How did the exterior product get its symbol?

As per the title: where did the notation $a\wedge b$ for the exterior product of $a$ and $b$ originate, and/or who popularised it? I'm especially interested in motivation for the choice of this symbol (which also means "and" in logic and "meet" in order theory).

I've moved this question here from MSE, where it got very little attention despite offering a bounty; the original version is pasted below the line because it includes a bit about my reason for asking.

It seems odd that something that looks so much like a join [see below] would get given "the wrong symbol". It's even worse when you dualise it and get something called (quite reasonably) the "meet product", but write that as $A\vee B$.

I'm aware the answer might be "it just turned out that way", but was this choice of symbol motivated by a natural way to think of $A\wedge B$ as a meet in some lattice or other?

If not, which historical figure should I blame for this situation? I understand Grassmann wrote $A\wedge B$ as $[A, B]$, which would let him off the hook.

Intuitively, if I have two vectors $a$ and $b$, $a\wedge b$ is the smallest object that contains both $a$ and $b$. If you order elements of the exterior algebra by inclusion, which seems very natural if you're doing geometry, you get a lattice where the join operation is the exterior product.

This mostly seems to come to the fore in Clifford-algebra-related approaches to geometry. Examples of people mentioning this relationship and complaining about / altering the standard notation:

While digging up these references I also turned up a claim that Peano wrote the exterior product using $\vee$, whereas Cartan used $\wedge$, but it's not clear to me whether Cartan originated (or popularised) this notation or whether he had a particular reason for choosing it [or indeed whether this claim is correct -- it was an offhand remark with no citation].

2 Answers

Cajori gives the early uses of logical symbols in volume 2 of History of Mathematical Notations. Neither Boole nor Schröder used $\wedge$ and $\vee$ in Boolean algebra, but rather $\cdot$ (or blank) and $+$, the idea was to make it as "algebra looking" as possible. Peirce in his 1865 lectures was apparently first to give Boolean algebra its modern form, by introducing the non-strict disjunction denoted $\vee$, and the material conditional. Peano uses $\smile$ for "the smallest class containing all the classes" and $\frown$ for "the greatest class contained in all the classes" (there was no sharp distinction between logical operations and operations on classes at the time) in Formulaire de Logique (1891). In fact, they already appear in his book Calcolo Geometrico Secondo l'Ausdehnungslhere di H. Grassmann (1888), along with the first modern axiomatization of vector spaces and exterior calculus. But they are for classes only, not as wedges or vees, for multivectors Peano writes concatenations occasionally preceded by $gr$. Russell and Whitehead went back to Peirce's use in Principia (1910-13), with $\vee$ for disjunction and $\cdot$ for conjunction. The meet/join notation for lattices came even later, so it was not established when Cartan was developing exterior calculus in 1899-1925.

Although it is common to assume that Cartan introduced wedges one finds none of them in his papers and monograph on the subject, linked in Where did Cartan introduce his notation for basis vectors and covectors? He mostly just concatenates forms without any symbol, and very occasionaly writes brackets. My guess is that the responsible party is his student Ehresmann, who came up with most of modern symbology and lingo in the calculus on manifolds in his 1935-1960 transformation of Cartan's work, see How did the exponential map of Riemannian geometry get its name?

As for the reason, while the use of $\wedge$ and $\vee$ in Boolean algebra is symmetric it is Boole's and Schröder's product that became Peano's $\frown$, reinforced as product by Russell and Whitehead. Since the exterior product is definitely a product and not a sum perhaps $\wedge$ seemed more suitable. Alexander and Kolmogorov used the "correct" $\cup$ for the corresponding cup product in cohomology in 1935. But your kindred spirit is Rota, who boldly reverses the direction even in exterior calculus, see On the Exterior Calculus of Invariant Theory.

• Thank you so much, this is exactly the kind of detail I was hoping for! – helveticat Mar 23 '16 at 7:19

Okay, this also came up in an interesting blog, Galileo unbound, by a physics prof at Purdue. https://galileo-unbound.blog In particular, his bit about Grassmann. https://galileo-unbound.blog/2019/12/02/hermann-grassmanns-nimble-wedge-product/ Thx, too, for your pointer to Ehresmann. So I looked at the Sem. Math. Julia 1936-1937 at numdam.org. According to the obituary for Cartan published in the Bull AMS, written by Chern and Chevalley, it was Anre Weil who got that semester's seminar devoted to Cartan's work, which was still under-appreciated, and Weil kicked it off. http://www.numdam.org/article/SMJ_1936-1937__4__A1_0.pdf Weil mostly uses juxtaposition for the exterior product, but remarks that he will use ^ when advisable to avoid confusion. Then come lectures by Henri Cartan, the son, who uses wedge systematically. Then comes Ehresmann, ditto. So we see it there, and I do not see it earlier. But I noticed something interesting. Boulignand, an associate of Cartan's, wrote a book meant as a prerequisite for general relativity. I can't access the book, but numdam.org has a correction http://www.numdam.org/article/NAM_1924_5_3__245_1.pdf he published. In that note, he uses ^, but only in a context where it can simply mean the vector cross product in three dimensions. And one sees from Cajori that both [vw] and v^w were common notations for the vector cross product. And, of course, Cartan's wedge product is, for three dimensions, the vector cross product when it gets applied to one-forms. So this seems to me to be the evolution of the exterior product from the older cross product, and why it was sometimes juxtaposition, sometimes (in Cartan) [$$\omega \eta$$], and with his son and others in the next generation, ^.

• Thanks for this -- that's good information. Makes me want to get back to this subject again! – helveticat Dec 31 '20 at 11:57