I remember that in highschool we were given an explanation of mathematical truth: it arises as “a consequence of our definitions”. Later I learned that pure mathematicians themselves do not actually use this as the foundation for their work. Instead, they use logic and set theory. Why? What is wrong with the intuitive notion we were taught in school?
The concept of a set seems clouded in mystery. I believe that of all words in the Oxford English Dictionary, ‘set’ occupies more pages and has more definitions than any other.
Logic also seems mysterious. There are many theories with a claim to being the correct logic, Many-valued logic, Quine’s system of logic, and Intuitionistic logic, to name a few.
The only foundation of mathematics that I could understand and believe in was based on definitions. A mathematical term would be defined in using other terms. I’d encountered this theory in books, where it was criticized as involving circular definitions. I believe this is just wrong, a silly misunderstanding of something most real mathematicians know. Definitions are relations, and like other relations, they can collectively specify an algebraic structure — but it is entirely wrong to suppose that this structure is either a cyclic group or cyclic graph. It may be a very complicated algebraic structure with several elements which have quite different properties.
That is a basic truth within mathematics, whether formalized with logic, set theory or any other purported foundation. It is simply wrong to criticize the notion that mathematics can be based on definitions alone by saying that leads to a circular definition.