Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Saunders Mac Lane reports that the contents of his 1942 paper (joint with Samuel Eilenberg), that first introduced categories, were then referred to (in the words of prominent representatives of the mathematical community of that time?) as "general abstract nonsense''. While today the term is mostly used, especially by practitioners, as an implicit recognition of deep mathematical perspectives (rather than in a derogatory sense), is it correct that the tone was actually sarcastic, to say the least, in the early days of the subject? Or is it a falsehood? Could you point me out some references (more focused on this than Mac Lane's article from the above link) in support of one or other of the two versions? Thanks in advance for any help.

Added later. According to the bibliography included in the Wiki article linked by Robert Israel in his answer below, the term general abstract nonsense is believed to have been coined by Norman Steenrod - and surely it was not intended by him as a putdown (in spite of what happens today, in some fringes of the mathematical community). On the other hand, I'm now particularly intrigued (and, I must really confess, a little bit puzzled) by P.A. Smith's, let's say, warm comments about Eilenberg and Mac Lane's General Theory of Natural Equivalences, as they are reported by Michael Barr in an old thread from the Category Theory mailing list (dating back to May 1998).

Does anybody know if Smith's comments are taken from a letter, review, or anything else appearing in a journal, book, etc.? If so, have they ever been "revised" by Smith?

share|improve this question
add comment

1 Answer

Norman Steenrod is generally credited with coining the phrase. I'd call it "tongue-in-cheek" rather than sarcastic. See http://en.wikipedia.org/wiki/Abstract_nonsense and references there.

share|improve this answer
5  
Of course, everyone of a certain age will remember the exercise in Lang's algebra (I think it is gone in later editions): pick up any book on homological algebra and prove all the theorems. –  Igor Rivin Oct 29 '12 at 17:06
3  
The Lang quote (from Algebra, 2nd Edition) in full reads: "Take any book on homological algebra, and prove all the theorems without looking at the proofs given in that book". Followed by, "Homological algebra was invented by Eilenberg-MacLane. General category theory (i.e., the theory of arrow-theoretic results) is generally known as abstract nonsense (the terminology is due to Steenrod)." I believe both paragraphs were, coming from Lang, meant as sneers. Although Rotman, who wrote a book on homological algebra, had a nice take on it (from the preface, page xi, continued next comment): –  Todd Trimble Oct 29 '12 at 17:59
3  
Homological Algebra presents a great pedagogical challenge for authors and for readers. At first glance, its flood of elementary definitions (which often originate in other disciplines) and its space-filling diagrams appear forbidding. To counter this first impression, S. Lang set the following exercise on page 105 of his book, Algebra: "Take any book on homological algebra and prove all the theorems without looking at the proofs given in that book." Taken literally, the statement of the exercise is absurd. But its spirit is absolutely accurate; the subject only appears difficult... (cont.) –  Todd Trimble Oct 29 '12 at 18:01
4  
However, having recognized the elementary character of much of the early material, one is often tempted to “wave one’s hands”: to pretend that minutiae always behave well. It should come as no surprise that danger lurks in this attitude. (End quote of Rotman) As for the quote from Paul Smith, Sammy Eilenberg remarked that his and Steenrod's much more admiring comment were both valid. To put another spin on Paul Smith's comment, let me quote Peter Freyd: "Perhaps the purpose of categorical algebra is to show that which is trivial is trivially trivial." A highly non-trivial observation! –  Todd Trimble Oct 29 '12 at 18:11
1  
Meaning: that which is formal is formally formal. –  Peter May Oct 29 '12 at 23:42
show 4 more comments

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.