# 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.

# “Principia” Notation in LaTeX

I recently had occasion to use the Peirce arrow, NOR (↓), in LaTeX! But I wanted to use Peirce’s notation from 1902. This takes the form of the symbol for strict implication, ⥽, turned 270 degrees so that the arrow’s head faces down, like so: ⥿. I was helped in typesetting this by Richard Zach’s helpful post. I have worked on Principia quite a bit, and I have needed to make ad hoc commands to rotate letters and other symbols to match the logical symbols used there. So I thought to share my work in case it saves others some time! I will update this as I devise further ad hoc commands.

All this requires is the package graphicx. With it, you can then use the command \text{\rotatebox[origin=c]{[DEGREES]}{[TEXT]}}. The reason for the \text{…} on the outside is to avoid conflicts in math mode and similar environments.

With that ado, here is LaTeX code I use for:

• \Runi for the universal set, V (an upside-down Lambda):
\newcommand{\Runi}{\text{\rotatebox[origin=c]{180}{$\Lambda$}}}
• \Dom for the domain, D’, of a relation:
\newcommand{\Dom}{$\text{D}$}
• \CDom for the converse domain of a relation (a ‘D’ rotated 180 degrees):
\newcommand{\CDom}{\text{\rotatebox[origin=c]{180}{$\text{D}$​}}}
• \Parrow for the Peirce arrow NOR in Peirce’s 1902 notation (⥿):
\newcommand{\Parrow}{\text{\rotatebox[origin=c]{270}{$\strictif$}}}

No doubt there will be more to come!

# Hi-Story Time x 2!

It’s story time! (Rather, hi-story time!) Here is a retelling of two truly terrific tales from the history of logic. I hope you find them fun and engaging!

• Story Time 1/2: Russell’s letter to Frege announcing the contradiction

On 16 June 1902, Bertrand Russell (1872-1970) wrote to Gottlob Frege (1848-1925). Russell had just discovered his famous paradox, and he wanted to inform Frege that an analogue was derivable in the logic of Grundgesetze (Basic Laws of Arithmetic, link). Meanwhile, Frege had just sent to press the second volume of Grundgesetze, his life’s work. Then Frege got the news of Russell’s contradiction.

Just to pump your sympathy for Frege: imagine you had just finished and sent off your book or dissertation, you were days from your publication or your defense, and then — somebody points out you contradicted yourself on some point that was fundamental to your argument. I reckon I would be pretty upset, and I reckon you would be, too.

How did Frege react? Like this:

“Dear colleague: Many thanks for your interesting letter of 16 June…Your discovery of the contradiction caused me the greatest surprise, and, I would almost say, consternation, since it has shaken the foundation on which I intended to build arithmetic…In any case your discovery is very remarkable and will perhaps result in a great advance in logic, unwelcome as it may seem at first glance.”

That is a powerful response to what must have been a monumental disappointment. And Russell thought as much, too. In a 23 November 1962 letter reflecting on the exchange about 50 years later, Russell commends Frege’s response to the contradiction:

“As I think about acts of integrity and grace, I realise that there is nothing in my knowledge to compare with Frege’s dedication to truth. His entire life’s work was on the verge of completion, much of his work had been ignored to the benefit of men infinitely less capable, his second volume was about to be published, and upon finding that his fundamental assumption was in error, he responded with intellectual pleasure clearly submerging any feelings of personal disappointment. It was almost superhuman and a telling indication of that of which men are capable if their dedication is to creative work and knowledge instead of cruder efforts to dominate and be known.”

Frege’s prediction was right, of course: the contradiction led Russell to collaborate with Whitehead to produce Principia Mathematica. And Principia was a direct inspiration of enormously fruitful work, including Gödel’s theorems, Church’s work on type theory, and computer scientists’ research on artificial intelligence and automated theorem-proving in the 1950s and 1960s (including Newell, Simon, and Shaw’s Logic Theorist). Speaking of Principia:

• Story Time 2/2: Whitehead and Russell had to self-publish Principia Mathematica

Imagine you spent ten years — say, your entire 30s — reworking and fixing the very foundations of mathematics and logic. You invent type theory, solve the contradictions, and put mathematics back on a firm foundations. You also prove a ton of interesting theorems about cardinals and ordinals just for kicks. An academic publisher would beg to print that work, right?

Wrong! It turns out Cambridge University Press estimated they would lose £600 in publishing Principia. (That amounts to almost $100,000 in today’s dollars.) So Cambridge University Press told Whitehead and Russell they would assume half that risk. That meant Whitehead and Russell, after working for ten years to fix math for everyone else, had to scrounge about for £300 to tell everyone what they did to fix it. Happily, they got a grant for £200. But Whitehead and Russell each had to fork over £50 (~$8,000 today) to self-publish their grand treatise on the foundations of mathematics. And Cambridge thought they would lose money: probably a copy now resides in every college and university library in the world.

A little more detail on the calculations from 1910 UK pounds to modern US dollars are given here (link). And you should definitely read Principia (link) — yes, all three volumes (or at least skip the introduction and skim **1-20, pp. 90-199). Enjoy!

# References for foundations of math!

If you are not interested in the foundations of mathematics (and of computer science, I think), then you may disregard this post. For those of you that are interested but are unsure where to start: I hope this will help you get started with the foundations movement at the turn of the twentieth-century!

• From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931 (Jean van Heijenoort, editor)

This source book is definitely worth purchasing for your own library and use. It contains highly influential primary texts in logic by, among others, Cantor, Peano, Frege, Russell, Zermelo (yes, the one of ZF fame), von Neumann, Hilbert, Brouwer, and, of course, Gödel. It is sorely missing Tarski and Church and Marcus, but you can make up for that elsewhere. Frege’s Begriffsschrift alone justify purchasing this, as do Gödel’s papers.

• The Search for Mathematical Roots, 1870-1940 (Ivor Grattan-Guinness, author)

Grattan-Guinness, sadly now deceased, was a highly regarded historian of mathematics (he also wrote a book on the history of calculus). This book offers a good sketch of the history of the foundations of mathematics movement. It is not exhaustive, but it covers a good enough selection to contextualize whatever else you go on to read.

A combination of those two should put you in a position to pursue whatever else you find interesting about the foundations of mathematics movement at the turn of the 20th century. I am, of course, happy to recommend further items. Just get in touch!