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,


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.

Being Corrected: The Best Part of Philosophy Conferences

Reflecting on the Central APA in Denver, I have to admit that one of the best parts of philosophy conferences is being corrected. This came up as follows: I indirectly pointed out to a speaker that their interpretation of an author’s notion of diagram should also be extended to natural deduction proofs. They thanked me after the talk for pointing out “a blind spot in their paper” that they wanted to think more about.

That got me thinking about the many times that I had been corrected at conferences. I confess that constructive criticism that improves my argument, or my presentation of it, is one of the most pleasurable experiences to come out of conferences (or the peer–review process). It really makes attending conferences entirely worthwhile.

The pleasant feeling of being corrected is perhaps best encapsulated in Socrates’ words to Theaetetus at the end of Plato’s Theaetetus (link). Their search for a definition of knowledge has apparently come up empty-handed. But Socrates makes clear that there has been some definite philosophical improvement by the elimination of ignorance:

SOCRATES: …And so, Theaetetus, knowledge is neither sensation nor true opinion, nor yet definition and explanation accompanying and added to true opinion?

THEAETETUS: I suppose not.

SOCRATES: And are you still in labour and travail, my dear friend, or have you brought all that you have to say about knowledge to the birth?

THEAETETUS: I am sure, Socrates, that you have elicited from me a good deal more than ever was in me.

SOCRATES: And does not my art show that you have brought forth wind, and that the offspring of your brain are not worth bringing up?

THEAETETUS: Very true.

SOCRATES: But if, Theaetetus, you should ever conceive afresh, you will be all the better for the present investigation, and if not, you will be soberer and humbler and gentler to other men, and will be too modest to fancy that you know what you do not know.

Much the same could be said of the experience of being corrected at conferences. We are all like Socratic midwives to one another, helping each other to “birth” philosophical ideas.

This makes Socrates’ goodbye to Theaetetus all the more applicable to my colleagues in academic philosophy:

SOCRATES: To-morrow morning, Theodorus, I shall hope to see you again at this place.

And indeed, I will hope to see you, my fellow philosopher, again at the next conference.

Philosophy Paper Guide

Over break, I did some work: I wrote a one-page guide for introductory philosophy students writing term papers. Feedback and criticism is welcome!

The goal in a philosophy paper is to give a good argument for your thesis. It is a stellar idea in making an argument outline to put your argument in one of the argument forms like we use in class. This format does not need to appear in your actual paper, but it can.

Giving a good argument for a philosophical thesis usually involves related tasks, like:

  • formulating and choosing a thesis that seems plausible and defensible in a term paper
  • clarifying what the issue is and bracketing other tangential issues
  • researching data and evidence to see if your premises and thesis are plausible
  • making your premises and methods clear and explicit
  • defending the premises that are used in your argument for your thesis
  • fairly presenting and critically discussing alternative theses
  • considering and critically replying to objections and implications of your thesis

In doing the above, you may find, as I often have, that a given thesis is not plausible and should be rejected. This—being wrong—is part of philosophizing, just as it is part of any scientic inquiry. As such, your paper may end with a devastating objection to your thesis or premises: it is perfectly acceptable to end your paper by rejecting your thesis with which you began. What I want to see, ultimately, is you critically arguing well.

Here are some further suggestions as to what a good philosophy paper does:

  1. Thesis Thesis is clear, concise, and at the beginning, perhaps after a brief introduction. The thesis is not trivial, like puppies are cute is. It is also defensible in a term paper.
  2. Issue The main issue is explicitly explained, perhaps by connecting it to the course content. No filler or bullshit is used. Why we should care about the issue is also explained.
  3. Premises The premises are clearly and explicitly stated, along with some basis for accepting them. The premises are first-blush plausible or well-supported by research.
  4. Clarification The thesis and premises, and the main issue, are claried with examples and explanations of key philosophical terms. Your non-philosopher friend should understand your thesis, premises, and the issue. Do not write as if I am the audience.
  5. Argument The thesis is cogently argued for using the premises. The argument for the thesis is persuasive to anyone who accepts your premises and methods.
  6. Alternatives Alternatives to your thesis or view on the main issue are fairly presented. Present them even better than their advocates do. Then critically discuss them. You might object to consequences of them, or agree with them to the extent that you do.
  7. Objections You fairly present and critically discuss concerns with your thesis and premises.
  8. Research Your sources and data are documented in a bibliography, and are acceptable in academic, scientic contexts wherein getting the facts right is the first, foremost goal.

Book Published!

I am happy to report that I have published my first book, The Philosophy of Logical Atomism: A Centenary Reappraisal! I co-edited this volume with Gregory Landini (University of Iowa), and contributed a chapter. You can see previews, buy chapters, or buy the whole thing at this link.

The volume collects fourteen original essays on Russell’s 1918 logical atomism lectures. The book is published with Palgrave Macmillan in their History of Analytic Philosophy series edited by Michael Beaney (Humboldt-Universität zu Berlin / King’s College–London). The other books in this series are also quite good.

Palgrave collection 2018 cover.png

Peer-Review in Intro Philosophy Papers!

I am really happy to report that I will be working with the University of Iowa’s Writing Center to set up a peer-review component of my introduction to philosophy course this spring!

The Writing Center has a Writing Fellows program in which honors students are trained to give feedback on term papers across the curriculum. They then give comments on rough drafts of two longer paper assignments. They do not grade assignments, but only provide feedback and tutoring on the papers.

The program is such that the rough drafts of longer papers are turned in at least two weeks before the final due date. This gives Writing Fellows time to read papers, provide detailed comments, and then meet with students to go over their papers. Then students have a chance to revise their papers before the final drafts in a focused manner with helpful guidance. (To that end, I actually gave students just three weeks from the submission of their rough draft to the submission of the final version.)

Though I will have to judge by the results, I expect that this program will benefits all parties involved—my students, the Writing Fellows, and me as a teacher:

  • This will help my students improve their writing in light of comments and give them the task of writing like I suggest they do—as if they were explaining philosophy to a peer without any philosophy background—by supplying them with such a peer.
  • This program also focuses my students’ attention on the revision process. Good philosophy often takes multiple drafts. This program makes revising earlier writing part of their grade (rather than submitting a paper only to never return to it, which perhaps makes the experience of reading feedback from me, say, less fruitful).
  • This program helps Writing Fellows by giving them more experience giving feedback on peers’ writing—and exposes them to philosophical content, to boot!
  • This program helps me in delivering guidance for paper feedback to others, and in collaborating with undergraduates.
  • This also incentivizes me to design a more detailed paper rubric—which will no doubt assist Writing Fellows in giving helpful feedback, assist my students in understanding what I look for in longer papers, and assist me in grading with more focus.

So this program seems like a win for everyone. I thank my colleagues, Professors Carol Severino (Director) and Megan Knight (Assistant Director), for providing me and other faculty at the University of Iowa, plus our students, with this golden opportunity.