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I'm just being lazy and quoting a claim from Wikipedia's entry on the Penrose-Lucas argument, allegedly referencing "The Emperor's New Mind", but without any sp

I'm just being lazy and quoting a claim from Wikipedia's entry on the Penrose-Lucas argument, allegedly referencing "The Emperor's New Mind", but without any specific location within that book. It has been a while since I read it, and I may attempt to find a more precise summary of Penrose's claim (he has written about it extensively elsewhere, and so have many other people) but I can't do that immediately.

benlivengood 4y ago
I think that Penrose basically thinks it's useful that humans can heuristically step up from a logic system S to S+"statement X is true" in an almost-grounded way, but in reality we are all actually constrained by Godel, Tarski, etc. We just trick ourselves into assuming we can inductively keep stepping up from S+"X is true" indefinitely for any practical problem we encounter, but that is actually an ungrounded assumption. What seems like an even better formal solution is something like logical induction[0] that works over something like probabilities instead of truth/falsehood. It has an existence proof but not a constructive formalization, afaik, but maybe some recent progress has been made. 0: https://arxiv.org/abs/1609.03543 https://arxiv.org/abs/1609.03543
- YeGoblynQueenne 4y ago
  Thanks for the paper, I need to read that. Much of my work is on inductive logic. >> I think that Penrose basically thinks it's useful that humans can heuristically step up from a logic system S to S+"statement X is true" in an almost-grounded way, but in reality we are all actually constrained by Godel, Tarski, etc. We just trick ourselves into assuming we can inductively keep stepping up from S+"X is true" indefinitely for any practical problem we encounter, but that is actually an ungrounded assumption. What you describe sounds to me more like logical abduction, rather than induction. Unfortunately, it's difficult to find simple explanations of what abduction is. One very rough and operational categorisation of modes of inference, as deductive, inductive or abductive, is as follows. Deduction is reasoning (as in automated theorem proving or rule-based systems), induction is learning (as in machine learning) and abduction is reasoning under uncertainty (as in probabilistic inference). There's a lot of confusion between what is abduction and what is induction, and the definitions of those in terms of "facts" and "rules" (coming from Peirce, who originally named "abduction" at least) is, for me, not very helpful. Another colloquial explanation is that abduction is a form of drawing conclusions from observations that isn't sound, so that conclusions are not necessarily true. I think that's where the association with probabilistic reasoning comes from, and that's why I think it describes better the kind of induction the linked paper is talking about. Anyway I'll have to read that, thanks.
YeGoblynQueenne 4y ago
Thanks for clarifying! Sorry for pouncing on that one little detail in your comment btw. Incompleteness vs. human reasoning is just something I think about on and off and your turn of phrase caught my attention.

benlivengood 4y ago

I think that Penrose basically thinks it's useful that humans can heuristically step up from a logic system S to S+"statement X is true" in an almost-grounded way, but in reality we are all actually constrained by Godel, Tarski, etc. We just trick ourselves into assuming we can inductively keep stepping up from S+"X is true" indefinitely for any practical problem we encounter, but that is actually an ungrounded assumption. What seems like an even better formal solution is something like logical induction[0] that works over something like probabilities instead of truth/falsehood. It has an existence proof but not a constructive formalization, afaik, but maybe some recent progress has been made. 0: https://arxiv.org/abs/1609.03543 https://arxiv.org/abs/1609.03543

YeGoblynQueenne 4y ago
Thanks for the paper, I need to read that. Much of my work is on inductive logic. >> I think that Penrose basically thinks it's useful that humans can heuristically step up from a logic system S to S+"statement X is true" in an almost-grounded way, but in reality we are all actually constrained by Godel, Tarski, etc. We just trick ourselves into assuming we can inductively keep stepping up from S+"X is true" indefinitely for any practical problem we encounter, but that is actually an ungrounded assumption. What you describe sounds to me more like logical abduction, rather than induction. Unfortunately, it's difficult to find simple explanations of what abduction is. One very rough and operational categorisation of modes of inference, as deductive, inductive or abductive, is as follows. Deduction is reasoning (as in automated theorem proving or rule-based systems), induction is learning (as in machine learning) and abduction is reasoning under uncertainty (as in probabilistic inference). There's a lot of confusion between what is abduction and what is induction, and the definitions of those in terms of "facts" and "rules" (coming from Peirce, who originally named "abduction" at least) is, for me, not very helpful. Another colloquial explanation is that abduction is a form of drawing conclusions from observations that isn't sound, so that conclusions are not necessarily true. I think that's where the association with probabilistic reasoning comes from, and that's why I think it describes better the kind of induction the linked paper is talking about. Anyway I'll have to read that, thanks.

YeGoblynQueenne 4y ago

Thanks for clarifying! Sorry for pouncing on that one little detail in your comment btw. Incompleteness vs. human reasoning is just something I think about on and off and your turn of phrase caught my attention.