Zero is important because it is the first abstraction. If your notion of numbers includes the idea of zero as a number, then you have broken through the first intellectual barrier of mathematics. Without this, it remains tempting to still consider numbers as existing for the purpose of counting _things_.
But with zero, this idea converges on the same thing. No matter what things you were counting, if you have zero of them, you have the same idea. And so you take a step towards the idea of a number being a concept in its own right, rather than existing purely for the purpose of counting or measurement.
It is the same sort of conceptual freedom that allows you to do things like add a number to a square. To deal with an equation like x + x ^ 2 = 0. If you're stuck with numbers "meaning" something beyond themselves, then you'll never add x to x^2. One is a length, the other an area. They are different objects.
This intellectual leap is one that must be made by all students of mathematics - and many young people do not.
> Zero is important because it is the first abstraction
I’m not sure what this even means, but Sumerians had abstract mathematics, in addition to art and literature which are abstract by their nature. They were using numbers in the abstract sense before the number zero was named, and so while it seems like a logical and tempting narrative that naming zero is what abstracted numbers, history doesn’t seem to support this particular post-facto rationalization. Naming zero is very important in the history of math, it just isn’t the first abstraction.
regardless of whether it makes sense to say that treating zero as a number was 'the first abstraction' or not†, ancient mesopotamians did not treat zero as a number. even noble fibonacci didn't consider it as a number or even a digit:
> The nine Indian figures are: 9 8 7 6 5 4 3 2 1. With these nine figures, and with the sign 0 (...) any number may be written.
i'm not sure when it became conventional to consider it a number rather than the absence of one; it might not have been until the early modern era
I am pretty sure early European mathematicians did treat polynomials as areas or volumes, x can be just a rectangle x units long in one dimension and 1 in the other, x^2 is a square, etc. This meant they had to go through contortions to avoid negative coefficients since they made no geometric sense. If a coefficient would otherwise be negative it would have to move to the other side of the equality, and be solved using a different method. Instead of a single quadratic formula they needed several different cases depending the exact form of the polynomial.
Why is that the first abstraction? I would think at least one of these three earlier developments would be:
1) deductive geometry of Thales, where we could now prove things independent of the physical world (abstracting away from the physical)
2) Plato’s remarks on incommensurability (proto irrational numbers) being something real but not physical, because no physical process could prove to the mathematician that two lengths really have no common unit measure. Here the abstraction is again away from physical means.
3) infinity of numbers. Abstracting away from large but finite collections. We can only ever survey finite collections physically, again is an abstraction.
Initially, no. They are a mechanism for counting, or a way of recording measurements. Many young people, when trying to learn mathematics, do not take the necessary step beyond these ideas.
They are not helped by educators insisting on presenting "word problems" when teaching maths. To get to the next level, you need to break the connection between numbers and the "real world". I've always felt that the number zero represented the first step that humanity took in this journey - and it's a step that every human also needs to individually take if they want to learn maths.
Yeah, a lot of these articles conflate use of zero as placeholder, numeral and number. But the real critical conceptual step is the last step of using zero like any other number (mostly) in arithmetic.
But with zero, this idea converges on the same thing. No matter what things you were counting, if you have zero of them, you have the same idea. And so you take a step towards the idea of a number being a concept in its own right, rather than existing purely for the purpose of counting or measurement.
It is the same sort of conceptual freedom that allows you to do things like add a number to a square. To deal with an equation like x + x ^ 2 = 0. If you're stuck with numbers "meaning" something beyond themselves, then you'll never add x to x^2. One is a length, the other an area. They are different objects.
This intellectual leap is one that must be made by all students of mathematics - and many young people do not.