[The following quotes are from Incompleteness: The Proof and Paradox of Kurt Gödel, 2005, by Rebecca Goldstein.]
Every error is due to extraneous factors (such as emotion and education); reason itself does not err.
It is as if one of the unwritten laws of his [i.e. Gödel’s] thought processes is: If reasoning and common sense should diverge, then . . . so much the worse for common sense! What, in the long run, is common sense, other than common?
Einstein, though not quite so strict with himself as Gödel, still shared the conviction that truly good science always keeps the larger philosophical questions in view: ‘Science without epistemology is—insofar as it is thinkable at all—primitive and muddled.’
Gödel professed himself a nonbeliever in evolution and topped this off by pointing out, as if this were additional corroboration for his own rejection of Darwinism: “You know Stalin didn’t believe in evolution either, and he was a very intelligent man.”
[Gödel] was a man of deep passions, as his life will bear out; but these passions were kept scrupulously hidden and they were rigourously intellectual.
An aspect of the Platonic vision is a rejection of the easy bifurcation between passion, on the one side, and reason, on the other. Plato is urging us toward impassioned reason, the higher intoxication. Of course, susceptibility to the higher intoxication is predicated on the ability to grasp the intellectual love object, the beauties of pure abstraction, “to look at that, to study it in the required way, and be together with it.” The young Kurt Gödel was singularly susceptible.
[Positivism: a philosophical system that holds that every rationally justifiable assertion can be scientifically verified or is capable of logical or mathematical proof, and that therefore rejects metaphysics and theism.]
By declaring the limits of knowability one and the same with the limits of meaningfulness, the positivists took the problematic aspect of such questions as the existence of God (or of moral values or of abstract entities) up a notch, so that now the unanswerability of certain questions no longer takes the measure of our cognitive inadequacies, but rather signals that the questions ought never to have been posed at all. Unknowability is regarded as a sign that a mistake in the use of language has been made.
The positivist transformation of the empiricist theory of knowledge into a theory of meaning meant that the single damning word “meaningless” was to be pronounced over the remains of much that had formerly passed for knowledge. Here was the single word with which to accomplish a program of cognitive hygiene such as the world had never seen.
The logical positivists believed that mathematics, just like logic, was devoid of any descriptive content. Mathematical propositions, if not quite tautologies, are analogous to them. (It’s hard to make out this middle ground, but never mind for now.) Another way to put this point is that mathematics is merely syntactic; its truth derives from the rules of formal systems, which are of three basic sorts: the rules that specify what the symbols of the system are (its “alphabet”); the rules that specify how the symbols can be put together into what are called well-formed formulas, standardly abbreviated “wff,” and pronounced “woof”; and the rules of inference that specify which wffs can be derived from which.
Russell’s famous paradox is of the self-referential variety. The liar’s paradox—this very sentence is false—is of the same variety.
He [Wittgenstein] is armour-plated against all assaults of reasoning. It is really rather a waste of time talking to him.
The traditional tool of the philosopher—the argument—is dispensed with [by Wittgenstein]; each assertion is put forth, as Russell once remarked, “as if it were a Czar’s ukase.” The poet’s obscurity of meaning is preserved....
Most of the propositions and questions to be found in philosophical works are not false but nonsensical. Consequently we cannot give any answer to questions of this kind, but can only point out that they are nonsensical. Most of the propositions and questions of philosophers arise from our failure to understand the logic of our language . . . And it is not surprising that the deepest problems are in fact not problems at all.
Wittgenstein (from the Tractatus)
All logic is ultimately tautological: “6.1262 Proof in logic is merely a mechanical expedient to facilitate the recognition of tautologies in complicated cases.” Because all logic is tautological, it says nothing: “5.43 But in fact all the propositions of logic say the same thing, to wit nothing.”
Gödel, of course, was poised to deliver the greatest surprise in the history of logic, one which, in the logician Jaakko Hintikka’s words, is “stranger than others by orders of magnitude.”
