Is addition more fundamental than subtraction?

When I followed an introductory! course group theory and throughout all my Math courses as a physicist, subtraction was always defined in terms of the inverse element and addition.

Is this the only way? I.e.: can subtraction be defined without addition?

If this is too broad a question, perhaps focus on groups and other physically relevant concepts.

Well, since you know a little from group theory, maybe this helps.

We always require a group operation to be associatve. But - is not: \begin{align*} (a-b)-c &= a-b-c\\ a-(b-c) &= a-b+c \end{align*}

Moreover but less importantly, + is abelian, while - is not.

Hence from a group theory perspective, - does not make much sense. It is not even an operation, its a short way of writing "add the inverse of the next element".

Tarski did this thing, where he defined "group" with a single operation called $x:y$ to be thought of as $x y^{-1}$, such that a single equation is satisfied identically. So, in fact, you can start with subtraction first and then define addition, zero, inverses from that.

Yes, you can define addition in terms of subtraction if you want to. But suppose I have a set $S$ which is non-empty and closed under subtraction as I would recognise it. Then:

• $S$ contains $0$, since it's just $x-x$ for some $x\in S$
• Hence, for all $x\in S$, $-x\in S$, since $-x = 0-x$.
• Hence, for all $x,y\in S$, $x+y\in S$, since $x+y=x-(-y)$.

Hence $S$ is a group under addition. The converse doesn't hold: there are structures, like the natural numbers, closed under $+$ but where $-$ doesn't make sense.

From this I conclude that $+$ is a strictly more general operation than $-$, hence more often applicable, and therefore (in addition[1] to it just generally being "nicer" in the sense of associativity and commutativity) more worthy of study and consideration. Wherever you find subtraction, it's because there's really some addition around, whereas addition need not come with a corresponding subtraction operation.

And one more thing: ask yourself what subtraction means without referring to addition. The simplest way to describe the concept of subtraction, rather than the implementation, is certainly to refer to addition, which itself is described in terms of the successor operation or something similar (the cardinality of a disjoint union, perhaps?)

[1]: Pun... sorta intended. Don't judge me.

In the natural numbers, there are no inverses, so you simply can't define subtraction as addition of the inverse element there. One way to define $a - b$ is as the unique solution, if it exists, of the equation $a = b + x$. But you can also avoid addition entirely and define subtraction recursively as $$\begin{align} a - 0 &= a, \\ S(a) - S(b) &= a - b. \end{align}$$ Then you declare $0 - S(a)$ to be undefined for all $a$, and that covers all the cases. So yes, in Peano arithmetic you can define subtraction without addition, but of course this may not generalize well to other number systems.