Roger Penrose argues that Gödel's incompleteness theorems demonstrate that human mathematical understanding transcends computation and therefore, strong AI, which posits that consciousness is computable, is fundamentally flawed. He asserts that humans can grasp the truth of Gödelian sentences (statements unprovable within a formal system yet demonstrably true outside of it), while a computer bound by algorithms within that system cannot. This, Penrose claims, illustrates a non-computable element in human consciousness, suggesting we understand truth through means beyond mere calculation.
Mathematicians are exploring the boundaries of provability using large language models (LLMs) and other automated theorem provers. While these tools can generate novel and valid proofs, they often rely on techniques too complex for human comprehension. This raises questions about the nature of mathematical truth and understanding. If a proof is too long or intricate for any human to verify, does it truly constitute "knowledge"? Researchers are investigating ways to make these computer-generated proofs more accessible and understandable, potentially revealing new insights into mathematical structures along the way. The ultimate goal is to find a balance between the power of automated proving and the human need for comprehensible explanations.
HN commenters discuss the implications of Gödel's incompleteness theorems and the article's exploration of concrete examples in Ramsey theory and Diophantine equations. Some debate the philosophical significance of undecidable statements, questioning whether they represent "true" mathematical facts or merely artifacts of formal systems. Others highlight the practical limitations of proof assistants and the ongoing search for more powerful automated theorem provers. The connection between computability and the physical universe is also raised, with some suggesting that undecidable statements could have physical implications, while others argue for a separation between abstract mathematics and the concrete world. Several commenters express appreciation for the article's clarity in explaining complex mathematical concepts to a lay audience.
Summary of Comments ( 128 )
https://news.ycombinator.com/item?id=43233420
Hacker News users discuss Penrose's argument against strong AI, with many expressing skepticism. Several commenters point out that Gödel's incompleteness theorems don't necessarily apply to the way AI systems operate, arguing that AI doesn't need to be consistent or complete in the same way as formal mathematical systems. Others suggest Penrose misinterprets or overextends Gödel's work. Some users find Penrose's ideas intriguing but remain unconvinced, while others find his arguments simply wrong. The concept of "understanding" is a key point of contention, with some arguing that current AI models only simulate understanding, while others believe that sophisticated simulation is indistinguishable from true understanding. A few commenters express appreciation for Penrose's thought-provoking perspective, even if they disagree with his conclusions.
The Hacker News post discussing Roger Penrose's video on Gödel's theorem and AI elicits a range of comments, mostly focused on the validity and interpretation of Penrose's argument. Several commenters express skepticism towards Penrose's stance. A recurring theme is the perceived gap between Gödel's incompleteness theorems, which deal with formal systems in mathematics, and the practical realities of AI development. Some commenters argue that Penrose misinterprets or overextends the implications of the theorems to suggest consciousness or non-computable aspects of human thought. They contend that even if human thought has non-computable elements, current AI systems are far from reaching that level of complexity, making the discussion somewhat irrelevant to the current state of the field.
Several users highlight the distinction between computational theory and physical implementation. They point out that while theoretical computational models might have limitations, physical systems could potentially bypass those limitations, suggesting that human brains, as physical entities, might not be bound by the same constraints as abstract Turing machines. This argument challenges Penrose's attempt to apply Gödel's theorems directly to the human mind.
Some commenters criticize Penrose's reliance on subjective experience and intuition as insufficient scientific evidence. They argue that claims about consciousness and the nature of understanding require more rigorous and empirical support than philosophical arguments. The notion of "understanding" itself is questioned, with some suggesting that it might be an illusion or an emergent property of complex computations.
A few comments offer alternative perspectives on consciousness and computation. One commenter suggests that while Gödel's theorem might not directly disprove the possibility of strong AI, it highlights the potential for unforeseen limitations in any computational system. Another comment mentions the concept of hypercomputation, suggesting the possibility of computational models beyond Turing machines that might be relevant to understanding the human mind.
While some comments express interest in Penrose's ideas, the overall tone is one of cautious skepticism. Many commenters find Penrose's arguments unconvincing, either due to perceived flaws in his reasoning, lack of empirical evidence, or the perceived irrelevance of Gödel's theorems to the current state of AI development.