The following remarks by Kurt Gödel are reproduced in Hao Wang’s From Mathematics lớn Philosophy, Routledge & Kegan Paul, 1974. Bracketed references are khổng lồ page numbers.
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The completeness theorem, mathematically, is indeed an almost trivial consequence of Skolem 1922.1 However, the fact is that, at that time, nobody (including Skolem himself) drew this conclusion. …
Hilbert & Ackerman, in the 1928 edition of their book2 on phường 68, state the completeness question explicitly as an unsolved problem. As far as Skolem is concerned, although in 1922 he proved the required lemma, neverthless, when in his 1928 paper3 (at the bottom of p. 134) he stated a completeness theorem (about refutation), he did not use his lemma of 1922 for the proof. Rather he gave an entirely inconclusive argument. (See p 134, line 10 from below to phường 135, line 3.)
This blindness (or prejudice, or whatever you may gọi it) of logicians is indeed surprising. But I think the explanation is not hard khổng lồ find. It lies in a widespread lack, at that time, of the required epistemological attitude toward metamathematics và toward non-finitary reasoning. …
How indeed could one think of expressing metamathematics in the mathematical systems themselves, if the latter are considered to lớn consist of meaningless symbols which acquire some substitute of meaning only through metamathematics?
1Some remarks on axiomatized mix theory, an address lớn the Fifth Congress of Scandinavian Mathematicians, August 1922, reproduced in Jean Van Heijenoort, From Frege to Gödel: a source book in mathematical logic, 1879–1931, Harvard University Press, 1967, phường 290–301.
2Grundzüge der theoretischen Logik, Springer.
3On mathematical logic, an address khổng lồ the Norwegian Mathematical Association, 22 October 1928, reproduced in From Frege lớn Gödel, p. 508–524.
Turing, in Proc. Lond. Math. Soc. 42 (1936), phường 250, gives an argument which is supposed khổng lồ show that mental procedures cannot carry any farther than mechanical procedures. However, this argument is inconclusive, because it depends on the supposition that that a finite mind is capable of only a finite number of distinguishable states. What Turing disregards completely is the fact that mind, in its use, is not static, but constantly developing. … Therefore, although at each stage of the mind’s development the number of its possible states is finite, there is no reason why this number should not converge khổng lồ infinity in the course of its development. Now there may exist systematic methods of accelerating, specializing, and uniquely determining this development, e.g. By asking the right questions on the basis of a mechanical procedure. But it must be admitted that the precise definition of a procedure of this kind would require a substantial deepening of our understanding of the basic operations of the mind.
The following remarks by Kurt Gödel are reproduced in Hao Wang’s biography Reflections on Kurt Gödel, MIT Press, 1987. Bracketed references are to lớn page numbers.
I don’t consider my work a “facet of the intellectual atmosphere of the early 20th century,” but rather the opposite. It is true that my interest in the foundations of mathematics was aroused by the “Vienna Circle,” but the philsophical consequences of my result, as well as the heuristic principles leading to them, are anything but positivistic or empiricistic. …
I was a conceptual and mathematical realist since about 1925. I have never held the view that mathematics is syntax of language. Rather this view, understood in any reasonable sense, can be disproved by my results. <20>
The more I think about language, the more it amazes me that people ever understand each other at all. <95>
Many symptoms show only too clearly, however, that the primitive concepts
Goödel wrote the following reply to lớn Russell’s assertion in his autobiography, “Gödel turned out khổng lồ be an unadulterated Platonist, và apparently believed that an eternal ‘not’ was laid up in heaven, where virtuous logicians might hope to meet it hereafter.”
Concerning my “unadulterated” Platonism, it is no more “unadulterated” than Russell’s own in 1921 when in the Introduction
There would be no danger of an atomic war if advances in history, the science of right & of state, philosophy, psychology, literature, art, etc. Were as great as in physics. But instead of such progress, one is struck by significant regresses in many of the spiritual sciences. <123>
What I điện thoại tư vấn the theological worldview is the idea that the world and everything in it has meaning & reason, and in particular a good and indubitable meaning. It follows immediately that our worldly existence, since it has in itself at most a very dubious meaning, can only be means lớn the kết thúc of another existence. The idea that everything in the world has a meaning
The following remarks, mostly verbal, by Kurt Gödel are recorded in Hao Wang’s supplemental biography of Gödel, A Logical Journey, MIT Press, 1996. Bracketed references are khổng lồ the quotation numbers in the text.
