One of the reasons I'm so interested in Gödel is that I believe his theorem bears on the popular topic of AI and the limits of AI, and whether AI can emulate human creativity. Gödel's theorem is an example of thinking outside the box, solving a problem from a frame of reference outside that in which the problem was posed. Humans can do this. I believe our ability to do this is related to our psychic connections, precognition and telepathy. I don't think that computers will be able to do this, even though I recognize that many of the most brain-intensive tasks that we are asked to perform do NOT require telepathy, and AI will soon surpass us.
The problem with AI is that people want to reify it. You can't, because consciousness doesn't emerge from complexity, it originates from somewhere else. It isn't "contained in in the brain". Brains appear to be transceivers.
For AI to ape human creativity, it will have to have a set of rules that draw associations between disparate ideas, and those associations will have to be firmly rooted in reality, otherwise we'll just continue to get the gibberish it puts on signs in lieu of actual cogent text, and extra fingers on hands. It's a "best guess as to what's next" machine, not a "thinking" machine. It can organize information in ways that's useful to humans, but can never "know" what's actually useful, because it has no consciousness behind it.
It's still super useful despite this, especially for translating masses of text. It's going to be very, very, interesting when they start translating (eg.) the huge mass of Sanskrit texts that have been previously unknown to us, and the previously secret library in that Tibetan monastery, and who knows what else that has been untranslated/unknown up to now.
Very nice write up. Though, it could be argued that Godel incompleteness is a very particular statement about very particular type of first order systems. There are axiomatic systems that are consistent and complete and include arithmetic. It is not clear that there are any implications of Godel incompleteness to reasoning or logic or philosophy in general. Interested to hear your take !
For example, drop the requirement that proofs have to be finite. The Godel numbering argument then does not work. You can then build an alternative axiomatic system that is complete and includes arithmetic. Godel incompleteness certainly has implications in computer science as it does illustrate the limits of any proof system that resembles a computer and thus illustrates the limits of computers. However, it is not clear it has any philosophical implications.
Thank you, David. The idea of an infinite proof is new to me, but I've found one and I'm working through it. When I'm done, I'm sure I'll have more to say.
Great! Logic is a rabbit hole, but you might want to check out Presburger arithmetic if you are interested in complete sysystem that include arithmetic.
I have been profoundly fond of Gödel's Theorum for decades ... re your last two paragraphs, surely it demonstrates that logic (rather than maths) is necessarily incomplete (or inconsistent)? And, by extension is it not the great wonder of his work that it sets us free from any totalizing attempt at integration?
I agree. "Totalizing" is not in the cards. But I'm optimistic about a next step that embeds science within a framework that includes intuition and extra-sensory, extra-logical ways of knowing.
... in my work (see https://www.hughwillbourn.com/book) I pick away, gently, at categorical thinking. I wonder what is it to understand more deeply what we already know.... so that leads off to a different discussion. Thank you very much for your lucid run-through of G's theorum. I have always been fond of its output and implications but lack the ability to parse the argument.
One of the reasons I'm so interested in Gödel is that I believe his theorem bears on the popular topic of AI and the limits of AI, and whether AI can emulate human creativity. Gödel's theorem is an example of thinking outside the box, solving a problem from a frame of reference outside that in which the problem was posed. Humans can do this. I believe our ability to do this is related to our psychic connections, precognition and telepathy. I don't think that computers will be able to do this, even though I recognize that many of the most brain-intensive tasks that we are asked to perform do NOT require telepathy, and AI will soon surpass us.
The problem with AI is that people want to reify it. You can't, because consciousness doesn't emerge from complexity, it originates from somewhere else. It isn't "contained in in the brain". Brains appear to be transceivers.
For AI to ape human creativity, it will have to have a set of rules that draw associations between disparate ideas, and those associations will have to be firmly rooted in reality, otherwise we'll just continue to get the gibberish it puts on signs in lieu of actual cogent text, and extra fingers on hands. It's a "best guess as to what's next" machine, not a "thinking" machine. It can organize information in ways that's useful to humans, but can never "know" what's actually useful, because it has no consciousness behind it.
It's still super useful despite this, especially for translating masses of text. It's going to be very, very, interesting when they start translating (eg.) the huge mass of Sanskrit texts that have been previously unknown to us, and the previously secret library in that Tibetan monastery, and who knows what else that has been untranslated/unknown up to now.
Very nice write up. Though, it could be argued that Godel incompleteness is a very particular statement about very particular type of first order systems. There are axiomatic systems that are consistent and complete and include arithmetic. It is not clear that there are any implications of Godel incompleteness to reasoning or logic or philosophy in general. Interested to hear your take !
Please tell me more, particularly about complete and consistent systems that include arithmetic.
For example, drop the requirement that proofs have to be finite. The Godel numbering argument then does not work. You can then build an alternative axiomatic system that is complete and includes arithmetic. Godel incompleteness certainly has implications in computer science as it does illustrate the limits of any proof system that resembles a computer and thus illustrates the limits of computers. However, it is not clear it has any philosophical implications.
Thank you, David. The idea of an infinite proof is new to me, but I've found one and I'm working through it. When I'm done, I'm sure I'll have more to say.
Great! Logic is a rabbit hole, but you might want to check out Presburger arithmetic if you are interested in complete sysystem that include arithmetic.
I have been profoundly fond of Gödel's Theorum for decades ... re your last two paragraphs, surely it demonstrates that logic (rather than maths) is necessarily incomplete (or inconsistent)? And, by extension is it not the great wonder of his work that it sets us free from any totalizing attempt at integration?
I agree. "Totalizing" is not in the cards. But I'm optimistic about a next step that embeds science within a framework that includes intuition and extra-sensory, extra-logical ways of knowing.
https://www.cia.gov/readingroom/docs/CIA-RDP96-00789R002600250001-6.pdf
... in my work (see https://www.hughwillbourn.com/book) I pick away, gently, at categorical thinking. I wonder what is it to understand more deeply what we already know.... so that leads off to a different discussion. Thank you very much for your lucid run-through of G's theorum. I have always been fond of its output and implications but lack the ability to parse the argument.
Thanks -- I'll check out your book.
Thank you! If you are outside the UK, you can find it on Amazon + ebook / audible ...