Noam Zeilberger

@gjm@mathstodon.xyz @dougmerritt@mathstodon.xyz but to be clear, I think the question of explaining the similarity here is curious but of really minor significance. I was delighted to receive such a kind response from Knuth, especially for all the extra historical context and interesting pointers.

@gjm@mathstodon.xyz @dougmerritt@mathstodon.xyz from Knuth's response I draw the inference that it is very likely that he *had* actually read Ulam's article in Scientific American, since he recalls reading Martin Gardner's article in the same issue, and it is normal not to remember every single thing you read 60 years ago (I have trouble remembering what I had for breakfast!). My running hypothesis is still that there could have been some subconscious incorporation of elements of Ulam's phrasing after such an exposure, or else that it is just a pure coincidence.

@MartinEscardo@mathstodon.xyz yes I already knew that Knuth had interacted with de Bruijn about the Automath project and mentioned it in my letter to him (which I guess I should have included for more context). I knew this from William Lovas, who got to chat with Knuth once when he was visiting at CMU. William told Knuth about his then thesis work on an extension of the LF logical framework with subtyping, and Knuth mentioned that he had proposed something similar to de Bruijn!

@mjd@mathstodon.xyz Doron is my uncle!

I wrote a letter to Knuth asking about this, and he wrote back! [Edited to add a link to the original letter I sent: https://noamz.org/misc/letter-to-knuth-11-feb-2026.pdf]

The arXiv's new "make everyone write in English to promote linguistic diversity" policy went into effect yesterday (https://blog.arxiv.org/2026/01/13/non-english-paper-submission-guidelines/), and they have now released a feedback survey:

https://cornell.ca1.qualtrics.com/jfe/form/SV_0rcozRH3ruvv54i

It's probably worth filling that out, even if the feedback is ultimately redirected to /dev/null.

I just checked and the original preface with these two paragraphs still appears in the second edition of Volume 1. On Knuth's TAOCP page [https://www-cs-faculty.stanford.edu/~knuth/taocp.html] in the "Errata for Volume 1 (2nd ed.)" he mentions that "In the third edition I will revise this preface extensively, so that it will reflect the actual scope of these books instead of the state they were in when Volume 1 was first published." But I don't see anything else about the preface on that page.

@gjm@mathstodon.xyz I agree that it looks unlikely to be a coincidence, but also extremely uncharacteristic of Knuth to lift a passage without attribution. A guess is that Knuth was subconsciously influenced to reconstruct this wording after having read Ulam's article a few years earlier. But that's just speculation, and probably the right person to ask is Knuth himself.

Here is the opening of Ulam's article:

Although to many people the electronic computer has come to symbolize the importance of mathematics in the modern world, few professional mathematicians are closely acquainted with the machine. Some, in fact, seem even to fear that individual scientific efforts will be pushed into the background or replaced by less imaginative, purely mechanical habits of research. I believe such fears to be quite groundless. It is preferable to regard the computer as a handy device for manipulating and displaying symbols. Even the most abstract thinkers agree that the simple act of writing down a few symbols on a piece of paper facilitates concentration. In this respect alone -- and it is not a trivial one -- the new electronic machines enlarge our effective memory and provide a marvelous extension of the means for experimenting with symbols in science. In this article I shall try to indicate how the computer can be useful in mathematical research.

Since the SciAm issue predates the TAOCP preface, I wonder if Knuth was influenced by reading Ulam's article, or whether it is just an example of great minds think alike?

(4/4)

Possibly the reason that neither paragraph is cited more often is that they were both eventually removed -- at least, they do not appear in the preface to the 3rd edition of Volume 1. I am not sure why, or whether they were still there in the 2nd edition. The version of the preface above is dated October 1967. Adding to the mystery, while trying to find references to the passage online, for fun I tried copying the text of the first paragraph into ChatGPT and asked it to identify the quote. It authoritatively answered that the quotation is from Stanisław Ulam's essay "Computers in mathematics" appearing in the book _Science, Computers, and People_ edited by M. C. Reynolds and Gian-Carlo Rota [https://doi.org/10.1007/978-1-4615-9819-0_6] Of course it isn't. But that essay is actually a reprint of an article by Ulam in the September 1964 issue of Scientific American [https://www.jstor.org/stable/24931634] which opens with strikingly similar language to the paragraph of Knuth's preface. (3/4)

I wish to show that the connection between computers and mathematics is far deeper and more intimate than these traditional relationships would imply. The construction of a computer program from a set of basic instructions is very similar to the construction of a mathematical proof from a set of axioms. Furthermore, pure mathematical problems historically have always developed from the study of practical problems arising in another field, and the advent of computers has brought a number of these with it. [...] Besides the interesting application of mathematical tools to programming problems, there are also interesting applications of computers to the exploration of mathematical conjectures, e.g., in combinatorial analysis and algebra; and in many of these cases there is considerable interplay between programming and classical mathematics. Attempts at mechanization of mathematics are also very important, since they lead to greater understanding of concepts we thought we know (until we had to explain them to a computer). I believe the connections between computers and pure mathematics which have been enumerated in this paragraph will become increasingly important.

I'm not sure whether I'll be able to use them in class, but I found these paragraphs interesting for multiple reasons. The first paragraph, which spells out a few legitimate concerns about the automation of mental tasks, feels very relevant today, with the difference perhaps that nowadays some mathematicians are starting to worry about losing their jobs. The second paragraph I found notable for its clear statement of the program-as-proof analogy, and I was surprised to not see it quoted more widely.

(2/4)