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      3天前 北大袁萌最新数学与教育的现代化

数学与教育的现代化

   今年223. 中共中央、国务院印发《中国教育现代化2035》开启了数学拉开了国内数学现代化的序幕。

  众所周知,超实数的引入,彻底简化微积分的证明过程,降低数学教育成本,节省教育资源。

   请见本文附件。

袁萌  陈启清   1017

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中共中央、国务院印发《中国教育现代化2035

2019-02-23  来源:新华网

  新华社北京223日电 近日,中共中央、国务院印发了《中国教育现代化2035》,并发出通知,要求各地区各部门结合实际认真贯彻落实。

  《中国教育现代化2035》分为五个部分:一、战略背景;二、总体思路;三、战略任务;四、实施路径;五、保障措施。

  《中国教育现代化2035》提出推进教育现代化的指导思想是:以习近平新时代中国特色社会主义思想为指导,全面贯彻党的十九大和十九届二中、三中全会精神,坚定实施科教兴国战略、人才强国战略,紧紧围绕统筹推进“五位一体”总体布局和协调推进“四个全面”战略布局,坚定“四个自信”,在党的坚强领导下,全面贯彻党的教育方针,坚持马克思主义指导地位,坚持中国特色社会主义教育发展道路,坚持社会主义办学方向,立足基本国情,遵循教育规律,坚持改革创新,以凝聚人心、完善人格、开发人力、培育人才、造福人民为工作目标,培养德智体美劳全面发展的社会主义建设者和接班人,加快推进教育现代化、建设教育强国、办好人民满意的教育。将服务中华民族伟大复兴作为教育的重要使命,坚持教育为人民服务、为中国共产党治国理政服务、为巩固和发展中国特色社会主义制度服务、为改革开放和社会主义现代化建设服务,优先发展教育,大力推进教育理念、体系、制度、内容、方法、治理现代化,着力提高教育质量,促进教育公平,优化教育结构,为决胜全面建成小康社会、实现新时代中国特色社会主义发展的奋斗目标提供有力支撑。

  《中国教育现代化2035》提出了推进教育现代化的八大基本理念:更加注重以德为先,更加注重全面发展,更加注重面向人人,更加注重终身学习,更加注重因材施教,更加注重知行合一,更加注重融合发展,更加注重共建共享。明确了推进教育现代化的基本原则:坚持党的领导、坚持中国特色、坚持优先发展、坚持服务人民、坚持改革创新、坚持依法治教、坚持统筹推进。

  《中国教育现代化2035》提出,推进教育现代化的总体目标是:到2020年,全面实现“十三五”发展目标,教育总体实力和国际影响力显著增强,劳动年龄人口平均受教育年限明显增加,教育现代化取得重要进展,为全面建成小康社会作出重要贡献。在此基础上,再经过15年努力,到2035年,总体实现教育现代化,迈入教育强国行列,推动我国成为学习大国、人力资源强国和人才强国,为到本世纪中叶建成富强民主文明和谐美丽的社会主义现代化强国奠定坚实基础。2035年主要发展目标是:建成服务全民终身学习的现代教育体系、普及有质量的学前教育、实现优质均衡的义务教育、全面普及高中阶段教育、职业教育服务能力显著提升、高等教育竞争力明显提升、残疾儿童少年享有适合的教育、形成全社会共同参与的教育治理新格局。

  《中国教育现代化2035》聚焦教育发展的突出问题和薄弱环节,立足当前,着眼长远,重点部署了面向教育现代化的十大战略任务:

  一是学习习近平新时代中国特色社会主义思想。把学习贯彻习近平新时代中国特色社会主义思想作为首要任务,贯穿到教育改革发展全过程,落实到教育现代化各领域各环节。以习近平新时代中国特色社会主义思想武装教育战线,推动习近平新时代中国特色社会主义思想进教材进课堂进头脑,将习近平新时代中国特色社会主义思想融入中小学教育,加强高等学校思想政治教育。加强习近平新时代中国特色社会主义思想系统化、学理化、学科化研究阐释,健全习近平新时代中国特色社会主义思想研究成果传播机制。

  二是发展中国特色世界先进水平的优质教育。全面落实立德树人根本任务,广泛开展理想信念教育,厚植爱国主义情怀,加强品德修养,增长知识见识,培养奋斗精神,不断提高学生思想水平、政治觉悟、道德品质、文化素养。增强综合素质,树立健康第一的教育理念,全面强化学校体育工作,全面加强和改进学校美育,弘扬劳动精神,强化实践动手能力、合作能力、创新能力的培养。完善教育质量标准体系,制定覆盖全学段、体现世界先进水平、符合不同层次类型教育特点的教育质量标准,明确学生发展核心素养要求。完善学前教育保教质量标准。建立健全中小学各学科学业质量标准和体质健康标准。健全职业教育人才培养质量标准,制定紧跟时代发展的多样化高等教育人才培养质量标准。建立以师资配备、生均拨款、教学设施设备等资源要素为核心的标准体系和办学条件标准动态调整机制。加强课程教材体系建设,科学规划大中小学课程,分类制定课程标准,充分利用现代信息技术,丰富并创新课程形式。健全国家教材制度,统筹为主、统分结合、分类指导,增强教材的思想性、科学性、民族性、时代性、系统性,完善教材编写、修订、审查、选用、退出机制。创新人才培养方式,推行启发式、探究式、参与式、合作式等教学方式以及走班制、选课制等教学组织模式,培养学生创新精神与实践能力。大力推进校园文化建设。重视家庭教育和社会教育。构建教育质量评估监测机制,建立更加科学公正的考试评价制度,建立全过程、全方位人才培养质量反馈监控体系。

  三是推动各级教育高水平高质量普及。以农村为重点提升学前教育普及水平,建立更为完善的学前教育管理体制、办园体制和投入体制,大力发展公办园,加快发展普惠性民办幼儿园。提升义务教育巩固水平,健全控辍保学工作责任体系。提升高中阶段教育普及水平,推进中等职业教育和普通高中教育协调发展,鼓励普通高中多样化有特色发展。振兴中西部地区高等教育。提升民族教育发展水平。

  四是实现基本公共教育服务均等化。提升义务教育均等化水平,建立学校标准化建设长效机制,推进城乡义务教育均衡发展。在实现县域内义务教育基本均衡基础上,进一步推进优质均衡。推进随迁子女入学待遇同城化,有序扩大城镇学位供给。完善流动人口子女异地升学考试制度。实现困难群体帮扶精准化,健全家庭经济困难学生资助体系,推进教育精准脱贫。办好特殊教育,推进适龄残疾儿童少年教育全覆盖,全面推进融合教育,促进医教结合。

  五是构建服务全民的终身学习体系。构建更加开放畅通的人才成长通道,完善招生入学、弹性学习及继续教育制度,畅通转换渠道。建立全民终身学习的制度环境,建立国家资历框架,建立跨部门跨行业的工作机制和专业化支持体系。建立健全国家学分银行制度和学习成果认证制度。强化职业学校和高等学校的继续教育与社会培训服务功能,开展多类型多形式的职工继续教育。扩大社区教育资源供给,加快发展城乡社区老年教育,推动各类学习型组织建设。

  六是提升一流人才培养与创新能力。分类建设一批世界一流高等学校,建立完善的高等学校分类发展政策体系,引导高等学校科学定位、特色发展。持续推动地方本科高等学校转型发展。加快发展现代职业教育,不断优化职业教育结构与布局。推动职业教育与产业发展有机衔接、深度融合,集中力量建成一批中国特色高水平职业院校和专业。优化人才培养结构,综合运用招生计划、就业反馈、拨款、标准、评估等方式,引导高等学校和职业学校及时调整学科专业结构。加强创新人才特别是拔尖创新人才的培养,加大应用型、复合型、技术技能型人才培养比重。加强高等学校创新体系建设,建设一批国际一流的国家科技创新基地,加强应用基础研究,全面提升高等学校原始创新能力。探索构建产学研用深度融合的全链条、网络化、开放式协同创新联盟。提高高等学校哲学社会科学研究水平,加强中国特色新型智库建设。健全有利于激发创新活力和促进科技成果转化的科研体制。

  七是建设高素质专业化创新型教师队伍。大力加强师德师风建设,将师德师风作为评价教师素质的第一标准,推动师德建设长效化、制度化。加大教职工统筹配置和跨区域调整力度,切实解决教师结构性、阶段性、区域性短缺问题。完善教师资格体系和准入制度。健全教师职称、岗位和考核评价制度。培养高素质教师队伍,健全以师范院校为主体、高水平非师范院校参与、优质中小学(幼儿园)为实践基地的开放、协同、联动的中国特色教师教育体系。强化职前教师培养和职后教师发展的有机衔接。夯实教师专业发展体系,推动教师终身学习和专业自主发展。提高教师社会地位,完善教师待遇保障制度,健全中小学教师工资长效联动机制,全面落实集中连片特困地区生活补助政策。加大教师表彰力度,努力提高教师政治地位、社会地位、职业地位。

  八是加快信息化时代教育变革。建设智能化校园,统筹建设一体化智能化教学、管理与服务平台。利用现代技术加快推动人才培养模式改革,实现规模化教育与个性化培养的有机结合。创新教育服务业态,建立数字教育资源共建共享机制,完善利益分配机制、知识产权保护制度和新型教育服务监管制度。推进教育治理方式变革,加快形成现代化的教育管理与监测体系,推进管理精准化和决策科学化。

