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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?

Frank
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  • Depends on whether you're a Platonist or a formalist. – user4894 Apr 02 '17 at 01:07
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    @user4894 - can you elaborate? – Frank Apr 02 '17 at 01:26
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    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 02 '17 at 02:54
  • @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 02 '17 at 03:49
  • @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 02 '17 at 04:38
  • @Frank I'll have to yield to the Kantians, but I've always found the analytic/synthetic distinction murky. – user4894 Apr 02 '17 at 04:52
  • @user4894 - it makes a lot of sense to me as a demarcation criterion between maths and the natural sciences. – Frank Apr 02 '17 at 14:44
  • this is precisely Frege's problem. see http://www.loyno.edu/~folse/Frege.html –  Apr 02 '17 at 19:39
  • Possible duplicate of [Was Locke right that analytic knowledge is vacuous?](http://philosophy.stackexchange.com/questions/24373/was-locke-right-that-analytic-knowledge-is-vacuous) – Conifold Apr 02 '17 at 20:21
  • @Conifold - no this is not a duplicate. I do not make the point that analytic knowledge is vacuous, only that mathematics is not synthetic a priori, and instead, completely analytic. My question would be perfectly answered by an explanation of why Kant believed mathematics was synthetic a priori. – Frank Apr 02 '17 at 21:34
  • You are simply using the words differently than Kant, your question will be answered once you spell out what "analytic" means *to you* (I am guessing something like "derivable from axioms by logic alone"). To him "analytic" was essentially synonymous to vacuous because his "logic" was Aristotle's syllogistic, and with no "axioms" allowed. Frege, Russell, etc., changed what "logic" means today, but that was a century after Kant. Aside from Kant though, the legitimacy of [epistemic closure](https://plato.stanford.edu/entries/closure-epistemic), which your viewpoint assumes, is highly dubious. – Conifold Apr 02 '17 at 21:49
  • @Conifold - I am using the word _analytic_ at least very closely to the meaning in Kant: _the predicate B belongs to subject A, it is contained in it_. I think that's different from _vacuous_, although it does feel somewhat _tautological_. I have no use for epistemic closure in my question, which is: how could Kant make maths _synthetic a priori_. I do not even talk about _knowlege_ at all. The status of mathematics with respect to knowledge is yet another question. – Frank Apr 02 '17 at 21:58
  • Sorry, but you are not. To him "contained" means contained conceptually, and you can not help yourself to some external axioms that relate concepts to other concepts, e.g. the axioms of arithmetic. But even if you could try deriving 5+7=12 using *only* syllogisms. When you say "was contained anyway in the original axioms" what you mean (to make it true) is was *entailed* by the original axioms, and under the full force of modern logic, not syllogistic. So your requirement on "new" (knowledge) is that it is not entailed by the "original", which is exactly epistemic closure. – Conifold Apr 02 '17 at 22:09
  • @Conifold - thanks for the correction. Let's drop any axioms from the question to simplify it. I don't think they are needed here. The question remains: is there anything "new" here. And what does synthetic a priori mean anyway? I am currently understanding _synthetic_ as "adding something new" and _analytic_ as "not adding something new, but better understanding something" (the predicate B belongs to A, _das Prädikat B gehört zum Subjekt A als etwas, was in diesem Begriffe A (versteckterweise) enthalten ist_). – Frank Apr 02 '17 at 22:39
  • "Adding something new" is too vague to mean anything. You can define "something new" as beyond epistemic closure under modern logic extended or not by special axioms, you can select any list of such axioms and/or weaken/strengthen the logic you allow at your leisure. On the austere end of it, with only syllogistic and no axioms, you'll find Kant's version of "something new" as "synthetic". The substantive question is how much is needed for the kind of elementary arithmetic you focus on (primitive recursive arithmetic will do), without specifying that the "question" is about choice of words. – Conifold Apr 02 '17 at 23:09
  • @Conifold - agreed on the vagueness - I will introduce a clarification in the question. I would like the question to stay focused on understanding _synthetic a priori_ though, more than being about mathematics - although I would love to ask questions about the epistemological status of mathematics too. – Frank Apr 02 '17 at 23:19
  • Since you already accepted an answer it might be a good idea to post a new question that focuses more precisely on what you are looking for rather than edit this one (in particular, elaborate on your informal notions of "synthetic" and "analytic"). It will go to the top of the queue and attract more attention. You can link to this question to provide context if necessary. – Conifold Apr 02 '17 at 23:31
  • @Conifold - let me see what I can do - but on second thought, I think the answer below is still appropriate. Sorry for lack of precision in the original question - at least we got many interesting pointers from my OP. I'll surely post more questions later. Let me ask one about the status of maths as "knowledge". – Frank Apr 02 '17 at 23:40
  • If you're dealing with octal it's just plain wrong! – Hot Licks Dec 22 '20 at 02:52

