G C3 B6del S Incompleteness Theorem Quotes

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 ~ John Von Neumann

[...] provability is a weaker notion than truth ~ Douglas R. Hofstadter

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. ~ Jordan Ellenberg

Explanation is always incomplete: we can always raise another Why-questions. And the new why-questions may lead to a new theory which not only "explains" the old theory but corrects it.
This is why the evolution of Physics is likely to be an endless process of correction and better approximation. And even if one day we should reach a stage where our theories were no longer open to correction, because they are simply true, they would still not be complete - and we should know it. For Godel's famous incompleteness theorem would come into play: in view of the Mathematical background of Physics, at best an infinite sequence of such true theories would be needed in order to answer the problems which any given (formalized) theory would be undecidable.
Such considerations do not prove that the objective physical world is incomplete, or undetermined: they only show the essential incompleteness of our efforts. But they also show that it's barely possible (if possible at all) for science to reach a stage in which it can provide genuine support for the view that the physical world is deterministic. Why, the, should we not accept the verdict of common sense- at least until these arguments have been refuted? ~ Karl Popper

We are used to things starting out small and simple and then progressing--evolving--to become ever more complex and sophisticated, so this is naturally what we expect to find on archaeological sites. It upsets our carefully structured ideas of how civilizations should behave, how they should mature and develop, when we are confronted by a case like Göbekli Tepe that starts out perfect at the beginning and then slowly devolves until it is just a pale shadow of its former self. ~ Graham Hancock

How I could make myself understand. The restlessness, the anxiety within me. The feeling of incompleteness and dissatisfaction, my frustration with my inability to execute my ideas on the page, either in words or in song. I could not catch my own mind, my thoughts racing past in a blur, like fingers rushing through sixteenth notes without regard to tempo. ~ S. Jae-Jones

Whereever you go we will share the same sky ~ Y.Odabasi

They rode through the quiet streets. The rain had stopped and an early morning mist fell around them under the streetlights. Victor remembered that his ancestors had believed this was a magical time when the gods walked the earth, Götterdämmerung, a time when men slept and divine creatures laid plans that ensnared or released them. He was not such a creature, no; he had to walk step by step on the hard earth beneath his feet and watch tragedies unfold, without shaping them. It was a disappointment to him. ~ J.J. Brown

A judgment is conscious of its own validity. This shows that the measure of validity to which it refers itself in this consciousness is inherent in the nature of judgment: a judgment is subject to this measure not in virtue of any circumstance in which it may find itself, but simply as judgment. Now when we think of an act simply as a judgment, we refer it to the power as an act of which it is a judgment: the power of judgment. Hence, the measure of validity of judgment is nothing other than the power of judgment. A judgment, being conscious of its validity, refers itself to the power from which it springs (as, e.g., St. Thomas Aquinas observes). ~ Sebastian Rödl

She heard Brannie call out to her, but she ignored her as well and kept going. Kept going with visions of a bloody and limb-missing Éibhear dancing through her head. Gods! What if they cut off his wings? Or removed his scales? Oh, by the gods, what if they shaved his head? What if they shaved his head? Nooooo! ~ G.A. Aiken

A personal relationship with God enhances life. First, it enables us to accept our limitations without being frustrated by them. It assures us that problems we can't solve are not necessarily insoluble. Second, when we need it, God offers us a sense of forgiveness, a sense of cleansing from our incompleteness ... Last and perhaps most important, a personal relationship with God redeems us from the fear of death. We needn't be afraid that all our good deeds will vanish when we die. ~ Harold S. Kushner

Any good theorem should have several proofs, the more the better ~ Michael Atiyah

Oh the joy of an idée fixe! The contentment of a life taken up some ideal, any ideal! A gentle trap to catch the infinite, like the sun in a piece of mirror in a child's hand. ~ Georges Rodenbach

Kepler's discovery would not have been possible without the doctrine of conics. Now contemporaries of Kepler - such penetrating minds as Descartes and Pascal - were abandoning the study of geometry ... because they said it was so UTTERLY USELESS. There was the future of the human race almost trembling in the balance; for had not the geometry of conic sections already been worked out in large measure, and had their opinion that only sciences apparently useful ought to be pursued, the nineteenth century would have had none of those characters which distinguish it from the ancien régime. ~ Charles Sanders Peirce

No part of mathematics is ever, in the long run, "useless." Most of number theory has very few "practical" applications. That does not reduce its importance, and if anything it enhances its fascination. No one can predict when what seems to be a most obscure theorem may suddenly be called upon to play some vital and hitherto unsuspected role. ~ C. Stanley Ogilvy

