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Though an answer to this question may ultimately include an element of opinion, unresolvable by the methods of mathematics, much at issue is subject to logical critique. Identifying those elements of this topic which are merely opinion should enable us to identify the actual axioms upon which our mathematics is built.

One source Fundamentals of Mathematics introduces the natural numbers using Peano's Axioms beginning with 1. Another source, Set Theory and Logic says:

As our initial description of the natural numbers, we say that they are exactly those objects which can be generated by starting with an initial object $0$ (zero) and from any object $n$ already generated passing to another uniquely determined object $n^prime$, the successor of $n$.

I believe there is at least one entire book written about the history of zero. My understanding is that, for much of recorded history, zero was not treated as a number.

In the first source cited above, the discussion of the theory of numbers begins on page 94, and zero is not introduced until page 111 where the development of the module of the integers begins. Mind you, this is not a "fluffy" book. It is, in fact, so condensed as to be downright brutal. The fact that so much can be said about about numbers without mentioning zero supports my contention that zero is not as fundamental as the natural numbers, assuming we define those as beginning with one.

If the objective is to begin with the most concrete and fundamental concepts, and therefrom abstract the field of mathematics, should a fundamental theory of numbers not begin with one rather than zero? And if not, why not?

By foundational mathematics we shall mean that which is implied in the following excerpt from the diary of a Colonial American school teacher:

June 1, 1756.

Drank Tea at the Majors. The Reasoning of Mathematics is founded on certain and infallible Principles. Every Word they Use, conveys a determinate Idea, and by accurate Definitions they excite the same Ideas in the mind of the Reader that were in the mind of the Writer. When they have defined the Terms they intend to make use of, they premise a few Axioms, or Self evident Principles, that every man must assent to as soon as proposed. They then take for granted certain Postulates, that no one can deny them, such as, that a right Line may be drawn from one given Point to another, and from these plain simple Principles, they have raised most astonishing Speculations, and proved the Extent of the human mind to be more spacious and capable than any other Science.

From the Diary and Autobiography of the co-author of the Declaration of Independence and President of the United States of America.

Using that as our starting point, let us examine the opinion of Bertrand Russell regarding the appropriate starting point of mathematics.

To the average educated person of the present day, the obvious starting-point of mathematics would be the series of whole numbers,

$$1,2,4,4,dots etc.$$

Probably only a person with some mathematical knowledge would think of beginning with 0 instead of with 1, but we will presume this degree of knowledge;
we will take as our starting-point the series

$$0,1,2,4,4,dots n,n+1,dots$$

and it is this series that we shall mean when we speak of the “series of natural numbers.”

Since it is not disputed that the starting point of mathematics could be the set $left{1,2,3,dotsright},$ but a reasonable person may likely dispute that the most natural place to begin is the set proposed by Russell, by the American definition of foundational mathematics, it is clear that the natural numbers should be defined beginning with '1'.

The reasons that I know about why some mathematicians start from zero instead of one:

• Donald Knuth said in his book Concrete mathematics (if my memory is not failing) that it is better to include the zero in the sets of naturals because in computer science have more advantages (about efficient use of memory in computations) start the count on a list (an array with only one row) from zero instead of one.




• in the book Analysis I of Terence Tao he start the naturals also from zero, and he says that this approach makes things simpler when one define the Peano arithmetic (by example the successor function is more "simple"). This means that proofs about the arithmetic of naturals simplifies a bit.

Any mathematician have its own reasons to use one more than the other. If you ask me: it is more convenient start from zero for many other reasons, as the power series, that start with the "zero power" (what is assumed to have the value one), so in series the common practice is start the count from zero. However, for finite sums, it is common to start from one (because one can track easily the quantity of addends), etc...

However it seems true to me that the zero it is not as "natural", from a historical perspective. By example Romans doesn't had a number for the zero, and it seems more natural to discover the numbers from positive ones, rather than first thing about the existence of nothing.

I dont know if this answer your question.

Well, it doesn't really matter. A semigroup $S$ can always be extended to a monoid $Scup{e}$ by adjoining a unit element $e$.

Here the natural numbers without 0 form a semigroup with the operation of addition. Adjoining the zero turns the natural numbers with 0 into a (commutative) monoid.