The following remarks by Kurt Gödel are reproduced in Hao Wang’s From Mathematics to Philosophy, Routledge & Kegan Paul, 1974. Bracketed references are to page numbers.
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 and Ackerman, in the 1928 edition of their book2 on p. 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 p. 135, line 3.)
This blindness (or prejudice, or whatever you may call it) of logicians is indeed surprising. But I think the explanation is not hard to find. It lies in a widespread lack, at that time, of the required epistemological attitude toward metamathematics and toward non-finitary reasoning. …
[M]y objectivistic conception of mathematics and metamathematics in general, and of transfinite reasoning in particular, was fundamental also to my other work in logic.
How indeed could one think of expressing metamathematics in the mathematical systems themselves, if the latter are considered to consist of meaningless symbols which acquire some substitute of meaning only through metamathematics? [emphasis his] [8–9]
1Some remarks on axiomatized set theory, an address to 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, p. 290–301.
2Grundzüge der theoretischen Logik, Springer.
3On mathematical logic, an address to the Norwegian Mathematical Association, 22 October 1928, reproduced in From Frege to Gödel, p. 508–524.
Turing, in Proc. Lond. Math. Soc. 42 (1936), p. 250, gives an argument which is supposed to 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 to 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. [emphasis his] [325]
The following remarks by Kurt Gödel are reproduced in Hao Wang’s biography Reflections on Kurt Gödel, MIT Press, 1987. Bracketed references are to 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 [of mathematical logic] need further elucidation. It seems reasonable to suspect that it is this incomplete understanding of the foundations which is responsible for the fact that mathematical logic has up to now remained so far behind the high expectations of Peano and others who (in accordance with Leibnitz’s claims) had hoped that it would facilitate theoretical mathematics to the same extent as the decimal system of numbers has facilitated numerical computations. For how can one expect to solve mathematical problems by mere analysis of the concepts occurring, if our analysis so far does not even suffice to set up the axioms? [110]
Goödel wrote the following reply to Russell’s assertion in his autobiography, “Gödel turned out to be an unadulterated Platonist, and 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 [to Mathematical Philosophy, 1919, p. 169] he said “[Logic is concerned with the real world just as truly as zoology, though with its more abstract and general features.]” At that time evidently Russell had met the “not” even in this world, but later on under the influence of Wittgenstein he chose to overlook it. [112]
There would be no danger of an atomic war if advances in history, the science of right and 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 call the theological worldview is the idea that the world and everything in it has meaning and 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 to the end of another existence. The idea that everything in the world has a meaning [reason] is an exact analogue of the principle that everything has a cause, on which rests all of science. [217]
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 to the quotation numbers in the text.
The notion of existence is one of the primitive concepts with which we must begin as given. It is the clearest concept we have. [4.4.12]
Power is a quality which enables one to reach one’s goals. … Yet a preoccupation with power distracts us from paying attention to what is at the foundation of the world … . [4.4.14]
In principle, we can know all of mathematics. It is given to us in its entirety and does not change. … That part of it of which we have a perfect view seems beautiful, suggesting harmony; that is that all the parts fit together although we see fragments of them only. … Mathematics is applied to the real world and has proved fruitful. This suggests that the mathematical parts and the empirical parts are in harmony and the real world is also beautiful. [4.4.18]
Positivists decline to acknowledge any a priori knowledge. They wish to reduce everything to sense perceptions. Generally they contradict themselves in that they deny introspection as experience. … They use too narrow a notion of experience and introduce an arbitrary bound on what experience is … . [5.4.5]
One bad effect of logical positivism is its claim of being intimately associated with mathematical logic. As a result, other philosophers tend to distance themselves from mathematical logic and therewith deprive themselves 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 of parts. [6.1.21]
The active intellect works on the passive intellect which somehow shadows what the former is doing and helps us as a medium. [6.1.22]
I don’t think the brain came in the Darwinian manner. In fact, it is disprovable. Simple mechanism can’t yield the brain. I think the basic elements of the universe are simple. Life force is a primitive element of the universe and it obeys certain laws of action. These laws are not simple, and they are not mechanical. [6.2.12]
Philosophy as an exact theory should do for metaphysics as much as Newton did for physics. I think it is perfectly possible that the development of such a philosophical theory will take place within the next hundred years or even sooner [7.3.6; portion misquoted, corrected p. 288, 332]
A set is a unity of which its elements are the constituents. It is a fundamental property of the mind to comprehend multitudes into unities. Sets are multitudes which are also unities. A multitude is the opposite of a unity. How can anything be both a multitude and a unity? Yet a set is just that. It is a seemingly contradictory fact that sets exist. It is surprising that the fact that multitudes are also unities leads to no contradictions: this is the main fact of mathematics. Thinking a plurality together seems like a triviality: and this appears to explain why we have no contradiction. But “many things for one” is far from trivial. [8.2.2]
