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My question is if there is some concrete symbolic logic at the foundation of human reasoning -something very rudimentary, but still formal? Question may be seen in context of the article given below.

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Refer to the following article by Wilfried Sieg : Godel on Computbility On Page 12 of 19, paragraph 2, author writes: "In a deep sense neither Church nor Godel recognized the genuinely distinctive character of Turing’s analysis, i.e., the move from arithmetically motivated calculations to general symbolic processes that underlie them. Most importantly in the given intellectual context, these processes have to be carried out programmatically by human beings: the Entscheidungsproblem had to be solved by us in a mechanical way; it was the normative demand of radical intersubjectivity between humans that motivated the step from axiomatic to formal systems. It is for this very reason that Turing most appropriately brings in human computers in a crucial way and exploits the limitations of their processing capacities, when proceeding mechanically".

Also, can somebody explain Sieg's claim that the move was from arithmetically motivated calculations to general symbolic processes that underlie them : It is clear Sieg is aware of μ-recursive functions, and their equivalence to Turing machines, then what exactly are these symbolic processes? And what is he trying to say by writing: "the Entscheidungsproblem had to be solved by us in a mechanical way"?

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It seems what you are referring to is what is usually known as the computational theory of the mind, which is the ontological framework behind cognitive science and their most precious toy, artificial intelligence. The answer to your question might be in this article:

https://plato.stanford.edu/entries/computational-mind/

The computational theory of the mind has often been heavily criticized due to problematic assumptions about the nature of the mind. The most well known critic of symbolic logic based AI systems is Hubert Dreyfus in his book What Computers Can't Do (1972). The limits of formal logic and the demise of the Hilbert Program and the logicist project of reducing the totality of mathematics to logic might also be a topic of interest to you:

https://plato.stanford.edu/entries/goedel-incompleteness/ https://plato.stanford.edu/entries/hilbert-program/

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  • Note that there's a difference between "symbolic A.I.", the idea that the AI would be programmed with rules about high-level conceptual thought, and the more general computationalist idea of identifying mental states with computations, including bottom-up simulations of individual neurons and synapses or even atoms in a brain. There's a proof that in quantum physics, any physical system can be simulated.
    – Hypnosifl
    Commented May 14, 2020 at 14:46

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