The notion of existence is one of the primitive concepts with which wemust begin as given. It is the clearest concept we have. <4.4.12>
power nguồn is a chất lượng which enables one to lớn reach one’s goals. …Yet a preoccupation with power distracts us from paying attention khổng lồ what isat the foundation of the world … . <4.4.14>
In principle, we can know all of mathematics. It is given to us in itsentirety & does not change. … That part of it of which we have aperfect view seems beautiful, suggesting harmony; that is that all theparts fit together although we see fragments of them only. …Mathematics is applied lớn the real world và has proved fruitful. Thissuggests that the mathematical parts & the empirical parts are in harmonyand the real world is also beautiful. <4.4.18>
Positivists decline lớn acknowledge any a priori knowledge. They wishto reduce everything to sense perceptions. Generally they contradictthemselves in that they deny introspection as experience. … They usetoo narrow a notion of experience và introduce an arbitrary bound on whatexperience is … . <5.4.5>
One bad effect of logical positivism is its claim of being intimatelyassociated with mathematical logic. As a result, other philosophers tendto distance themselves from mathematical xúc tích và ngắn gọn and therewith deprivethemselves of the benefits of a precise way of thinking. <5.4.7>
The brain is a computing machine connected with a spirit. <6.1.19>
Consciousness is connected with one unity. A machine is composed ofparts. <6.1.21>
The active intellect works on the passive intellect which somehowshadows what the former is doing và helps us as a medium. <6.1.22>
I don’t think the brain came in the Darwinian manner. In fact, itis disprovable. Simple mechanism can’t yield the brain. I think thebasic elements of the universe are simple. Life force is a primitive elementof the universe và it obeys certain laws of action. These laws are notsimple, và they are not mechanical. <6.2.12>
Philosophy as an exact theory should vị for metaphysics as much asNewton did for physics. I think it is perfectly possible that thedevelopment of such a philosophical theory will take place within thenext hundred years or even sooner <7.3.6; portion misquoted, corrected phường 288, 332>
A phối is a unity of which its elements are the constituents. It is afundamental property of the mind to lớn comprehend multitudes into unities.Sets are multitudes which are also unities. A multitude is the opposite ofa unity. How can anything be both a multitude and a unity? Yet a mix isjust that. It is a seemingly contradictory fact that sets exist. It issurprising that the fact that multitudes are also unities leads khổng lồ nocontradictions: this is the main fact of mathematics. Thinking aplurality together seems like a triviality: và this appears khổng lồ explainwhy we have no contradiction. But “many things for one” is farfrom trivial. <8.2.2>
khổng lồ arrive at the totality of integers involves a jump. Overviewing itpresupposes an infinite intuition. What is given is a psychologicalanalysis. The point is whether it produces objective conviction. <8.2.11>
Reason & understanding concern two levels of concept. Dialectics andfeelings are involved in reason. <8.4.9>
Religion may also be developed as a philosophical system built onaxioms. In our time rationalism is used in an absurdly narrow sense… . Rationalism involves not only logical concepts. Churches deviatedfrom religion which had been founded by rational men. The rationalprinciple behind the world is higher than people. <8.4.10>
General philosophy is a conceptual study, for which method isall-important. <9.0.1>
Whole and part—partly concrete parts and partly abstractparts—are at the bottom of everything. They are most fundamental inour conceptual system. Since there is similarity, there are generalities.Generalities are just a fundamental aspect of the world. It is a fundamentalfact of reality that there are two kinds of reality: universals andparticulars. <9.1.24>
Whole & unity; thing or entity or being. Every whole is a unity andevery unity that is divisible is a whole. For example, the primitiveconcepts, the monads, the empty set, & the unit sets are unities but notwholes. Every unity is something và not nothing. Any unity is a thing oran entity or a being. Objects and concepts are unities & beings.<9.1.25>
Philosophical thinking differs from thinking in general. It leaves outattention to lớn objects but directs attention to inner experiences. (It isnot so hard if one also directs attention to lớn objects.) to lớn developthe skill of introspection and correct thinking
