Math traditionally has had some bad notation from a formal point of view, because humans are good at coping with bad notations (or going back and forth between variants), unlike machines and formal systems. Computer science being a (more) formal science (vs math which is overwhelmingly not done formally), it has criticized some traditional math notation which are ad-hoc and not nicely formalizable (and put forward variants that are actually better behaved in terms of mathematical structures).
For indices: indices are about referencing elements of finite ordered sets, say of size N. Hence the 'abstract' indexing set for N elements is the ordinal N. The most canonical way to represent it is to take the length-N prefix of the natural numbers (eg 0-based indexing, von neumann ordinals), which happen to have all sorts of additional structure (eg mod-N arithmetic). This is also consistent with the offset view (the i-th element is at offset i). The fact that people tend to start ordinal numbers at 1 doesn't change anything that mathematicians working with ordinal numbers take them to start at 0, for the same reason we start naturals at 0.
See also: notation for higher derivatives https://arxiv.org/abs/1801.09553; a bit further but in the same vein: notations for free variables in programs as de-bruijn indices (or some variant thereof) (it's further because it's practical for doing proofs, but not for writing concrete terms). There are probably other instances.
For indices: indices are about referencing elements of finite ordered sets, say of size N. Hence the 'abstract' indexing set for N elements is the ordinal N. The most canonical way to represent it is to take the length-N prefix of the natural numbers (eg 0-based indexing, von neumann ordinals), which happen to have all sorts of additional structure (eg mod-N arithmetic). This is also consistent with the offset view (the i-th element is at offset i). The fact that people tend to start ordinal numbers at 1 doesn't change anything that mathematicians working with ordinal numbers take them to start at 0, for the same reason we start naturals at 0.
See also: notation for higher derivatives https://arxiv.org/abs/1801.09553; a bit further but in the same vein: notations for free variables in programs as de-bruijn indices (or some variant thereof) (it's further because it's practical for doing proofs, but not for writing concrete terms). There are probably other instances.