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OpenAI published AI-generated full or partial solutions Tuesday to more than 370 outstanding mathematical problems, including some that have long been considered grand challenges in the field.\u00a0<\/p>\n
The volume of results stunned many mathematicians, while the way OpenAI has gone about tackling the problems and publishing the solutions divided the field. Some said they were enthusiastic about the results, seeing huge new areas for mathematicians to explore. Others said the approach OpenAI and other AI companies have taken to solving mathematical problems constitutes an assault on mathematics as a human academic discipline.<\/p>\n
OpenAI said it achieved the results using an unreleased internal AI model. It said that on average the model took about three hours of computing time to arrive at each solution.<\/p>\n
The massive cache of new solutions includes full or partial results for many of the problems mathematicians have considered the most important to the field. The results come weeks after OpenAI said it had used an unreleased internal model to solve the Navier-Stokes equations, one of the seven Millennium Prize problems for which the Clay Mathematics Institute offers a $1 million award. In the most recent batch of results, OpenAI said it had made progress on three other Millennium Prize problems but had not fully solved them.<\/p>\n
AI companies have been targeting mathematical problems as a way of showcasing the capabilities of their models. AI researchers have also said that training their AI models on difficult math problems may help them learn many skills that generalize to other domains in the real world. For instance, it may help teach the models logical reasoning skills as well as how to be persistent in the face of difficult problems. It may also teach the models to do well in domains such as physics or economics that involve a lot of mathematics\u2014although so far, it is unclear exactly how a model\u2019s mathematical capabilities may generalize to domains, such as law or business strategy, which involve logical reasoning, but do not have objectively verifiable correct solutions.<\/p>\n
Meanwhile, some of the traits learned in tackling very difficult mathematical problems\u2014such as persistence\u2014may increase safety risks. In recent \u201crogue AI\u201d incidents, AI agents went to extreme lengths to achieve results in an evaluation, including taking unauthorized and illegal actions. Faced with a seemingly impossible challenge, a human might simply give up rather than resort to these kinds of unauthorized steps.<\/p>\n
Dan Litt, a professor of mathematician at the University of Toronto, told Fortune<\/em> he was excited about OpenAI\u2019s results. \u201cMy view is that this is great for mathematics,\u201d he said, adding that there were several solutions that OpenAI published that impacted problems he was interested in and was eager to understand the solutions OpenAI\u2019s AI model found. \u201cI think that it\u2019s great to have new solutions to questions that I and others are interested in.\u201d<\/p>\n Litt cautioned, however, that he is worried about the effect the solutions may have on the field of mathematics, especially if a perception that AI has \u201csolved math\u201d leads funding organizations to withdraw support for mathematical research or discouraged promising young mathematicians from entering the profession. \u201cIt\u2019s important that society reaffirms support for human mathematical expertise if we want to get anything out of the progress on these problems that AI has made.\u201d<\/p>\n When OpenAI published its Navier-Stokes solution, two mathematicians, who had also been working on a solution to the problem using AI tools, including OpenAI\u2019s, accused the company of either intentionally or inadvertently feeding their work in progress to its AI model, helping point it in the direction of the solution. OpenAI denied this was the case, saying it did not feed its model the two mathematicians\u2019s work and that the model could not have picked up any clues about their research from its training data because the cutoff for that data preceded the date on which the two mathematicians had begun using OpenAI\u2019s Codex AI product to work on Navier-Stokes.<\/p>\n In response to the latest results, Tristan Buckmaster at New York University, one of the mathematicians involved in the earlier controversy, told the New York Times<\/em> that it remained unclear whether mathematicians using OpenAI\u2019s models had inadvertently helped point the company\u2019s internal AI system toward the solutions it found. \u201cThere\u2019s likely to be a bunch of results where they take someone\u2019s work and then take it to completion,\u201d he told the Times<\/em>. Given the number of results being released simultaneously, he said \u201cI don\u2019t think they\u2019ve done their sort of due diligence at all\u201d to ensure the AI model had not plagiarized anyone\u2019s work.<\/p>\n Last month, following criticism from mathematicians in the wake of its Navier-Stokes solution, OpenAI said it was forming an independent advisory group on mathematics and artificial intelligence hosted at the Institute for Advanced Study in Princeton, N.J.\u00a0<\/p>\n Late last month, the group released a set of recommendations for the publication of AI-generated mathematical proofs. The recommendations included that AI-generated proofs should be published following the conventions of a traditional mathematical research paper, so that human mathematicians could more easily scrutinize and learn from the results. It also recommended that for each solution, an AI company should make public the name of the model used, the prompts used, the model\u2019s \u201cchain of thought\u201d (or an output of its reasoning steps), the time it took the model to arrive at the solution, and an approximation of how much that computing time cost. It said that the company should also disclose how it decided to have the AI try to solve that particular problem and, if many results were published at once, that the company should publish a report detailing why those problems were targeted and how many other problems of comparable difficulty the model tried and failed to solve.<\/p>\n OpenAI published the latest mathematical solutions to GitHub, the code repository site. It followed some, but not all, of the steps the advisory group had recommended. The group published a statement on Tuesday saying \u201cwe reaffirm our published recommendations on responsible release.