Wednesday, August 24, 2005

The Unnatural Naturals

In a comment from my previous post, vito got me thinking about the naturals. Most mathematicians use the term 'natural' to refer to numbers in the set {1,2,3,...}. However, I prefer to think of the set {0,1,2,3,...} as the naturals. As the precise definition of 'natural' is still debatable I can get away with my usage, as long as I define exactly what I mean. My gripe with the more usual definition of 'natural' is that it's not natural at all.

Consider some of the natural ways to build the integers. For example there are the Peano axioms and the finite ordinals. These define sequences that start at zero, not one. In fact, both of those web pages define zero to be natural. Countless theorems are more naturally written as statements about {0,1,2,...} rather than {1,2,3,...}. I can't even see why the set {1,2,3,...} needs a name any more than the set {2,3,4,5,6,7,...}. Having said that, when I was a student at Cambridge there was a formal debate over the definition of the naturals. Things suddenly became more complicated when a faction suddenly appeared arguing for the set {2,3,4,...} on the grounds that if numbers are for counting then the number one doesn't exactly do much counting. I think I should have formed a pro-{3,4,5,6,...} faction at this point, after all, the ancient greeks, who are noted for their wisdom, only marked their nouns as plural for numbers strictly greater than 2.

Anyway, there is some discussion of the debate on Wikipedia and the conclusion is probably the same one I would come to: the set {0,1,2,3,...} is more popular among logicians and set theorists (I'd probably add category theorists) and {1,2,3,4,...} is more popular among number theorists.

And of course, when programming, I think arrays should start at zero. You won't catch me using Fortran or matlab. Come to think of it, as the first ordinal is zero, shouldn't we actually start counting at zero?



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