Applications of set theory to abstract harmonic analysis

数学非凡的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。一个不容易验证的公理怎么可能对实际的数学问题产生影响，比如人们在物理学或金融学中遇到的那些问题？但是，任何认为不可能有这种影响的人，应该考虑一个在数学中经常使用而不加进一步考虑的公理，即无穷公理。无穷公理说的是所有自然数的集合存在；并不是说每个自然数都是独立存在的，而是说所有自然数的集合作为一个数学对象而存在。******我们不应该过早地将其视为一个没有实际应用价值的哲学问题。每次我们使用无理数时，比如根号2，我们隐含地使用了无穷公理，因为我们同时处理了这个数的所有无穷多位，作为一个单一的集合。无限是无法避免的，即使在几何学中也是如此；实数轴的完备性要求它。******大多数数学家接受无限公理是一个有效的公理是没有困难的，尽管必须承认，有一些人觉得数学的应用调用无限是不可信的。但是对于无限公理之外的公理，也就是通常不会出现在数学论证中的公理，我们该说些什么呢？虽然这种情况很少见，但这些公理在量子物理学、甚至经济学和金融数学等应用领域都有一些分支。这样的论点可信吗？需要这些公理吗？如果不假设这些公理，我们能在数学中证明什么（并因此对其真实性有信心）？******提出的研究计划的目的是帮助描绘那些容易受到普遍接受的公理影响的数学领域。确定这个边界是理解哪些数学论证可能因为依赖额外公理而存在问题的重要的第一步。

***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.