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Let's go back to the original meanings of addition and multiplication back in ancient Sumer when arithmetic was primarily used as a tool in the trade of sheep and beer. Addition meant something like this: Adamen sold me 30 pots of beer and Biluda sold me 50 pots of beer. So I'll use this new-fangled technology to save me the time of counting all of my pots of beer in my warehouse. Instead, I'll just calculate 30+50=80; I have 80 pots of beer.

Now consider how the law of commutation applies to this application. What is A+B=B+A saying? It's not saying that if I count A pots first and then B pots, I'll get the same as if I count B pots first and then A pots, because I never contemplated counting the pots in any particular order. I have a warehouse full of beer and all I contemplated was counting all of the pots in the warehouse. There's no reason to think that I would count all of the Adamen pots together and all of the Biluda pots together; the pots could be counted in any convenient order, so the notation A+B doesn't imply any order, that's just an artifact of the notation. A+B means exactly the same things as B+A. It's an analytic judgement.

By contrast AB=BA is not at all obvious. It means something like this: Adamen sells beer in lots of 3, and I bought 5 lots. Biluda sells beer in lots of 5 and I bough 3 lots. How does my number of Adamen pots compare to my number of Biluda pots? Well, 5 lots of 3 is the same as 3 lots of 5. Really? That's surprising! When I said 5 lots of 3, I didn't at all mean the same as when I said 3 lots of 5. The knowledge that these two numbers are equal is not analytic, but synthetic.

I'm looking for any authors who have discussed this issue specifically with respect to algebra and algebraic notation.

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  • Have you considered the additive representation of whole number multiplication? 3x5 = 5+5+5 = 1+1+1+1+1+1+1+1+1+1+1+1+1+1+1 = 3+3+3+3+3 = 5x3
    – g s
    Commented Aug 13, 2023 at 0:13
  • @gs, It's not an issue of what can be figured out; it's an issue of what I mean when I say 3+5 vs 3*5. When I say 3+5, no order of evaluation is implied. I write the 3 first only because I have to write one of them first. There is no significance to which one comes first. With 3*5, there is significance to the order. 3*5 means something different from 5*3. When I say "meaning" I'm speaking of the concept represented in my mind, not the mathematical result. Commented Aug 13, 2023 at 0:18
  • That the pots can be counted in any convenient order, and the same result is always obtained, is neither a consequence of meanings nor an artifact of the notation. So no, A+B = B+A is not analytic, unless you fix the order and redefine their usual meanings as counting in the same order, not in some order. That you did not contemplate the order at all simply makes it ill-defined. See Kant's argument that 7+5=12 is synthetic.
    – Conifold
    Commented Aug 13, 2023 at 5:43
  • @Conifold, the fact that the counting process always produces the same answer, regardless of order, is not analytic, but the fact that when I write A+B I mean the same thing as when I write B+A is analytic. It's strictly a matter of what I meant when I wrote it, so even if counting order did make a difference, it wouldn't matter because I didn't intend the order in which I wrote the letters to specify any counting order or to signify anything at all. Commented Aug 13, 2023 at 6:42
  • @DavidGudeman Since arithmetic is in common use it is not a matter of what someone chooses to mean. The standard interpretation relates A+B and B+A to two different ways of counting. That they coincide is the statement of arithmetic, A+B=B+A, and it is not analytic. If you did not mean to specify the order then neither A+B nor B+A is defined, and A+B=B+A is not a statement. If you did specify the same order for A+B and B+A then A+B=B+A is analytic, but not a statement of arithmetic as commonly interpreted.
    – Conifold
    Commented Aug 13, 2023 at 7:05

2 Answers 2

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Kant seems to have held that even, "Twice 2 is 4," is synthetic, but since the unfolding of the hyperoperator sequence via, "nm n = nm+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.

So firstly:

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).

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  • Lovely insight in your note XX. Did you just come up with that or did yo read it somewhere? Commented Aug 13, 2023 at 19:23
  • @DavidGudeman I think I half-misread the Wikipedia bit about the factorial of zero. It doesn't say 0! = 1 is set to be true by definition, exactly, but the first explanation refers to an empty product. I know that n - (n - 1) = 1 in general so I think that must be involved somehow in how the empty product itself is defined relative to the case? Commented Aug 13, 2023 at 19:55
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What I think about when I contemplate A+B or A*B, and what others think about when they contemplate them are more or less irrelevant to the question of what those expressions mean. Maybe I visualise things. Maybe I visualise A*B as a rectangular box with A rows and B columns and I picture myself rotating the box 90 degrees and see that the product does not change. Maybe I have an intuitive grasp of how multiplication works that is non-visual. Maybe I have synaesthesia and I can sense how arithmetic operations work in some weird way.

Such things may be relevant in understanding how I come to learn arithmetic, but they are of little relevance if any as to what arithmetic expressions mean. As a matter of fact, I have completely forgotten how I learned to count and add up. No doubt it involved counting toy blocks or something, but whatever it was, it does not inform my current understanding of arithmetic. The meaning of words, expressions and sentences is public property, not something in a private language of my own.

It may help to remind outselves of the distinction between reality and models of reality, or between the map and the territory. Mathematics and logic are part of the map; they are not the territory. Whether a given expression such as A*B applies to anything in the world is an empirical question. Some things add up using the ordinary sense of A+B, while others don't. If I put an apple in my pocket and then another apple, I have two apples, but only because it is an empirical fact about apples that they do not coalesce or react together to make something else. If I add one drop of water to another drop of water, I do not get two drops of water. Mass is an extensive property, so the mass of the sum is equal to the sum of the masses; but temperature is intensive and not additive. Again, both are empirical facts.

The mathematical properties of expressions such as A+B may be specified formally, and there may be rules that work in an a priori fashion for manipulating such expressions, but they are still just part of the map. We can make up whatever rules we like for our maps. If we want numbers where A*B is not equal to B*A then we can: quarternions are an example. If we want formal languages where "A and B" is different from "B and A" we can do that too. In most modern computer programming languages, conjunction and disjunction are not commutative.

But the map is not the territory. The fact that I can make a priori deductions about some feature of my map from some other features of my map tells me nothing about whether my map correctly describes the territory. That question will always be empirical.

I risk failing to answer your question, because you have couched it using the terms 'analytic' and 'synthetic'. For myself, I have little or no use for these terms. I'm choosing my words carefully here: I'm not saying it is impossible to define these terms or give them a meaning, only that I don't believe they have any useful explanatory value, so I prefer just not to use them. So in attempting a response to your question, I have spoken instead of what is empirical versus a priori, but hopefully it is close enough to be of value.

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