Is $\pi$ a property of mathematical expressions or algorithms? I was trying to think about what $\pi$ actually is. There are a lot of ways to get $\pi$ for example $4(1-\frac{1}{3}+\frac{1}{5}-\cdots)$.
But there is no one way to define it. 
On the other hand a fraction like $\frac{1}{2}$ also has multiple definitions e.g. $\frac{2}{4}$ or $\frac{3}{6}$. Although we generally take the simplest case as the definition. But it is really the class of all pairs $(n,2n)$. We might say that a tuple of the form $(n,2n)$ has the property of half-ness. Two fractions that both have this half-ness property can be set equal. 
In the same way, there is not one algorithm to define $\pi$. So $\pi$ must be the class of all algorithms that define it(?). Or the class of all mathematical expressions that define it.
We might say that an expression $4\sum_{n=0}^\infty (-1)^n\frac{1}{2n+1}$ has the property of $\pi$-ness and if two expressions both have this property then they can be set equal.
The difficultly I see is that some expressions, it might be unknown if they have the property of $\pi$-ness. 
Also, if this definition was true we could not write that anything is equal to $\pi$ as $\pi$ is just a property. How would we write this? Perhaps we would write that the expression belongs to the set of expressions that have the $\pi$-ness property. Perhaps:
$$\text{Expression}\left\{4\sum_{n=0}^\infty (-1)^n\frac{1}{2n+1} \right\}\subset \pi$$
And then this would work for other irrational numbers too:
$$ \text{Expression}\left\{\sum_{n=0}^\infty \frac{1}{n!}\right\} \subset e$$
But then this is not very convenient if we want to express the numerical value of one of these algorithms we would have to write something like:
$$\text{Expression}\left\{x\right\} \subset \pi \implies x \approx 3.14159$$
The alternative would simply be to have $\pi$ set equal to one of the expressions with the $\pi$-ness property but this seems a bit like cheating.
 A: This is a philosophical musing more so than a question.
It asks less about the nature of $\pi$ than about the nature of numbers themselves.
I would say the general stance is that we don't particularly care what numbers are.
Do you use a set-theoretic definition?
Do you build upon it with Cauchy sequences?
Are they Dedekind cuts?
Regardless of the answer, what we are most concerned is with their properties and behavior, more so than which particular construction gave rise to the objects that exhibit these properties.

In this sense, $\pi$ is a real number (however you've come to define it), and it exhibits properties that make it equal to the result of various other expressions, including the series you include in your OP.
A: Just as rationals are defined by the equivalence class of ordered pairs of integers under the equivalence relation $\langle a, b\rangle \sim \langle c, d\rangle$ iff $a \cdot d = b \cdot c$, reals are defined as the equivalence class of Cauchy sequences of rational numbers under the equivalence relation $f : \mathbb{N} \rightarrow \mathbb{Q} \sim g : \mathbb{N} \rightarrow \mathbb{Q}$ iff $h : \mathbb{N} \rightarrow \mathbb{Q}$ defined by $h(2n) = f(n)$ and $h(2n + 1) = g(n)$ is also a Cauchy sequence.
For the definition of a Cauchy sequence see here: https://en.wikipedia.org/wiki/Cauchy_sequence
Thus $\pi$ being a real number is defined to be the equivalence class of Cauchy sequences of rational numbers under the above equivalence relation that converges to the ratio of the circumference of a circle to its diameter.
A: The answer is that with $\pi$ as with many mathematical concepts, there is no objective reason to single out one characterization as being "the definition". One establishes a definition within a context; for example one may write "For the purposes of this paper [or this textbook, or this web page, etc.] we take $\pi$ to be the ratio of the circumference of a circle to the diameter", but in another context, one might define it to be the ratio of the area of a circle to the area of the square on its radius, and in another as the sum of a certain infinite series, and another as the value of a certain integral, and in another (in Baby Rudin, I think?) as $2$ times the smallest positive zero of the solution of $y''=-y$ and $y(0)= 1$ and $y'(0)=0.$
Likewise a parabola can be characterized as the graph of $y=x^2,$ or by means of a focus and a directrix, or as a section of a cone by a plane, or as what's left of an ellipse in a projective plane when you deleted one of its tangent lines from the projective plane to get an affine plane, etc. Which characterization you take to be the definition within a context is just a matter of which one best serves the purpose that you have on the particular occasion.
