# When was the first commutative diagram?

Thanks to @LogicalAnalysis, I saw an interesting post from Tom Leinster at The n-Category Café. The post considers whether Russell’s 1919 Introduction to Mathematical Philosophy contains the first printed occurrence of a commutative diagram. I have two remarks on this suggestion: (1) pretty much the same diagram occurs in Principia some eight years earlier; (2) while it is controversial whether this diagram really should count as a commutative diagram in the modern sense of the phrase “commutative diagram”, I will argue that it is one.

The diagram in question occurs on page 54 in the public domain text hosted on Archive.org. Here is a screenschot: Now for remark (1): this diagram is not the first such to occur in print. Whitehead and Russell’s Principia Mathematica contains an earlier occurrence. It is given on page 321 in the PDF (page 295 in the text) in the public domain text of Volume II hosted on Archive.org. Here is a screenshot: The upshot of this is that it is not Russell, but Whitehead and Russell, who printed this commutative diagram. I do not know of an earlier source for it. No such diagram occurs in Volume 1 of Principia. Readers: do you have an earlier text in which such a diagram occurs?

There is a kind of anticipation of this diagram in Whitehead’s 1911 An Introduction to Mathematics. It is given on page 94 in this public domain text hosted on Archive.org. Here is a screenshot: This diagram is purposefully coordinatized, as the context and reference to Descartes’ Discourse makes clear. But as the context, especially the subsequent discussion on page 96, also makes clear, Whitehead is abstracting from this diagram the commutativity and associativity of addition. Given that this diagram is also published in 1911, like the diagram above, we can just focus on the Principia diagram, which Whitehead of course also had a hand in.

(Similar uses of coordinatized diagrams to represent mathematical properties occur in Whitehead’s 1898 A Treatise on Universal Algebra, but nothing so distinctly used to represent algebraic properties like commutativity and associativity: most of these earlier diagrams occur as force diagrams or as representing geometric properties. I am open to correction on that score from someone who is more knowledgeable about Whitehead.)

Now for remark (2): the phrase “commutative diagram” is often used specifically to maps between categories. Categories are a specific kind of mathematical object, and if a map must be between such objects to count as a commutative diagram, then Whitehead and Russell most definitely did not print the first commutative diagram. However, it certainly seems to be a commutative diagram if we merely want the following feature, namely, that is commutes in the usual mathematical sense of “commutes”:

Generally, a diagram is said to commute if whenever there are two paths from an object X to an object Y, the map from X to Y obtained by composing along one path is equal to the map obtained by composing along the other. (Leinster, 2014, Basic Category Theory, page 11, link)

This is exemplified by the notation in the above screenshot from Principia,

P=S|Q|Š,

which is clearly the analogue of gfjih in the screenshot from Leinster’s text (ibid.): So I am comfortable claiming that Whitehead and Russell did indeed print the first commutative diagram in the usual sense in their 1911 Principia Mathematica: Volume II. I am always open to correction on this from someone who knows of an earlier occurrence.