  九是开创教育对外开放新格局。全面提升国际交流合作水平,推动我国同其他国家学历学位互认、标准互通、经验互鉴。扎实推进“一带一路”教育行动。加强与联合国教科文组织等国际组织和多边组织的合作。提升中外合作办学质量。优化出国留学服务。实施留学中国计划,建立并完善来华留学教育质量保障机制,全面提升来华留学质量。推进中外高级别人文交流机制建设,拓展人文交流领域,促进中外民心相通和文明交流互鉴。促进孔子学院和孔子课堂特色发展。加快建设中国特色海外国际学校。鼓励有条件的职业院校在海外建设“鲁班工坊”。积极参与全球教育治理,深度参与国际教育规则、标准、评价体系的研究制定。推进与国际组织及专业机构的教育交流合作。健全对外教育援助机制。

  十是推进教育治理体系和治理能力现代化。提高教育法治化水平,构建完备的教育法律法规体系,健全学校办学法律支持体系。健全教育法律实施和监管机制。提升政府管理服务水平,提升政府综合运用法律、标准、信息服务等现代治理手段的能力和水平。健全教育督导体制机制,提高教育督导的权威性和实效性。提高学校自主管理能力,完善学校治理结构,继续加强高等学校章程建设。鼓励民办学校按照非营利性和营利性两种组织属性开展现代学校制度改革创新。推动社会参与教育治理常态化,建立健全社会参与学校管理和教育评价监管机制。

  《中国教育现代化2035》明确了实现教育现代化的实施路径:一是总体规划,分区推进。在国家教育现代化总体规划框架下,推动各地从实际出发,制定本地区教育现代化规划,形成一地一案、分区推进教育现代化的生动局面。二是细化目标,分步推进。科学设计和进一步细化不同发展阶段、不同规划周期内的教育现代化发展目标和重点任务,有计划有步骤地推进教育现代化。三是精准施策,统筹推进。完善区域教育发展协作机制和教育对口支援机制,深入实施东西部协作,推动不同地区协同推进教育现代化建设。四是改革先行,系统推进。充分发挥基层特别是各级各类学校的积极性和创造性,鼓励大胆探索、积极改革创新,形成充满活力、富有效率、更加开放、有利于高质量发展的教育体制机制。

  为确保教育现代化目标任务的实现,《中国教育现代化2035》明确了三个方面的保障措施:

  一是加强党对教育工作的全面领导。各级党委要把教育改革发展纳入议事日程,协调动员各方面力量共同推进教育现代化。建立健全党委统一领导、党政齐抓共管、部门各负其责的教育领导体制。建设高素质专业化教育系统干部队伍。加强各级各类学校党的领导和党的建设工作。深入推进教育系统全面从严治党、党风廉政建设和反腐败斗争。

  二是完善教育现代化投入支撑体制。健全保证财政教育投入持续稳定增长的长效机制,确保财政一般公共预算教育支出逐年只增不减,确保按在校学生人数平均的一般公共预算教育支出逐年只增不减,保证国家财政性教育经费支出占国内生产总值的比例一般不低于4%。依法落实各级政府教育支出责任,完善多渠道教育经费筹措体制,完善国家、社会和受教育者合理分担非义务教育培养成本的机制,支持和规范社会力量兴办教育。优化教育经费使用结构,全面实施绩效管理,建立健全全覆盖全过程全方位的教育经费监管体系,全面提高经费使用效益。

  三是完善落实机制。建立协同规划机制、健全跨部门统筹协调机制,建立教育发展监测评价机制和督导问责机制,全方位协同推进教育现代化,形成全社会关心、支持和主动参与教育现代化建设的良好氛围。

(责任编辑:刘潇翰)






 

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4天前 北大袁萌基础数学研究的好教材!

基础数学研究的好教材!

荷兰基础理论研究型名校格罗宁根大学知名数学J.Ponstein教授为不的熟悉数理逻辑数学工作者精心撰写了一部介绍“非标准分析”科普专著,对于我国高校微积分教学改革很有借鉴意义。

请见本文附件。 袁萌 陈启清 1015

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本世纪最好的NSA

2081025

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6天前 北大袁萌创建研究型大学,指日可待

创建研究型大学,指日可待

  近日,国家教育部发布《关于深化本科教育教学改革 全面提高人才培养质量的意见》的通知.

  该《意见》分为三个部分,共计22个条款.                                           十分重要;

  其中第二款为:“2.激励学生刻苦学习。高校要切实加强学风建设,教育引导学生爱国、励志、求真、力行。要提升学业挑战度,强化人才培养方案、教学过程和教学考核等方面的质量要求,科学合理设置学分总量和课程数量,增加学生投入学习的时间,提高自主学习时间比例,引导学生多读书、深思考、善提问、勤实践。合理增加学生阅读量和体育锻炼时间,以适当方式纳入考核成绩。积极组织学生参加社会调查、生产劳动、志愿服务、公益活动、科技发明和勤工助学等实践活动。”

如果认真贯彻、落实这一条款,创建研究型大学,指日可待!

  《通知》全文,.请见本文附件。

袁萌  陈启清   1013

附件:

教育部关于深化本科教育教学改革 全面提高人才培养质量的意见

教高〔20196

各省、自治区、直辖市教育厅(教委),新疆生产建设兵团教育局,有关部门(单位)教育司(局),部属各高等学校、部省合建各高等学校:

为深入贯彻全国教育大会精神和《中国教育现代化2035》,全面落实新时代全国高等学校本科教育工作会议和直属高校工作咨询委员会第二十八次全体会议精神,坚持立德树人,围绕学生忙起来、教师强起来、管理严起来、效果实起来,深化本科教育教学改革,培养德智体美劳全面发展的社会主义建设者和接班人,现提出如下意见。

一、严格教育教学管理

1.把思想政治教育贯穿人才培养全过程。坚持把立德树人成效作为检验高校一切工作的根本标准,用习近平新时代中国特色社会主义思想铸魂育人,加快构建高校思想政治工作体系,推动形成“三全育人”工作格局。把思想政治理论课作为落实立德树人根本任务的关键课程,推动思想政治理论课改革创新,建设一批具有示范效应的思想政治理论课,不断增强思想政治理论课的思想性、理论性和亲和力、针对性。把课程思政建设作为落实立德树人根本任务的关键环节,坚持知识传授与价值引领相统一、显性教育与隐性教育相统一,充分发掘各类课程和教学方式中蕴含的思想政治教育资源,建成一批课程思政示范高校,推出一批课程思政示范课程,选树一批课程思政优秀教师,建设一批课程思政教学研究示范中心,引领带动全员全过程全方位育人。

2.激励学生刻苦学习。高校要切实加强学风建设,教育引导学生爱国、励志、求真、力行。要提升学业挑战度,强化人才培养方案、教学过程和教学考核等方面的质量要求,科学设置学分总量和课程数量,增加学生投入学习的时间,提高自主学习时间比例,引导学生多读书、深思考、善提问、勤实践。合理增加学生阅读量和体育锻炼时间,以适当方式纳入考核成绩。积极组织学生参加社会调查、生产劳动、志愿服务、公益活动、科技发明和勤工助学等实践活动。

3.全面提高课程建设质量。立足经济社会发展需求和人才培养目标,优化公共课、专业基础课和专业课比例结构,加强课程体系整体设计,提高课程建设规划性、系统性,避免随意化、碎片化,坚决杜绝因人设课。实施国家级和省级一流课程建设“双万计划”,着力打造一大批具有高阶性、创新性和挑战度的线下、线上、线上线下混合、虚拟仿真和社会实践“金课”。积极发展“互联网+教育”、探索智能教育新形态,推动课堂教学革命。严格课堂教学管理,严守教学纪律,确保课程教学质量。

4.推动高水平教材编写使用。高校党委要高度重视教材建设,落实高校在教材建设中的主体责任,健全教材管理体制机制,明确教材工作部门。做好马克思主义理论研究和建设工程重点教材统一使用工作,推动教材体系向教学体系转化。鼓励支持高水平专家学者编写既符合国家需要又体现个人学术专长的高水平教材,充分发挥教材育人功能。

5.改进实习运行机制。推动健全大学生实习法律制度,完善各类用人单位接收大学生实习的制度保障。充分考虑高校教学和实习单位工作实际,优化实习过程管理,强化实习导师职责,提升实习效果。加大对学生实习工作支持力度,鼓励高校为学生投保实习活动全过程责任保险,支持建设一批共享型实习基地。进一步强化实践育人,深化产教融合、校企合作,建成一批对区域和产业发展具有较强支撑作用的高水平应用型高等学校。

6.深化创新创业教育改革。挖掘和充实各类课程、各个环节的创新创业教育资源,强化创新创业协同育人,建好创新创业示范高校和万名优秀创新创业导师人才库。持续推进国家级大学生创新创业训练计划,提高全国大学生创新创业年会整体水平,办好中国“互联网+”大学生创新创业大赛,深入开展青年红色筑梦之旅活动。

7.推动科研反哺教学。强化科研育人功能,推动高校及时把最新科研成果转化为教学内容,激发学生专业学习兴趣。加强对学生科研活动的指导,加大科研实践平台建设力度,推动国家级、省部级科研基地更大范围开放共享,支持学生早进课题、早进实验室、早进团队,以高水平科学研究提高学生创新和实践能力。统筹规范科技竞赛和竞赛证书管理,引导学生理性参加竞赛,达到以赛促教、以赛促学效果。

8.加强学生管理和服务。加强高校党委对学生工作的领导,健全学生组织思政工作体系,坚持严格管理与精心爱护相结合。加强学生诚信教育和诚信管理,严格校规校纪刚性约束。配齐建强高校辅导员队伍,落实专职辅导员职务职级“双线”晋升要求,积极探索从时代楷模、改革先锋、道德模范、业务骨干等群体中选聘校外辅导员。积极推动高校建立书院制学生管理模式,开展“一站式”学生社区综合管理模式建设试点工作,配齐配强学业导师、心理辅导教师、校医等,建设师生交流活动专门场所。