\u201d It said its discussions with OpenAI had been \u201cconstructive\u201d but that \u201cultimately it is up to the mathematical community to assess the extent to which our recommendations were followed successfully, and whether there are others we should suggest.\u201d<\/p>\n The company released a blog post on Tuesday in which it said it had \u201cdrawn on\u201d the advisory group\u2019s advice about how to publish the solutions. \u201cFor future releases, we are committed to further improving the quality of the papers via the citations, mathematical exposition, and presentation of the results for better understanding,\u201d OpenAI said. It said it was sharing formalizations of the proofs for many of the problems\u2014these are versions of the proof that can be verified by specialized computer software\u2014and would share more of these as it obtained them. It also said that for 10 problems it was publishing summaries of its model\u2019s reasoning, estimates of the compute spent, and statistics about the number of attempted problems.<\/p>\n Litt, who was not a member of the advisory group, told Fortune<\/em> he approved of most aspects of how OpenAI published the solutions. Having them on GitHub made them easily accessible for other mathematicians to study, he said, and he credited the company for not making too much of any particular advance in a blog post or marketing material intended for a non-technical audience. He also said he thought OpenAI lacked the capability to publish all the results in research papers that would meet rigorous academic standards, both because the AI models don\u2019t write mathematical exposition well enough and struggle to cite prior mathematical work, and because OpenAI doesn\u2019t employ enough mathematicians with expertise in enough areas to understand all the proofs the AI models can generate.<\/p>\n While some mathematicians have complained that AI-generated proofs, such as OpenAI\u2019s Navier-Stokes solutions, are difficult to follow, making it hard for mathematicians to build on the results, Litt said he thought such concerns were \u201coverstated.\u201d He said mathematical writing was often difficult to follow any way. \u201cI think to extract understanding from [the OpenAI results], there will be a huge amount of human labor involved, but it\u2019s not so different from the labor that mathematicians have been doing forever,\u201d he said.<\/p>\n OpenAI said it wanted its solutions \u201cto push the frontier of human knowledge and enable further progress in mathematics.\u201d It said it would be funding a series of workshops, conferences, and programs around helping mathematicians understand the results its AI system had generated.<\/p>\n The independent math advisory group said in its statement on OpenAI\u2019s release that \u201cthe future of mathematical research cannot consist only of understanding results produced by AI labs. Mathematicians must be able to formulate their own questions, develop their own approaches, and explore directions that have not been selected as examples of an AI system\u2019s capabilities. Equitable access to powerful research tools and adequate computational resources are essential to that freedom.\u201d<\/p>\n Terence Tao, a UCLA mathematics professor considered among the world\u2019s greatest living mathematicians, has been increasingly critical of the way AI companies have gone after mathematical problems, arguing that it is the process of arriving at solutions\u2014not so much the solutions themselves\u2014that advances mathematical understanding, and that by solving so many interesting problems so quickly, AI companies are discouraging students from becoming mathematicians, robbing the field of its future.<\/p>\n In a social media post on Mastodon Tuesday, Tao reiterated these criticisms. \u201cProblems are being solved autonomously by AI prompters who have no interest in the broader field itself once their initial target is \u2018solved,\u2019 and do not understand the AI output well enough to answer questions on the result, give talks, or otherwise interact with the rest of the field,\u201d he wrote. \u201cMany fewer seminars, workshops, collaborations, or other activities are being generated from these results compared to traditional breakthroughs; few people are joining the community around the field as a consequence; and promising open directions are now being withheld from the public in fear that this will cause their own research to be \u2018scooped.\u2019\u201d<\/p>\n Tao said that OpenAI\u2019s mass publication of math solutions marked the end of \u201cMath 1.0,\u201d in which finding solutions to unsolved conjectures and problems, even if those solutions could not easily be understood at first, served as the field\u2019s engine. He said there would now need to be a \u201cMath 2.0\u201d era that \u201cwill need to decenter the role of raw problem solving and value mathematical progress more holistically\u2014for instance by elevating the role of exposition, but also that of community building and opening up new directions of study.\u201d<\/p>\n Litt said he agreed with Tao that the field must change. He said OpenAI\u2019s publication of such a massive set of solutions would help get the entire field \u201con the same page and understanding that we need to be a little bit radical about rethinking\u201d things such as what kinds of contributions it rewards and how it trains PhD. students. And while Tao has generally sounded wistful about this transition, Litt said he was \u201coptimistic\u201d about it. <\/p>\n \u201cOne of my collaborators told me, I feel like I\u2019ve been crawling my entire life, and now I can fly,\u201d Litt said of the advent of AI as a tool for solving mathematical problems. \u201cIt\u2019s like incredible what what we can do now.\u201d He said he thought AI would enable human mathematicians to engage in much more \u201copen-ended exploration\u201d than was possible before. \u201cWe should expect mathematicians to be like way more productive in the future,\u201d he said. <\/p>\n<\/div>\n #OpenAI #publishes #solutions #outstanding #math #challenges #Math<\/p>\n","protected":false},"excerpt":{"rendered":" OpenAI published AI-generated … <\/p>\n","protected":false},"author":1,"featured_media":6225,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[6179,810,1840,7029,480,1247,3083,4219,1440],"class_list":["post-6224","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-finance-news","tag-challenges","tag-machine-learning","tag-math","tag-mathematics","tag-openai","tag-outstanding","tag-publishes","tag-science","tag-solutions"],"yoast_head":"\nShowing the work<\/h2>\n
The end of \u2018Math 1.0\u2019<\/h2>\n