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: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759135
• 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，但是当使用额外的数学公理时，这个问题变得更加困难。一个真理性不容易证实的公理怎么可能对物理学或金融学中遇到的实际数学问题产生任何影响呢？但是，任何认为不可能有这种影响的人都应该考虑一个在数学中经常被使用而没有进一步考虑的公理，即无穷大公理。无穷大公理所说的只是所有自然数的集合都存在;不是说每个自然数都独立存在，而是说所有自然数的集合都作为数学对象存在，我们不应该太快地把它当作一个没有实际应用的哲学问题而不予考虑。每次我们使用一个无理数，比如2的平方根，我们都隐含地使用了无穷大公理，因为我们一次操作这个数的所有无穷多位，作为一个单一的集合。即使在几何学中，无穷大也是不可避免的，因为真实的数线的完备性要求无穷大。大多数数学家都毫不费力地接受无穷大公理为有效的公理，尽管必须承认，有些人认为数学应用无穷大是不可信的。但是，对于超越无穷大公理的公理，那些通常不会出现在数学论证中的公理，我们应该说些什么呢？虽然这是罕见的，但在量子物理甚至经济学和金融数学等应用领域，这些公理有一些分支。这样的说法可信吗？这些公理是必要的吗？如果不假设这些公理，我们在数学中能证明什么（并因此对真理有信心）？拟议的研究计划的目的是帮助划定那些容易受到公理超出普遍接受的数学领域。确定这个边界是理解哪些数学论证可能有问题的重要的第一步，因为它们依赖于额外的公理。