9.严把考试和毕业出口关。完善过程性考核与结果性考核有机结合的学业考评制度,综合应用笔试、口试、非标准答案考试等多种形式,科学确定课堂问答、学术论文、调研报告、作业测评、阶段性测试等过程考核比重。加强考试管理,严肃考试纪律,坚决取消毕业前补考等“清考”行为。加强学生体育课程考核,不能达到《国家学生体质健康标准》合格要求者不能毕业。科学合理制定本科毕业设计(论文)要求,严格全过程管理,严肃处理各类学术不端行为。落实学士学位管理办法,健全学士学位管理制度,严格学士学位标准和授权管理,严把学位授予关。

二、深化教育教学制度改革

10.完善学分制。学分制是以学分作为衡量学生学习质量和数量,为学生提供更多选择余地的教学制度。支持高校进一步完善学分制,扩大学生学习自主权、选择权。建立健全本科生学业导师制度,安排符合条件的教师指导学生学习,制订个性化培养方案和学业生涯规划。推进模块化课程建设与管理,丰富优质课程资源,为学生选择学分创造条件。支持高校建立与学分制改革和弹性学习相适应的管理制度,加强校际学分互认与转化实践,以学分积累作为学生毕业标准。完善学分标准体系,严格学分质量要求,建立学业预警、淘汰机制。学生在基本修业年限内修满毕业要求的学分,应准予毕业;未修满学分,可根据学校修业年限延长学习时间,通过缴费注册继续学习。支持高校按照一定比例对特别优秀的学士学位获得者予以表彰,并颁发相应的荣誉证书或奖励证书。

11.深化高校专业供给侧改革。以经济社会发展和学生职业生涯发展需求为导向,构建自主性、灵活性与规范性、稳定性相统一的专业设置管理体系。完善人才需求预测预警机制,推动本科高校形成招生计划、人才培养和就业联动机制,建立健全高校本科专业动态调整机制。以新工科、新医科、新农科、新文科建设引领带动高校专业结构调整优化和内涵提升,做强主干专业,打造特色优势专业,升级改造传统专业,坚决淘汰不能适应社会需求变化的专业。深入实施“六卓越一拔尖”计划2.0,全面实施国家级和省级一流本科专业建设“双万计划”,促进各专业领域创新发展。完善本科专业类国家标准,推动质量标准提档升级。

12.推进辅修专业制度改革。促进复合型人才培养,逐步推行辅修专业制度,支持学有余力的全日制本科学生辅修其它本科专业。高校应研究制定本校辅修专业目录,辅修专业应与主修专业归属不同的专业类。原则上,辅修专业学生的遴选不晚于第二学年起始时间。辅修专业应参照同专业的人才培养要求,确定辅修课程体系、学分标准和学士学位授予标准。要结合学校定位和辅修专业特点,推进人才培养模式综合改革,形成特色化人才培养方案。要建立健全与主辅修制度相适应的人才培养与资源配置、管理制度联动机制。对没有取得主修学士学位的学生不得授予辅修学士学位。辅修学士学位在主修学士学位证书中予以注明,不单独发放学位证书。

13.开展双学士学位人才培养项目试点。支持符合条件高校创新人才培养模式,开展双学士学位人才培养项目试点,为学生提供跨学科学习、多样化发展机会。试点须报省级学位委员会审批通过后,通过高考招收学生。试点坚持高起点、高标准、高质量,所依托的学科专业应具有博士学位授予权,且分属两个不同的学科门类。试点人才培养方案要进行充分论证,充分反映两个专业的课程要求、学分标准和学士学位授予标准,不得变相降低要求。高校要推进试点项目与现有教学资源的共享,促进不同专业课程之间的有机融合,实现学科交叉基础上的差异化、特色化人才培养。本科毕业并达到学士学位要求的,可授予双学士学位。双学士学位只发放一本学位证书,所授两个学位应在证书中予以注明。高等学历继续教育不得开展授予双学士学位工作。

14.稳妥推进跨校联合人才培养。支持高校实施联合学士学位培养项目,发挥不同特色高校优势,协同提升人才培养质量。该项目须报合作高校所在地省级学位委员会审批。该项目相关高校均应具有该专业学士学位授予权,通过高考招收学生。课程要求、学分标准和学士学位授予标准,不得低于联合培养单位各自的相关标准。实施高校要在充分论证基础上签署合作协议,联合制定人才培养方案,加强学生管理和服务。联合学士学位证书由本科生招生入学时学籍所在的学士学位授予单位颁发,联合培养单位可在证书上予以注明,不再单独发放学位证书。高等学历继续教育不得开展授予联合学士学位工作。

15.全面推进质量文化建设。完善专业认证制度,有序开展保合格、上水平、追卓越的本科专业三级认证工作。完善高校内部教学质量评价体系,建立以本科教学质量报告、学院本科教学评价、专业评价、课程评价、教师评价、学生评价为主体的全链条多维度高校教学质量评价与保障体系。持续推进本科教学工作审核评估和合格评估。要把评估、认证等结果作为教育行政部门和高校政策制定、资源配置、改进教学管理等方面的重要决策参考。高校要构建自觉、自省、自律、自查、自纠的大学质量文化,把其作为推动大学不断前行、不断超越的内生动力,将质量意识、质量标准、质量评价、质量管理等落实到教育教学各环节,内化为师生的共同价值追求和自觉行动。全面落实学生中心、产出导向、持续改进的先进理念,加快形成以学校为主体,教育部门为主导,行业部门、学术组织和社会机构共同参与的中国特色、世界水平的质量保障制度体系。

三、引导教师潜心育人

16.完善高校教师评聘制度。高校可根据需要设立一定比例的流动岗位,加大聘用具有其它高校学习和行业企业工作经历教师的力度。出台高校教师职称制度改革的指导意见,推行高校教师职务聘任制改革,加强聘期考核,准聘与长聘相结合,做到能上能下、能进能出。高校教师经所在单位批准,可开展多点教学并获得报酬。引导高校建立兼职教师资源库,开展兼职教师岗前培训,为符合条件的兼职教师、急需紧缺人才申报相应系列专业技术职务。研究出台实验技术系列职称制度改革的指导意见,优化高校实验系列队伍结构。

17.加强基层教学组织建设。高校要以院系为单位,加强教研室、课程模块教学团队、课程组等基层教学组织建设,制定完善相关管理制度,提供必需的场地、经费和人员保障,选聘高水平教授担任基层教学组织负责人,激发基层教学组织活力。支持高校组建校企、校地、校校联合的协同育人中心,打造校内外结合的高水平教学创新团队。要把教学管理队伍建设放在与教师队伍建设同等重要位置,制定专门培养培训计划,为其职务晋升创造有利政策环境。

18.完善教师培训与激励体系。推动教师培训常态化,探索实行学分管理,将培训学分作为教师考核和职务聘任的重要依据。加强高校教师发展中心建设,重点面向新入职教师和青年教师,以提升教学能力为目的,开展岗前和在岗专业科目培训。推进高校中青年教师专业发展,建立高校中青年教师国内外访学、挂职锻炼、社会实践制度。完善校企、校社共建教师企业实践流动岗(工作站)机制,共建一批教师企业实践岗位。鼓励高校为长期从事教学工作的教师设立荣誉证书制度。鼓励社会组织对教师出资奖励,开展尊师活动,营造尊师重教良好社会风尚。

19.健全教师考核评价制度。加强师德师风建设,将师德考核贯穿于教育教学全过程。突出教育教学业绩在绩效分配、职务职称评聘、岗位晋级考核中的比重,明确各类教师承担本科生课程的教学课时要求。切实落实教授全员为本科生上课的要求,让教授到教学一线,为本科生讲授基础课和专业基础课,把教授为本科生的授课学时纳入学校教学评估指标体系。教师日常指导学生学习、创新创业、社会实践、各类竞赛展演以及开展“传帮带”等工作,计入教育教学工作量,纳入年度考核内容。

20.建立健全助教岗位制度。助教岗位承担课堂教辅、组织讨论、批改作业试卷、辅导答疑、协助实习实践等教学辅助任务,主要由没有教学经历的新入职教师、研究生、优秀高年级本科生等担任。高校应建立健全助教岗位制度,完善选拔、培训、评价、激励和反馈的全流程助教岗位管理制度。新入职教师承担的助教工作应纳入教师工作量考核,对于表现优秀的应在职称评聘、职务晋升中予以优先考虑。加强对担任助教工作学生的岗前培训和规范管理,合理确定补贴标准,提供必要条件保障,确保教学工作质量。

四、加强组织保障

21.加强党对高校教育教学工作的全面领导。地方党委教育工作部门、高校各级党组织要坚持以习近平新时代中国特色社会主义思想为指导,全面贯彻党的教育方针,坚定社会主义办学方向,落实“以本为本、四个回归”的要求,加强对本科教育教学改革的领导。高校党委会、常委会和校长办公会要把本科教育教学改革工作纳入重要议题研究部署,高校主要领导、各级领导干部、广大教师要把主要精力投入教育教学工作,深入党建和思政、教学和科研一线,切实把走进学生、关爱学生、帮助学生落到实处。高校的人员、经费、物质资源要聚焦本科教育教学改革,强化人才培养质量意识,形成全员、全方位支持教育教学改革的良好氛围。

22.完善提高人才培养质量的保障机制。各地教育行政部门要增强工作针对性和实效性,结合区域实际,明确深化本科教育教学改革总体目标、重点内容、创新举措、评价考核和保障机制,加强政策协调配套,调整教育经费支出结构,加大对教育教学改革的投入力度。要进一步落实高校建设主体责任和办学自主权,提升高校治理能力和治理水平,加强内部统筹,着力解决建设难点和堵点问题。要加强对高校教育教学改革成效的督导检查,加大典型做法的总结宣传力度,推动形成狠抓落实、勇于创新、注重实效的工作局面。

教育部

2019929

 



 

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7天前 北大袁萌1.8莱布尼兹的天才发明

1.8 莱布尼兹的天才发明

1.8莱布尼兹的天才发明

   从数学发展的历史长河中来看,在我们国内发表此文(1.8莱布尼兹的天才发明)具有标志性意义。

  注:法国数学家柯西(分析数学奠基人)竟然也是莱布尼兹的信徒。

  请见本文附件。

  感叹国内无穷小“痴迷者”太少了。

袁萌  陈启清  1010

附件:

1.8 Innitesimals in the 17th to the 19th century

There can be no doubt that in the 1670’s, some 1900 years after

Archimedes lived, innitesimals were conceived by Leibniz. Moreover, he formulated their main properties, and many contemporary mathematicians as well as mathematicians after him, among them Euler and Cauchy, were able to successfully work with them. But the theory of the innitesimals lacked a rigorous basis, and during some 200 years all trials to improve this situation were in vein, so that at last one gave up, the more so because in the 1870’s Weierstrass came up with a rigorous theory of limits and continuity, which became the basis of what now is known as classical analysis, and where there was and is no need to consider innitesimals any more.

