There is a different calculability: what Hilbert and Gödel discovered

There is a different calculability: what Hilbert and Gödel discovered

In the XX century, the German mathematician David Hilbert was giving a lot of focus on the question of calculability, attempting to axiomise the entire mathematics. He believed that in order to achieve this goal, it was necessary to prove the consistency and logical completeness of the arithmetic of natural numbers.

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Hilbert introduced several mandatory requirements for an ideal axiomatic theory: completeness, consistency, independence of axioms. He gave it all the collective title of logical computability, implying that if concepts were properly introduced and explained, all mathematics could be proven consistently.

As a result, Hilbert was never able to completely bring calculations down to logical explanations and formulations of all terms and values used in calculations. Hilbert was "halted" by another brilliant logician, mathematician and philosopher of mathematics, Kurt Gödel. In contrast to Hilbert’s problems, he developed incompleteness theorems that demonstrated the impossibility of realizing Hilbert’s problems — and the latter accepted this.

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Science offers its own principles of calculation. AI uses a combination of methods and principles to train its models and solve a range of problems, but without a coherent "scientific" explanation of how this occurs. However, neither scientific principles nor AI principles can solve truly complex problems with the efficiency the human brain demonstrates when doing it.

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The process of creating an ideal axiomatic theory was called "Hilbert's formalism." Other mathematicians saw it as pointless playing with formulas that wasn’t rooted in natural science, that is, in the real world.

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Incompleteness theorems speak about the fundamental limitations of formal arithmetic, and, therefore, also of the limitations of any formal system in which basic arithmetic concepts can be defined: natural numbers, including 0 and 1, addition and multiplication.

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Gödel explained that the principles, the foundations of mathematics, raised numerous questions, as well as the very concept of calculation did. Calculation in the broad sense of the word is not finite. It cannot be argued that the calculations that are used today, describe everything completely, accurately, and in the only way possible. It turned out that different ways of calculation existed, as well as different principles of calculations.