Every error is caused by emotions and education (implicit andexplicit); intellect by itself (not disturbed by anything outside) couldnot err. <9.2.5>
Don’t collect data. If you know everything about yourself, youknow everything. There is no use burdening yourself with a lot of data.Once you understand yourself, you understand human nature & then the restfollows. <9.2.6>
Intuition is not proof; it is the opposite of proof. We vì chưng not analyzeintuition lớn see a proof but by intuition we see something without a proof.<9.2.46>
It is a mistake to lớn argue rather than report. This is the same mistakethe positivists make: to prove everything from nothing. A large part isnot lớn prove but to call attention to certain immediately given but notprovable facts. It is futile khổng lồ try lớn prove what is given. <9.3.2>
khổng lồ explain everything is impossible: not realizing this fact producesinhibition. <9.3.7>
khổng lồ be overcritical và reluctant to use what is given hampers success.To reach the highest degree of clarity và general philosophy, empiricalconcepts are also important. <9.3.8>
Analysis, clarity & precision all are of great value, especially inphilosophy. Just because a misapplied clarity is current or the wrong sortof precision is stressed, that is no reason to give up clarity ofprecision. Without precision, one cannot bởi anything in philosophy.<9.3.9>
Practical reason is concerned with propositions about what oneshould do. For example, stealing does not pay. Will is theopposite of reason. This world is just for us lớn learn. <9.3.15>
Learn khổng lồ act correctly: everybody has shortcomings, believes insomething wrong, và lives to lớn carry out his mistakes. <9.3.16>
Rules of right behavior are easier to lớn find than the foundations ofphilosophy. <9.3.17>
True philosophy is precise but not specialized. <9.3.21>
The meaning of the world is the separation of wish and fact. Wish is aforce as applied khổng lồ thinking beings, to realize something. A fulfilledwish is a union of wish & fact. The meaning of the whole world is theseparation & the union of fact and wish. <9.4.3>
By definition, wish is directed to lớn being something. Love is wishdirected lớn the being of something, and hate is wish directed to thenonbeing of something. <9.4.7>
The maximum principle for the fulfilling of wishes guides the buildingup of the world by requiring that it be the best possible. <9.4.8>
In materialism all elements behave the same. It is mysterious to lớn thinkof them as spread out và automatically united. For something khổng lồ be awhole, it has to have an additional object, say, a soul or a mind.“Matter” refers to one way of perceiving things, and elementaryparticles are a lower size of mind. Mind is separate from matter. <9.4.12>
When an extremely improbable situation arises, we are entitled to drawlarge conclusions from it. <9.4.14>
My philosophical viewpoint
The world is rational. Human reason can, in principle, be developed more highly (through certain techniques). There are systematic methods for the solution of all problems. There are other worlds và rational beings of a different và higher kind. The world in which we live is not the only one in which we shall live or have lived. There is incomparably more knowable a priori that is currently known. The development of human thought since the Renaissance is thoroughly one-dimensional. Reason in mankind will be developed in every direction. Formal rights comprise a real science. Materialism is false. The higher beings are connected to lớn the others by analogy, not by composition. Concepts have an objective existence. There is a scientific (exact) philosophy và theology, which deals with concepts of the highest abstractness; and this is also most highly fruitful for science. Religions are, for the most part, bad—but religion is not.<9.4.17>
Our total reality & total existence are beautiful và meaningful . .. . We should judge reality by the little which we truly know of it.Since that part which conceptually we know fully turns out to lớn be sobeautiful, the real world of which we know so little should also bebeautiful. Life may be miserable for seventy years & happy for a millionyears: the short period of misery may even be necessary for the whole.
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<9.4.20>
If it were true
Time is no specific character of being. … I do not believe in theobjectivity of time. The concept of Now never occurs in scienceitself … .