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.”
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