible only on occasion, a voice that (and this is about the most fundamental polemic) never had a thing to do with the substantial interests of political life, never said a word about them, but only dealt with Socrates’ and at most his friends’ completely private and particular affairs—to substitute this for the Greek life permeated, even in the most insignificant manifestations, by a god-consciousness, to substitute a silence for this divine eloquence echoed in everything.” Hegel comments on this birth of human agency, “In the ‘divine sign’ of Socrates we see the will which formerly had simply transferred itself beyond itself now beginning to apply itself to itself and so to recognize its own inward nature. This is the beginning of a self-knowing and so of a genuine freedom.” Kierkegaard continues, “Socrates’ position, then, is that of subjectivity, of inwardness, which reflects upon itself and in its relation to itself detaches and volatilizes the established [Bestaaaende] in the flood of thought that surges over it and carries it away while it itself recedes again into thought. Replacing the grace to be ashamed, which powerfully but mysteriously kept the individual tied to the lead string of the state, there now came the decisiveness and self-assurance of subjectivity.”

Another aspect of Socrates’ philosophy that Kierkegaard focusses on is the supremacy of ignorance. Socrates “gave up the study of nature for the study of man. To say, therefore, that Socrates did not accept the gods accepted by the state does not mean that he was an atheist. On the contrary, Socrates’ nonacceptance of the national gods was essential to his whole position, which he himself theoretically characterized as ignorance…. His accusers really ought to have charged him precisely with this ignorance, since particularly in the Greek state and to a certain extent in every state there is indeed an ignorance that must be regarded as a crime…. In the philosophic sense… he was ignorant. He was ignorant of the ground of all being, the eternal, the divine—that is, he knew that it was, but he did not know what it was.” Here Kierkegaard turns back to the newly emerging concepts of subjectivity and individuality. “When subjectivity by means of its negative power has broken the spell in which human life lay in the form of substantiality, when it has emancipated man from his relation to God just as it freed him from his relation to the state, then the first form in which this manifests itself is ignorance…. This ignorance is in turn quite consistently called human wisdom, because here man has come into his own right.”

Socrates freed himself from the bounds of Greek society through irony. “The ironic freedom he enjoyed because no relationship was strong enough to bind him and he continually felt himself free above it, the enjoyment of being sufficient unto himself, to which he abandoned himself—all this suggests something aristocratic…. True freedom, of course, consists in giving oneself to enjoyment and yet preserving one’s soul unscathed.” The preeminent form of freedom was moral freedom. “Here the moral individual is the negatively free individual. He is free because he is not bound by another, but he is negatively free precisely because he is not limited in another. When the individual by being in his other is in his own, then for the first time he is in truth (i.e., positively) free, affirmatively free. Therefore, moral freedom is arbitrariness; it is possibility of good and evil.” Hegel comments, “The conscience as formal subjectivity is simply to be on the verge of slipping into evil.” Kierkegaard continues, “Socrates brought this about, but not in the sense of the Sophists, who taught the individual to constrict himself in his own particular interests; he brought the individual to this by universalizing subjectivity, and to that extent he is the founder of morality…. The individual should no longer act out of fear of the law but with a conscious knowledge of why he acted.”

Going back to Socrates’ trial, Kierkegaard repeats why Socrates was anathema to the Athenian polis. “The whole substantial life of Greek culture had lost its validity for him, which means that to him the established actuality was unactual, not in this or that particular aspect but in its totality as such; that with regard to this invalid actuality he let the established order of things appear to remain established and thereby brought about its downfall; that in the process Socrates became lighter and lighter, more and more negatively free.” Kierkegaard concludes by bringing it back to irony. “It cannot really be said that the ironist places himself outside and above morality and ethics, but he lives far too abstractly, far too metaphysically and esthetically to reach the concretion of the moral and the ethical. For him, life is drama, and what absorbs him is the ingenious complication of this drama. He himself is a spectator, even when he is the one acting.”

“Sugar Street” by Nagub Mahfouz (translated by William Maynard Hutchins, Olive E. Kenny, Lorne M. Kenny, & Angele Botros Samaan)

This final novel of Mahfouz’s trilogy finds the Jawad household having aged and grown through marriages and births. Al-Sayyid Ahmad’s grandchildren are by now already attending college and, through ill health, he has had to reform his debauched ways. “Every time he had exceeded the limits, he had paid the price. He had finally been forced to give in, eating or drinking only what he was supposed to and coming home by nine. His heart had not given up hope that, by whatever means, he would regain his health and enjoy a pleasant, quiet existence, even though his past life had disappeared forever.” His youngest son, Kamal, has settled into a lowly existence, as a humble grade school teacher, who moonlights writing philosophic scribblings in an unread magazine. “The happiest part of his day was the period he devoted to philosophy. Lasting until midnight, it was the time—as he put it—when he felt like a human being. The rest of his day spent as a teacher in al-Silahdar School or in satisfying various needs of daily life was the stamping ground of the animal concealed in him. The creature’s goals were limited to self-preservation and the gratification of desires.”

Meanwhile, Al-Sayyid Ahmad’s son-in-law, the layabout of independent means, Ibrahim Shawkat, bemoans how best to raise his two rebellious sons, Abd al-Muni’m, a member of the Muslim Brethren, and Ahmad, a budding Communist, “We rear our children, guide them, and advise them, but each child finds his way to a library, which is a world totally independent of us. There total strangers compete with us. So what can we do?” Soon enough, Ahmad, himself, is forced to reflect on the subversive paths he and his brother have taken in life. “Ahmad asked himself why the three older men had been arrested. Had the charges been theft, fighting, drunkenness, or rowdy behavior? Clad in his overcoat, he had often written about “the people” in his beautiful study. Here they were—cursing or snoring in their sleep. For a few seconds by the light of the torch he had seen their wretched sullen faces, including that of the man who was scratching his head and his armpits. At this very moment his lice might be advancing resolutely toward Ahmad and his brother…. “You are devoting your life to people like this,” he told himself…. Ahmad advised himself, “Without regard to the differences of taste between us, our common human condition has united us in this dark and humid place: the Muslim Brother, the Communist, the drunkard, and the thief. Despite dissimilarities in our luck and success at looking after ourselves, we are all human beings.””