Does 5 + 7 = 12 really say anything new?

Question

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?

Answer 1

You seem to have hit upon the paradox of analysis, or at least issues in the vicinity. The whole SEP article on Conceptions of Analysis in Analytic Philosophy is worth a read, but the section on G.E. Moore is particularly relevant.

A little snippet:

Consider an analysis of the form ‘A is C’, where A is the analysandum (what is analysed) and C the analysans (what is offered as the analysis). Then either ‘A’ and ‘C’ have the same meaning, in which case the analysis expresses a trivial identity; or else they do not, in which case the analysis is incorrect. So it would seem that no analysis can be both correct and informative.

As the same paragraph notes, the paradox is also discussed in Plato's Meno and various writings of Frege.

Since you tagged the question "Kant", I'll note that the same page has some quotations from Kant on analysis. Famously, Kant thought of math as synthetic -- and so not engaged in analysis -- and he is taken as the inspiration for Intuitionism as well as Frege's later views on geometry. I've never been able to make much sense of Kant, though, so I won't venture to explain his views or comment on whether those inspired by him "got the right idea" from his writings.

As for why Kant thought math was synthetic a priori, I refer you to the SEP article on Kant's philosophy of mathematics. In particular, the section "Kant's theory of the construction of mathematical concepts in 'The Discipline of Pure Reason in Dogmatic Use'" contains the most relevant information:

The central thesis of Kant's account of the uniqueness of mathematical reasoning is his claim that mathematical cognition derives from the “construction” of its concepts: “to construct a concept means to exhibit a priori the intuition corresponding to it”.... Kant claims further that the pure concept of magnitude is suitable for construction because, unlike other pure concepts, it does not represent a synthesis of possible intuitions, but “already contains a pure intuition in itself.” But since the only candidates for such “pure intuitions” are space and time (“the mere form of appearances”), it follows that only spatial and temporal magnitudes can be exhibited in pure intuition, i.e., constructed. Such spatial and temporal magnitudes can be exhibited qualitatively, by displaying the shapes of things, e.g. the rectangularity of the panes of a window, or they can be exhibited merely quantitatively, by displaying the number of parts of things, e.g., the number of panes that the window comprises. In either case, what is displayed counts as a pure and “formal intuition”, inspection of which yields judgments that “go beyond” the content of the original concept with which the intuition was associated. Such judgments are paradigmatically synthetic a priori judgments (to be discussed at greater length below) since they are ampliative truths that are warranted independent of experience (Shabel 2006).

In 2.2 they discuss the argument you start off with and some of the disagreements in interpretation.