You're fighting a straw man; that isn't the argument. 1-based indexing is elegant for many mathematical uses. Technical computing often implements algorithms that are best written down using mathematical notation. In such a case the largest conceptual difficulty is not the origin of the index but the successful translation of the algorithm. To minimize the possibility of error, the original indexing (especially when it involves nontrivial mathematical maps) is often preserved.
> Dijkstra makes all of the convincing arguments for why 0-based is better than 1-based.
He essentially makes only one point. Namely that, when using 0-based indexing we can easily determine the length of a sequence by only its upper index. Though it's not a profound observation since the index bias is zero, it often gets treated as such.
Either way, it's a convention. A competent programmer should be able to handle any well-specified indexing convention (including those starting at negative indices). Different circumstances will confer different benefits on different conventions, and you should use the one that's most appropriate for the task at hand rather than religiously promoting The One True Way (TM).