Whereas the early Wittgenstein had laboured hard with Russell on problems of logic, the later Wittgenstein came to regard the entire field as a “curse” (while Russell, disheartened by his earlier labours with Wittgenstein—his inability to understand him—withdrew from the field and wrote bestsellers.)
Wittgenstein’s attitude toward the inherent contradiction of the Tractatus is perhaps more Zen than positivist. He deemed the contradiction unavoidable. Unlike the scientifically minded philosophers who took him as their inspiration, he was paradox-friendly. Paradox did not, for Wittgenstein, signify that something had gone deeply wrong in the processes of reason, setting off an alarm to send the search party out to find the mistaken hidden assumption. His insouciance in the face of paradox was an aspect of his thinking that it was all but impossible for the very un-Zenlike members of the Vienna Circle to understand.
Ironically, the Vienna Circle, united by their core distaste for mystery, were embracing a thinker committed to mystery, at least in so far as questions of ethics, aesthetics, metaphysics, and the meaning of life—all the matters they had banished from the realm of reasonable consideration—were concerned.
The more I think about language the more it amazes me that people ever understand each other.
Gödel had become a mathematical realist in 1925, had attended the Vienna Circle’s meeting between 1926 and 1928, and by 1928 had begun to work on the proof for the first incompleteness theorem which he interpreted as disproving a central tenet of the Vienna Circle, the very tenet that had caused them to append “logical” to the Machian viewpoint of positivism. He had used mathematical logic, beloved of the positivists, to wreak havoc on the positivist antimetaphysical position. Yet here he was, in 1974, still having to explain, in missives that he never mailed, that he was not a positivist, that the intended import of his celebrated theorems had been in fact, to prove the positivists wrong. The positivists had endorsed the Sophist’s man-measurement of truth. Gödel had sought to vindicate the Sophist’s implacable antagonist, Plato. Gödel, unlike his friend Einstein, did not have a well-developed sense of the ironic, which is, all things considered, a shame.
This conviction of the solvability of every mathematical problem is a powerful incentive to the worker. We hear within us the perpetual call: There is the problem. Seek its solution. You can find it by pure reason, for in mathematics there is no ignorabimus.
Hilbert regarded “the situation with respect to the paradoxes” with dismay: “Admittedly, the present state of affairs where we run up against the paradoxes is intolerable. Just think, the definitions and deductive methods which everyone learns, teaches, and uses in mathematics, lead to absurdities! If mathematical thinking is defective, where are we to find truth and certitude?”
Logicism, intuitionism, formalism, Wittgenstein: there was no representative of Platonism to argue that point of view on the first day [7 October 1930] in Königsberg. All the views represented there that day were committed to the claim that the notion of mathematical truth was reducible to provability; the disagreements between them were on the conditions of provability.
Gödel’s announcement, delivered during the summarizing session on the third and last day of the conference, was so understated and casual—so thoroughly undramatic—that it hardly qualified as an announcement, and no one present, with one exception, paid it any mind at all.
A logically true proposition, or a tautology, is one that is true no matter what meanings we substitute for the nonlogical terms. (Since “logically true” thus makes reference to meanings—something is logically true if it’s true no matter what meanings we assign to its nonlogical terms—it’s a semantic, rather than syntactic, notion.)
Completeness is exactly what one would like from one’s formal system of logic, and it was one of the problems for which Hilbert had demanded a solution.
Gödel gave no indication of the revolution he was hiding up his sleeve until the last day of the conference, which had been reserved for general discussion of the papers of the two preceding days. He waited until quite late in the general discussion and then he mentioned, in a single immaculately worded sentence, that it was possible that there might be true, though unprovable, arithmetical propositions, and moreover that he had proved that there are:
One can (assuming the [formal] consistency of classical mathematics) even give examples of propositions (and indeed of such a type as Goldback and Fermat) which are really contextually [materially] true but unprovable in the formal system of classical mathematics.
