Applications of set theory to abstract harmonic analysis

集合论在抽象调和分析中的应用

• 批准号: RGPIN-2017-05712
• 负责人: Steprans, Juris
• 金额: $ 2.19万
• 依托单位: York University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=683352
• 关键词: Applications; set; theory; abstract; harmonic

***The reasons for the remarkable effectiveness and applicability of mathematics have been the subject of various philosophical considerations, but the question becomes even more difficult when the use of extra mathematical axioms come into play. How could an axiom whose truth is not easily verifiable have any influence on practical mathematical matters such as those one encounters in physics or finance? But anyone who thinks there can be no such influence should consider an axiom that is often used without further consideration in mathematics, the axiom of infinity. All that the axiom of infinity says is that the set of all natural numbers exists; not that each natural number exists on its own, but that the set of all of them exists as a mathematical object.******One should not be too quick to dismiss this as a philosophical issue with no practical applications. Every time we use an irrational number, such as the square root of 2, we are implicitly using the axiom of infinity since we are manipulating all of the infinitely many digits of that number at once, as a single set. Infinity cannot be avoided, even in geometry; the completeness of the real number line requires it. ******Most mathematicians have no difficulty accepting the axiom of infinity as a valid axiom, although it must be admitted that there are some who feel that applications of mathematics that invoke infinity cannot be trusted. But what is to be said about axioms beyond the axiom of infinity, axiom that do not usually appear in mathematical arguments? While it is rare, there are some ramifications of such axioms in applied areas such as quantum physics and even economics and financial mathematics. Can such arguments be trusted? Are these axioms needed? What can we prove (and, hence, be confident in the truth of) in mathematics without assuming these axioms?******The purpose of the proposed research program is to help delineate those areas of mathematics that are susceptible to axioms beyond the commonly accepted ones. Determining this boundary is an important first step in understanding which mathematical arguments might be problematic because of their reliance on extra axioms.

数学的显著有效性和适用性的原因一直是各种哲学思考的主题，但当额外的数学公理的使用开始发挥作用时，这个问题变得更加困难。一个真理不易验证的公理怎么会对人们在物理或金融领域遇到的实际数学问题产生任何影响呢？但任何认为不可能存在这种影响的人，都应该考虑一条在数学中经常未经进一步考虑就被使用的公理，即无穷大公理。无穷无尽的公理所说的就是所有自然数的集合是存在的；不是每个自然数独立存在，而是所有自然数的集合作为一个数学对象而存在。人们不应该太快地将这视为一个没有实际应用的哲学问题而不屑一顾。每次我们使用无理数，比如2的平方根，我们就是在隐含地使用无穷大公理，因为我们一次操作该数字的所有无穷多位数字，作为一个集合。无限是无法避免的，即使在几何学中也是如此；实数线的完备性要求无限。*大多数数学家毫不费力地接受无穷大公理是一个有效的公理，尽管必须承认，有些人觉得调用无穷大的数学应用是不可信的。但是，关于无穷公理之外的公理该说些什么呢？无穷公理是数学论证中不常出现的公理。尽管这很少见，但在量子物理甚至经济和金融数学等应用领域，这种公理也有一些后果。这样的论点可信吗？需要这些公理吗？在不假设这些公理的情况下，我们能在数学中证明什么(并因此对其真理有信心)？*拟议的研究计划的目的是帮助描绘那些容易受到公理的影响的数学领域，这些领域超出了普遍接受的公理。确定这一界限是理解哪些数学论点可能有问题的重要第一步，因为它们依赖于额外的公理。