Why is zero important?

Question

I am not sure whether this question is more appropriate here or in theoretical computer science. I leave it to the wisdom of moderators.

On the computer science site I came across the following (possibly unintended) question:

is the empty string a necessary concept?

This reminded me of the fact that the ancient Greeks did good work in arithmetics without having zero, syntactically (as a place holder, as in positional number systems) or semantically.

Having zero (resp. the empty string) will of course simplify analysis and reasoning about integers (resp. strings and formal languages), and the way definitions and theorems can be stated.

However, I came to wonder whether it changes in any significant way the kind of results that can be proved, or whether we get essentially the same theories.

In the case of arithmetics, I guess this would mean that 0 is replaced by 1 in the nine Peano's axioms, and that for addition, the axiom $a+0=a$ is replaced by $a+1=S(a)$. Indeed, it has to give the same theory as the original axiom $a+0=a$ can be proved from the new one if we add a new integer 0 such that $S(0)=1$. But I am not sure things are always that simple, and the lack of zero could cause considerable complexity in expressing other concepts.

So my question is whether this issue has been formally analyzed, or whether it even makes sense of asking that question. Would the power of mathematics be impaired (other than in expressive perspicuity) if the concept of zero were somehow not available (assume that some church will deal harshly with you if you even only hint at the possibility of the concept)?

In the case of the empty string, another user asserted that not having the empty string would change some properties of strings, such as prefixing. I asserted that it is only a matter of consistent definitions, and that the only change is in expressiveness (which is far from negligible).

Is there any text discussing formally or informally this issue? I would guess it is important, at least for epistemology and history of sciences.

Answer 1

I believe zero is not needed at all, in the sense that you get an equivalent structure whether it's included or not.

A little more precisely, if you consider the category $\mathcal C$ of semigroups, and the category $\mathcal D$ of monoids (semigroups with identity) satisfying the condition that $x\cdot y=e\implies x=y=e$ (where $e$ is the identity; note that ${\bf N}$ with addition is in $\mathcal D$, as is the semigroup of strings including the empty string, but there are no nontrivial groups in there), then the two categories are equivalent: you have a functor $F\colon \mathcal C\to \mathcal D$ which, given a semigroup $S$, returns $S\cup \{e\}$, where $e$ is a new element with operations defined in the natural way, and a functor $G\colon \mathcal D\to \mathcal C$ which given a monoid $S\cup \{e\}\in \mathcal D$ returns the subsemigroup $S$, and $F\circ G$ and $G\circ F$ are both isomorphic to identity functors on the respective categories.

This means that a semigroup and en element of $\mathcal D$ are essentially the same thing, as we can translate anything between the two categories mostly seamlessly.

More model-theoretically, if you have a nontrivial semigroup $S$ with two named (or definable) elements $s_1\neq s_2$ (like $1$ and $2$, both of which are definable in the positive natural numbers, or two single character strings, for which you assume you have names for anyway), then it is naturally bi-interpretable with $\overline S=S\cup \{e\}$: you can define $\overline S$ as the set $\{ (s_1,s_1), (s,s_2)\mid s\in S\}$ which is definable in $(S,s_1,s_2)$ where $(s_1,s_1)$ is intended to be the identity, and the operation which makes this set isomorphic to $\overline S$ is definable.

This means that at least for semigroups with two named (or definable) constants, a semigroup and the same semigroup with an adjoined identity are essentially the same from logical point of view. This is problematic only if we really don't want to add constants (but in that case, you can't really say too much interesting stuff anyway), or are working with the trivial group (in that case, you absolutely can't say anything interesting).

In fact, the stuff in the preceding paragraphs (both the “category-theoretical” ones and the “logical” ones) doesn't really need associativity, you can to the same in any set with some binary operation.

In short, adding identity doesn't really make it possible to say anything meaningful you couldn't say without it, but it just might make it a little easier to say some things.

Answer 2

All of mathematics can be done without 0. Many techniques would become more difficult, but it is completely possible. This happens in topology. For instance, to show that the set where $f=g$ is closed in a Hausdorff space, you cannot subtract and set equal to 0, because there is no subtraction and no 0. So you work around it using other properties.

Math is just a way of organizing human thought, so you can make any restrictions you want.

Edit addressing comments: you just make subtraction super complicated. Subtraction is only defined if one number is strictly greater than the other, and all groups are multiplicative. Although getting rid of 1 as well is even more fun.

As for superhuge cardinals, I oppose overfeeding animals.

Answer 3

Here's my take on the matter.

Let $X$ denote a set. Then we certainly want to allow empty subsets of $X$. Otherwise, the intersection of two subsets of $X$ needn't exist; furthermore, the maximum subset of $X$ won't have a complement. Basically, getting rid of empty subsets would destroy all that nice symmetry of the powerset lattice.

Therefore, empty subsets are really nice. But, what is a subset? In category theory, a subset of a set $X$ would be defined as an injective function $f : A \rightarrow X$ modulo some largely irrelevant details. So if we want every set to have an empty subset, then there had better be an empty set $\emptyset$ for the domain of $f$. Therefore, the empty set $\emptyset$ is really important.

Finally, since natural numbers are primarily used to count the number of elements in a finite set, it would be better to include $0$ in our list of natural numbers.

There's many other reasons to include $0$ in our list, but I'll leave it here for now.