The great Soviet child psychologist Lev Vygotsky identified the 'zone of proximal development' in which learning takes place through tasks just beyond current skills but typically attainable. Seems relevant to many math and other science/humanities thesis topics from the point of view of the advisor-‘parent’. Looks like recent AI is taking away much PhD ZPD in math and elsewhere, or at least the math/humanities as practiced to date. That's one way to destroy a cultural tradition.
> If it’s true that the tech industry has been relying on the word of mathematicians, in their outreach to investors, to vouch for the competence and reliability of their models, the Declaration’s most consequential result may turn out to be the replacement of the the mathematical community’s support by one more reason to be nervous.
I very much doubt that the mathematical community has been vouching for the competence and reliability of most of these models. Quite the opposite: I believe it has been well understood in that community that the theories, techniques, tools and tradecraft do not yet exist at a level of maturity and general understanding to do anything of the kind.
By "vouching" I refer to the press releases and positive news coverage featuring quotations by distinguished mathematicians. This is sufficient to communicate with the target audience, in which the mathematical community is merely a bystander.
Ah yes, that's different of course. But I believe there has been some creative confusion between what distinguished mathematicians having been saying and the message the AI industry at large would like to convey.
Distinguished mathematicians have been saying things like "AI is surprisingly good at producing solutions to problems in mathematics". What the AI industry would like their customers to hear is "the mathematical sciences community endorses the accuracy, reliability, etc of AI models". These are quite different: indeed, one is true and one is false. But one has been promoted as if it were the other.
I wonder if this is one of the factors that has led to the -- to my mind ivery interesting -- problems in the mathematical foundations of ML/AI attracting so little interest from the mathematical sciences community. It's as if no-one has been working in that area since the 1990s. But if people have been encouraged to believe by AI marketing that it's all sorted out, then of course they wouldn't want to work there, or advise their colleagues and students to do so, or apply for, or support grants, ...
Michael - In the unit distance problem has anyone mentioned Herbrand’s process of Skolemization which finds a counterexample when one exists? Polya recommends trying to prove and disprove a problem so an AI can do that too, and easily understanding Herbrand could be running that process while doing other stuff. This method was part of automated theorem proving decades ago. I am surprised not to have heard this mentioned. It would apply to all problems of this nature, ie there’s a counterexample.
I haven't heard of anything like that. I'm no expert but it doesn't seem to me that manipulation of quantifiers would be much help with questions like the unit distance problem.
I share your concern about the motives and modus operandi of OpenAI and its peer competitors. But suppose that after "solving math" (motivated to some extent if not primarily by their pre-IPO PR campaign) they make the "tool" they built in the process publicly available (say akin to DeepMind making AlphaFold publicly available to the scientific community). In this scenario too "it's hard to see how the all the social structures that support the subject will be able to avoid major disruption over the next few years" (to quote Timothy Gowers reacting to the OpenAI internal model disproof of the unit distance conjecture).
This is an interesting thought. AlphaFold was trained "on over 170,000 protein structures from the Protein Data Bank, a public repository of protein sequences and structures," according to Wikipedia. The Protein Data Bank is in turne administered by the Worldwide Protein Data Bank, an organization consisting of five member organizations based in the US, Europe, and Japan, which meets annually.
I was unable to find information on funding of these public resources but I didn't try very hard. When the Protein Data Bank was established in the 1970s it must have been already understood that the data would be of interest to the large pharmaceutical labs, and the decision to share it with them freely must have been made consciously. You might want to argue that the arXiv or the public repositories where mathematicians receiving public support are required by law to share their articles are analogous structures.
The source code for AlphaFold is freely available for non-commercial purposes. I assume this means it can be run on local computers without going through DeepMind. It seems to me unlikely that the same can be said of the OpenAI, Gemini, and Anthropic models that were used by the teams that competed on the First Proof second batch. If you could actually run something like ChatGPT 5.5 on a local computer without communicating with OpenAI, and thus to have mechanical access to the entire history of mathematics, then it might be comparable to the major disruption of the social structures in the vicinity of 34th St and 5th Avenue that took place around 400 years ago. But on a much smaller scale than the major disruptions to universities that are announced practically every week in the Chronicle of Higher Education.