It is quite interesting to see how Euler [2] shows the well-known product formula for the sine function. He begins his proof with the equality, 2·sinh x = (1 + x/n)n −(1−x/n)n, valid for – in Eulers’s own words – ‘innitely large values’ of n. Obviously, this is only true up to an innitesimal. Then the right-hand side is treated as if n were

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a classical natural number. Th



 

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10天前 北大袁萌1.4无穷小放飞互联网,究竟为了什么?

1.4 无穷小放飞互联网,究竟为了什么?

    坦率地说,七年前,我们放飞无穷小到国内互联网的目的是:进一步改进国内高校微积分的教学改革,提高数学的创新能力。

   国内数学不能故步自封。请见本文附件。

袁萌  陈启清  107

附件:

1.4 The purpose of nonstandard analysis

After the digression of the preceding section let us now contemplate the purpose of nonstandard analysis. Starting from IN, the sets ZZ,Q and IR (andC, but below complex numbers will be ignored) have been introduced in classical mathematics in order to enrich mathematics with more tools and to rene existing tools. The introduction of negative numbers, of fractions, and of irrational numbers is felt as a strong necessity, and without it mathematics would only be a small portion of what it actually is. The introduction of IN, ZZ, Q, and IR, however, was not meant at all to enrich mathematics (at least not when it all started), but only to simplify doing mathematics. For as soon as notions like limit and continuity are involved, denitions in nonstandard analysis can be given a simpler form, and theorems can be proved in a simpler way. Often the simplications are considerable. In one case the proof of a classical conjecture was found by means of nonstandard analysis, after which a classical proof was found as well. Moreover, both denitions and proofs receive a more natural appearance. This may even enhance the discovery of new facts.

In the mean time nonstandard analysis has also been applied in a more traditional way, namely to introduce new mathematical notions and models.

 

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17天前 北大袁萌实数系统为何还需要保序扩张?

实数系统为何还需要保序扩张?

   这个问题是荷兰格罗宁根大学知名数学家J.Ponstein教授在其专著“非标准分析”(1996年发表的PDF电子版)中明确提出的,并且做出了自己的=

解答方法。

注:本文附件是此书的前言。前言的第一段落就明确提出了这一问题(即非阿基米德伤序扩张)。

袁萌  陈启清  103

附件:

Preface

An innitesimal is a ‘number’ that is smaller then each positive real number and is larger than each negative real number, so that in the real number system there is just one innitesimal, i.e. zero. But most of the time only nonzero innitesimals are of interest. This is related to the fact that when in the usual limit denition x is tending to c, most of the time only the values of x that are dierent from c are of interest. Hence the real number system has to be extended in some way or other in order to include all innitesimals.

This book is concerned with an attempt to introduce the innitesimals and the other ‘nonstandard’ numbers in a naive, simpleminded way. Nevertheless, the resulting theory is hoped to be mathematically sound, and to be complete within obvious limits. Very likely, however, even if ‘nonstandard analysis’ is presented naively, we cannot do without the axiom of choice (there is a restricted version of nonstandard analysis, less elegant and less powerful, that does not need it). This is a pity, because this axiom is not obvious to every mathematician, and is even rejected by constructivistic mathematicians, which is not unreasonable as it does not tell us how the relevant choice could be made (except in simple cases, but then the axiom is not needed).

The remaining basic assumptions that will be made would seem to be acceptable to many mathematicians, although they will be taken partly from formalistic mathematics – i.e. the usual logical principles, in particular the principle of the excluded third – as well as from constructivistic mathematics – i.e. that at the start of all of mathematics the natural numbers (in the classical sense of the term) are given to us. Not only the natural number, but also the set and the pair will be taken as primitive notions. The net eect of this is a version of mathematics that, except for truly nonstandard results, would seem to produce the same theorems as produced by classical mathematics.

One of the consequences of combining ideas from the two main schools of mathematical thinking is that the usual axioms of set theory, notably those due to Zermelo and Fraenkel, will be ignored. First of all, there will be elements that are

17

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not sets, the natural numbers to begin with, only then sets will be formed from them in stages (or day by day), whereas when starting from the Zermelo-Fraenkel axioms each mathematical entity, in particular each natural number, is some set. From a formal point of view the latter has the advantage that there is just one primitive notion, but from a naive point of view it is not so obvious why numbers should be sets (in formalistic mathematics after the natural numbers come to life in the form of sets, this fact is concealed as soon as possible). Moreover, aren’t we presupposing at least the order of the natural numbers already when writing down axioms by means of suitable symbols?

To a certain extent nonstandard analysis is superuous! For if a theorem of classical mathematics has a nonstandard proof, it also has a classical proof (this follows from what in nonstandard analysis is known as the ‘transfer’ theorem). Often the nonstandard proof is intuitively more attractive, simpler and shorter, which is one of the reasons to be interested in nonstandard analysis at all. Another reason is that totally new mathematical models for all kinds of problems can be (and in the mean time have been) formulated when innitesimals or other nonstandard numbers occur in such models. A trivial example is a problem involving a heap of sand containing very many grains of sand, but where the number of grains of sand must not be innite. Then taking the inverse of some positive innitesimal and rounding the result up or down produces a so-called innitely large ‘natural number’ that is larger than each ordinary natural number, but is smaller than innity. It can be manipulated in much the same way as the ordinary numbers, which cannot, of course, be said of innity. As a consequence the mathematics of innitely large sets is essentially simpler than that of innite sets. A peculiarity, however, is that the ‘selected’ innitesimal and hence the innitely large natural number are not specied the way the number of elements of a set of, say, 25 elements is specied. On the other hand, if ω is that innitely large natural number, it makes sense to consider another heap of sand with ω2 grains of sand, that can be thought of as the result of combining ω heaps of sand each containing w grains of sand. But in what follows the analysis of practical models containing nonstandard numbers will not be stressed.

Chapter 1

Generalities

 

 



 

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18天前 北大袁萌1.2超实数的诞生:-变换

1.2 超实数的诞生: -变换

    在数学发展史上,超实数的诞生是一个里程碑事件。

   实际上,超实数系统是无穷小微积分的理论基础。

袁萌  陈启清   102

附件:

1.2 Other -transforms; generating new numbers The -transform not only can be obtained for IR but also for IN, ZZ,Q, and in fact any set X of classical mathematics (and for much more, see Section 1.5). Their -transforms are indicated by IN, ZZ, Q, and X, respectively. Throwing all nonnite numbers out of IN and ZZ we obtain again IN and ZZ, but something similar is not true for Q (for IR we know this already), simply because Q (just as IR) contains nite non-classical numbers. Yet there is a striking dierence between Q and IR in this respect: the ‘standard part theorem’ discussed at the end of the preceding section does not hold for Q, that is to say, there are nite elements t of Q that cannot be written as t = x + ε, with x Q, ε Q, ε 0. For let c be any irrational number, say c = 2, and let (r1,r2,...) be some Cauchy sequence of rationals converging to c. Later on it will become clear that then the sequence (r1−c,r2−c,...) ‘generates’ an innitesimal δ in IR (because this sequence converges to zero). On the other hand (r1,r2,...) generates an element r Q IR, and r is nite (because the ri are rational, and this sequence converges), but it has no standard part in Q, for otherwise r = x + ε for some x Q and some ε Q, ε ' 0. But (r1 −c,r2 −c,...) also generates the nite number r −c IR, so that r −c = δ ' 0. It follows that x−c = δ −ε ' 0, hence x−c = 0 (as x−c is an ordinary real), which would mean that c Q, a contradiction. On the other hand, in IR we have that st(r) = c. (Carrying this argument further it turns out that there exists a 1−1 mapping between IR and the set of all nite elements of Q modulo the set of all rational innitesimals, preserving addition and multiplication; i.e. the mapping is an isomorphism. In other words, IR (not IR) can in a sense be produced by Q.)

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There are various ways to introduce the new numbers. Below this will be done by means of innite sequences of classical numbers. In particular, the elements of IR will be generated by means of innite sequences of reals, and it will be necessary to consider all such sequences. (Recall that the elements of IR can be generated by means of rather special innite sequences of rationals, i.e. the Cauchy sequences.) More generally, given any classical set X the elements of its -transform X will be generated by means of innite sequences of elements of X, and again all such sequences must be taken into account. Each such sequence ‘generates’ an element of X, and in case X is a set of numbers (or n-tuples of numbers) special sequences generate the elements of X itself. For example, (1,2,3,...) generates a hyperlarge element of IN, and (3/2,5/4,9/8,...) generates a nite element of Q, that is equal to the sum of 1, generated by (1,1,1,...) and an innitesimal, generated by (1/2,1/4,1/8,...). Dierent sequences may generate the same element of X. In fact, given any x X there are many (uncountably many) dierent sequences that generate x (if X contains at least two elements). For example, changing nitely many terms of a generating sequence has no eect on the element generated. But there are many more variations on this theme. Wouldn’t it be possible to restrict ourselves to a suitable subset of all sequences? Unless we are satised with some sort of mutilated nonstandard analysis, most likely the answer is ‘no’. See Section 4.4.

