Applications of set theory to abstract harmonic analysis

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是所有自然数的集合存在；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存在。

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.