I am only an amateur mathematician, and I have recently been interested in the foundations of mathematics. Upon reading about predicate logic and finding language defined in terms of sets and functions (function symbols and relation symbols and arity, respectively), I found myself questioning the circularity of mathematics in its foundation. These questions have, for the most part, been answered by the following thread: Does mathematics become circular at the bottom? What is at the bottom of mathematics?.
In particular, this section of the top answer by user21820 really resonated with me:
There are two main parts to the 'circularity' in Mathematics (which is, in fact, a sociohistorical construct). The first is the understanding of logic, including the conditional and equality. If you do not understand what "if" means, no one can explain it to you because any purported explanation will be circular. Likewise for "same". (There are many types of equality that philosophy talks about.) The second is the understanding of the arithmetic of natural numbers, including induction. This boils down to the understanding of "repeat". If you do not know the meaning of "repeat" or "again" or other forms, no explanation can pin it down.
It makes a lot of sense to me that you need a preliminary understanding of implication and equality to understand any mathematics in a meaningful way, and it also seems to me that you would need an innate understanding of collection or plurality since sets are such a ubiquitous concept in this topic (What do others think of that?). The part I want to ask more about is the mention of the natural numbers and repetition in this answer. Basically, what needs to be taken as a primitive concept in regard to natural numbers? The ability to count seems pretty necessary since arity is needed for predicate logic, but what about induction? Can we formally define induction in a meaningful way with just an informal understanding of counting? Do we need an informal understanding of arithmetic as the original answer suggests since arithmetic can be defined axiomatically (supposing we are using PA and ZFC)?
This is quite a difficult question for me to word, so I apologise if it is not quite what is supposed to be asked on this forum, and I would appreciate it if people could point me to a more suitable platform/format if this is the case.