You wrote Say we did define $S(n)=n+1$... What definition of $+$ are you using? Am I correct in guessing that you are implicitly assuming that the natural numbers and their operations of addition and multiplication have already been defined and so you are making use of them?

@LeeMosher I meant that the only definition of addition is the one which was given- and that we take our set of numbers to be the usual $\mathbb{N}$, as stated.

"the one which was given...* Which one is that? What is the definition of addition that was given?

@LeeMosher Also, I wasn't thinking of multiplication at all as I didn't consider it relevant to the circular definition between succession and addition presented here, although maybe I was wrong

@LeeMosher in "for any natural $m$, we define $0+m = m$, and if for some natural $n$, we have defined $n+m$, then we can define $S(n) + m = S(n+m)$", which was in my paragraph following the one in which I quoted Tao. Such was the definition of addition which Tao uses in Analysis I

Well, there it is. That definition assumes that $S$ is given. That's all there is to it.

@LeeMosher and $S$ was given, right? Only it was given in terms of addition. Why can't it be like mutual recursion in programming, where 2 functions are defined in terms of each other, and due to the base cases everything works out? (here I perceive something other than base cases possibly making things work out)

"I am assuming that the number system is defined as the usual $\mathbb{N}$..." I don't have the book in question; what properties are assumed to be satisfied by "the usual $\mathbb{N}$" here?

@BrianTung I mean $\{0,1,2,3,. . . \}$; I edited to clarify. there are no assumptions about this structure beyond what we know; we just know that this is the set of numbers, which element thereof is the distinguished $\mathbf{0}$, how succession is defined, and how addition is defined

Well, in your post you gave $S$ in terms of addition, but that's not how it is done in Tao's book. Instead, $S$ is assumed to be given, and it is assumed to satisfy some simple axioms, and that's it. Everything else is derived from that. This is the essence of the axiomatic method: something has to be given from which everything else is derived; mathematics does not rise unborn from a giant scallop shell. Hopefully what is given is as simple as possible.

@LeeMosher thanks- are operations like addition supposed to be derived though? I thought they fall more into the realm of being 'invented' by us, rather than shown to be true from the axioms?

If you have given a valid definition of $+$ and $S$, then you should be able to compute them. Let's say that we use von Neumann numerals so that $$0:=\{\},\quad1:=\{0\},\quad2:=\{0,1\},\dotsc.$$ Show us step-by-step how you compute $S(3)$ with your definition. This means $S$ has to take in $$3 = \{\{\},\{\{\}\},\{\{\},\{\{\}\}\}\}$$ and output $$4=\{\{\},\{\{\}\},\{\{\},\{\{\}\}\},\{\{\},\{\{\}\},\{\{\},\{\{\}\}\}\}\}.$$

I should say that there are other ways to develop analysis axiomatically than to start from Peano's Axioms. Another traditional way, which you can see carried out in Fitzpatrick's "Advanced Calculus", is to start from axioms of the real numbers themselves.

@PrincessMia addition of the natural numbers is derived from counting which is how we created addition. That's why this approach is taken, it's the way we all learned as children.

@LeeMosher I mean, according to the methodologies of axiomatic theory (rather than some metaphysical morality), are operations like addition supposed to be derived? I am trying to understand why the circular scheme I outlined above is consistent with axiomatic theory as you have described it

I would not say that circularity is inherently inconsistent*, but it is more prone to inconsistency, and it's not a good way to do exposition even if it turns out to be consistent in a given instance. Perhaps the successor function and the addition operation could be defined circularly, in a way that is consistent, but there's not really a good reason to do so, and it would still give people an uneasy feeling just because of the circularity. ¶ *Now someone will mention So-and-So's Theorem that says that from any circular definition can be derived both a statement and its negation. <shrug>!

"there are no assumptions about this structure beyond what we know; we just know that this is the set of numbers, which element thereof is the distinguished 0, how succession is defined, and how addition is defined": The second part is rather a lot of what we know about the natural numbers! And if succession and addition are already defined, then what is left to be circular?

@BrianTung I didn't mean to say that we include among our assumptions that succession and addition are defined- I mean that we include among our assumptions the way in which they were attempted to be defined, namely we include among our assumptions that $S(n) = n + 1$. I edited for more clarity

I'll probably have to stop considering this question for now, since even after all your clarifications, I'm still having trouble making out what situation you're setting up, and what it is exactly that you're asking. I don't mean to suggest that it's your fault or anything (whatever that would mean), but I suspect it's not in my interests for now to continue thinking about this. Sorry!

With your statements about $S$ and addition, are you asking "can we show these functions exist and have these properties", or "is it possible these functions exist and have these properties", or "if functions exist having these properties, can we deduce the other classic properties of addition"?

@aschepler the former 2 questions are what I meant- I am concerned with whether it is possible the functions exists or not- but in order to know whether they exist or not, I would think we would have to prove that it is possible for them exist with those properties, so both the first and second question you enumerated are in the scope of what I'm asking

If you can define addition and derive its properties starting from simpler axioms, then you may do so.

When you have mutually recursive functions in software, the role of the base case is similar to the role of a primitive notion in a mathematical theory. It's a fact that isn't derived from any other fact. Can each function have a base case that needs to evaluate the base case of the other function? How do you think that would work out?

That was the genius of the Peano axioms: something seemingly so simple, and yet you could use them to define all of arithmetic.

@LeeMosher I certainly see that it works to define addition in terms of succession and that we could derive many further things herefrom- I am still trying to understand why circularly defining them in terms of each other doesn't make sense, though. Do you have a way to explain this which appeals to our intuition about logic?

@PrincessMia didn't you already give yourself an example where circularly defining them doesn't make sense? You're stuck in infinite loop where $S(1)$ means $1+1$ and $1+1$ means $S(1)$

@ioverri I still don't 100% understand why we can't reason in another way to deduce the value of $S(1)$- I see that that specific line of reasoning leads to nonsense, but I don't see why all lines of reasoning about what $S(1)$ lead to nonsense

@ioveri I appreciate your answer and have been trying to understand it so far regarding the $S(1)$ example

Just take it as an axiom of logic: any concepts used in a definition must be either previously defined or previously axiomatized.

@LeeMosher thanks:) 2 questions though 1. Do we regard the axioms and the operations defined on them as being in a certain order- like axiom 1 defined before axiom 2, all the axioms always defined before all the operations- and 2. What about mutually recursive functions? Doesn't the definition of one function use a concept not previously defined or axiomatized- it is only in tandem that the 2 functions can be defined?

@PrincessMia consider this question: What it means to be "the value" of $S(1)$? What qualifies as "a value"?

@PrincessMia as for mutual recursion, that's already addressed in my answer. In short, you need a proof of existence and uniqueness, which is usually not mentioned.

Yes, the development of mathematics is like a textbook, it occurs in order. When one states a definition, or an axiom, or a theorem, that statement may only use concepts that have been defined or axiomatized somewhere earlier in the textbook.

By the way, you might be interested in reading the recursion theorem. You'll find that recursion is, at its root, not a circular process either.

But "can we show these functions exist" and "is it possible these functions exist" are two different questions. Usually in this sort of development of mathematics, we start assuming a certain list of axioms, but for everything else we'll want to satisfy the first question defining things in a strict enough way to prove existence. Just allowing existence is not so useful if we want to lay the groundwork for basic things like addition.