The incompleteness of knowledge systems

In classical logic, a correctly formulated statement is either true or false. “A dog has four legs. Bello is a dog, so Bello has four legs” is true. However “A dog has four legs. Bello is a dog, but Bello is missing a leg” is a clear story in everyday life, but logically untrue. Missing a leg conflicts with the claim that Bello is a dog, because dogs have four legs.

Producing properly worded claims is not easy. This is evidenced by a statement such as “The barber of Seville shaves everyone who does not shave himself”. It is now unclear whether he shaves himself. If he shaves himself, he doesn’t shave himself and vice versa. This paradox arises from an unclear formulation. It is an example of Russell’s paradox, in short: Does the set of all sets that do not contain themselves contain itself of not? If it does, it does not. If it does not, it does.

However, even if the wording is sharp and clear, there can be ambiguity about whether or not a statement is true. Evidence is sometimes hard to find. For example, it took a very long time to find evidence for the four color theorem: “Every map can be colored with four colors without two neighboring countries getting the same color”. The proof, from 1976, is not accepted as proof by some, because it can only be verified by computer.

Very interesting is a theorem of the mathematician Kurt Gödel. He proved that in every system of a certain minimum complexity (for example, the rules of calculation with natural numbers) there are statements that cannot be proven or disproved. So there are truths that can be stated but cannot be formally justified. An example could be Goldbach’s conjecture: every even number is the sum of two primes. For an introduction and discussion see Nagel and Newman.

The consequences of Gödel’s thesis are large and not yet fully understood. For instance, sciences based on mathematics and logic, such as physics and computer science, are incomplete: they may contain correct statements that cannot be proven. There is thereby a need for a logic that is not binary (statements that are not true are false) but ternary (statements are true, false, or indeterminate).

Even in the physical, sharply defined world, it is not fully provable or calculable what will or may happen. Complete knowledge of the laws of nature and the exact observation of the initial situation together are not sufficient to fathom the unfolding reality. Events are possible that do not conflict with the laws of nature, but cannot be deduced from them either. The future cannot therefore be determined unequivocally by the laws of nature derived from past observations.

Print Friendly, PDF & Email
Scroll to Top