Austrian-born American Mathematician, Logician
"Every system of knowledge contains meaningful tenets whose truth or falsity cannot be established if one remains completely within that system."
"I don’t see any reason why we should have less confidence in this kind of perception, I.e., in mathematical intuition, than in sense perception, which induces us to build up physical theories and to expect that future sense perceptions will agree with them and, moreover, to believe that a question not decidable now has meaning and may be decided in the future."
"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. "
"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. "
"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."
"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."
"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 then people."
"Reason and understanding concern two levels of concept. Dialectics and feelings are involved in reason."
"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."
"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."
"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."
"Every error is caused by emotions and education (implicit and explicit); intellect by itself (not disturbed by anything outside) could not err."
"Learn to act correctly: everybody has shortcomings, believes in something wrong, and live to carry out his mistakes."
"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."
"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."
"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."
"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."
"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. "
"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."
"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 farther 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. "
"A consistency proof for [any sufficiently powerful] system… can be carried out only by means of modes of inference that are not formalized in the system… itself."
"Anything you can draw a circle around cannot explain itself without referring to something outside the circle – something you have to assume but cannot prove. [Gödel’s Incompleteness Theorem]"
"Syllogism: 1. All non-trivial computational systems are incomplete. 2. The universe is a non-trivial computational system. 3. Therefore the universe is incomplete."
"Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, but not provable in the theory. [Gödel’s theorem] "