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AI\uac00 \uc218\ucc9c \ub144\uac04 \uc778\uac04 \uc218\ud559\uc790\ub9cc\uc758 \uc601\uc5ed\uc774\uc5c8\ub358 \uc218\ud559\uc801 \uc99d\uba85\uc5d0 \uc2e4\uc9c8\uc801\uc73c\ub85c \uae30\uc5ec\ud558\uae30 \uc2dc\uc791\ud588\ub2e4. \uc804\ubb38\uac00\ub4e4\uc740 2030\ub144 AI\uc640 \uc778\uac04\uc774 \uacf5\ub3d9\uc73c\ub85c \ud544\uc988\uc0c1\uc744 \ubc1b\uc744 \uc218\ub3c4 \uc788\ub2e4\uace0 \uc804\ub9dd\ud55c\ub2e4. \uac8c\ud2f0\uc774\ubbf8\uc9c0\ubc45\ud06c \uc81c\uacf5<\/p>\n
Artificial intelligence (AI) is beginning to fundamentally transform how mathematical research is conducted, going beyond simple calculation to propose new proof strategies and connect previously separate areas of mathematics. Many experts now judge that computers have begun to make substantive contributions to rigorous mathematical proofs, which for millennia were the exclusive domain of human mathematicians.\u00a0<\/p>\n
The international journal Nature reported on the 19th (local time) that academic researchers and AI company scientists are conducting large-scale projects to test both the limits and the potential of AI\u2019s mathematical capabilities.\u00a0<\/p>\n
A representative case is Liam Price, a non-specialist in the United Kingdom. Not yet enrolled at a university, Price solved Hungarian mathematician Paul Erd\u0151s\u2019s long-standing open problem \u201cErd\u0151s problem 1196\u201d from his home in southwest England with the help of \u201cGPT 5.4 Pro.\u201d The problem concerns \u201cprimitive sets,\u201d in which no number divides another in the set, and the conjecture that a certain sum over such sets never exceeds 1. Even prominent mathematicians had failed to resolve it for 60 years.\u00a0<\/p>\n
Since Erd\u0151s\u2019s 1935 paper, previous researchers had taken a probabilistic approach as the obvious starting point. GPT did not accept that premise and instead solved the problem in its original formulation. Terence Tao, a professor at the University of California, Los Angeles (UCLA), assessed that GPT\u2019s solution implicitly created a link between numbers and probability. Because this connection between numbers and probability was such a natural and attractive direction, it may actually have obscured other possibilities for decades.\u00a0<\/p>\n
General-purpose large language models (LLMs) such as GPT, Gemini, and Claude are rapidly improving their abilities in logical reasoning and proof generation even without specialized training in mathematics. S\u00e9bastien Bubeck, a researcher at OpenAI, said, \u201cA year ago, we thought LLMs could not go beyond their training data,\u201d adding that the current progress is \u201chard to believe.\u201d Luong Thang, head of the Superhuman Reasoning team at Google DeepMind, predicted, \u201cAround 2030, AI and mathematicians may jointly receive the Fields Medal.\u201d<\/p>\n
Daniel Litt, a professor at the University of Toronto in Canada, remains critical of the hype surrounding AI achievements but still rates its future potential very highly. \u201cAI systems already hold existing mathematical knowledge at a superhuman level, have demonstrated powerful reasoning abilities, and never suffer from fatigue or loss of motivation,\u201d he said. \u201cIt is, if anything, more puzzling that AI has not yet made any major discoveries.\u201d He added that it remains unclear what makes human mathematicians truly exceptional, and whether there is a \u201csecret ingredient\u201d of originality that only humans possess.<\/p>\n
There are also clear challenges. Current AI models can at best generate proofs about three to four pages long. Internal Google models have already exceeded this and are expected soon to reach around ten pages, but proofs on the order of 100 pages remain a distant prospect.\u00a0<\/p>\n
AI-generated proofs also frequently contain errors, increasing the burden of verification. Lauren Williams, a professor at Harvard University, pointed out that \u201cAI produces proofs that look plausible, but it takes a lot of time to find the errors.\u201d Editors of mathematical journals are likewise struggling with an influx of low-quality papers produced by AI.<\/p>\n