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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.

Dennis
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  • 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 02 '17 at 04:20
  • @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 02 '17 at 04:25
  • 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 02 '17 at 04:30
  • 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 02 '17 at 04:36
  • @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 02 '17 at 04:37
  • @Frank yea, the relevance of non-Euclidean geometry to his thought is a substantial area of debate. I've been told that it's not as devastating as might be thought that he was ignorant of those matters, but that's where my interest and patience for Kant wears out. – Dennis Apr 02 '17 at 04:39
  • @Frank: Apart from Gauss probably being influenced by Kant (too lazy to search for the question it is mentioned in), the real question is what is to be understood under synthetic. If 7+5=12, or, for that matter, 12=5+7 could be understood solely out of the understanding (or, more specifically, *experience*) of 12, this would totally be an analytic truth. But is this really the case? Do we, by experiencing "twelveness", really analitically have to subsume 5 and 7? Or is it more some kind of *additional operation of the mind* that brings 5, 7 and 12 together, for that matter? – Philip Klöcking Apr 02 '17 at 04:54
  • @PhilipKlöcking - there is no "experience" of 12. There is an intellectual construction of arithmetic from a few arbitrary axioms and the use of logic. The words "experience" and "intuition" in my opinion have nothing to do here, and even more, we want to eradicate them to construct arithmetic. The only "operation of the mind" involved would be logical inference. To me, 5, 7, and 12 are just convenient names we give to |||||, ||||||| and ||||||||||||, each mark obtained by adding another "|" to the previous one, which is allowed by the axioms. "+ 1" only means "adding one more |". – Frank Apr 02 '17 at 14:50
  • @Frank: Dewey would (and I think rightfully so) criticise that this is intellectualistic reconstruction out of concepts that are abstracted out of experience much later - genealogically speaking - taking these abstract concepts as primary. The fact that the axioms have - historically speaking - been formulated very late supports this view. – Philip Klöcking Apr 02 '17 at 19:12
  • @PhilipKlöcking - this is not a very impressive critic in general. It is not true that all of mathematics was abstracted out of experience after the fact. Today, it is possible to posit axioms a priori, ex nihilo, and apply logic to infer by deduction. That goes back to the discussion about non-euclidean geometries, which Kant could historically not know about. In any case, mathematics _should_ be free to posit arbitrary axioms, this is a liberating move, it would be quite uninteresting if mathematics had to follow experience. – Frank Apr 02 '17 at 19:32
  • Isn't the statement "5+7=12" new to a student just learning arithmetic? If this is so, at what point does "5+7=12" cease to be new to the student? If we ask whether "5+7=12" SAYS anything new, don't we need to consider the word SAY. Does it mean SIGNIFY or COMMUNICATE? If it means COMMUNICATE, it clearly can say something new. If it means signify, does the idea of NEW actually mean anything? Who is there for it to be new to? – Philip Roe Apr 15 '17 at 20:31