I'm not sure why 5 + 7 = 12 should say anything new: I take it to be a shorthand notation to give a name to 5 + 7, which anyway is nothing but 5 times the unit + 7 times the unit, so there is not really anything "new" here. Or is there?
If we see it the other way around, as 12 = 5 + 7, maybe 5 + 7 is not contained in 12: we have a set of 12 objects, and we say that we can see it as a subset of 5 objects + a subset of 7 objects, so we say that that subdivision of the original set of cardinality 12 exists. But is that really "new"?
Of course, 5 + 7 = 12 is cited as an example at the beginning of Kant's Critique of Pure Reason, when he explains his view of analytic and synthetic knowledge. The question here though is not meant to criticize analytic judgments, but rather to criticize synthetic a priori judgments. I believe (for the sake of argument) that mathematics is composed entirely of analytic judgments, not synthetic ones, and I am trying to understand why Kant could argue at all that mathematics was in the synthetic camp.
More precisely. To define analytic and synthetic judgments, Kant writes at the beginning of the Critique of Pure reason, 2nd ed (emphasis mine):
Entweder das Prädikat B gehört zum Subjekt A als etwas, was in diesem Begriffe A (versteckterweise) enthalten ist; oder B liegt ganz außer dem Begriff A, ob es zwar mit demselben in Verknüpfung steht. Im ersten Fall nenne ich das Urteil analytisch, in dem andern synthetisch.
Which can be translated as:
Either the predicate B belongs to the subject A, as somewhat which is contained (though covertly) in the conception A; or the predicate B lies completely out of the conception A, although it stands in connection with it. In the first instance, I term the judgement analytical, in the second, synthetical.
So my question is why does Kant further say that Der arithmetische Satz ist also jederzeit synthetisch (Arithmetical propositions are therefore always synthetical): what is ganz außer in 5 + 7 = 12, if we see 12 as just a convenient short name for 5 + 7?