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In most cases, we want to discern between days first, them months, then years.That really, really depends on particulars on your life. While I strongly dislike the US format of putting the month first, the irony is, the month is the most important datum for most of the dates I read with. Year is usually obvious given particular context (it's typically either the current one, the next one, or sometimes the previous one). Days don't matter for anything that isn't in the same month. I'd go as far as saying that months are typically the most important for most people in XXI century, then days, then years.
But that's no problem for punctuated date formats in general. People don't read things linearly at letter level, they deconstruct tokens in a random-access way. Separators like - or / help clearly identify which part is which (compare how easy it is to read 03/01/2021 with 03012021), and there's no extra cost associated with reading off the part in the middle.
The benefits of YYYY-MM-DD notation on top of the above are:
- With all terms being fully expanded, dates line up vertically as well, which helps when dealing with stacks of documents or tabular data.
- Being ordered by magnitude like normal numbers provides a Schelling point for interpretation[0]. That is, when a random person who wasn't exposed to much computing or internationalization before sees 2021-01-03, they will automatically assume that 01 must be the month, because writing YYYY-DD-MM would be dumb and inconsistent with how numbers are written in general.
Yes, both of these apply to DD/MM/YYYY, but, YYYY-MM-DD also gets you sane lexicographic sorting, which the UK/European format doesn't. It's super useful feature not just when dealing with computers, but also when perusing any kind of date index. ISO 8601 dates work much like words in a dictionary, you can binary-search through them.
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[0] - https://en.wikipedia.org/wiki/Focal_point_(game_theory)