A perfect unison is no interval at all, and should be labelled zero rather than one. I really don't see how there's any room for argument other than "It's always been done that way".
First of all mathematic is just a language more ore less useful to describe an empirical reality and a tool (see my previous link to Galileo's father). Nothing obeys to mathematical laws, mathematical laws successfully describe things, as long as they work.
About zero, I am not sure that Pitagora knew it when he first described harmonic intervals.
Among the lucky coincidences in the western music system, the 7th, 8th, 9th, 10th 11th 12th 13th harmonics, are the 7th, 8th, 9th, 10th 11th 12th 13th grades of a scale.
As the Greek mathematician pointed out, if you divide a string in seven parts, you get (roughly) what is now called "7th"
If you want to be really logical, do like a friend of mine and divide the octave in 6 equal steps. His piano (he could not patent it, because the original idea dates back to 16th century) has a white key followed by a black key. All intervals are simpler, as is the system (common in computer music) of dividing the octave in 12 steps. In the good old days we used to say that a third is "5 strings", as is a diminished fourth, and a plus-that-augmented second.
You just need to learn two fingerings and you have all the major scales (a mark is needed on C).
Quite useless in performing western music.
edit: I did an error.
The harmonic numbers are the denominator of the original string length. 2nd harmonic is 1/2 string, 3rd is 1/3 and so on. so they start from 1.