This seems naive. When we count, we implicitly start with zero, not one. By similar reasoning, the 21st century begins on January 1 of the year 2001, not of 2000. Thus goes the unassailable argument of the mathematical purists.
Mathematical logic is no match for human custom, however, which greatly complicates matters.
Over the centuries, the calendar has been revised repeatedly. Strictly speaking, then, the 18th century was 11 days shorter than previous and subsequent centuries. Moreover, the packaging of years into decades, centuries, millenia or other units is more often a convention of convenience than of precision, and so may not be bound by strict rules of mathematical rigor. So although the purists have reason on their side, the 21st century will probably begin when most people decide to celebrate it. Yet even that popular decision may not affect the date historians choose.