Mind vs. Machine: A Philosophical Corollary of Gödel’s Incompleteness Theorem | Philosophy Of Mathematics - Medium

Canadian mathematician Simon Kochen recalled in his tribute to Kurt Gödel how during his PhD exam, he was asked to name five of Gödel’s theorems by Jan Gronwald in Cantor’s Paradise. 

Albert Einstein (left) presenting the first Albert Einstein Award for achievement in natural sciences to Austrian mathematician Kurt Gödel (second from right) and American physicist Julian Schwinger (right), with Lewis L. Strauss looking on, March 14, 1951.

Photo: New York World-Telegram and the Sun Newspaper/Library of Congress, Washington, D.C. (Digital ID cph 3c33518)

The essence of the question was that each of the theorems either gave birth to a new branch of, or revolutionized, modern mathematical logic. “Proof theory, model theory, recursion theory, set theory, intuitionistic logic - all had been transformed by, or, in certain cases, had gotten their inception from, Gödel’s work” (Goldstein, 2005). But among the brilliant achievements of Kurt Gödel one stands out exceptionally.

One need not to be a practicing mathematician in order to grasp the basic idea and message of the Incompleteness Theorem. And maybe that is why this result gained so much audacity in the popular scientific debate. But, of course, this ingenious simplicity is only one of many aspects of the 1931 work that distinguish it from other outstanding works of the Austrian intellectual giant.

It seems to me that what strikes us from the beginning when we come across the Incompleteness Theorem for the first time is the observation that it is not merely one of many mathematical results...

Some thinkers, such as John Lucas and celebrated physicist Roger Penrose (yes, the 2020 Nobel Prize winner) believed Gödel’s and Turing’s work to show with mathematical precision that the human mind “infinitely surpasses the machine”...

 Of course, there is a lot to clarify in the discussion. The notion of the “human mind”, the “abstracted mind” and also of the “machine” still need some explanations (although serious developments in the automata theory and theoretical linguistics seem to give some first results). Not to mention Turing’s concepts of “ingenuity” and “intuition”, and Gödel’s “mathematical intuition” that play a vital role in the debate, but are still very vague.

We go deeper and deeper into the forrest of questions. Some time ago the Incompleteness Theorem seemed to me as a decisive argument that closed a number of discussions. But lately I tend to see the opposite: how many questions it inspires and how philosophically fertile this artwork is.

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Source: Medium