Thinking when the new Millenium starts has IMO nothing to do with math, but with "emotions" and "feelings" and "common sense".
Let those math geniouses argue what ever they want, no one with at least "some" common sense would tell that the year 1000 did belong to the first Millenium ... but the second ..... why then should 2000 be regarded as the last year of the second ???? !!!!