The great Soviet child psychologist Lev Vygotsky identified the 'zone of proximal development' in which learning takes place through tasks just beyond current skills but typically attainable. Seems relevant to many math and other science/humanities thesis topics from the point of view of the advisor-‘parent’. Looks like recent AI is taking away much PhD ZPD in math and elsewhere, or at least the math/humanities as practiced to date. That's one way to destroy a cultural tradition.
> If it’s true that the tech industry has been relying on the word of mathematicians, in their outreach to investors, to vouch for the competence and reliability of their models, the Declaration’s most consequential result may turn out to be the replacement of the the mathematical community’s support by one more reason to be nervous.
I very much doubt that the mathematical community has been vouching for the competence and reliability of most of these models. Quite the opposite: I believe it has been well understood in that community that the theories, techniques, tools and tradecraft do not yet exist at a level of maturity and general understanding to do anything of the kind.
By "vouching" I refer to the press releases and positive news coverage featuring quotations by distinguished mathematicians. This is sufficient to communicate with the target audience, in which the mathematical community is merely a bystander.
Ah yes, that's different of course. But I believe there has been some creative confusion between what distinguished mathematicians having been saying and the message the AI industry at large would like to convey.
Distinguished mathematicians have been saying things like "AI is surprisingly good at producing solutions to problems in mathematics". What the AI industry would like their customers to hear is "the mathematical sciences community endorses the accuracy, reliability, etc of AI models". These are quite different: indeed, one is true and one is false. But one has been promoted as if it were the other.
Yes, that's exactly the point.
I wonder if this is one of the factors that has led to the -- to my mind ivery interesting -- problems in the mathematical foundations of ML/AI attracting so little interest from the mathematical sciences community. It's as if no-one has been working in that area since the 1990s. But if people have been encouraged to believe by AI marketing that it's all sorted out, then of course they wouldn't want to work there, or advise their colleagues and students to do so, or apply for, or support grants, ...
Michael - In the unit distance problem has anyone mentioned Herbrand’s process of Skolemization which finds a counterexample when one exists? Polya recommends trying to prove and disprove a problem so an AI can do that too, and easily understanding Herbrand could be running that process while doing other stuff. This method was part of automated theorem proving decades ago. I am surprised not to have heard this mentioned. It would apply to all problems of this nature, ie there’s a counterexample.
I haven't heard of anything like that. I'm no expert but it doesn't seem to me that manipulation of quantifiers would be much help with questions like the unit distance problem.
the translation from Dutch is good enough; perhaps a minor point "marketing" is "mooi" - in the context "nice/cheesy/cheap" rather than "clever".
Thank you!
I share your concern about the motives and modus operandi of OpenAI and its peer competitors. But suppose that after "solving math" (motivated to some extent if not primarily by their pre-IPO PR campaign) they make the "tool" they built in the process publicly available (say akin to DeepMind making AlphaFold publicly available to the scientific community). In this scenario too "it's hard to see how the all the social structures that support the subject will be able to avoid major disruption over the next few years" (to quote Timothy Gowers reacting to the OpenAI internal model disproof of the unit distance conjecture).
This is an interesting thought. AlphaFold was trained "on over 170,000 protein structures from the Protein Data Bank, a public repository of protein sequences and structures," according to Wikipedia. The Protein Data Bank is in turne administered by the Worldwide Protein Data Bank, an organization consisting of five member organizations based in the US, Europe, and Japan, which meets annually.
I was unable to find information on funding of these public resources but I didn't try very hard. When the Protein Data Bank was established in the 1970s it must have been already understood that the data would be of interest to the large pharmaceutical labs, and the decision to share it with them freely must have been made consciously. You might want to argue that the arXiv or the public repositories where mathematicians receiving public support are required by law to share their articles are analogous structures.
The source code for AlphaFold is freely available for non-commercial purposes. I assume this means it can be run on local computers without going through DeepMind. It seems to me unlikely that the same can be said of the OpenAI, Gemini, and Anthropic models that were used by the teams that competed on the First Proof second batch. If you could actually run something like ChatGPT 5.5 on a local computer without communicating with OpenAI, and thus to have mechanical access to the entire history of mathematics, then it might be comparable to the major disruption of the social structures in the vicinity of 34th St and 5th Avenue that took place around 400 years ago. But on a much smaller scale than the major disruptions to universities that are announced practically every week in the Chronicle of Higher Education.