Anyway, the nuisance of having to use generating sequences is only temporary. Once the new numbers have been introduced (as well as new functions, etc.) in most cases it is not necessary at all to know that they came about by means of innite sequences. The situation is entirely analogous to that of introducing the real numbers: most of real analysis can be developed without the interference of Cauchy sequences. Most of the time an irrational such as 2 is treated as just a number, not as a sequence. Although IN, ZZ, Q and IR are extensions of IN, ZZ,Q and IR, respectively, in general X is not always an extension of X. If, for example X = {IN}, then X = {IN}, and since IN 6= IN, X is not contained in X.



 

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18天前 北大袁萌1.1无穷小是一个数

 1.1无穷小是一个数
 祖国可爱!
 祖国伟大!
 
 袁萌  陈启清  10月1日
 附件?
 Chapter 1
 Generalities(概论)
 1.1 Innitesimals and other nonstandard numbers: getting acquainted
 An innitesimal is a number that is smaller than every positive real number and is larger than every negative real number, or, equivalently, in absolute value it is smaller than 1/m for all m ∈ IN = {1,2,3,...}. Zero is the only real number that at the same time is an innitesimal, so that the nonzero innitesimals do not occur in classical mathematics. Yet, they can be treated in much the same way as can the classical numbers. For example, each nonzero innitesimal ε can be inverted and the result is the number ω = 1/ε. It follows that | w |> m for all m ∈ IN, for which reason ω is called (positive or negative) hyperlarge (or innitely large). Hyperlarge numbers too do not occur in classical mathematics, but nevertheless can be treated like classical numbers. If, for example, ω is positive hyperlarge, we can compute √ω, ω/2, ω −1, ω + 1, 2ω, ω2, etc., and we have (ω−1) + (ω + 1) = 2ω, (ω−1)•(ω + 1) = ω2 −1, etc. Also, for all m ∈ IN, m < √ω < ω/2 < ω−1 < ω < ω + 1 < 2ω < ω2 giving seven dierent hyperlarge numbers. The positive hyperlarge numbers must not be confused with innity (∞), which should not be regarded a number at all, and which anyway does not satisfy these inequalities, except the rst one.
 Regrettably, there does not seem to exist a synonym for ‘hyperlarge number’ that would make a nice pair with ‘innitesimal’, so let us introduce the synonym ‘hypersmall number’ for the latter.
 If ε is hypersmall, if δ too is hypersmall but nonzero, and if ω is positive hyperlarge, so that −ω is negative hyperlarge, we write, ε ' 0, δ ∼ 0, ω ∼∞, −ω ∼−∞ respectively.
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 It would be wrong, of course, to deduce from ω ∼∞ that the dierence between ω and ∞, or that between −ω and −∞ would be hypersmall. Given any x ∈ IR, x 6= 0, and any δ ' 0, let t = x + δ, then, ε <| t |< ω, for all ε ∼ 0 and all ω ∼∞. The number t is called appreciable (as it is not too small and not too large).
 Three nonoverlapping sets of numbers (old or new) can now be presented:
 a) the set of all innitesimals, to which zero belongs, b) the set of all appreciable numbers, to which all nonzero reals belong, and c) the set of all hyperlarge numbers, containing no classical numbers at all.
 Together these three sets constitute the set of all numbers of ‘real nonstandard analysis’. This set, which clearly is an extension of IR is indicated by,
 ∗IR and is called the ∗-transform of IR. The elements of ∗IR are called hyperreal. The use of the prex ‘hyper’ here is not entirely defendable, as, say, 5, which obviously is an element of ∗IR, is just an ordinary real.
 Abbreviating hypersmall, appreciable, and hyperlarge to s, a and l, respectively, and assuming that x and y are positive numbers, for addition and multiplication the following holds, y\x s a l y\x s a l s s a l s s s ? a a a l a s a l l l l l l ? l l addition multiplication
 where the quotation marks stand for s or a or l. Examples for the lower left quotation mark are x ∼ 0 and y = √x−1, or 1/x, or 1/x2. For x−y the results are the same as for x + y (if still x,y > 0), except that if both x and y are appreciable, then x−y is either hypersmall or appreciable, and that if both x and y are hyperlarge, then x−y is either hyperlarge (positive or negative), or appreciable, or hypersmall, as is shown by the following examples: y = x/2, or 2x, or x−1, or x + ε, with ε ' 0. If a number is not hyperlarge it is called nite or limited.
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 Remark: Elsewhere in the literature, any element of ∗IR is called nite. Clearly, t is nite if and only if t = x + ε for some x ∈ IR and some ε ' 0. Given such a t, both x and ε are unique, for, x + ε = y + δ, x,y ∈ IR, ε,δ ' 0 implies that x−y = δ−ε ' 0, so that (as x−y ∈ IR), x−y = 0, hence x = y and ε = δ. By denition x is called the standard part of t, and this is written as,
 x = st(t).
 The standard part function st provides an important (mainly one-way) bridge between the nite numbers of nonstandard analysis and the classical numbers. Trivially, if t is itself a classical number, then st(t) = t.
 
 


 

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19天前 北大袁萌1.9无穷小新生五十年

1.9无穷小新生五十年

荷兰千年名校格罗宁根大学知名数学J.Ponstein教授为不的熟悉数理逻辑的数学工作者精心撰写了一部介绍“非标准分析”科普专著,对于我国高校微积分教学改革很有帮助。
   为此,我们将这部著作分为章节发表,加上适当小标题与评论。请读者注意。

袁萌  陈启清  929

附件:

1.9 Innitesimals in the 20th century

When in the 1870’s Weierstrass formulated the well-known ε−δ denitions of limit and continuity, denitions that completely ignore nonstandard numbers, the dispute regarding innitesimals quickly settled in their disadvantage, but only temporarily, for in 1961 Robinson [6,7] presented a mathematically sound theory of the nonstandard numbers. These works embody the rst fairly complete analysis of the nonstandard numbers. Not only are they based on work of forerunners, but also on an amount of mathematical logic that hitherto was unusual in mathematics. Only a few references should suce here, see [8–12].

Robinson starts from the axioms of set theory due to Zermelo and Fraenkel, and the axiom of choice (called together the ZFC axioms), derives IR in a classical kind of way, and then extends IR to IR by applying a rather considerable amount of mathematical logic, as indicated before. Another way to dene IR was already indicated by Hewitt [10] and worked out by Luxemburg [13]. Here the ZFC axioms are again the point of departure, but the more usual line of mathematical thinking is followed. (Except for the ZF axioms, this way is also followed in the next chapter.) Still another way to introduce IR was found by Nelson [14]. Nelson adds three more axioms to the ZFC axioms, as well as a new symbol, st (for ‘standard’) that is used as a kind of label to distinguish standard constants from nonstandard constants. This leads directly to the set of all standard as well as all nonstandard constants, without the intermediary step of rst introducing IR; consequently

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in internal set theory IR is denoted by IR, and similarly, IN is denoted by IN, etc. Actually, the point of view of internal set theory is that the IN of classical mathematics is the same as the IN of nonstandard analysis; and that all that happens is that unexpected elements of IN are discovered, elements that had always been there. In other words, according to this point of view, 0, 1, 2, etc. do not at all ll up IN (see Robert [15] and F. Diener et G. Reeb [16]). The additional axioms make sure that transfer is guaranteed (axiom of ‘transfer’), that nonstandard numbers exist (axiom of ‘idealization’), and that unique standard sets can be derived from given sets (axiom of ‘standardization’). Even though internal set theory uses relatively little of mathematical logic, the new axioms require some study, and do not seem to be as obvious as, for example, the axioms of Greek geometry: Transfer: stt1 ...sttk : [stx : P(x,t1,...,tk) ⇒∀x : P(x,t1,...,tk)]. Idealization: [st nx : x : y z : P(x,y)] [x : sty : P(x,y)]. Standardization: stx : sty : stz : [z y z xP(z)]. Here stu means that the variable u must be standard, and similarly the label n means that the corresponding variable must be nite (but beware, in internal set theory any hyperlarge natural number is nite, only the combination of standard and nite amounts to the classical notion of niteness). Note that whereas stu means that the variable u is standard, means the variable is standard, because st is a label butis a mapping. P(...) denotes a given internal statement, except in the last axiom, where P(...) may even be external (see Section 1.6).

In naive nonstandard analysis these three additional axioms are not assumed but derived from the existence of the natural numbers and the axiom of choice. Transfer has already been discussed; and idealization is used to prove the existence of nonstandard elements in any internal set with an innite number of elements. Perhaps standardization is the most intriguing of the three because it contains a statement P(z) that may be external. Reformulated naively it means that, ∀∗x : ∃∗y : ∀∗z : [z y ⇔∗z xP(z)], where x, y and z are, of course, classical. Since always s S if and only if s S, it follows that, y = {z x : P(z)}, or equivalently,

y = {z x : P(x)},

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which in internal set theory are illegal set formations. Here are a few examples, where x and y are still classical, but z need not be classical. 1) P(z) z INz is standard; then x = {1,2,3} gives y = x = 1,2,3}, x = IN gives y = x = IN, and x = IR also gives y = IN.

In fact IN is the largest y that is possible for variable x. 2) P(z) z INz < n, with n IN given such that n ; then the results are as under 1). 3) P(z) z IRz ' 0; then y = {0} if 0 x and y = if 0 6 x. For other details the reader should consult more adequate treatments of internal set theory.

