# Does 5 + 7 = 12 really say anything new?

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?

• Depends on whether you're a Platonist or a formalist. – user4894 Apr 2 '17 at 1:07
• @user4894 - can you elaborate? – Frank Apr 2 '17 at 1:26
• If you're a Platonist, you believe that 5 + 7 = 12 expresses a truth about the world. If you're a formalist, you see it as a definition in a game of formal symbol manpulation. It's like asking if the way the knight moves in chess is meaningful in the real world. Of course not, it's a formal game. To a formalist, so is math. – user4894 Apr 2 '17 at 2:54
• @user4894 I don't see how this touches upon the issues presented by OP. You point to a distinction in truth, but truth was not what was at stake. The question is one of something like "sameness of meaning (or definition)". Even a formalist can distinguish between the definitions presented by arithmetic and those presented by set theory, e.g., they'd just tend to make the distinction in something like syntactic or deductive terms. – Dennis Apr 2 '17 at 3:49
• @user4894 - I agree with Dennis - I think the problem touched upon here is quite different from a platonic/formalist problem. And 5 + 7 = 12 is from the beginning of the Critique of Pure Reason by Kant, when he talks about analytic v synthetic knowledge. – Frank Apr 2 '17 at 4:38

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.

• Yes, this is taken from Kant, who uses exactly that example to show that mathematics has to be synthetic a priori, which I do not subscribe to. I like the analytic/synthetic distinction, but I believe that mathematics is on the analytical side. So I was trying to make sense of Kant's argument to make mathematics synthetic a priori. It seems to me he missed important things about mathematics. – Frank Apr 2 '17 at 4:20
• @Frank from the little I know it has to do with knowledge of math deriving from our intuition of space and time and thus being more akin to observation -- but "observation" delivered by pure intuition as opposed to the empirical sciences (hence it being a priori). I'll update the answer to include links relevant to that issue. – Dennis Apr 2 '17 at 4:25
• Thanks a lot. Hmmm. It sounds like I need to dig into this one. My view is that mathematics is a game of logical deduction from arbitrary axioms, where we have left any kind of intuition by the wayside, as well as any kind of connection to "the real world" which is the domain of empirical (synthetic for me) sciences. I also think that derivations from axioms using logic do not contain more that what is already in the axioms. – Frank Apr 2 '17 at 4:30
• I think that, historically, Kant's view of mathematics could not be sophisticated enough: this would require the emergence of non-euclidean geometries, and other modern developments, that make it more obvious that you can do a lot of (modern) mathematics without a priori intuitions of space and time. Topology or algebraic geometry extend light-years beyond any intuitive idea we might have of space, by now. – Frank Apr 2 '17 at 4:36
• @Frank yea, Kant is VERY difficult due to the systematicity of his thought and its relative unclarity without substantial study. I've not made those efforts, myself, but I find that every time I think I've made sense of something he says an expert comes along and shows how I've failed to take something crucial into account. Good luck! – Dennis Apr 2 '17 at 4:37