There seems to be a general push to make the API self-documenting so these constants would likely end up as localized JSON data and version as the API does.
The server can't really do the lookups because of that localization - users of API's like this one speak a variety of languages and so client end mapping allows the API to get used by more than just English speaking users. I.e. as a developer I may wish to provide the results from this API in some obscure language, or map a variety of responses to just a general purpose "Error" message. Constant mapping allows me to do that.
As a side note, some of the best designed APIs structure the constants into a simple tree, for example:
With the English text available as e.g:
Where 'types' originates as a JSON object from some API resource, e.g. at /types/en/. It would be great if CH took an approach like this as it makes working with localizable APIs a breeze.