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:

Russell IMP Commutative Diagram.png

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:

Russell PM Commutative Diagram.png

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:

Whitehead IM Commutative Diagram.png

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.):

Leinster BCT Commutative Diagram.png

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.

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!

Digging In In A Library Basement (Pt. 1)

In summer 2017, I was awarded a T. Anne Cleary Dissertation Research Fellowship to travel to the Bertrand Russell Archives at McMaster University. My dissertation focuses on Russell’s philosophy of logical atomism. So visiting the Archive was a real treat, and truly helpful in getting a more complete picture of Russell’s logical atomism. I am grateful to the University of Iowa Graduate College for their generous support of my first archival adventure! In this series of three posts, I will give an overview of my trip, talk about what it was like to work in the Russell Archives, and then discuss the historical data I found that aided my research.


First, I had a delightful time on my first car trip to Canada! I unfortunately did discover much too late that, when driving across a border with enough suitcases full of books crammed into a 2007 Honda Accord to start a new life, you should bring documentation that shows why you are traveling there: Canadian border security was suspicious of me, a young summer traveler bringing what looked like all their Earthly possessions. It took a good bit of midnight dialogue to persuade them to let me pass. That was lesson one: bring documentation evidencing that someone in the country you are visiting expects you to visit.

I followed most of the other lessons pretty well. I contacted the Archival staff at the Mills Library, where the Bertrand Russell Archives are housed, and I did this well in advance. (In an exciting development, the Russell Archives will have their own dedicated Russell House in spring 2018!) Similarly, I contacted the Russell Archivist, Kenneth Blackwell, who met Russell and was responsible for their documentation and growth since their arrival at McMaster University. (This interesting chapter in the history of the history of analytic philosophy is detailed here.) I also got in touch with some well-known historians of analytic philosophy in McMaster’s philosophy department: Nick Griffin and Sandra Lapointe. My interactions with all these folks were superbly helpful. The Archival staff in the William Ready Division was also terrific – more on that next time!

I also followed a tip from some academics in history that helpfully blogged about their archival research experiences. That was lesson two: bring a nice camera and a hands-free stand for taking photographs. I spent $200 on the Nikon Coolpix L840. (Of course an academic would buy their first camera for doing archival work!) I also dropped $40 on a 32GB memory card, rechargeable batteries, a battery charger, a case, and – most importantly of all – a tripod for the camera.

Why did I do all this? That was lesson threemake the most of your time in the archive. The Russell Archives were open Monday through Friday from 9 to 5. I only had a few weeks there, and I wanted to spend as much time as I practically could documenting materials that I could not access from home. So I spent almost all the time I could in the Archive taking as many photographs as I could. I would only organize photographs on my computer when I got home for the evening. (And lesson four: organizing photos is a must! Otherwise you might be unable to know what on Earth you are looking at when you return home. For my part, I followed the documentation system of the Russell Archives and organized photos by box number, item number, and document title.)

So I had two sets of four rechargeable batteries: as my camera needed four batteries at a time, that let me charge one set while I was using the other, so I could photograph without interruption almost the entire time the Russell Archives were open. I also chose my camera because it had WiFi networking capacities. That let me download a free Android application that I used to take photos by pressing a button on my cell phone while seated. If you are taking photos by holding your arms above your head while seated, or by standing, that gets uncomfortable fast. It also distracts your hands from arranging primary documents by making them fiddle with or hold a camera. My archiving arrangement let me take photographs while comfortably seated and freed my hands to focus on laying the documents before the camera. (See the picture.) The arrangement was highly successful: it resulted in around 7,400 photos over three weeks, which amounts to about 500 photos daily over fifteen working days.

That is enough for now. In Part 2, I will talk more about what it was like to work in there and about the wonderful folks I met.

Featured Content

Last summer, I had the distinct pleasure of directing a week-long conference with my advisor, Gregory Landini. This was “The Philosophy of Logical Atomism: A Centenary Celebration” and was held on 12-16 June, 2017. The conference celebrated the upcoming centenary of Bertrand Russell’s 1918 lectures on logical atomism.

The University of Iowa’s Obermann Center for Advanced Studies funded and supported the seminar. The seminar was co-sponsored by the Bertrand Russell Society (Iowa Chapter). The participants are listed below. It was a wonderful time! And stay tuned for a resulting publication in Palgrave Macmillan’s History of Analytic Philosophy series!

Eric C. Banks (Wright State University)
Landon D. C. Elkind (University of Iowa, Director)
David Fisher (Indiana University)
Richard Fumerton (University of Iowa)
Sebastien Gandon (Blaise Pascal University)
Pieranna Garavaso (University of Minnesota – Morris)
Andrew Irvine (University of British Columbia)
Kevin C. Klement (University of Massachusetts – Amherst)
Gülberk Koç Maclean (Mount Royal University)
Anssi Korhonen (University of Helsinki)
Gregory Landini (University of Iowa, Director)
James Levine (Trinity College Dublin)
Bernard Linsky (University of Alberta)
David Charles McCarty (Indiana University)
Francesco Orilia (University of Macerata)
Katarina Perovic (University of Iowa)
Peter M. Simons (Trinity College Dublin)
David G. Stern (University of Iowa)
Russell Wahl (Idaho State University)