To arrive at the totality of integers involves a jump. Overviewing it presupposes an infinite intuition. What is given is a psychological analysis. The point is whether it produces objective conviction. [8.2.11]
Reason and understanding concern two levels of concept. Dialectics and feelings are involved in reason. [8.4.9]
Religion may also be developed as a philosophical system built on axioms. In our time rationalism is used in an absurdly narrow sense … . Rationalism involves not only logical concepts. Churches deviated from religion which had been founded by rational men. The rational principle behind the world is higher than people. [8.4.10]
General philosophy is a conceptual study, for which method is all-important. [9.0.1]
Whole and part—partly concrete parts and partly abstract parts—are at the bottom of everything. They are most fundamental in our conceptual system. Since there is similarity, there are generalities. Generalities are just a fundamental aspect of the world. It is a fundamental fact of reality that there are two kinds of reality: universals and particulars. [9.1.24]
Whole and unity; thing or entity or being. Every whole is a unity and every unity that is divisible is a whole. For example, the primitive concepts, the monads, the empty set, and the unit sets are unities but not wholes. Every unity is something and not nothing. Any unity is a thing or an entity or a being. Objects and concepts are unities and beings. [9.1.25]
Philosophical thinking differs from thinking in general. It leaves out attention to objects but directs attention to inner experiences. (It is not so hard if one also directs attention to objects.) To develop the skill of introspection and correct thinking [is to learn] in the first place what you have to disregard. The ineffectiveness of natural thinking comes from being overwhelmed by an infinity of possibilities and facts. In order to go on, you have to know what to leave out; this is the essence of effective thinking. [9.2.4]
Every error is caused by emotions and education (implicit and explicit); intellect by itself (not disturbed by anything outside) could not err. [9.2.5]
Don’t collect data. If you know everything about yourself, you know everything. There is no use burdening yourself with a lot of data. Once you understand yourself, you understand human nature and then the rest follows. [9.2.6]
Intuition is not proof; it is the opposite of proof. We do not analyze intuition to see a proof but by intuition we see something without a proof. [9.2.46]
It is a mistake to argue rather than report. This is the same mistake the positivists make: to prove everything from nothing. A large part is not to prove but to call attention to certain immediately given but not provable facts. It is futile to try to prove what is given. [9.3.2]
To explain everything is impossible: not realizing this fact produces inhibition. [9.3.7]
To be overcritical and reluctant to use what is given hampers success. To reach the highest degree of clarity and general philosophy, empirical concepts are also important. [9.3.8]
Analysis, clarity and precision all are of great value, especially in philosophy. Just because a misapplied clarity is current or the wrong sort of precision is stressed, that is no reason to give up clarity of precision. Without precision, one cannot do anything in philosophy. [9.3.9]
Practical reason is concerned with propositions about what one should do. For example, stealing does not pay. Will is the opposite of reason. This world is just for us to learn. [9.3.15]
Learn to act correctly: everybody has shortcomings, believes in something wrong, and lives to carry out his mistakes. [9.3.16]
Rules of right behavior are easier to find than the foundations of philosophy. [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 a force as applied to thinking beings, to realize something. A fulfilled wish is a union of wish and fact. The meaning of the whole world is the separation and the union of fact and wish. [9.4.3]
By definition, wish is directed to being something. Love is wish directed to the being of something, and hate is wish directed to the nonbeing of something. [9.4.7]
The maximum principle for the fulfilling of wishes guides the building up 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 think of them as spread out and automatically united. For something to be a whole, it has to have an additional object, say, a soul or a mind. “Matter” refers to one way of perceiving things, and elementary particles are a lower form of mind. Mind is separate from matter. [9.4.12]
When an extremely improbable situation arises, we are entitled to draw large 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 and rational beings of a different and 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 the others by analogy, not by composition.
- Concepts have an objective existence.
- There is a scientific (exact) philosophy and 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 and total existence are beautiful and 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 be so beautiful, the real world of which we know so little should also be beautiful. Life may be miserable for seventy years and happy for a million years: the short period of misery may even be necessary for the whole. [9.4.20]
If it were true [that there are mathematical problems undecidable by the human mind] it would mean that human reason is utterly irrational in asking questions it cannot answer, while asserting emphatically that only reason can answer them. Human reason would then be very imperfect and, in some sense, even inconsistent, in glaring contradiction to the fact that those parts of mathematics which have been systematically and completely developed show an amazing degree of beauty and perfection. In these fields, by entirely unexpected laws and procedures, means are provided not only for solving all relevant problems, but also solving them in a most beautiful and perfectly feasible manner. [9.4.22]
Time is no specific character of being. … I do not believe in the objectivity of time. The concept of Now never occurs in science itself … .
