"Soon I think you can also get an AI chatbot sitting next to you while you’re going through the proof, and they can take questions and they can explain each step as if they were the author. I think we’re already very close to that. "
Assuming this is possible, would it then mean that the chatbot understands the proof? More on this in a few weeks.
I prefer to say the chat bot has learned the semantics. That’s absolutely impressive. We also have the semantics, perhaps also internalized ‘sub-symbolically’ like a neural net. But our ‘understanding’ overlays that semantics with consciousness, which is presumably biologically based. So no to ‘understanding’ for the bot. Given that, Dennett eg has anyway argued (inspired by Wittgenstein) for decades that many mental terms like ‘understanding’ are way over-determined , basically onto-theological projections in the spirit of Derrida (my take). Caution is therefore advised in these bot-human comparisons. What’s being compared to what? I think Geoff Hinton makes ambiguous statements about bot ‘understanding’ for these reasons.
Thank you for these comments. All of these terms are terribly ambiguous and not suitable for incorporation in a formal language, although "semantics" does have a more or less well-defined meaning in some branches of logic. On the other hand, every natural language with which I am familiar has a word or words translating "understanding" whose use is analogous to ours. When we reflect on how we use the word it does appear that consciousness is a precondition. Though perhaps the word "understanding" is still meaningful when used in the context of a dream.
All of this to say that when specialists apply the word to AI they are assigning a meaning to it that is at some remove from ordinary language. Over the past month I have been exploring what mathematicians mean when we say we understand a proof. We definitely do mean something! It is in fact one of the most characteristic experiences of mathematics, for professional mathematicians as well as for children learning arithmetic. The experience in the two cases is the same but the content is very different.
I realized after hitting 'send' that I was confusing language understanding with the conceptual understanding and novelty which your genus example provides.
So I take back what I wrote about understanding, especially regarding mathematics or arithmetic as you point out.
Another problem like yours might give the bot Paul Halmos' book Naive Set Theory, or something similar, and see if it could formulate or discover Russell's paradox, or can determine where 'set' should be something like 'proper class', or create something like Zermelo's limiting of definitions of new sets to subsets of existing sets.
A difficulty right off is that all that might be known from training on the internet.
As your post indicated , Lakatos’ Proofs & Refutations has historical examples of concept-stretching to improve Euler's theorem, or excluding counterexamples without improving the theorem. Those and 19th century definitions of function, continuity and series might be a source of setups , but would again have to avoid over-training.
That's a nice example. If some future "AlphaLogic" could learn some version of the axioms and independently come up with Russell's paradox, I would definitely take note; and if it then displayed symptoms of depression I would begin to believe in machine consciousness.
Current ML models lack any substantial mechanism for self-reflection or meta-reasoning which makes them particularly poorly suited torward mathematics. I don't think this is an insurmountable problem but it will require some important advances in representing things like beliefs and propositions that current models are unable to perform.
I predict we are going to see a ML models quickly master the kind of tasks that people do effortlessly (speak, walk etc) and this will revolutionize media and art but then we'll hit a plateau while we wait for new conceptual advances in ML that will allow for more systematic reasoning combined with this kind of intuition training.
There are solid reasons to confirm the critics of Alpha Geometry included in this post- not only matters of energy footprint - ,and more generally of all recent announcements tending to announce big steps towards '' AGI''
- The cheating mentioned of the geometric construction of bisector is a general technique used in most recent successes of AI. Just one example : the protein folding success which led the bosses of Deep Mind to announce their machine will soon get Nobel Prizes , is in fact the cooperation of AI with many years of scientific research and results connected with the study of protein folding .
- Not only solving IMO problems is different from what most mathematicians believe of what is '' creative mathematics '' but an essential part of mathematics is the discovery or creation of concepts and important questions- conjectures- in the same time ( Euler 's caracteristics here ,or Poincaré's conjecture in a recent post) .All of recent successes of AI in scientific domains involve a cooperation of computer based technology with scientific works . Even the extension to medical successes ( see the history of AI applied to radiology ) and other examples show it would have been probably better not to choose the term ''intelligence '' but a more modest name when creating the new discipline .
Might you be impressed enough were AlphaGeometry 2.0 to use Euler's formula (never mind Euler's characteristic) to prove, say, Erdős–Szemerédi sum-product inequality with a reasonable constant to boot? To wit: why would this not be tantamount to a just claim to "rediscovery", suitably conceived?
If that's geometry I don't see it. Even in this situation that involves nothing resembling visual intuition, the ground rules are that an "inequality with a reasonable constant" would have to optimize a search criterion that the system will have generated spontaneously. The actual result would impress me less than the generation of the search criterion.
Touche MH. I will be very impressed if the AI discovers the concept of a hole. Then it can go find one and fall into it forever.
I just read Terence Tao's interview with Steven Strogatz on the Quanta website, at "https://quantamagazine.us1.list-manage.com/track/click?u=0d6ddf7dc1a0b7297c8e06618&id=638cad0e61&e=a83f490494" At one point Tao said
"Soon I think you can also get an AI chatbot sitting next to you while you’re going through the proof, and they can take questions and they can explain each step as if they were the author. I think we’re already very close to that. "
Assuming this is possible, would it then mean that the chatbot understands the proof? More on this in a few weeks.
