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.
Kurt Gödel was a 20th century Austrian mathematician, logician and philosopher and one of the most unique thinkers in human history. He made significant contributions to the study of formalized mathematical systems and mathematical logic.
Gödel is best known for his Incompleteness Theorems which demonstrate that there are built-in limitations to all axiomatic systems that can describe the arithmetic of the natural numbers. An axiomatic system is consistent if it doesn’t produce any contradictory statements. The First Incompleteness Theorem states that for any consistent system of axioms that can be represented as a collection of mechanical rules, there will always be true statements about the natural numbers that can’t be proved within the system.
To prove this (*), Gödel started with a formal equivalent of the sentence: “this statement is unprovable”. If the sentence is true, then it can’t be proved, but if you can prove that it’s true, then it’s false. What’s more, if the sentence is false, then it can be proved, and is true. Therefore the sentence must be true and unprovable. By developing a code to convert statements like this into numbers, he was able to show that it could not be proved using the axioms of number theory.
The Second Incompleteness Theorem states that any such system of axioms can’t prove its own consistency. We can refer to the sentence “this statement is unprovable” as ‘p’. The proof of the first theorem shows that if the system is consistent, then ‘p’ is true but not provable. If we assume that consistency can be proved from within the system, then the statement ‘p is not provable’ can also be proved in the system. Of course, the statement ‘p is not provable’ is the same statement as ‘p’ itself, so we can prove ‘p’, producing a contradiction. Therefore if the system can prove its own consistency, it is inconsistent.
The Incompleteness Theorems were a major blow to the quest for a single all-encompassing formal system from which all mathematical truth could be derived. Gödel himself, however, believed these results demonstrated the importance of intuition in mathematics. The Incompleteness Theorems would also later be influential in the development of computer science.
Consciousness is connected with one unity. A machine is composed of parts.
After moving to the US, Gödel became friends with Albert Einstein and, as a gift on Einstein’s 70th birthday, demonstrated the existence of solutions to Einstein’s field equations in general relativity. Gödel’s closed curve solutions, or “rotating universes”, would allow for time travel into the past. Einstein helped Gödel through his US citizenship exam, where Gödel tried to explain to the judge that he had found a dangerous loophole in the US Constitution that could allow the US to turn into a Nazi-like dictatorship.
I don’t believe in empirical science. I only believe in a priori truth.
Gödel also produced a formal proof of the existence of God, called Gödel’s ontological proof based on St. Anselm of Canterbury’s ontological argument and similar arguments by René Descartes and Gottfried Leibniz. Recently, two scientists were able to formalize this proof on a laptop, showing that Gödel’s argument was correct.
Gödel eventually starved himself to death because of his fear of being poisoned.
Kurt Gödel was also a philosopher, bringing the precision and clarity of his mind to the ‘spiritual sciences’ as well. Below are some selections from his work.
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.
The notion of existence is one of the primitive concepts with which we must begin as given. It is the clearest concept we have.
The brain is a computing machine connected with a spirit.
The more I think about language, the more it amazes me that people ever understand each other at all.
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…
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.
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.
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.
The formation in geological time of the human body by the laws of physics (or any other laws of similar nature), starting from a random distribution of elementary particles and the field is as unlikely as the separation of the atmosphere into its components. The complexity of the living things has to be present within the material [from which they are derived] or in the laws [governing their formation].
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.
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.
Every error is caused by emotions and education (implicit and explicit); intellect by itself (not disturbed by anything outside) could not err.
(*) Note that this is an extreme simplification of the actual method of proof.