Research on solutions is also active. The open-source, math-specific formal language \u201cLean\u201d is drawing attention. Lean converts mathematical proofs that people usually write in prose into a language that computers can read and check logically. Proofs are expressed in code form, and the computer automatically verifies every logical step.\u00a0<\/p>\n
A team led by Professor Bin Dong at Peking University demonstrated Lean\u2019s practicality by applying it to algebra problems. The startup Math, Inc. used dedicated software to formalize the work of 2022 Fields Medal laureate Maryna Viazovska in Lean. It was the first time a Fields Medal-winning result had been converted into Lean.\u00a0<\/p>\n
Google DeepMind has developed \u201cAletheia,\u201d a multi-agent AI system equipped with verification modules, and \u201cAlphaProof\u201d has pioneered an approach that writes proofs directly in the Lean language from the outset.<\/p>\n
However, the scope of mathematics that can be written or translated into Lean remains limited. In the \u201cFirst Proof\u201d AI mathematical capability evaluation test, which began trial operation in February this year, most submitted solutions were written in natural language that had to be checked directly by mathematicians, and only one was verified in Lean. In June, a full-scale test is scheduled, in which various AI systems will be presented with new problems and their solutions will be manually checked.<\/p>\n
Most researchers believe human mathematicians will retain the initiative for the time being. Mark Selke, a researcher at OpenAI, said, \u201cDeciding which problems to study is a matter of judgment, and for now that decision will be made by humans.\u201d Javier G\u00f3mez-Serrano, a professor at Brown University, commented, \u201cI no longer dare to imagine what the world will look like five years from now,\u201d adding, \u201cThings are moving so fast that anything could happen.\u201d<\/p>\n
Some also warn that leaving machines to develop ideas that humans themselves cannot understand may be pointless or even dangerous. Jeremy Avigad, a professor at Carnegie Mellon University, emphasized, \u201cThe ultimate goal of mathematics is to understand mathematical phenomena,\u201d and said, \u201cWe do not want an AI that merely spits out results and says \u2018the theorem is true.\u2019 Humans must be part of the process.\u201d<\/p>\n
Copyright \u24d2 DongA Science. All rights reserved.<\/p>\n","protected":false},"excerpt":{"rendered":"AI\uac00 \uc218\ucc9c \ub144\uac04 \uc778\uac04 \uc218\ud559\uc790\ub9cc\uc758 \uc601\uc5ed\uc774\uc5c8\ub358 \uc218\ud559\uc801 \uc99d\uba85\uc5d0 \uc2e4\uc9c8\uc801\uc73c\ub85c \uae30\uc5ec\ud558\uae30 \uc2dc\uc791\ud588\ub2e4. \uc804\ubb38\uac00\ub4e4\uc740 2030\ub144 AI\uc640 \uc778\uac04\uc774 \uacf5\ub3d9\uc73c\ub85c \ud544\uc988\uc0c1\uc744 \ubc1b\uc744…\n","protected":false},"author":2,"featured_media":49346,"comment_status":"","ping_status":"","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[9],"tags":[25,5044,29389,2408,132,7543,11380,3570,5809,866,17122,29383,29384,29385,29386,12764,29387,29388],"class_list":["post-49345","post","type-post","status-publish","format-standard","has-post-thumbnail","category-google","tag-artificial-intelligence","tag-deepmind","tag-erdos","tag-gemini","tag-google","tag-google-deepmind","tag-gpt","tag-lean","tag-mathematics","tag-nature","tag-terence-tao","tag-29383","tag-29384","tag-29385","tag-29386","tag-12764","tag-29387","tag-29388"],"_links":{"self":[{"href":"https:\/\/www.europesays.com\/ai\/wp-json\/wp\/v2\/posts\/49345","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.europesays.com\/ai\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.europesays.com\/ai\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.europesays.com\/ai\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.europesays.com\/ai\/wp-json\/wp\/v2\/comments?post=49345"}],"version-history":[{"count":0,"href":"https:\/\/www.europesays.com\/ai\/wp-json\/wp\/v2\/posts\/49345\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.europesays.com\/ai\/wp-json\/wp\/v2\/media\/49346"}],"wp:attachment":[{"href":"https:\/\/www.europesays.com\/ai\/wp-json\/wp\/v2\/media?parent=49345"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.europesays.com\/ai\/wp-json\/wp\/v2\/categories?post=49345"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.europesays.com\/ai\/wp-json\/wp\/v2\/tags?post=49345"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}