In the mean time other versions of nonstandard analysis have been developed. In one of them external sets are ‘legalized’ by means of still other axioms, and another label, ext (for ‘external’).

By now many hundreds of publications have been devoted to nonstandard analysis: it is an established branch of mathematics.

No matter how innitesimals are introduced, with or without the axioms of set theory, with or without extra axioms and new undened symbols (st and ext), always the axiom of choice seems indispensible. If one tries to develop innitesimal calculus without this axiom, it seems that one should be satised with a mutilated theory, as will be explained later on in Section 4.4. Here attempts by Chwistek [17,18] in this direction should be mentioned. In his 1926 paper Chwistek introduces new numbers by means of innite sequences of classical numbers. These new numbers are called Progressionszahlen (‘sequence numbers’), and equality for them is dened as follows. Let Ni(αi) and Ni(βi) be two new numbers, then, Ni(αi) = Ni(βi) if and only if αi = βi for i > n for some n IN. Something similar is done to dene inequality, and an operation like addition is dened by,

Ni(αi) + Ni(βi) = N(αi + βi).

A classical function f is extended by means of,

f(Ni(αi)) = Ni(f(αi)).

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The extended function happens to be quite similar to f, the -transform of f. Even so not much new calculus is developed. An extension of IR that includes all sequence numbers could be introduced, however.

In his 1948 book Chwistek spends less then ten pages on the subject, but nevertheless shows that he is well aware of the fact that ‘innitely small’ numbers can be introduced, and he also introduces internal functions (called normal functions by him). Again there is no fully expanded calculus. Most likely, the deeper reason for this is that Chwistek denes (in)equality for his sequence numbers as indicated above. This denition has the advantage that the axiom of choice is not needed, but leads to rather serious problems, as will become clear in Section 4.4. It remains to remark that working with sequences is a technique used by Hewitt [10] and Luxemburg [13], and will be the technique of the next chapter, which is based on assumptions that from a naive, intuitive point of view are understandable, obvious, and acceptable, except perhaps the axiom of choice, and where everything that is not so obvious, such as transfer and all the rest, will be proved, rather than assumed.

 



 

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20天前 北大袁萌1.8无穷小病态三百年

1.8无穷小病态三百年

荷兰千年名校格罗宁根大学知名数学J.Ponstein教授为不的熟悉数理逻辑的数学工作者精心撰写了一部介绍“非标准分析”科普专著,对于我国高校微积分教学改革很有帮助。
   为此,我们将这部著作分为章节发表,加上适当小标题与评论。请读者注意。

袁萌  陈启清  929

附件:

1.8 Innitesimals in the 17th to the 19th century

There can be no doubt that in the 1670’s, some 1900 years after Archimedes lived, innitesimals were conceived by Leibniz. Moreover, he formulated their main properties, and many contemporary mathematicians as well as mathematicians after him, among them Euler and Cauchy, were able to successfully work with them. But the theory of the innitesimals lacked a rigorous basis, and during some 200 years all trials to improve this situation were in vein, so that at last one gave up, the more so because in the 1870’s Weierstrass came up with a rigorous theory of limits and continuity, which became the basis of what now is known as classical analysis, and where there was and is no need to consider innitesimals any more.

It is quite interesting to see how Euler [2] shows the well-known product formula for the sine function. He begins his proof with the equality, 2·sinh x = (1 + x/n)n −(1−x/n)n, valid for – in Eulers’s own words – ‘innitely large values’ of n. Obviously, this is only true up to an innitesimal. Then the right-hand side is treated as if n were

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a classical natural number. This leads after a purely classical reasoning to,

(1 + x/n)n −(1−x/n)n = (8x/n)·

m Y k=1

sin2(kπ/n)·{1 + x2/n2 tan(kπ/n)}, where m = (n−1)/2, taking n odd (the details of the reasoning do not matter here, and the case for n even is similar). So,

sinh x = (4x/n)·

m Y k=1

sin2(kπ/n)·{1 + x2/n2 tan2(kπ/n)}. Taking x 6= 0, and dividing by x, and then taking x = 0, gives, 1 = (4/n)· m Y k=1 sin2(kπ/n), and hence,

sinh x = x·

m Y k=1{1 + x2/n2 tan2(kπ/n)}. Now for k nite, n2 tan2(kπ/n) is ‘innitely close’ to (kπ)2, so (?)

sinh x = x·

Y k=1{1 + x2/k2π2}, and putting x = iz, this gives the desired result,

sinz = z·

Y k=1{1−z2/k2π2}. Obviously, at the question mark the argument goes a little too fast, and a number of steps must be included here (see e.g. Luxemburg [3]).

Another famous example is Cauchy’s proof ([4], p. 131), that a convergent series of continuous functions has a continuous limit function. To many this theorem was not correct, because it would seem that all kinds of counter-examples could be given. One of them is the series with the partial sums,

sn(x) = (4/π)·

n X k=1

sin(2k + 1)x 2k + 1

,

that is periodic modulo 2π and converges to, f(x) = −1 if −π < x < 0 0 if x = 0 or x = π +1 if 0 < x < π

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as can be shown by classical Fourier analysis. Since the sine function is everywhere continuous and sn(x) converges to f(x) for n tending to, according to Cauchys theorem f ought to be continuous, which it isnt. But sofar, everything takes place within IR, and Cauchy let everything happen in what we have indicated by IR.

For him continuity of f at c meant that, x IR, x ' c : f(x) ' f(c), where, however, f : IR IR and f need not be a standard function, and c IR, not only c IR, which is why his continuity is not continuity (in nonstandard analysis it is called S-continuity; recall denition (1.1) in Section 1.4, where c IR and a standard function was involved, so that there S-continuity was the same as continuity).

And by convergence of sn(x) to f(c) he meant that, n : sn(c) ' f(c), where again everything is in IR. Note that the Weierstrassian denitions of limit and continuity appeared half a century after Cauchy’s book, so Cauchy in a sense ‘had to’ work with denitions of the kind given here.

Now, by transfer,

sn(x) = (4/π)·

n X k=1

sin(2k + 1)x 2k + 1

,n IN, x IR,

and

f(x) = −1, or 0, or + 1, x IR, since if the range of a classical function f is nite, the range of its transform is the same as that of f. Let m be xed, and let x = c = 1/(2m), and dt = 1/m, so that x 0, dt 0. Then, sm1 2m= (2/π)· m X k=1 sin(2k + 1)dt/2 (2k + 1)dt/2 ·dt. If we had that m IN, then the sum to the right would be an approximation of the Riemann-integral, J =Z1 0 sint t ·dt,

36

and it should therefore not come as a surprise that it can be shown that the standard part of the right-hand side is exactly equal to 2J/π, and hence,

sm1 2m−2J/π ' 0. But by direct calculation it follows that 2J/π 6= −1, 0, and +1, and since in particular for c = 1/(2m), f(c) = −1, or 0, or +1 (−1 is in fact impossible), it follows that sn(c) does not converge to f(c). Also, since for all n, sn(0) = 0, sm(x) is not continuous at c = 0, so that the ‘counter-example’ does not satisfy the assumptions of Cauchy’s theorem, and this is why Cauchy maintained his theorem against all criticism, but without basing his proof (and much of his other work) on a rigorous theory of the innitesimals and other nonstandard numbers. For many interesting details, see Lakatos [5].

 

 

 



 

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  • 匿名人士 江苏省南通市 (153.37.114.*): 不做翻译,感觉普及不了   (2019-10-02 22:31:43)

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22天前 北大袁萌珍贵数学文献(II)

珍贵数学文献(II

   荷兰千年名校格罗宁根大学知名数学J.Ponstein教授为不的熟悉数理逻辑的数学工作者精心撰写了一部介绍“非标准分析”科普专著,对于我国高校微积分教学改革很有帮助。

为此,我们将此文分为两个部分顺序转发。请见本文附件。

对此专著的分析与评论随后再发、

袁萌  陈启清  928

附件Non-standard analysis

Cnapter  3

Some applications

3.1 Introduction and least upper bound theorem

The aim of this chapter is to show how many denitions and proofs of elementary calculus can be simplied by means of nonstandard analysis. Only a number of important examples will be considered. A much more complete treatment is Keisler [26], where the existence of nonstandard numbers is taken for granted, however, and a simplied form of transfer is introduced in an axiomatic kind of way.

Theorem 3.1.1 (The least upper bound theorem.) Let S be a nonempty subset of IR that is bounded above by some (classical) real number. Then S has a least upper bound in IR.

Proof:

Taking any c S, instead of S we may consider {s : s S, s c}, that is to say we may assume that s c for all s S. Then c, b IR, c < b, exist such that s S : c s b, so that, by transfer, s S : c s b. Let ω IN, ω be arbitrary and divide [c,b] in ω equal subintervals of length δ = (b−c)/ω, so that δ 0, and consider the points a, a + δ, a + 2δ, ..., a + ωδ = b. Then, j IN : [s S : s a + jδ] [s0 S : s0 > a + jδ−δ].

Let β =st(a+jδ), which is well dened as a+jδ is limited. Then β is a (hence the) least upper bound of S. For rst of all if s S then s S, hence s a+jδ = β+ε for some ε ' 0, but since s, β IR this means that s β. And secondly, if β0 were a smaller upper bound of S, then β > β0 + 1/m for some m IN, hence

101

102



 

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24天前 北大袁萌关于二十世纪的无穷小概念

关于二十世纪的无穷小概念

1025日,我们转发了一篇冠名,,名为“本世纪最好的DNA!(J.Ponsteni著,发表于 1996年)一文(原文系英文)

此文第一部分概论的第1.9节名为“二十世纪的无穷小概念”,

很有参考价值,值值得一阅。

请见  文件本文附件。

袁萌   陈启清   924

1.9 Innitesimals in the 20th century

When in the 1870’s Weierstrass formulated the well-known ε−δ denitions of limit and continuity, denitions that completely ignore nonstandard numbers, the dispute regarding innitesimals quickly settled in their disadvantage, but only temporarily, for in 1961 Robinson [6,7] presented a mathematically sound theory of the nonstandard numbers. These works embody the rst fairly complete analysis of the nonstandard numbers. Not only are they based on work of forerunners, but also on an amount of mathematical logic that hitherto was unusual in mathematics. Only a few references should suce here, see [8–12].

