This snippet is from Smale's paper Smale, Steve (1999). "Mathematical problems for the next century". In Arnold, V. I.; Atiyah, M.; Lax, P.; Mazur, B. (eds.). Mathematics: frontiers and perspectives. American Mathematical Society. pp. 271–294. ISBN 978-0821820704.

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Problem 18: Limits of Intelligence

What are the limits of intelligence, both artificial and human?

Penrose (1991) attempts to show some limitations of artificial intelligence. His argumentation brings in the interesting question of whether the Mandelbrot set is decidable (dealt with in [Blum and Smale, 1993]) and implications of the Gödel incompleteness theorem.

However, a broader study is called for, one which involves deeper models of the brain and of the computer, in a search of what artificial and human intelligence have in common, and how they differ. I would look in a direction where learning, problem-solving, and game theory play a substantial role, together with the mathematics of real numbers, approximations, probability, and geometry.

I hope to expand on these thoughts on another occasion.

Did he follow up on these thoughts (hopefully) in a mathematical way? Since his list of $18$ problems is intended to be in mathematics, I was hoping this last question on his list has a mathematical exposition!


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Smale and Felipe Cucker published "On the mathematical foundations of learning" in the Bulletin of the AMS in 2002.

I remember Smale gave a talk at MIT in late 2001 or early 2002 about this work, at which there was some dissatisfaction expressed by members of the audience with his formalism's lack of transparency. In particular, it is not easy to see how to incorporate previously known (or, worse, incompletely known) information into the paper's model of a learning process. Smale made the rather obvious remark that assuming some learning had already taken place meant working in a different Hilbert space, but he admitted that it was not necessarily straightforward to identify that smaller (?) Hilbert space.


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