Question: I wondered whether notation in mathematics can lead to actual mistakes in proofs and deductions or whether notation is just style that doesn't matter logically. I read that there are formal rules to introduce new notation that can be found here: [How could we formalize the introduction of new notation?](https://math.stackexchange.com/questions/1863150/how-could-we-formalize-the-introduction-of-new-notation) or <https://en.wikipedia.org/wiki/Extension_by_definitions>.

I assume that these rules are the way they are because using arbitrary notation can actually lead to mistakes, which is why one is ristricting new notation to those rules. However, I can't really come up with examples where it might lead to mistakes and I haven't seen this talked about in any of my lectures thus far. I have always seen notation as a form of style of authors which doesn't make a difference, but since I read about those rules I am quite confused when seeing notation and often think about whether that notation can actually harm. Something that comes to my mind is the following. In peano arithmetic, there exists an object $S(x)$ for all $x \in \mathbf{N}$, called the successor of $x$. Suppose $n$ is a natural number, then one could (informally, this is none of the rules above) suppress the dependency of $n$ and call this element $m$, or in short: $m:=S(n)$. All one does here is assign a new name to an existing object, but by doing so suppress one property. This is done when defining $1:=S(0)$ for example. Something that is also done sometimes is introducing a new symbol for an object that is not unique. Again, this is not captured by the rules above. There are probably more examples, but I hope that this is already illustrating what I think about. Also, I have read that "$:=$" is informal, and not allowed formally, why is that? I feel like this might be connected.

In short, should I think about notation or is it rather irrelevant, besides being a good way to increase intuition and understanding? Can it ever lead to mistakes? Thanks!

Answer: 
Here's an example:
We introduce a notation for the *sum* of finitely many numbers $a\_1,a\_2,\ldots, a\_n$ as
$$ \sum\_{k=1}^na\_k.$$
We introduce a strikingly similar notation for *series*:
$$ \sum\_{k=1}^\infty a\_k,$$
but this is a totally different kind of beast. For example,
$$ \sum\_{k=1}^na\_k=a\_n+\sum\_{k=1}^{n-1}a\_k,$$
whereas
$$ \sum\_{k=1}^\infty a\_k=a\_\infty+\sum\_{k=1}^{\infty-1}a\_k$$
makes no sense. (Not to mention that the sum is always defined in a ring, allows permutation of the summands, and whatnot).

Another example:
We introduce the notation
$$f(n)=O(g(n))$$
as a shorthand for "$f$ does not grow faster than $g$", or formally.
$$ \exists c\in\Bbb R\colon \exists m\in\Bbb N\colon \forall n>m\colon |f(n)|<c|g(n)|$$
In spite of the equals sign being part of the notation, this must not be considered as an equality! Normally, from $a=b$ and $c=b$, we can conclude $a=c$, but
$$ n^2=O(2^n)\quad\text{and}\quad 2n^4=O(2^n)$$
you should not conclude
$$ n^2=2n^4.$$
Here is a frequent source of errors, especially under the tag Boolean algebra on this site.

In Boolean algebra, it is common practice to use the notation $x+y$ and $xy$ for the addition and product of $x$ and $y$, respectively, and $\overline x$ for the negation of $x$. This is not a bad notation, but it becomes ambiguous when you write $\overline x\overline y$. It is actually mostly a $\LaTeX$ problem, easily solved by writing $\bar x$ ($\texttt{\bar x}$) instead of $\overline x$ ($\texttt{\overline x}$), but even so, $\bar x \bar y$ is frequently confused with $\overline{xy}$.