In all those sentences, academy/academies is generic, of course. If it were the name of the school, it would read something like this: The Math and Science Academy.
When you put a/an in front of a word, yes, you're indicating one, although perhaps not a particular one.
1. He goes to an academy to study math and English.
[Is 'an academy' in generic use? Or Does 'an' mean 'one'?]
2. He goes to academies to study math and English.
[Is the word 'academies' in generic use?]
3. He goes to the academy to study math and English.
[Does 'the academy' mean the specific academy? Or is 'the academy' in generic use?]
6 answers
Thank you for your help.
4. He goes to academies to study math and English after school.
5. He goes to an academy to study math and English after school.
6. He goes to academy to study math and English after school.
[Then, you mean we can use both 4 and 5. In #4 'academies' are in generic use. In #5, is 'an academy' in generic use? In #6, is 'goes to academy' ungrammatical?]
4. He goes to academies to study math and English after school.
5. He goes to an academy to study math and English after school.
6. He goes to academy to study math and English after school.
[Then, you mean we can use both 4 and 5. In #4 'academies' are in generic use. In #5, is 'an academy' in generic use? In #6, is 'goes to academy' ungrammatical?]
That's right. You can use 4 and 5, but 6 shouldn't be used as it's written.
If the word "academy" is the name of the school, Academy, it is a proper noun and should be capitalized. As a common noun, you really need "an" or "the".
Thank you for your help. By the way, do you mean that 'an academy' is in generic use?
Yes.