What do Gödel’s incompleteness theorems and Artificial Intelligence have in common?
Gödel and Artificial Intelligence address fundamental questions about knowledge and data processing limits.
Gödel showed with his incompleteness theorems that any consistent formal system strong enough to describe the fundamental properties of arithmetic is necessarily incomplete; This means there will always be true mathematical statements that cannot be proved within the system.
“The purpose of computing is insight, not numbers.”
~ Richard Hamming
In the field of artificial intelligence, the development of machine learning algorithms and the increasing ability of artificial systems to perform tasks previously thought to require human intelligence has raised questions about the limits of what computers can do and what it means for a method to understand or know something truly. Aligning AI to ensure that AI systems act in ways consistent with human values is one way to explore these questions.
Gödel’s incompleteness theorems and the development of AI highlight the need for a deeper understanding of the relationship between knowledge, truth, and computation. Both have important implications for the future of technology and humanity.
A simple and telling example of the connection between Gödel’s incompleteness theorems and AI:
Suppose we want to develop an AI system that can prove mathematical theorems. In this case, we can train the system with a large corpus of mathematical knowledge and then let it generate its mathematical proofs. However, according to Gödel’s incompleteness theorems, there will always be true mathematical statements that cannot be proven in any formal system, including the one implemented by our AI.
This means that there will always be some mathematical truths that our AI cannot prove, no matter how sophisticated its training is or how extensive its mathematical knowledge base is. The whole thing raises questions about the limits of what can be computed and known and underscores the need for a more sophisticated understanding of knowledge and truth.
In this sense, the example shows how Gödel’s incompleteness theorems and AI intersect and inform each other and underscores the importance of considering both the mathematical foundations and the implications of new technologies in our quest for knowledge and understanding.
Note:
It is a very superficial view of the subject but primarily about sensitizing the mind to it ;-)
Murat
Author of the books “MINDFUL AI — Reflections on Artificial Intelligence” and “A Primer to the 42 Most commonly used Machine Learning Algorithms (With Code Samples)”