Robinson starts from the axioms of set theory due to Zermelo and Fraenkel, and the axiom of choice (called together the ZFC axioms), derives IR in a classical kind of way, and then extends IR to IR by applying a rather considerable amount of mathematical logic, as indicated before. Another way to dene IR was already indicated by Hewitt [10] and worked out by Luxemburg [13]. Here the ZFC axioms are again the point of departure, but the more usual line of mathematical thinking is followed. (Except for the ZF axioms, this way is also followed in the next chapter.) Still another way to introduce IR was found by Nelson [14]. Nelson adds three more axioms to the ZFC axioms, as well as a new symbol, st (for ‘standard’) that is used as a kind of label to distinguish standard constants from nonstandard constants. This leads directly to the set of all standard as well as all nonstandard constants, without the intermediary step of rst introducing IR; consequently

37

in internal set theory IR is denoted by IR, and similarly, IN is denoted by IN, etc. Actually, the point of view of internal set theory is that the IN of classical mathematics is the same as the IN of nonstandard analysis; and that all that happens is that unexpected elements of IN are discovered, elements that had always been there. In other words, according to this point of view, 0, 1, 2, etc. do not at all ll up IN (see Robert [15] and F. Diener et G. Reeb [16]). The additional axioms make sure that transfer is guaranteed (axiom of ‘transfer’), that nonstandard numbers exist (axiom of ‘idealization’), and that unique standard sets can be derived from given sets (axiom of ‘standardization’). Even though internal set theory uses relatively little of mathematical logic, the new axioms require some study, and do not seem to be as obvious as, for example, the axioms of Greek geometry: Transfer: stt1 ...sttk : [stx : P(x,t1,...,tk) ⇒∀x : P(x,t1,...,tk)]. Idealization: [st nx : x : y z : P(x,y)] [x : sty : P(x,y)]. Standardization: stx : sty : stz : [z y z xP(z)]. Here stu means that the variable u must be standard, and similarly the label n means that the corresponding variable must be nite (but beware, in internal set theory any hyperlarge natural number is nite, only the combination of standard and nite amounts to the classical notion of niteness). Note that whereas stu means that the variable u is standard, means the variable is standard, because st is a label butis a mapping. P(...) denotes a given internal statement, except in the last axiom, where P(...) may even be external (see Section 1.6).

In naive nonstandard analysis these three additional axioms are not assumed but derived from the existence of the natural numbers and the axiom of choice. Transfer has already been discussed; and idealization is used to prove the existence of nonstandard elements in any internal set with an innite number of elements. Perhaps standardization is the most intriguing of the three because it contains a statement P(z) that may be external. Reformulated naively it means that, ∀∗x : ∃∗y : ∀∗z : [z y ⇔∗z xP(z)], where x, y and z are, of course, classical. Since always s S if and only if s S, it follows that, y = {z x : P(z)}, or equivalently,

y = {z x : P(x)},

38

which in internal set theory are illegal set formations. Here are a few examples, where x and y are still classical, but z need not be classical. 1) P(z) z INz is standard; then x = {1,2,3} gives y = x = 1,2,3}, x = IN gives y = x = IN, and x = IR also gives y = IN.

In fact IN is the largest y that is possible for variable x. 2) P(z) z INz < n, with n IN given such that n ; then the results are as under 1). 3) P(z) z IRz ' 0; then y = {0} if 0 x and y = if 0 6 x. For other details the reader should consult more adequate treatments of internal set theory.

In the mean time other versions of nonstandard analysis have been developed. In one of them external sets are ‘legalized’ by means of still other axioms, and another label, ext (for ‘external’).

By now many hundreds of publications have been devoted to nonstandard analysis: it is an established branch of mathematics.

No matter how innitesimals are introduced, with or without the axioms of set theory, with or without extra axioms and new undened symbols (st and ext), always the axiom of choice seems indispensible. If one tries to develop innitesimal calculus without this axiom, it seems that one should be satised with a mutilated theory, as will be explained later on in Section 4.4. Here attempts by Chwistek [17,18] in this direction should be mentioned. In his 1926 paper Chwistek introduces new numbers by means of innite sequences of classical numbers. These new numbers are called Progressionszahlen (‘sequence numbers’), and equality for them is dened as follows. Let Ni(αi) and Ni(βi) be two new numbers, then, Ni(αi) = Ni(βi) if and only if αi = βi for i > n for some n IN. Something similar is done to dene inequality, and an operation like addition is dened by,

Ni(αi) + Ni(βi) = N(αi + βi).

A classical function f is extended by means of,

f(Ni(αi)) = Ni(f(αi)).

39

The extended function happens to be quite similar to f, the -transform of f. Even so not much new calculus is developed. An extension of IR that includes all sequence numbers could be introduced, however.

In his 1948 book Chwistek spends less then ten pages on the subject, but nevertheless shows that he is well aware of the fact that ‘innitely small’ numbers can be introduced, and he also introduces internal functions (called normal functions by him). Again there is no fully expanded calculus. Most likely, the deeper reason for this is that Chwistek denes (in)equality for his sequence numbers as indicated above. This denition has the advantage that the axiom of choice is not needed, but leads to rather serious problems, as will become clear in Section 4.4. It remains to remark that working with sequences is a technique used by Hewitt [10] and Luxemburg [13], and will be the technique of the next chapter, which is based on assumptions that from a naive, intuitive point of view are understandable, obvious, and acceptable, except perhaps the axiom of choice, and where everything that is not so obvious, such as transfer and all the rest, will be proved, rather than assumed.

 

 



 

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28天前 北大袁萌超实数探索浪潮波及中国数学界

超实数探索浪潮波及中国数学界

   近日,“超实数探索五十年”一文已经清楚地表明:当前,超实数探索浪潮已经兴起,而且已经波及我国数学教育界。为什么这么根据何在?

   在中国,超实数探索浪潮已经波及数学教育界界的事实证据是:经过7年的不断努力,在中国互联网上已经积累、存在2000多篇超实数探索文章,而且有独立的超实数中心专业网站。换句话说,在国内数学界几乎无人不知超实数无穷小的大名。

   从本质上来看,“无穷小微积分”专业网站是我国数学教育界的一个不可缺少的组成部分。

袁萌  陈启清  922



 

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28天前 北大袁萌超实数探索五十载,成果颇丰,名声大振

超实数探索五十载,成果颇丰,名声大振
  回顾历史,1889年,意大利数学家Peano第一次给出算术的公理系统(阿基米德系统)。1899年,德国数学家希尔伯特在其《几何基础》中,第一次指出非阿基米德数学系统的兼容性(也不会导致矛盾)。1934年,Skolem证明非阿基米德算术确实存在,换句话说,无穷大非准自然数存在。  
   人类数学的这一发展是本质性的创新。从此,“无穷数”进入了人的视线。
  上世纪六十年代,鲁宾逊在前人研究的基础上创立了非标准分析(NSA)。数学进入了一个全新的发展时代。
   事实上,在过去的这五十年里面,非标准数学专著发表了五十余本(请见本文附件)。在此期间,超实数研究工作发表了研究论文一百多篇。
  反观我们国内,有人闭眼睛说:超实数很玄乎,将来必定会销声匿迹。此言差矣!
袁萌  陈启清  9月20日
附件:
Popular on the web
Lectures on
the hyperreals
罗伯特•戈德布拉特, 1998 年
Nonstandard
Analysis for the Working Mathematician
2000 年
Applied
Nonstandard Analysis
马丁•戴维斯, 1977 年
Nonstandard
Analysis: Theory and Applications
1997 年
Nonstandard
Analysis in Practice
1995 年
Elementary
Calculus: An Infinitesimal Approach
H•傑爾姆•基斯勒, 1976 年
Nonstandard
Methods in Stochastic Analysis and Mathematical Physics
Sergio Albeverio, 1986 年

An Introduction to Nonstandard Real Analysis
1985 年

The Strength
of Nonstandard Analysis
2007 年
Nonstandard
Analysis
阿兰•M•罗伯特, 1988年
Nonstandard
Analysis and Its Applications
尼格尔•卡特兰, 1988 年
Nonstandard
Analysis, Axiomatically
2004 年
Abraham
Robinson: The Creation of Nonstandard Analysis, A Personal and Mathematical Odyssey
道本周, 1995 年

Non-standard
analysis
亚伯拉罕•鲁滨逊, 1966 年

A primer of
infinitesimal analysis
约翰•洛讷•贝尔, 1998 年

Models for Smooth Infinitesimal Analysis
1991 年
Nonstandard
Methods for Stochastic Fluid Mechanics
1995 年

Lectures on Non- Standard Analysis
1969 年

Introduction to
the theory of infinitesimals
基思•斯特罗扬

Nonstandard
Asymptotic Analysis
Imme van den Berg, 1987 年

Selected
papers of Abraham Robinson
亚伯拉罕•鲁滨逊

Foundations
of infinitesimal calculus
H•傑爾姆•基斯勒, 1976 年

Nonstandard
Analysis.: A Practical Guide with Applications.
1981 年

Applications
of Model Theory to Algebra, Analysis, and Probability
1969 年

Optimization
and Nonstandard Analysis
J. E. Rubio,
1994 年

An infinitesimal approach to stochastic analysis
H•傑爾姆•基斯勒, 1984 年

Infinitesimal Analysis

Victoria
Symposium on Nonstandard Analysis: University of Victoria
1972
1974 年

Nonstandard
Methods of Analysis

Advances in
Analysis, Probability and Mathematical Physics: Contributions of Nonstandard Analysis
1995 年