The following are excerpts from “The modern development of the foundations of mathematics in the light of philosophy”, a paper written Kurt Gödel in 1961 but not publshed during his lifetime. It was published posthumously in Kurt Gödel, Collected Works, Volume III: Unpublished Essays and Lectures, Oxford University Press, 1995, p. 374–387.
… [T]he most fruitful principle for gaining an overall view of the possible world-views will be to divide them up according to the degree and the manner of their affinity to or, respectively, turning away from metaphysics (or religion). In this way we immediately obtain a division into two groups: scepticism, materialism and positivism stand on one side, spiritualism, idealism and theology on the other. …
As far as the rightness and wrongness, or, respectively, truth and falsity, of these two directions is concerned, the correct attitude appears to me to be that the truth lies in the middle or consists of a combination of the two conceptions. …
[T]here is no reason to trust blindly in the spirit of the time … [T]his means that the certainty of mathematics is to be secured not by proving certain properties by a projection onto material systems—namely, the manipulation of physical symbols but rather by cultivating (deepening) knowledge of the abstract concepts themselves which lead to the setting up of these mechanical systems, and further by seeking, according to the same procedures, to gain insights into the solvability, and the actual methods for the solution, of all meaningful mathematical problems. …
[T]here exists today the beginning of a science which claims to possess a systematic method for such a clarification of meaning, and that is the phenomenology4 founded by Husserl. Here clarification of meaning consists in focusing more sharply on the concepts concerned by directing our attention in a certain way, namely, onto our own acts in the use of these concepts, onto our powers in carrying out our acts, etc. But one must keep clearly in mind that this phenomenology is not a science in the same sense as the other sciences. Rather it is or in any case should be a procedure or technique that should produce in us a new state of consciousness in which we describe in detail the basic concepts we use in our thought, or grasp other basic concepts hitherto unknown to us. I believe there is no reason at all to reject such a procedure at the outset as hopeless. …
If one considers the development of a child, one notices that it proceeds in two directions: it consists on the one hand in experimenting with the objects of the external world and with its own sensory and motor organs, on the other hand in coming to a better and better understanding of language, and that means—as soon as the child is beyond the most primitive designating of objects—of the basic concepts on which it rests. With respect to the development in this second direction, one can justifiably say that the child passes through states of consciousness of various heights. …
[O]ne may view the whole development of empirical science as a systematic and conscious extension of what the child does when it develops in the first direction. The success of this procedure is indeed astonishing and far greater than one would expect a priori: after all, it leads to the entire technological development of recent times. …
[E]ven without the application of a systematic and conscious procedure, but entirely by itself, a considerable further development takes place in the second direction, one that transcends ‘common sense’. … [I]n the systematic establishment of the axioms of mathematics, new axioms, which do not follow by formal logic from those previously established, again and again become evident. It is not at all excluded by the negative results mentioned earlier that nevertheless every clearly posed mathematical yes-or-no question is solvable in this way. For it is just this becoming evident of more and more new axioms on the basis of the meaning of the primitive notions that a machine cannot imitate. …
4Phenomenology: A philosophy or method of inquiry based on the premise that reality consists of objects and events as they are preceived or understood in human consciousness and not of anything independent of human consciousness. (The American Heritage Dictionary of the English Language, 4th ed., Houghton Mifflin, 2000)
The following remark by Kurt Gödel is reproduced in Abraham Robinson’s Non-Standard Analysis, revised edition, North-Holland, 1974, in the author’s preface on p. x. Robinson states that Gödel made these remarks after a talk that Robinson gave at the Institute for Advanced Study in Princeton in March 1973.
I would like to point out a fact that was not explicitly mentioned by Professor Robinson, but seems quite important to me; namely that non-standard analysis frequently simplifies substantially the proofs, not only of elementary theorems, but also of deep results. This is true, e.g., also for the proof of the existence of invariant subspaces for compact operators, disregarding the improvement of the result; and it is true in an even higher degree in other cases. This state of affairs should prevent a rather common misinterpretation of non-standard analysis, namely the idea that it is some kind of extravagance or fad of mathematical logicians. Nothing could be further from the truth. Rather there are good reasons to believe that non-standard analysis, in some version or other, will be the analysis of the future.