In any event, Socrates' proof of prenatal immortality is that one of Meno's uneducated slave boys actually comes up with the Pythagorean theorem without ever having studied geometry! Therefore, he must be remembering it. You recall that theorem: in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Huh? We can barely remember that from tenth grade, let alone from before we were born. ~ Thomas Cathcart

Our emotional life maps our incompleteness: A creature without any needs would never have reasons for fear, or grief, or hope, or anger. ~ Martha C. Nussbaum

To admit the existence of a need in God is to admit incompleteness in the divine Being. Need is a creature-word and cannot be spoken of the Creator. God has a voluntary relationg to everything He has made, but He has no Necessary relation to anything outside of Himself. His interest in His creatures arises from His sovereign good pleasure, not from any need those creatures can supply nor from any completeness they can dring to Him who is complete in himself. ~ A.W. Tozer

We have been accustomed to make this existence worth-while by the belief that there is more than the outward appearance
that we live for a future beyond this life here. For the outward appearance does not seem to make sense. if living is to end in pain, incompleteness, and nothingness, it seems a cruel and futile experience for being who are born to reason, hope, create, and love. man, as a being of sense, wants his life to make sense, and he has found it hard to believe that it does so unless there is more that what he see
unless there is an eternal order and an eternal life behind the uncertain and momentary experience of life-and-death. ~ Alan W. Watts

I'm a very ordinary man who's worked and fed like everyone else. I'm no longer afraid of dying, but death doesn't seem to want anything to do with me, now that I can see no point in living. I'm afraid he's forgotten me. ~ Emile Zola

A reply to Olbers' attempt in 1816 to entice him to work on Fermat's Theorem. I confess that Fermat's Theorem as an isolated proposition has very little interest for me, because I could easily lay down a multitude of such propositions, which one could neither prove nor dispose of. [] ~ Carl Friedrich Gauss

I carried this problem around in my head basically the whole time. I would wake up with it first thing in the morning, I would be thinking about it all day, and I would be thinking about it when I went to sleep. Without distraction I would have the same thing going round and round in my mind.
(Recalling the degree of focus and determination that eventually yielded the proof of Fermat's Last Theorem.) ~ Andrew John Wiles

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Anything may happen when womanhood has ceased to be a protected occupation. ~ Virginia Woolf

Freedom to leave the persecutors are not so worthless. ~ Mehmet Kececi

And not only narrativity but the quality of the writing is of the first importance to me. Rightly or not, I believe a dull, inept style signals poverty or incompleteness of thought. I see the accuracy, scope, and quality of Darwin's intellect directly expressed in the clarity, strength, and vitality of his writing - the beauty of it. ~ Ursula K. Le Guin

At least you'll never be a vegetable - even artichokes have hearts. ~ Amélie Poulain

Number is the ruler of forms and ideas, and the cause of gods and demons. ~ Pythagoras

There was a seminar for advanced students in Zürich that I was teaching and von Neumann was in the class. I came to a certain theorem, and I said it is not proved and it may be difficult. Von Neumann didn't say anything but after five minutes he raised his hand. When I called on him he went to the blackboard and proceeded to write down the proof. After that I was afraid of von Neumann. ~ George Polya

The fact that scientists do not consciously practice a formal methodology is very poor evidence that no such methodology exists. It could be said - has been said - that there is a distinctive methodology of science which scientists practice unwittingly, like the chap in Molière who found that all his life, unknowingly, he had been speaking prose. ~ Peter Medawar

About Thomas Hobbes: He was 40 years old before he looked on geometry; which happened accidentally. Being in a gentleman's library, Euclid's Elements lay open, and "twas the 47 El. libri I" [Pythagoras' Theorem]. He read the proposition "By God", sayd he, "this is impossible:" So he reads the demonstration of it, which referred him back to such a proposition; which proposition he read. That referred him back to another, which he also read. Et sic deinceps, that at last he was demonstratively convinced of that truth. This made him in love with geometry. ~ John Aubrey

The deepest level of obsession is obsession with a sense of self. A sense of self, generated as a reaction to non-referential space, lies at the core of every habituated pattern. A self is felt to be a permanent, independent unit. The feeling of permanence manifests in life as a feeling of dullness, of not being quite present. The illusion of independence arises as a feeling of separation. The feeling of being one thing arises as a feeling of incompleteness or dissatisfaction. Together, these three qualities obscure the mystery of being. ~ Ken McLeod