Standard and
Nonstandard Analysis: Fundamental Theory, Techniques, and Applications
R•F•霍斯金斯,
1990 年

Developments
in Nonstandard Mathematics

Infinitesimal
Methods of Mathematical Analysis
J. Sousa Pinto

A Combination of Geometry Theorem Proving and Nonstandard Analysis with Application to Newton’s Principia
Jacques Fleuriot, 2001 年

Nonstandard
Methods in Functional Analysis: Lectures and Notes
Siu-Ah Ng,
2010 年


 

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1月前 北大袁萌超实数发现的历史真相

超实数发现的历史真相

    根据最新历史研究表明:1931年,哥德尔不完全性定理导出模型论紧致性定理,据定理此,Skolem1934年证明非标准算术的存在性。

1960年,鲁宾逊

天才地领悟到,利用Skolem证明思路创立非标准分析.由此成功地引入了超实数系统。

本文附件文章,参阅60余历史资料没证明了上述论断。      请读者参阅。

袁萌  陈启清  919

附件:2012 6

Hyperreals and Their Applications

Sylvia Wenmackers

Formal Epistemology Project Faculty of Philosophy University of Groningen Oude Boteringestraat 52 9712 GL Groningen The Netherlands E-mail: s.wenmackers@rug.nl URL: http://www.sylviawenmackers.be/

Overview

Hyperreal numbers are an extension of the real numbers, which contain innitesimals and innite numbers. The set of hyperreal numbers is denoted by R or R; in these notes, I opt for the former notation, as it allows us to read the -symbol as the prex ‘hyper-’. Just like standard analysis (or calculus) is the theory of the real numbers, non-standard analysis (NSA) is the theory of the hyperreal numbers. NSA was developed by Robinson in the 1960’s and can be regarded as giving rigorous foundations for intuitions about innitesimals that go back to Leibniz (at least). This document is prepared as a handout for two tutorial sessions on “Hyperreals and their applications”, presented at the Formal Epistemology Workshop 2012 (May 29–June 2) in Munich. It is set up as an annotated bibliography about hyperreals. It does not aim to be exhaustive or to be formally precise; instead, its goal is to direct the reader to relevant sources in the literature on this fascinating topic. The document consists of two parts: sections 1–3 introduce NSA from dierent perspectives and sections 4–9 discuss applications, with an emphasis on topics that may be of interest to formal epistemologists and to philosophers of mathematics or science.

1

Part 1: Introducing the hyperreals

Abstract

NSA can be introduced in multiple ways. Instead of choosing one option, these notes include three introductions. Section 1 is best-suited for those who are familiar with logic, or who want to get a avor of model theory. Section 2 focuses on some common ingredients of various axiomatic approaches to NSA, including the star-map and the Transfer principle. Section 3 explains the ultrapower construction of the hyperreals; it also includes an explanation of the notion of a free ultralter.

1 Existence proofs of non-standard models

1.1 Non-standard models of arithmetic

The second-order axioms for arithmetic are categoric: all models

 

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1月前 北大袁萌哥德尔预言无穷小微积分是未来的数学分析

 

哥德尔预言无穷小微积分是未来的数学分析

 

    二十世纪世界伟大的数学家哥德尔预言非标准分析是未来的数学分析。

 

   哥德尔1974年预言的原文如下:

 

There are good reasons to believe that non-standard analysis, in some version or other, will be the analysis of the future” [33]. Kurt G¨odel, 1974.请见本文附件1

 

   注:本文附件2是发表于201328日的非标准分析论文,此文附有44篇珍贵的非标准分析论文。

 

袁萌  陈启清  917

 

附件1

 

[33] T. Runge. Hypernite probability theory and stochastic analysis within Edward Nelsons internal set theory. 2011. URL http://www10.informatik. uni-erlangen.de/Publications/Theses/2010/Runge_DA10.pdf.

 

附件2

 

Eoghan Staunton

 

ID Number: 09370803

 

Final Year Project

 

National University of Ireland, Galway

 

Supervisor: Dr. Ray Ryan

 

February 8, 2013

 

I hereby certify that this material, which I now submit for assessment on the programme of study leading to the award of degree is entirely my own work and has not been taken from the work of others save and to the extent that such work has been cited and acknowledged within the text of my work.

 

Author:

 

Eoghan Staunton

 

ID No:

 

09370803

 

Contents

 

1 Introduction   1

 

2 Construction of the Hyperreals  2

 

2.1 Our aim . . . . . . 2

 

2.2 Z, Q and R from N . . . . . . . . 2

 

.3 Free Ultralters . . . . .. . 4

 

2.4 Generating elements of R . . . . . . . . . . . . . . . . . . . . . . . 6 2.5 Arithmetic operations and inequalities in R . . . . . . . . . . . . . 7 2.6 Some Notation & Denitions . . . . . . . . . . . . . . . . . . . . . 9 2.7 Other Ultrapower Constructions . . . . . . . . . . . . . . . . . . . 9 2.8 The -transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.9Internal vs. External constants . . . . . . . . . . . . . . . . . . . . 11 2.10 Innitesimals and Hyperlarge numbers in R . . . . . . . . . . . . 11

 

3 The Transfer Principle 14 3.1 History and Importance . . . . . . . . . . . . . . . . . . . . . . . . 14

 

3.2 Mathematical Logic . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.3 L o´s’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.4 The Transfer Principle . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.5 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

 

3.6 Nonstandard Analysis as a Tool in Classical Mathematics . . . . . 23

 

4 The History of Innitesimals 25 4.1 Use in Ancient Greek Mathematics . . . . . . . . . . . . . . . . . . 25 4.2 Geometers of the 17th century and Indivisibles . . . . . . . . . . . 26

 

4.3 The Development of Calculus . . . . . . . . . . . . . . . . . . . . . 26

 

4.4 Modern Nonstandard Analysis . . . . . . . . . . . . .. . 29

 

5 Applications of Nonstandard Analysis 29 5.1 Economics and Finance . . . . . . . . . . . . . . . . . . . . . . . . 30 5.2

 

Selected Other Applications . . . . . . . . . . . . . . . . . . . . . . 33

 

6 Appraisal and Conclusion    34

 

1 Introduction

 

There are good reasons to believe that non-standard analysis, in some version or other, will be the analysis of the future” [33]. Kurt G¨odel, 1974.

An innitesimal is a number that is smaller in magnitude than every positive real number. The word innitesimal comes from the Latin word innitesimus and was coined by the German mathematician Gottfried Wilhelm Leibniz around 1710 [1]. We learn early on in our study of standard analysis that nonzero innitesimals cannot exist. It is also true however that many people use the intuitive notion when trying to understand basic concepts in a


 

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  • 匿名人士 江苏省南通市 (153.37.114.*): 现代非标准分析   (2019-10-02 22:34:08)

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1月前 北大袁萌柯西导数与鲁宾逊导数之比较

柯西导数与鲁宾逊导数之比较

   给定函数

y = f( x )

在点a’的邻域内有定义。

大家知道:

在点a处的柯西导数为

1)                   S = lim(y/x)

x 0 , x 0

相应地,鲁宾逊导数为

2)  S  = st(y/x)

x≈ 0, x 0

 

两者之差别在于:柯西导数需要用极限(ε,δ)方法来说明(使用三个量词),而鲁宾逊导数只需要一个全称量词,但是x是无穷小,“st”是取超实数标准部分。运算。

   两者的主要差别就在于此。

  注:符号“ ”表示无限趋向的意思,而符号“”表示无限接近的意思。

“趋向”与“接近”的含义不同。

袁萌  陈启清 916

 



 

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1月前 北大袁萌关于数学极限定义的量词组合复杂度

关于数学极限定义的量词组合复杂度 

   在数学中 ,使用的量词只有两大类:“”与“”。   

实际上,在现代数学中使用的量词并不算多。所以,任何高校合格数学老师应该都会使用它们。

注:

表示命题P ( x ) 对于所有 x x为真

表示存在至少一个 x 使 命题P ( x ) 为真。

传统微积分定义极限概念使用“量词组”(),而无穷小微积分只需要一个量词(),其量词组合复杂度远远低于前者。

数学理论量词组合复杂度 概念是J.Keisler教授于2006年首先提出的。

   请见本文附件文章。

袁萌   陈启清  913

附件:

Comparison with infinitesimal definition

Keisler proved that a hyperreal definition of limit reduces the quantifier complexity by two quantifiers.] Namely,

f ( x )  converges to a limit L as x  tends to a if and only if for every infinitesimal e, the value

f ( x + e )  is infinitely close to L; see microcontinuity for a related definition of continuity, essentially due to Cauchy. Infinitesimal calculus textbooks based on Robinson's approach provide definitions of continuity, derivative, and integral at standard points in terms of infinitesimals. Once notions such as continuity have been thoroughly explained via the approach using microcontinuity, the epsilon–delta approach is presented as well. Karel Hrbáek argues that the definitions of continuity, derivative, and integration in Robinson-style non-standard analysis must be grounded in the ε–δ method in order to cover also non-standard values of the input. Baszczyk et al. argue that microcontinuity is useful in developing a transparent definition of uniform continuity, and characterize the criticism by Hrbáek as a "dubious lament".Hrbáek proposes an alternative non-standard analysis, which (unlike Robinson's) has many "levels" of infinitesimals, so that limits at one level can be defined in terms of infinitesimals at the next level.

 



 

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