In the 2009 Star Trek movie Spock is learning in the Vulcan academy what it seems chemistry (alcohols and aldehyde, plus benzine) and mathematics/physics (perhaps a Laplacian) from the Earth, but its in English idiom, so isn’t supposed to be Vulcan the language in Vulcan academy? or Spock was learning Earth languages?
There is no human language visible in the scene, but there are of course human symbols on the screen.
Apart from simple ignorance of the producers (let's face it: we all know this is the case), the only in-universe explanations I can think of are:
- Humans play a very predominant role within the Federation and they were the driving force of its inception. One can assume based on this one of two things:
- Instead of relying on the universal translator, there is a common language for the whole of the Federation, spoken by people from all member worlds. This language is either English or another earth language that uses latin letters or alternatively, a new language has been crafted that uses latin letters. Given the relatively human-like physiology of the species in the early Federation their vocal folds might all be able to produce the same sounds easily, allowing them all to speak the same language without much trouble.
- There may not be a common language in the Federation. However for scientific purposes the already used latin letters in formulas and symbols on earth were adopted by all other member worlds for a common framework for scientific purposes. So the latin letter "d" to mark a derivative may have been adopted as a symbol independently from the spoken language (you can draw a d without knowing that it is connected to a specific vocal sound in a language). Similarly the capital letters used to describe elements may have been adopted as a symbol too, void of their meaning as a letter in a language.
- Spock is in fact learning a language at the moment and the best way to do so is to use it naturally while doing and/or learning other things. So he may be learning chemistry and mathematics, but at the same time also the latin alphabet or the earth symbols by looking at the formulas and diagrams in earth language. The fact that his mother is human and his father is the Vulcan ambassador to Earth makes it very plausible that he is learning an Earth language.
I prefer the second of the three explanations. The universal translator makes a common language somewhat unnecessary (one may still argue that one does not want to rely on a technical device for basic communication) and a common language would be difficult to be easily spoken by all member world species. At the same time a common framework for denoting scientific meanings like an element or a mathematical expression is necessary and can easily be adopted by others: Chinese people use the same numbers and mathematical expressions that people in western cultures use, even though the actual letters are from a western culture alphabet.
That said, I wonder why you picked this example. Why not the all-English LCARS screens in basically all else of Trek? The only possible explanation for this would be that there is in fact a common language across the Federation and it is the same English we speak today. Or alternatively, that Federation citizen speak and read their mother tongue only and because humans are so predominant they just discriminate against all non-humans by displaying information on a screen in English.