Today I had a very interesting discussion with my brother concerning the nature of the number 0, which has been long interested both of us.
My brother is reading the book “From Zero to Infinity” from Constance Reid; she is of the opinion that 0 is a natural number, because it is a valid answer to the question “How many… ?”, like “How many cows are there on the meadow?”
This has long been my position, and I could never understand how one could not say that 0 were a natural number; at my University Maths courses, we often heard that “0 is not a natural number because a natural number is only that with which you can count … one cow, two cows, three cows….”
I always responded: yes, but I count with 0 too: suppose I am a farmer, and I have 2 cows on the meadow. Unfortunately, one gets sick and dies soon thereafter; I go to the meadow and count my cows: only one left. Now this one gets sick too, and dies. I go to the meadow and count: 0 cows. In no way can I ever count -1 or so cows on the meadow 
But I can count 0 cows.
Constance Reid seems also to argue for this position in her book. So far so good. But Mario Bunge destroyed all my highly held opinions of 0 in one fell swoop. In his book “Chasing Reality” he makes a fine distinction between facts and propositions. Facts are states or events which occur in the physical world. What we say about the world are propositions. So, i.e. there are no scientific facts, there are only scienctific propositions. These can be true if they correspond to facts in the physical world (I’ll write about truth in a later entry).
Why is this important? Because, while there are negative and disjunctive propositions (“There is no cow on the meadow”; “These people are either twins or something fishy is going on here”), there are only positive facts. There are only states or events in the world, never “negative” states or a disjunction of events.
Numbers are cognitive constructs, and for the sake of this discussion we may call them propositions (they are actually fictions which become part of propositions). So, if I make a proposition “There is one cow on the meadow”, this can correspond to the physical fact that there is one cow on the meadow. But the proposition “There are 0 cows on the meadow”, which is just another way of saying “There is no cow on the meadow” is a negative proposition. But there is no fact which could make this true. There is no negative fact in the physical world, no no-cowness, that could make such a propositon true (how “true” negative propositions can be assigend a truth value is not unproblematic, but will be reserved for future discussion). Therefore, this difference is a strong case for assuming that 0 is, indeed, not a natural number.
Constance Reid also admits this in her book, by stating that 0 has a special position: there is only one empty set, but an infinite number of sets of the cardinality of the other natural numbers. There is the set of “one cat”, “one dog” etc. etc. But only one empty set!
This comes as not very surprising when confronted with the above observation that there are no negative facts. There are many cats or dogs which can make the set of one true. But there are no negative facts in the physical world, and therefore only one empty set – the one constructed by our cognitive ability for categorization.
Conclusion: if we define the natural numbers as those which are used to correspond to physical objects in the world, 0 is not a natural number.
If we concentrate on countability, which belongs more to the domain of cognition than to physics (apart from practical problems, that is ), then 0 might just as well be a natural number. It depends on the frame of reference you use.
