Kant seems to have held that even, "Twice 2 is 4," is synthetic, but since the unfolding of the hyperoperator sequence via, "n ↑m n = n ↑m+1 2," seems to make, "2 + 2 = 2 × 2 = 22 = ..." into an analytic matter instead,X we can start with some of his associated comments to get at whether philosophers have directly, or at least indirectly, passed judgment on the analytic or synthetic character of algebraic laws.
For example, the propositions: "If equals be added to equals, the wholes are equal"; "If equals be taken from equals, the remainders are equal"; are analytical, because I am immediately conscious of the identity of the production of the one quantity with the production of the other; whereas axioms must be a priori synthetical propositions. On the other hand, the self-evident propositions as to the relation of numbers, are certainly synthetical but not universal, like those of geometry, and for this reason cannot be called axioms, but numerical formulae. That 7 + 5 = 12 is not an analytical proposition. For neither in the representation of seven, nor of five, nor of the composition of the two numbers, do I cogitate the number twelve. (Whether I cogitate the number in the addition of both, is not at present the question; for in the case of an analytical proposition, the only point is whether I really cogitate the predicate in the representation of the subject.)
Much later in the same document he writes:
The method of algebra, in equations, from which the correct answer is deduced by reduction, is a kind of construction—not geometrical, but by symbols—in which all conceptions, especially those of the relations of quantities, are represented in intuition by signs; and thus the conclusions in that science are secured from errors by the fact that every proof is submitted to ocular evidence.
He also says somewhere that "our numeration (and this is more observable in large numbers [emphasis added]) is a synthesis according to conceptions, because it takes place according to a common basis of unity (for example, the decade)." Taking these remarks together, one might think that Kant would have held that A + B = B + A might be analytic in at least some cases but that A × B = B × A might not.
I'm not sure about that, however. But consider the commutativity of addition relative to subtraction. 1 - 2 does not equal 2 - 1, but 1 + -2 does equal -2 + 1. Yet then when we attend to division, 0/1 does not equal 1/0, and then 0 × 1/1 does not equal 1 × 0/0. Perhaps for such reasons, commutative algebra proceeds from the assumption that it is possible to construct noncommutative multiplications, so that commutativity depends as much on the items being multiplied as on the nature of multiplication. Note that this will apply to addition also, however: addition of transfinite ordinals is not always commutative, since 1 + ω does not equal ω + 1 unless we go to surreal ordinals (at least, the notation scheme for surreals allows for a waiver of the distinction between 2ω and ω × 2 in turn, taking both to potentially mean the same thing as ω + ω).
When doing arithmetic, and thinking through algebraic laws for this, are we proceeding from stipulative definitions by-the-by, then? In fact, sometimes we simply define base cases in specified ways: 0! = 1 even though almost nothing else about the factorial operation would make us think that this was a valid expression, seeing as n! = n × (n - 1) × ... × 1.XX Or sometimes 00 is left undefined, sometimes equated to 1. Then consider Saul Kripke's skeptical argument about meaning:
Suppose that I’ve never dealt with numbers larger than 57. (Given our finite nature and the infinitude of the natural number series, there will always in fact be such a number.) I’m asked to perform the computation ‘68 + 57’, and I arrive at the answer ‘125’, which I take to be right. However, a “bizarre skeptic” (Kripke 1982: 8) questions my certainty. She suggests that in the past I used ‘plus’ and ‘+’ to mean a different function, which she calls “quaddition”. Quaddition yields the same result as addition if the numbers are lower than 57, and 5 otherwise, so the correct result of the aforementioned computation is ‘5’, not ‘125’. I should answer ‘5’ if I intend to use ‘plus’ in the same way in which I have been using it in the past, or so the sceptic suggests.
Is it enough to intend to mean the same thing by A + B as B + A for the equivalence to hold, even for natural numbers, then? Or, then, one might adapt Kripke's speculation to the question of adding elements of ℕ vs. multiplying them. Still, though the argument might bear on that question, it doesn't seem to be a clear example of someone debating whether the commutativity of ℕ's elements modulo addition is analytic but synthetic for multiplication.
But so another context in which the issue is at least implicitly addressed is proof-theoretic ordinal analysis. If a theory T has the proof-theoretic ordinal ωωω... = ϵ0, this means that T has exponentiation resolved up to a certain limit. Expressions of exponentiation (and, by backwards implication, multiplication and addition) would be taken as analytic in the sense of deductively provable from the T-axioms given. The remainder of the analytic apriority in play would depend on whether the axioms are considered analytic, at least for the kind of number, e.g. the elements of ℕ, at stake.
Summary: W. Tait's "Kant on 'Number'" in Cambridge University Press's anthology on Kant's philosophy of mathematics attributes to Kant the belief that the commutativity of multiplication is analytic, but this is given as an interpretative assumption:
Some of the intermediate steps in these calculations (e.g., “equals
added to or subtracted from equals are equal”) he explicitly regarded as
analytic and it is reasonable to assume that he (as opposed to Schultz)
regarded others, such as commutativity and associativity of addition and
multiplication, likewise as analytic.
At any rate, it appears that J. Schultz did, then, ask about whether additive/multiplicative commutation is analytic or synthetic. With respect to Frege and then Gentzen (the progenitor of ordinal analysis), and debates over the reducibility of multiplication to iterated addition, we have some implicit reflection on the question, tending towards commutativity being analytic given the appropriate elements (e.g. elements of ℕ). If analytic judgment is read through intended meanings, Kripke's hypothetical skepticism touches upon the issue to some extent, and we might look into his broader consideration of analytic aposteriority in this vein.
XThough see about the debate over whether multiplication is really "just" repeated addition.
XXAlthough if instead of 1 at the end, we have n - (n - 1), then for 0! we have 0 - (0 - 1) = 0 - -1 = 0 + 1 = 1 (I hadn't noticed this beforehand, so I've edited my post accordingly).