One reason is the just mentioned simplification of proofs, since simplifications facilitates discovery. Another, even more convincing reason, is the following: Arithmetic starts with the integers and proceeds by successively enlarging the number system by rational and negative numbers, irrational numbers, etc. But the next quite natural step after the reals, namely the introduction of infinitesimals, has simply been omitted. I think, in coming centuries it will be considered a great oddity in the history of mathematics that the first exact theory of infinitesimals was developed 300 years after the invention of the differential calculus. I am inclined to believe that this oddity has something to do with another oddity relating to the same span of time, namely the fact that such problems as Fermat’s, which can be written down in ten symbols of elementary arithmetic, are still unsolved 300 years after they have been posed.5 Perhaps the omission mentioned is largely responsible for the fact that, compared to the enormous development of abstract mathematics, the solution of concrete numerical problems was left far behind.
5Gödel is apparently referring to Fermat’s Last Theorem, which was proved in 1995 but was unproved at the time he made this remark.
The following are from “Conversations with Gödel”, in Rudy Rucker’s Infinity and the Mind, Princeton University Press, 1995. Rucker is speaking unless otherwise noted. Bracketed references are to page numbers.
Gödel certainly impressed me as a man who had freed himself from the mundane struggle. I visited him in the Institute office three times in 1972, and if there is one single thing I remember most, it is his laughter. … The conversation and laughter of Gödel were almost hypnotic. Listening to him I would be filled with the feeling of perfect understanding. He, for his part, was able to follow any of my chains of reasoning to its end almost as soon as I had begun it. What with his strangely informative laughter and his practically instantaneous grasp of what I was saying, a conversation with Gödel was very much like direct telepathic communication. [165–6]
Despite his vast knowledge, he still could discuss ideas with the zest and openness of a young man. If I happened to say something particularly stupid or naive, his response was not mockery, but rather an amused astonishment that anyone could think such a thing. [166–7]
Gödel seemed to believe that not only is the future already there, but worse, that it is, in principle, possible to predict completely the actions of some given person. I objected that if there were a completely accurate theory predicting my actions, then I could prove the theory false—by learning the theory and then doing the opposite of what it predicted. According to my notes, Gödel’s response went as follows:
It should be possible to form a complete theory of human behavior, i.e., to predict from the hereditary and environmental givens what a person will do. However, if a mischievous person learns of this theory, he can act in a way so as to negate it. Hence I conclude that such a theory exists, but that no mischievous person will learn of it. In the same way, time travel is possible, but no person will ever manage to kill his past self.
Gödel laughed his laugh then, and concluded,
The a priori is greatly neglected. Logic is very powerful.
Apropos of the free will question, on another occasion he said:
There is no contradiction between free will and knowing in advance precisely what one will do. If one knows oneself completely then this is the situation. One does not deliberately do the opposite of what one wants. [168]
There is one idea truly central to Gödel’s thought that we discussed at some length. This is the philosophy underlying Gödel’s credo, “I do objective mathematics.” By this, Gödel meant that mathematical entities exist independently of the activities of mathematicians, in much the same way that the stars would be there even if there were no astronomers to look at them. For Gödel, mathematics, even the mathematics of the infinite, was an essentially empirical science. According to this standpoint, which mathematicians call Platonism, we do not create the mental objects we talk about. Instead, we find them, on some higher plane that the mind sees into, by a process not unlike sense perception. [168–9]
The last time I spoke with Kurt Gödel was on the telephone, in March 1977. I had been studying the problem of whether machines can think, and I had become interested in the distinction between a system’s behavior and the underlying mind or consciousness, if any.
… I had begun to think that consciousness is really nothing more than simple existence. By way of leading up to this, I asked Gödel if he believed there is a single Mind behind all the various appearances and activities of the world.
He replied that, yes, the Mind is the thing that is structured, but that the Mind exists independently of its individual properties.
I then asked if he believed that the Mind is everywhere, as opposed to being localized in the brains of people.
Gödel replied, “Of course. This is the basic mystic teaching.” [170]
… [T]hen I asked him my last question: “What causes the illusion of the passage of time?” … Finally he said this:
The illusion of the passage of time arises from the confusing of the given with the real. Passage of time arises because we think of occupying different realities. In fact, we occupy only different givens. There is only one reality. [170–1]
I wanted to visit Gödel again, but he told me that he was too ill. In the middle of January 1978, I dreamed I was at his bedside.
There was a chess board on the covers in front of him. Gödel reached his hand out and knocked the board over, tipping the men onto the floor. The chessboard expanded to an infinite mathematical plane. And then that, too, vanished. There was a brief play of symbols, and then emptiness—an emptiness flooded with even white light.
The next day I learned that Kurt Gödel was dead. [171]