Hilbert’s Foundations of Geometry in 1899 made Poincaré think of “reasoning machines” before Hilbert did. Poincaré found the idea “deadly for teaching, and desiccating for researchers” but indispensable for telling when intuitions have been fully expressed. A machine will use stated axioms without the vague intuitions Poincaré considered vital to learning and research. Years of famously intuitive creativity, plus boundless faith in technology, as well as the impact of Hilbert, led Poincaré to see that machines could aid human intuition but not replace it, precisely because machines have no intuition. This relates to recent machine achievements in Lean and HoTT, and to the issues in Akshay Venkatesh’s essay.
{"title":"Poincaré on the value of reasoning machines","authors":"Colin McLarty","doi":"10.1090/bull/1822","DOIUrl":"https://doi.org/10.1090/bull/1822","url":null,"abstract":"Hilbert’s Foundations of Geometry in 1899 made Poincaré think of “reasoning machines” before Hilbert did. Poincaré found the idea “deadly for teaching, and desiccating for researchers” but indispensable for telling when intuitions have been fully expressed. A machine will use stated axioms without the vague intuitions Poincaré considered vital to learning and research. Years of famously intuitive creativity, plus boundless faith in technology, as well as the impact of Hilbert, led Poincaré to see that machines could aid human intuition but not replace it, precisely because machines have no intuition. This relates to recent machine achievements in Lean and HoTT, and to the issues in Akshay Venkatesh’s essay.","PeriodicalId":9513,"journal":{"name":"Bulletin of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141128009","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Machine learning is making significant contributions to many fields but how can it be used as a tool for mathematicians? This article explores the emerging role of machine learning in mathematical research, highlighting how its perceptual capabilities can augment human intuition and lead to new discoveries.
{"title":"Working with machines in mathematics","authors":"Alex Davies","doi":"10.1090/bull/1843","DOIUrl":"https://doi.org/10.1090/bull/1843","url":null,"abstract":"Machine learning is making significant contributions to many fields but how can it be used as a tool for mathematicians? This article explores the emerging role of machine learning in mathematical research, highlighting how its perceptual capabilities can augment human intuition and lead to new discoveries.","PeriodicalId":9513,"journal":{"name":"Bulletin of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141128131","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Neural networks have demonstrated impressive success in various domains, raising the question of what fundamental principles underlie the effectiveness of the best AI systems and quite possibly of human intelligence. This perspective argues that compositional sparsity, or the property that a compositional function have “few” constituent functions, each depending on only a small subset of inputs, is a key principle underlying successful learning architectures. Surprisingly, all functions that are efficiently Turing computable have a compositional sparse representation. Furthermore, deep networks that are also sparse can exploit this general property to avoid the “curse of dimensionality”. This framework suggests interesting implications about the role that machine learning may play in mathematics.
{"title":"Compositional sparsity of learnable functions","authors":"Tomaso Poggio, M. Fraser","doi":"10.1090/bull/1820","DOIUrl":"https://doi.org/10.1090/bull/1820","url":null,"abstract":"Neural networks have demonstrated impressive success in various domains, raising the question of what fundamental principles underlie the effectiveness of the best AI systems and quite possibly of human intelligence. This perspective argues that compositional sparsity, or the property that a compositional function have “few” constituent functions, each depending on only a small subset of inputs, is a key principle underlying successful learning architectures. Surprisingly, all functions that are efficiently Turing computable have a compositional sparse representation. Furthermore, deep networks that are also sparse can exploit this general property to avoid the “curse of dimensionality”. This framework suggests interesting implications about the role that machine learning may play in mathematics.","PeriodicalId":9513,"journal":{"name":"Bulletin of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141127692","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Computational tools might tempt us to renounce complete certainty. By forgoing of rigorous proof, we could get (very) probable results for a fraction of the cost. But is it really true that proofs (as we know and love them) can lead us to certainty? Maybe not. Proofs do not wear their correctness on their sleeve, and we are not infallible in checking them. This suggests that we need help to check our results. When our fellow mathematicians will be too tired or too busy to scrutinize our putative proofs, computer proof assistants could help. But feeding a mathematical argument to a computer is hard. Still, we might be willing to undertake the endeavor in view of the extra perks that formalization may bring—chiefly among them, an enhanced mathematical understanding.
