Quotes About Gödel

The AIs claimed to have worked it out, then announced they couldn't explain it to us. Gödel was right after all: No system can fully understand itself.

Einstein's and Gödel's metaconvictions were addressed to the question of whether their respective fields are descriptions of an objective reality—existing independent of our thinking of it—or, rather, are subjective human projections, socially shared intellectual constructs.

Our world is eagerly awaiting the posthumous publication of his works, which are rumored to contain an a priori proof of God's existence—a situation which has prompted me to flirt with the idea of a symbolism-heavy play entitled Waiting for Gödel.)

If Nietzsche's existential relativism be accepted, then there will always be true things that do not fit any existing reality-tunnel, just as in mathematics Godel demonstrated that there will always be true theorems not deducible from any set of axioms. (See Chapter Two.)

Gödel's incompleteness theorems brought the second ending of the Grundlagenstreit. Where Hilbert had won the conflict in the social sense, he had lost it in the scientific sense.

The mole is a creature that I admire," Gödel said, before listing several of its most salient virtues, ranging from industry to persistence. "And it does not call attention to itself or its work. It works in secret. That, too, is to be admired.

Gödel freely admitted that the intuition of a concept was not proof; he argued that it was the opposite. "We do not analyze intuition to see a proof, but by intuition we see something without a proof." Recently, however, he'd gone beyond that conclusion, too, and asserted that there must then logically be a realm unknowable to our simple senses, where ultimate truth resided.

Thus, Godel appears to have taken it as evident that the physical brain must itself behave computationally, but that the mind is something beyond the brain, so that the mind's action is not constrained to behave according to the computational laws that he believed must control the physical brain's behavior.

Support for the Platonic viewpoint (as opposed to the formalist one) was an important part of Godel's initial motivations. On the other hand, the arguments from Godel's theorem serve to illustrate the deeply mysterious nature of our mathematical perceptions. We do not just 'calculate' in order to form these perceptions, but something else is profoundly involved-something that would be impossible without the very conscious awareness that is, after all, what the world of perceptions is all about.

What Godel and Rosser showed is that the consistency of a (sufficiently extensive) formal system is something that lies outside the power of the formal system itself to establish.

Mathematical truth is not determined arbitrarily by the rules of some 'man-made' formal system, but has an absolute nature, and lies beyond any such system of specifiable rules. Support for the Platonic viewpoint ...was an important part of Godel's initial motivations.

If a 'religion' is defined to be a system of ideas that contains unprovable statements, then Gödel taught us that mathematics is not only a religion, it is the only religion that can prove itself to be one.

Kurt Gödel's achievement in modern logic is singular and monumental – indeed it is more than a monument, it is a landmark which will remain visible far in space and time. ... The subject of logic has certainly completely changed its nature and possibilities with Gödel's achievement." —John von Neumann

There's a kind of Gödel's Theorem in human affairs: Every attempt to systemize life or to govern it by a set of axioms rich enough to encompass the totality of experience leads to a contradiction.

In 1931, Kurt Godel proved in his famous second incompleteness theorem that there could be no finitary proof of the consistency of arithmetic. He had killed Hilbert's program with a single stroke. So should you be worried that all of mathematics might collapse tomorrow afternoon? For what it's worth, I'm not. I do believe in infinite sets, and I find the proofs of consistency that use infinite sets to be convincing enough to let me sleep at night.

Gödel saw beyond the surface level of number theory, realizing that numbers could represent any kind of structure.

you can build an organ which can do anything that can be done, but you cannot build an organ which tells you whether it can be done."9 "This is connected with the theory of types and with the results of Gödel," he continued. "The question of whether something is feasible in a type belongs to a higher logical type.

Where does meaning come in? If everything is assigned a number, does this diminish the meaning in the world? What Gödel (and Turing) proved is that formal systems will, sooner or later, produce meaningful statements whose truth can be proved only outside the system itself. This limitation does not confine us to a world with any less meaning. It proves, on the contrary, that we live in a world where higher meaning exists.

In PM, as Gödel said, "one can prove any theorem using nothing but a few mechanical rules.

There must be truths, that is, that cannot be proved—and Gödel could prove it.

The incompleteness theorem is a mathematical theorem precisely because the relevant notions of truth and provability are mathematically definable. Nonmathematical "Gödel sentences" and Liar sentences give rise to prolonged (or endless) discussions of just what is meant by a proof, by a true statement, by sound reasoning, by showing something to be true, by convincing oneself of something, by believing something, by a meaningful statement, and so on.

BOTH GÖDEL'S AND COHEN'S arguments proceed by constructing a model of set theory, though I will not explain them in detail.

For Einstein, as for Gödel, philosophy without ontology was an illusion, and physics without philosophy reduced to engineering.

Godel continued drawing conclusions beyond the point where Einstein stopped.