This question is similar to this one; the difference is that I'd like to find out about quantitative measurements that are not based solely on the vocabulary.
Are there any such objective quantitative measurements in existence? The Levenshtein distance mentioned in the top answer to the other question seems to be vocabulary based; if two languages with the same vocabulary differ grammatically then Levenshtein distance wouldn't measure that.
I wonder whether a comprehensive linguistic distance was ever proposed.
I think such measurement is possible. Here's one possibility: consider an autoencoder trained to translate between the given languages. An auto-encoder, as any neural network (NN), has a definite size: the number of nodes. And, because it translates between the given languages, the NN encapsulates the difference between languages.
Now the numerical distance between languages can be based on the smallest possible size among all the auto-encoders translating between those languages. One could measure that by training a bunch of NNs of different depths and sizes, qualifying their accuracy, and defining a the distance between languages as the smallest size among all the qualified auto-encoders between them.
Prompted by the first comment, I'd like to explain where this question is coming from and how this question is relevant to learning languages.
I speak a couple of languages and want to learn another one or two. I'd like to pick a language that's relatively easy given my background. One way to do that is go choose a language that is, in some sense, "close" to the ones I know. To my surprise, I couldn't find any objective comprehensive measure of language distance.
Yes, there are some language difficulty classifications for English speaker: category 1 (Latin derivatives), category 2 (most of them), and category 3 (eastern Asia). However, this classification seems subjective and, more importantly, is based on English speakers. Since I also speak Russian I find Ukrainian much easier for me than, say, Spanish, contrary to the table.
Yes, one can google numeric language distance tables. Unfortunately, all those I found so far are purely vocabulary based: they only measure differences between words and pay no attention to grammar. This is contrary to language learning experience, where grammar is often the most challenging thing to learn.
Linguistic is not exactly a new science, and I would expect that such basic things as measuring distances between languages should be well developed by now. Alas, the mighty google led to nothing satisfactory. Thus the question.