Reading it again I was reminded of Gödel’s extraordinary theorems. Accepted by mathematicians, they have not only modernised mathematics, showing that mathematical truth is more than logic and computation but also how we might interpret the universe itself. One once said “Does Godel's work imply that someone or something transcends the universe?”.
The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an "effective procedure" is capable of proving all facts about the natural numbers. For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system. The second incompleteness theorem shows that if such a system is also capable of proving certain basic facts about the natural numbers, then one particular arithmetic truth the system cannot prove is the consistency of the system itself.
For over 2000 years Euclidean geometry past the test of time. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other theorems from these. An example is the idea that you can add one to any number and get a bigger number. If Euclid's work had a weak link, it was his fifth axiom, the axiom about parallel lines. Euclid said that if you were given a straight line, you could draw only one other straight line parallel to it through a set point somewhere outside it.
However, in the 19th century Riemann and others created new geometries by saying that there could be two parallel lines through the outside point or no parallel lines. It turned out that Riemann's geometry is better at describing the curvature of space than Euclid's and Einstein in the early 1900s incorporated Riemann's ideas into relativity theory.
Now not only had Riemann created a system of geometry which stood commonsense notions on its head, but the philosopher-mathematician Bertrand Russell had bumped into a serious paradox for set theory. Russell did not feel that this paradox was insurmountable and felt he could create a single self-consistent, self-contained mathematical system. He and Whitehead produced the 3-volume Principia Mathematica. However, even before it was complete, Russell's expectations were dashed.
Godel’s paper "On Formally Undecidable Propositions of Principia Mathematica and Related Systems." In it he showed that a statement in a system could be made to refer to itself in such a way that it said about itself that it was unproveable and shattered naive expectations that human thinking could be reduced to algorithms or that our thoughts can be a mechanical process.
Christians, for example, see man as a spiritual being with understanding that springs not just from the physical organ of the mind but also from soul and spirit.
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