[kragen]

Hmm, I'm not sure whether operator-like notation has any special advantage for commutativity and distributivity other than brevity. a + b and add(a, b) are equally easy to rewrite as b + a and add(b, a).

Maybe there is an advantage for associativity, in that rewriting add(a, add(b, c)) as add(add(a, b), c) is harder than rewriting a + b + c as a + b + c. Most of the time you would have just written add(a, b, c) in the first place. That doesn't handle a + b - c (add(a, sub(b, c)) vs. sub(add(a, b), c)) but the operator syntax stops helping in that case when your expression is a - b + c instead, which is not a - (b + c) but a - (b - c).

Presumably the notorious non-associativity of floating-point addition is what you're referring to with respect to fadd() and fmul()?

I guess floating-point multiplication isn't quite commutative either, but the simplest example I could come up with was 0.0 * 603367941593515.0 * 2.9794309755910265e+293, which can be either 0 or NaN depending on how you associate it. There are also examples where you lose bits of precision to gradual underflow, like 8.329957634267304e-06 * 2.2853928075274668e-304 * 6.1924494876619e+16. But I feel like these edge cases matter fairly rarely?

On my third try I got 3.0 * 61.0 * 147659004176083.0, which isn't an edge case at all, and rounds differently depending on the order you do the multiplications in. But it's an error of about one part in 10⁻¹⁶, and I'd think that algorithms that would be broken by such a small amount of rounding error are mostly broken in floating point anyway?

I am pretty sure that both operators are commutative.

[zozbot234]

We do often find add(a, b, c), just written as Σ(a, b, c). Similar for mul and Π. The binary sub operator can be simply rewritten in terms of add and unary minus; the fact that we write (a - b) instead of (a + [-b]) or perhaps Σ(a, [-b]) is ultimately a matter of notational convenience, but comes at some cost in mathematical elegance. Considering operators that are commutative yet not associative is not very useful; ultimately we want more from our expression rewriting than just flipping left and right subexpressions within an expression tree while keeping the overall complexity unchanged.

[kragen]

Usually you'd have to write that as \sum_{v \in \{a, b, c\}} v; one of the ways I think conventional math notation could in fact be improved would be by separating the aggregate function of summation from the generation of the items, allowing you to write \sum \{a, b, c\}, at the minor cost of having to write \sum_{i = 1}^N i^2 as something like \sum |_{i=1}^N i^2.

It's not conventional to write commutative-but-not-associative functions as infix operators, but I don't think that's due to some principled reason, but just because they're not very common; non-associative operators such as subtraction and function application are almost universally written with infix operators, even the empty-string operator in the case of function application. The most common one is probably the Sheffer stroke for NAND (although Sheffer himself used it to mean NOR in his 01913 paper: https://www.ams.org/journals/tran/1913-014-04/S0002-9947-191...).

You can go a bit further in the direction of logical manipulability, as George Spencer Brown did with "Laws of Form" (LoF): his logical connective, the "cross", is an N-ary negation function whose arguments are written under the operation symbol without separators between them, and he denotes one of the elementary boolean values as the empty string (let's call it false, making the cross NOR). ASCII isn't good at reproducing his "cross" notation, but if we use brackets instead, we can represent his two axioms as:

    [][] = []  (not false or not false is not false)
    [[]] =     (not not false is false)

In this way Spencer Brown harnesses the free monoid on his symbols: the empty string is the identity element of the free monoid, so appending it to the arguments of a cross doesn't change them and thus can't change the cross's value. Homomorphically, false is the identity element of disjunction, which is a bounded semilattice, and thus a monoid.

This allows not only the associative axiom but also the identity axiom to be simple string identity, which seems like a real notational advantage. (Too bad there isn't any equivalent for the commutative axiom.) It allows Spencer Brown to derive all of Boolean logic from those two simple axioms.

However, so far, I haven't found that the LoF notation is an actual improvement over conventional algebraic notation. Things like normalization to disjunctive normal form seem much more confusing:

    a(b + c)  → ab + ac          (conventional notation, rewrite rule towards DNF)
    [[a][bc]] → [[a][b]][[a][c]] (LoF notation)

It's a little less noisy in Spencer Brown's original two-dimensional representation (note that the vertical breaks between the U+2502 BOX DRAWINGS LIGHT VERTICAL characters are not supposed to be there; possibly if you paste this into a text editor or terminal it will look better)

    ┌─────    ┌────┌────
    │┌─┌──  → │┌─┌─│┌─┌─
    ││a│bc    ││a│b││a│c

but not, to my eye, any less confusing.

[kragen]

A thing I didn't appreciate the first time I read Spencer-Brown's book is that he actually cites Sheffer's 01913 paper, and proves Sheffer's postulates within his system in an appendix. This situates him significantly closer to the mathematical mainstream than I had thought previously, however flawed his proof of the four-color theorem may have been.

Also, the axioms I cited above are written in his notation on his gravestone: https://en.wikipedia.org/wiki/G._Spencer-Brown#/media/File:G... but I have evidently reversed left and right in my rendering of the DNF rewrite rule above. It should be:

    ─────┐   ────┐────┐
    ─┐──┐│ → ─┐─┐│─┐─┐│
    a│bc││   a│b││a│c││

His first statement of the first axiom in the book is a little more general than the version I reproduced earlier and which is inscribed on his gravestone; rather than his "form of condensation"

    [][] = []

his "law of calling" is general idempotence, i.e.,

    AA = A

although the two statements are equipotent within the system he constructs. Similarly, before stating his "form of cancellation"

    [[]] =

he phrases it as the "law of crossing", which I interpret as

    [[A]] = A