That was it. The proof that was to become the “famous Incompleteness Proof” had apparently been accomplished the year before, when Gödel was 23, and it was to be submitted in 1932 as his Habilitationsschrift, the last stage in the prolonged process of becoming an Austrian or German Dozent. It is one of the most astounding pieces of mathematical reasoning ever produced, astounding both in the simplicity of its main strategy and in the complexity of its details, the painstaking translating of metamathematics into mathematics by way of what has come to be called Gödel numbering. It is a thoroughly ordered blending of several layers of “voices,” both mathematical and metamathematical, counterpoint merging into harmonic chords never before heard. Music does seem to provide a particularly apt metaphor, which is why Ernest Nagel and James R. Newman in their classic explicatory work Gödel’s Proof, described the proof as an “amazing intellectual symphony.”
The idea that the criteria for semantic truth could be separated from the criteria for provability was so unthinkable from a positivist point of view that the substance of the theorem simply could not penetrate.
In science . . . novelty emerges only with difficulty, manifested by resistance, against a background provided by expectation. Initially only the anticipated and usual are experienced even under circumstances where anomaly is later to be observed.
The syntactic features of formal systems—which were meant to obviate intuitions, those breeders of paradox—can’t capture all the truths about the system, including the truth of its own consistency.
The possibility of paradox, meant to be forever eliminated by Hilbert’s program, reasserted itself. And one of the strangest things about the odd and beautiful proof that subverted Hilbert’s defense against paradox was the way in which paradox itself was incorporated into the very structure of the proof.
Mathematics cannot be incomplete; any more than a sense can be incomplete. Whatever I can understand, I must completely understand. This ties up with the fact that my language is in order just as it stands, and that logical analysis does not have to add anything to the sense present in my propositions in order to arrive at complete clarity.
Just as Gödel demonstrated that our formal systems cannot exhaust all that there is to mathematical reality, so the early Wittgenstein argued that our linguistic systems cannot exhaust all that there is to non-mathematical reality. All that can be said can be said clearly, according to the Tractatus; but we cannot say the most important things.
The formalists had tried to certify mathematical certitude by eliminating intuitions. Gödel had shown that mathematics cannot proceed without them. Restricting ourselves to formal syntactic considerations will not even secure consistency.
The mathematical knowledge that we possess cannot be captured in a formal system. That is what Gödel’s first incompleteness theorem seems to tell us. But formal systems are precisely what captures the computing of computers, which is why they are able to figure things out without having any recourse to meanings.
Gödel himself was far more reserved about drawing conclusions concerning the nature of the human mind from his famous mathematical theorems. What is rigourously proved, he suggested in his conversations with Hao Wang as well as in the Gibbs lecture that he gave in Providence, Rhode Island, 26 February 1951 (which he never published), is not a categorical proposition as regards mind. Rather what follows is a disjunction, an “either-or” sort of proposition. That is, he was admitting that nonmechanism doesn’t follow, clean and simply, from his incompleteness theorem. There are possible outs for the mechanist.
According to Wang, Gödel believed that what had been rigourously proved, presumably on the basis of the incompleteness theorem, is: Either the human mind surpasses all machines or else there exist number theoretical questions undecidable for the human mind.
What exactly did Gödel have in mind with this second disjunct? I think that what he is considering here is the possibility that we are indeed machines—that is, that all of our thinking is mechanical, determined by hard-wired rules—but that we are under the delusion that we have access to unformalizable mathematical truth. We could possibly be machines who suffer from delusions of mathematical grandeur.
Of course there is no proof that we know all that we think we know, since all that we think we know can’t be formalized; that, after all, is incompleteness. This is why we can’t rigourously prove that we’re not machines. The incompleteness theorem, by showing the limits of formalization, both suggest that our minds transcend machines and makes it impossible to prove that our minds transcend machines. Again, an almost-paradox.
For Gödel, the distinction between intuitions and rigourous proof was always vividly clear. After all, it was the unavoidability of that very distinction that had been so strongly suggested by his famous proof.
Gödel’s theorems are darkly mirrored in the predicament of psychopathology: Just as no proof of the consistency of a formal system can be accomplished within the system itself, so, too, no validation of our rationality—of our very sanity—can be accomplished using our rationality itself. How can a person, operating within a system of beliefs, including beliefs about beliefs, get outside that system to determine whether it is rational? If your entire system becomes infected with madness, including the very rules by which you reason, then how can you ever reason your way out of your madness?
