There's a famous story about an exercise from Lang's Algebra that says something along the lines of "pick up a homological algebra book and prove all of the theorems yourself". I cannot find it in the third revised edition, and I'm wondering if it's still in the third revised edition, if it's only in the older editions, or if it's an urban legend.

  • $\begingroup$ FWIW I once had a prof tell me that a standard thing was to ask students to prove everything in Spanier. But not everything there is a diagram chase... $\endgroup$ Jan 6 '10 at 14:21
  • $\begingroup$ @YCor, could you explain why are you making these minor edits for questions that have been there for a decade? $\endgroup$
    – GTA
    Feb 28 '20 at 16:13
  • $\begingroup$ @GTA this can be (and has already been) discussed at the dedicated chat: chat.stackexchange.com/rooms/10243/conversation/… $\endgroup$
    – YCor
    Feb 28 '20 at 16:27

It's real, but only in the first and second editions. (I don't have any electronic proof, but I've seen it in my copy of the second edition and someone else's copy of the first edition.) It's the only exercise in the chapter.

The full quote in the second edition is:

Take any book on homological algebra, and prove all the theorems without looking at the proofs given in that book.

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).

If I'm not mistaken, the quote is the same in the first edition. First edition: page 105; Second edition: page 175; so you can look in your library to see if I messed up the quote!

And I do have an electronic copy of the third edition, which I've searched to confirm it is not there. The historical remarks were expanded, written less dismissively and put at the intro to Part Four.

Update: I've scanned the evidence. First, Second.

  • $\begingroup$ If you have the means, the time, and the will, is there any chance you could scan that page? $\endgroup$ Jan 6 '10 at 10:07
  • $\begingroup$ Done! $ $ $\endgroup$ Jan 6 '10 at 16:07
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    $\begingroup$ Joseph Rotman was the first to tell me about this exercise in an email correspondence.I told him this is why everyone hates Lang except the do-it-yourself crew. I seriously doubt anyone but the most brillant of students ever succeeded in proving everything fom scratch in any homological algebra text worth looking at. $\endgroup$ Apr 12 '10 at 20:40
  • $\begingroup$ My memory isn't very precise, but Lang's attitude was that the intention of the exercise was essentially psychological: There are times when a certain kind of mathematical apparatus needs to be absorbed without too much fuss. Those weren't his exact words, of course. $\endgroup$ Aug 5 '10 at 1:42
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    $\begingroup$ @Minhyong: you and Joe Rotman (in his homological algebra book) are rather generous to Lang. I am very glad Lang had a change of mind and actually wrote something responsible on homological algebra in the third edition; the original exercise strikes me as an obnoxious joke, a kind of then-fashionable sneer at categorical subjects. $\endgroup$
    – Todd Trimble
    Dec 13 '10 at 16:50

In the Russian 1968 translation of Lang's Algebra (which is done from the Addison-Wesley 1965 edition), this exercise is there on page 126, exactly as quoted in Jonas' answer. It is complemented by two footnotes by the editor of the tranlation.

The footnotes say: "1) We suggest to skip these exercises on the first reading." and "2) It should be pointed out that the term `abstract nonsense' in this book has a positive character and is used below in the serious sense."

  • $\begingroup$ That is interesting! $\endgroup$ Jan 6 '10 at 16:06
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    $\begingroup$ I've also seen the footnotes in the Polish translation I used. $\endgroup$ Jan 30 '10 at 16:37

It is not an urban legend at all. It's certainly in the second edition, which is the one I studied from when I was a grad student. However it seems to be missing from the Springer 2002 edition. Certainly a loss, in my opinion.

  • $\begingroup$ Probably because in almost 30 years since the 1st edition,no student succeeded in doing it,Jose.I'm sure students were able to get big chunks of the task done,but I doubt the whole thing. $\endgroup$ Apr 12 '10 at 20:42

I got to this list since I was trying to remember whether the quote from Lang was true or not. Indeed, my copy, first edition I guess, does have that quote on p. 105. I don't know exactly how "positive"(ly) the term "abstract nonsense" is meant in that quote, though. I would imagine that Lang would have been annoyed had someone suggested they do that with any of his books.

Re: Spanier. Yes, the book is terse, and hard to read. I sincerely doubt that the intent was primarily to make a teaching text for beginning grad students; this was, and is, the standard reference book for basic algebraic topology. What makes it hard to read is the fact that every statement is made is as great a generality as he could. This is an advantage for a reference text. I sincerely doubt, however, that there was anything in that book that Spanier could not prove completely.

That being said, I certainly struggled through a course using it. I chose to take algebraic topology from someone else, to avoid taking it from Spanier himself, since I thought he would teach it as generally as was in the book. It turns out, though, that he did back down from that level of abstraction when teaching the course, and perhaps I missed something not taking the course from him.

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    $\begingroup$ I did take algebraic topology from Spanier at Chicago, and it was the graduate course that most excited me (though I wound up working in quite different fields). $\endgroup$ Aug 4 '10 at 20:45
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    $\begingroup$ Spanier's book was one of the textbooks for the algebraic topology course I took as a grad student. I hardly used it at all during that year, but afterward I found it to be an excellent resource. (The course I took was taught by Albrecht Dold, and his lectures were so amazingly clear and well thought-out, that one didn't really need a textbook.) $\endgroup$ Aug 4 '10 at 21:59
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    $\begingroup$ Interestingly, Dold's Algebraic topology text is also rather formal. $\endgroup$ Aug 5 '10 at 2:04

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