Why does 7+5=12 is not considered "analytic a priori" by Kant?
According to Kant, analytic statements really tell us nothing new about the world: they simply assert a structural relationship between subject and predicate, i.e., a predicate that is already "contained" in the subject as part of its meaning. They are "analytic" in the sense that the predicate can be found by analyzing the meaning of the subject. They merely clarify information that is already contained in our concepts of things. "All bodies are extended" is analytic because part of the definition of a body is that it occupies space (which is what "extended" means here). This statement clarifies what we already understand a body to be, but does not tell us anything more about them.
See that "All bodies are heavy" is synthetic because "having weight" is not part of the definition of "occupying space." While we cannot conceive of an unextended body (any more than we can conceive of a married bachelor), we can conceive of a spatial object that has no weight. Perhaps it is false that there are any weightless bodies. But the claim that there are is not contradictory. In addition,
Take a look at this:
7+5= 12 (one had to add them)
24 - (2x6) = 12 (less obvious)
Actually I can have an infinte number of combinations on the right hand side of the equation that equals 12.
Same with "straight line" and "two points" and "shortest line" and "two points", neither of which can be analytically "extracted" from the other.
* "The principle of contradiction is a formal criteria of logic which determines the logical possibility...of a cognition." Namely, it is the principle that a cognition "be logically possible, that is, not contradict itself".