Gödel, like Leibniz, believed that some version of the infamous “ontological proof for God’s existence” was valid. This is an argument that tries to deduce the existence of God from the right definition of God.
We found it hilarious that the greatest logician since Aristotle deluded himself into believing that God’s existence could be proved a priori, that he was perhaps contemplating the day when atheists would be brought round by a good stiff course in quantificational logic.
Gödel’s position at the University of Vienna was rather a lowly one. He became Privatdozent in Vienna in 1933. A Privatdozent is granted the right to lecture, though he receives no salary. For the honour, a candidate has to write a second dissertation.
However, this wasn’t the last hoop a candidate had to sump through before being declared a Privatdozent. The entire faculty had to take a vote, not only on the candidate’s scientific worth but also on his personal worthiness as well.
Gödel’s world back in Vienna was now thoroughly Nazified. Menger had written to Veblen, while still in Vienna, that “whereas I . . . don’t believe that Austria has more than 45 % Nazis, the percentage at the universities is certainly 75 % and among the mathematicians I have to do with . . . [apart from] some pupils of mine, not far from 100 %.”
Menger’s affection for Gödel considerably cooled, not to return until decades later, at the end of Gödel’s life, when he came to understand more completely the deep and abiding strangeness of the logician.
His profound isolation wasn’t only a matter of his intellectual estrangement from the philosophical positivism that he felt had trailed him to the New World from Vienna (which in some sense it had). On a personal level, as well, Gödel was quite completely alienated from his mathematical colleagues at the Institute. Unlike Einstein, they weren’t amused by his “strange axiom,” his version of Leibnitz’s principle of sufficient reason, which disposed him to believe that everything that happens has a thoroughly logical explanation—especially since Gödel’s application of his axiom led him to believe that those in authority are indeed in authority for a sufficiently good reason. Gödel’s axiom inclined him to give the Powers That Be the benefit of the doubt—they must have their good reasons for their decisions and actions, even if all empirical evidence seems to indicate that they don’t—and such reasoning served to deeply divide the logician from his colleagues at the Institute.
Kochen told me that during the troubles over Bellah, Gödel would sometimes call to speak to him about it. “He was very distressed at the incivility of the atmosphere.”
Wang did manage to gain entry into the Gödel house on 17 December and seems to have been reassured by Gödel’s manner and presence (though he must have been emaciated). “His mind remained nimble and he did not appear sick. He said, ‘I’ve lost the faculty for making positive decisions. I can only make negative decisions.’ ”
Karl Menger contributed one last anecdote: In one of his last telephone calls before his own death (in July, 1977) Morgenstern described an event that evoked in me memories that long ago had somewhat estranged me from Gödel—but it evoked them by its contrast to those memories, so that Morgenstern’s story moved me very much. Once again it was a question of Gödel’s rights, where his punctiliousness knew no bounds. What had happened was that Gödel, apparently suffering severely, sought and was granted admission to a Princeton hospital, but soon thereafter insisted that he had no right to one of the benefits proffered, since his insurance policy did not provide for it. He therefore refused to accept the benefit. The details of the case escape me now, though of course I am convinced that Gödel’s logic in interpreting the insurance contract was superior to the hospital’s. But be that as it may, in his juridical precision, Gödel unshakably maintained his ground.
The experience of the now means something special for man, something essentially different from the past and the future, but this important difference does not and cannot occur within physics.
Philosophy had inspired Kurt Gödel’s formidable mathematical career from the beginning. It had been his focus ever since his first course at the University of Vienna in the history of philosophy, when Gödel, like so many lovers of abstraction, had found in Plato a vision of reality that answered to his intellectual love. As philosophy had been his end, so, too, it was by philosophy’s light that he judged his life, finally, incomplete. No longer believing that it was possible to change other people’s minds, not even by way of a priori proof, he awaited the epiphany that would change his own.
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