Objections based on the principles of mathematics

Thanks to the work of Turing and Gödel, it is widely known that some mathematical questions cannot even be answered in principle within the framework of concrete formal systems. In this case, the most famous was the Gödel incompleteness theorem. Briefly, this theorem can be formulated in the following way: for any formal axiomatic system F that is powerful enough for arithmetic to be represented in it, it is possible to construct the so-called “Gödel sentence” G (F) with the properties described below.

• G (F) is a sentence of the system F, but it cannot be proved in the framework of F.
• If system F is consistent, then the sentence G (F) is true.

Philosophers, including J. R. Lucas, argued that this theorem shows that machines as thinking subjects will always stand below people, because machines are formal systems limited by the incompleteness theorem (they are not able to establish the truth about Gödel’s own proposals), and people have no such limitation.

The debate around this statement lasted for several decades and generated a huge amount of literature, including two books by the mathematician Sir Roger Penrose, who repeated this statement with some new twists (such as the hypothesis that a person is different from a machine, because his brain acts on quantum gravity ).

First, the Gödel incompleteness theorem applies only to formal systems that are powerful enough to represent arithmetic in them. Such formal systems include Turing machines, so Lucas’s statement is partly based on the assumption that computers are Turing machines. This is a good approximation, but not entirely justified. Turing machines are infinite, and computers are finite, so any computer can be described as a (very large) system in propositional logic that is not covered by the Gödel incompleteness theorem.

Secondly, the agent should not be ashamed of the fact that he cannot determine the truth of a certain statement, while other agents may. Consider the suggestion below. J.R. Lucas cannot conclusively assert that this sentence is true.

If Lucas had confirmed the truth of this sentence, he would have contradicted himself, therefore Lucas cannot unquestionably confirm the truth of this sentence, which means that it must be true. (This proposal cannot be false, because if Lucas could not unquestionably confirm it, then it would be true.) Thus, we demonstrated that there is such a proposal that Lucas cannot undeniably confirm, while other people (and cars) can. But because of this, no one has the right to change his opinion about Lukas for the worse.

As another example, let us point out that no one person in his entire life can calculate the sum of 10 billion ten-digit numbers, and a computer is able to perform such an operation in seconds. However, we do not view this fact as evidence of a fundamental limitation of a person’s ability to think. People behaved intellectually thousands of years before they invented mathematics, so it is unlikely that the ability to form mathematical reasoning plays a more than peripheral role in what is meant by the concept of intellectuality.

Thirdly (and this is the most important objection), even if one accepts the assumption that computers are limited in what they are able to prove, there is no reason to believe that these restrictions do not apply to people. It is too easy to argue this dispute by rigorously proving that the formal system cannot perform action X, and then announcing that people can perform action X using their human informal methods, but without giving any evidence in favor of this statement. Indeed, it is impossible to prove that Godel's theorem on incompleteness does not apply to formal reasoning conducted by people, since any rigorous proof itself must contain the formalization of the abilities of human genius, which many claim to be non-formalizable, and therefore must refute itself.

Thus, we can only resort to the intuitive notion that people are sometimes able to show the superhuman traits of mathematical insight. Such statements are expressed in the form of arguments: "we must be sure that we think correctly, because otherwise thinking becomes impossible at all." But if we are talking about this, then the propensity of people to make mistakes has long been known.