{"title":"Proofs for a price: Tomorrow’s ultra-rigorous mathematical culture","authors":"Silvia De Toffoli","doi":"10.1090/bull/1823","DOIUrl":"https://doi.org/10.1090/bull/1823","url":null,"abstract":"Computational tools might tempt us to renounce complete certainty. By forgoing of rigorous proof, we could get (very) probable results for a fraction of the cost. But is it really true that proofs (as we know and love them) can lead us to certainty? Maybe not. Proofs do not wear their correctness on their sleeve, and we are not infallible in checking them. This suggests that we need help to check our results. When our fellow mathematicians will be too tired or too busy to scrutinize our putative proofs, computer proof assistants could help. But feeding a mathematical argument to a computer is hard. Still, we might be willing to undertake the endeavor in view of the extra perks that formalization may bring—chiefly among them, an enhanced mathematical understanding.","PeriodicalId":9513,"journal":{"name":"Bulletin of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141128172","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Book Review: Regularity Theory for Elliptic PDE","authors":"Connor Mooney","doi":"10.1090/bull/1840","DOIUrl":"https://doi.org/10.1090/bull/1840","url":null,"abstract":"","PeriodicalId":9513,"journal":{"name":"Bulletin of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141127743","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
I argue that since technology is already changing the way we do mathematics, we can use this technology to allow mathematics to be more congressive without human mathematicians becoming redundant in the face of technological advances. In thinking about what it means to “do mathematics,” I examine the following aspects of technology in mathematics: teaching and learning, asking questions, collaboration, dissemination, and the act of doing research. This is not intended to be a rigorous analysis but an informed reflection based on my experience as a mathematician.
{"title":"How machines can make mathematics more congressive","authors":"Eugenia Cheng","doi":"10.1090/bull/1827","DOIUrl":"https://doi.org/10.1090/bull/1827","url":null,"abstract":"I argue that since technology is already changing the way we do mathematics, we can use this technology to allow mathematics to be more congressive without human mathematicians becoming redundant in the face of technological advances. In thinking about what it means to “do mathematics,” I examine the following aspects of technology in mathematics: teaching and learning, asking questions, collaboration, dissemination, and the act of doing research. This is not intended to be a rigorous analysis but an informed reflection based on my experience as a mathematician.","PeriodicalId":9513,"journal":{"name":"Bulletin of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140446637","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
M. Fraser, Andrew Granville, Michael Harris, Colin McLarty, Emily Riehl, Akshay Venkatesh
{"title":"Will machines change mathematics?","authors":"M. Fraser, Andrew Granville, Michael Harris, Colin McLarty, Emily Riehl, Akshay Venkatesh","doi":"10.1090/bull/1836","DOIUrl":"https://doi.org/10.1090/bull/1836","url":null,"abstract":"","PeriodicalId":9513,"journal":{"name":"Bulletin of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140446580","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
I discuss how mathematicians come to a shared notion of what is important, and how automated reasoning might affect that process.
我将讨论数学家如何就什么是重要的达成共识，以及自动推理可能会如何影响这一过程。
{"title":"Some thoughts on automation and mathematical research","authors":"Akshay Venkatesh","doi":"10.1090/bull/1834","DOIUrl":"https://doi.org/10.1090/bull/1834","url":null,"abstract":"I discuss how mathematicians come to a shared notion of what is important, and how automated reasoning might affect that process.","PeriodicalId":9513,"journal":{"name":"Bulletin of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139962116","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Continued fractions have a long history in number theory, especially in the area of Diophantine approximation. The aim of this expository paper is to survey the main results on the theory of p p -adic continued fractions, i.e., continued fractions defined over the field of p p -adic numbers Q p mathbb {Q}_p , which in the last years has recorded a considerable increase of interest and research activity. We start from the very first definitions up to the most recent developments and open problems.
续分数在数论，尤其是在 Diophantine approximation 领域有着悠久的历史。本文的目的是考察 p p -adic 续分数理论的主要成果，即定义在 p p -adic 数域 Q p mathbb {Q}_p 上的续分数，在过去几年中，人们对它的兴趣和研究活动大大增加。我们将从最初的定义开始，直到最新的发展和悬而未决的问题。
{"title":"Continued fractions in the field of 𝑝-adic numbers","authors":"Giuliano Romeo","doi":"10.1090/bull/1819","DOIUrl":"https://doi.org/10.1090/bull/1819","url":null,"abstract":"Continued fractions have a long history in number theory, especially in the area of Diophantine approximation. The aim of this expository paper is to survey the main results on the theory of \u0000\u0000 \u0000 p\u0000 p\u0000 \u0000\u0000-adic continued fractions, i.e., continued fractions defined over the field of \u0000\u0000 \u0000 p\u0000 p\u0000 \u0000\u0000-adic numbers \u0000\u0000 \u0000 \u0000 \u0000 Q\u0000 \u0000 p\u0000 \u0000 mathbb {Q}_p\u0000 \u0000\u0000, which in the last years has recorded a considerable increase of interest and research activity. We start from the very first definitions up to the most recent developments and open problems.","PeriodicalId":9513,"journal":{"name":"Bulletin of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139960443","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A note from the chief editor","authors":"Alejandro Adem","doi":"10.1090/bull/1837","DOIUrl":"https://doi.org/10.1090/bull/1837","url":null,"abstract":"","PeriodicalId":9513,"journal":{"name":"Bulletin of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139835095","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：