A. Gödel's theorems, the move from the perfection of mathematics to the perfection of reality.
During my three years of intensive scientific studies in preparatory classes, I learned key scientific results which would lead me to lose my almost religious confidence that science could solve all the major problems of this world. This would also open my heart and mind to the possibility of spirituality.
In the universe of science, you have a kind of implicit 'hierarchy'. The 'hard sciences,' like mathematics and physics, are considered as sound and reliable. Mathematics give the models that physics can use to describe reality, and the accurate predictions that physics give then support the value and importance of mathematics - an efficient tandem between mathematics and physics.
When you move toward less absolute sciences, like chemistry or biology for instance, there is less respect or trust in the scientific arena. This is perhaps because you cannot really predict precisely results.
If you move to domains like human sciences, like sociology, anthropology or psychology there can be a greater distance and even a defiance. Scientists from the 'hard sciences' will have less respect for what is not precisely predictable and considered such domains at best as fragile models for experimentations but not really as 'science'.
This means that the most solid foundations of science are to be found in mathematics and physics.
By discovering results that severely limited both the mathematical models and the the physics models of this world, I would be led to an almost religious crisis.
A crack in the wall of logic: Kurt Gödel's incompleteness theorems
The first result that shook my belief in science came from a domain that I thought perfect: Logic, in some ways the heart of mathematics.
In 1900, a German mathematician, David Hilbert, had highlighted 23 problems to be solved, in order to find a complete and consistent set of axioms for all mathematics.
(An axiom is a statement accepted as true, that serves as a foundation for a branch of mathematics, like this axiom in Euclidean geometry: In a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the point).
Hilbert's second problem was: to find a proof that the arithmetic is consistent (= free of any internal contradictions).
In 1931, a young Austrian mathematician, Kurt Gödel, would prove that Hilbert's second problem could never be solved. After the amazing successes of the scientific revolution, this result would bring a new awareness of the limitations of any scientific model.
Gödel's incompleteness theorems demonstrate the inherent limitations of every formal axiomatic system containing basic arithmetic. Hilbert's program would never be completed.
In mathematical language, Gödel proved that every non-trivial formal system is either incomplete or inconsistent:
1. For a given (non-trivial) formal system, there will be statements that are true in that system, but which cannot be proved to be true inside the system.
2. If a system can be proved to be complete using its own logic, then there will be a theorem in the system that is contradictory.
In order to attempt a 'simple' illustration of this result, let us consider heuristically the field of human laws. A law is like an axiom in mathematics. In law, you have cases and you want to find who is guilty and not guilty, as in mathematics you have statements that you want to prove as either true or false.
In law, the adaptation of Gödel's theorem would then imply that:
1. either you don't have enough laws and some cases cannot be solved (a person cannot be proved innocent or guilty) -> an incomplete system, or
2. you have too many laws, and you can prove that a person is both guilty and innocent -> a contradictory system
By continuing to add laws we continue to move toward a less incomplete system, but also toward contradictions inside this very legal system. Contradictions would thus allow persons with more time and more lawyers to find more ways to prove a case, even if with the same time and effort the opposite could also be proven.
In simple language, Gödel's demonstration highlighted that logic and therefore mathematics were not perfect, they had inherent limits. Either they could not describe perfectly a situation (incomplete), or they would provide conflicting statements (contradictory).
Gödel's results challenged directly the claim of science to provide a perfect model of reality, through the powerful alliance of mathematics and physics.
Therefore, instead of seeing reality as an external manifestation of a perfect mathematical model, I began to see mathematics as the abstract foundation of an imperfect physical model of reality.
I hope you can sense the radical change this provoked in me. I would no more focus on mathematics as the perfect foundation of reality. I would then begin to focus on reality as more complex and rich than any mathematical or scientific model.
My passion to understand and master science then began to be replaced by a budding passion to understand reality. This was my first step to become open to the existence of a spiritual being.
In the next post, I will describe the impact that the results of quantum mechanics had on my vision of the world, and how, in relationship with 'chaos physics', they provided an even broader opening to the possibility of a spiritual world.
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interesting!