I prefer to say the chat bot has learned the semantics. That’s absolutely impressive. We also have the semantics, perhaps also internalized ‘sub-symbolically’ like a neural net. But our ‘understanding’ overlays that semantics with consciousness, which is presumably biologically based. So no to ‘understanding’ for the bot. Given that, Dennett eg has anyway argued (inspired by Wittgenstein) for decades that many mental terms like ‘understanding’ are way over-determined , basically onto-theological projections in the spirit of Derrida (my take). Caution is therefore advised in these bot-human comparisons. What’s being compared to what? I think Geoff Hinton makes ambiguous statements about bot ‘understanding’ for these reasons.
Thank you for these comments. All of these terms are terribly ambiguous and not suitable for incorporation in a formal language, although "semantics" does have a more or less well-defined meaning in some branches of logic. On the other hand, every natural language with which I am familiar has a word or words translating "understanding" whose use is analogous to ours. When we reflect on how we use the word it does appear that consciousness is a precondition. Though perhaps the word "understanding" is still meaningful when used in the context of a dream.
All of this to say that when specialists apply the word to AI they are assigning a meaning to it that is at some remove from ordinary language. Over the past month I have been exploring what mathematicians mean when we say we understand a proof. We definitely do mean something! It is in fact one of the most characteristic experiences of mathematics, for professional mathematicians as well as for children learning arithmetic. The experience in the two cases is the same but the content is very different.
Thanks very much for that reply Michael.
I realized after hitting 'send' that I was confusing language understanding with the conceptual understanding and novelty which your genus example provides.
So I take back what I wrote about understanding, especially regarding mathematics or arithmetic as you point out.
Another problem like yours might give the bot Paul Halmos' book Naive Set Theory, or something similar, and see if it could formulate or discover Russell's paradox, or can determine where 'set' should be something like 'proper class', or create something like Zermelo's limiting of definitions of new sets to subsets of existing sets.
A difficulty right off is that all that might be known from training on the internet.
As your post indicated , Lakatos’ Proofs & Refutations has historical examples of concept-stretching to improve Euler's theorem, or excluding counterexamples without improving the theorem. Those and 19th century definitions of function, continuity and series might be a source of setups , but would again have to avoid over-training.
That's a nice example. If some future "AlphaLogic" could learn some version of the axioms and independently come up with Russell's paradox, I would definitely take note; and if it then displayed symptoms of depression I would begin to believe in machine consciousness.
Current ML models lack any substantial mechanism for self-reflection or meta-reasoning which makes them particularly poorly suited torward mathematics. I don't think this is an insurmountable problem but it will require some important advances in representing things like beliefs and propositions that current models are unable to perform.
I predict we are going to see a ML models quickly master the kind of tasks that people do effortlessly (speak, walk etc) and this will revolutionize media and art but then we'll hit a plateau while we wait for new conceptual advances in ML that will allow for more systematic reasoning combined with this kind of intuition training.
I know from personal experience that mastering walking has helped make me more attentive to holes, especially since I moved to New York City (https://www.nydailynews.com/2020/10/24/man-hospitalized-after-falling-into-sinkhole-on-bronx-sidewalk/) but I can't say whether this alone would suffice to abstract their topological as opposed to metric properties.
There are solid reasons to confirm the critics of Alpha Geometry included in this post- not only matters of energy footprint - ,and more generally of all recent announcements tending to announce big steps towards '' AGI''
- The cheating mentioned of the geometric construction of bisector is a general technique used in most recent successes of AI. Just one example : the protein folding success which led the bosses of Deep Mind to announce their machine will soon get Nobel Prizes , is in fact the cooperation of AI with many years of scientific research and results connected with the study of protein folding .
- Not only solving IMO problems is different from what most mathematicians believe of what is '' creative mathematics '' but an essential part of mathematics is the discovery or creation of concepts and important questions- conjectures- in the same time ( Euler 's caracteristics here ,or Poincaré's conjecture in a recent post) .All of recent successes of AI in scientific domains involve a cooperation of computer based technology with scientific works . Even the extension to medical successes ( see the history of AI applied to radiology ) and other examples show it would have been probably better not to choose the term ''intelligence '' but a more modest name when creating the new discipline .
Jean-Michel KANTOR
Might you be impressed enough were AlphaGeometry 2.0 to use Euler's formula (never mind Euler's characteristic) to prove, say, Erdős–Szemerédi sum-product inequality with a reasonable constant to boot? To wit: why would this not be tantamount to a just claim to "rediscovery", suitably conceived?
If that's geometry I don't see it. Even in this situation that involves nothing resembling visual intuition, the ground rules are that an "inequality with a reasonable constant" would have to optimize a search criterion that the system will have generated spontaneously. The actual result would impress me less than the generation of the search criterion.